Progress in Optics : Volume 43

EDITORIAL ADVISORY BOARD

G.S. Agarwal

Ahmedabad, India

G. Agrawal

Rochester, USA

T. Asakura

Sapporo, Japan

A. Aspect

Orsay, France

M.V Berry

Bristol, England

A.T. Friberg

Stockholm, Sweden

VL. Ginzburg

Moscow, Russia

E Gori

Rome, Italy

A. Kujawski

Warsaw, Poland

L.M. Narducci

Philadelphia, USA

J. Pefina

Olomouc, Czech Republic

R.M. Sillitto

Edinburgh, Scotland

H. Walther

Garching, Germany

Preface The seven review articles presented in this volume cover a broad range of subjects. The first article, by L. Noethe, is concerned with the use of active optics in modem, large telescopes. Active optics is a branch of optics which deals with the control of the shape and the alignment of the components of optical systems. For modern large telescopes with flexible monolithic or segmented primary mirrors and with flexible structures, this technique is indispensable for attainment of performance which is either diffraction-limited for operations in space or is limited by the effects of the atmosphere for operations on the ground. This article describes first the theory of active optics relating to the wavefront analysis and for the correction mechanism. The method is then illustrated by the design of several active systems. The article concludes with an account of practical experience with systems of this kind. The second article, by B.A. Malomed, discusses variational methods used in nonlinear fiber optics and in related fields. A systematic review is given of the analytic and semi-analytic methods, which have been developed for the use with numerous static and dynamical models, based on nonlinear differential equations that have Lagrangian representation. The article deals mainly with one-dimensional models which describe light propagation in fibers and in waveguides. The article by O. Keller which follows deals with a topic of historical interest, which one might perhaps not expect to find in a volume in this series. However, I consider myself fortunate to have an opportunity to include the article. It seems that the majority of scientists are too preoccupied these days to be able to reflect on the way a particular discipline has developed. The article presents a fascinating account of researches of the Danish physicist L.V Lorenz who in 1867 established the electrodynamic theory of light, independently of the work of James Clerk Maxwell. Later it turned out that Lorenz' theory was equivalent to Maxwell's theory, written in covariant form. Lorenz' electrodynamic theory represents the culmination of his effort to achieve a unified understanding of the propagation of light

VI

Preface

in inhomogeneous media, surface optical phenomena, double refraction and optical activity. In 1890, the year before his death, Lorenz published a paper on scattering of light by a spherical particle. His results are equivalent to those which G. Mie presented in his classic 1908 paper. This work of Lorenz is, however, less well known than his contribution to the famous LorentzLorenz formula, which relates the refractive index to the polarizability of a medium. The fourth article, by A. Luks and V Pefinova, is concerned with the canonical quantum description of light propagation in dielectric media. The subject is related to the role of non-classical light in various applications. The spectral-temporal description of the electromagnetic field in linear and nonlinear dielectrics is discussed, both within the framework of microscopic and macroscopic theories. Attention is mainly paid to canonical quantum descriptions of light propagation in nonlinear dispersion-free dielectric media and in dispersive dielectric media, both linear and nonlinear. Phenomenological macroscopic theories, which account for light absorption in the medium, are also discussed. A simplified description is also presented for one-dimensional propagation and is illustrated by some optical processes. The fifth article, by D. Dragoman, describes the similarities and the differences between classical optics and quantum mechanics in phase space. The similarity has its origin in the bilinear nature of the Wigner distribution fimction, both in the quantum and the classical descriptions. The phase-space approach provides a formally similar treatment of interference phenomena, even though superposition of the wave fiinction in the quantum description and superposition of wavefields in the classical description reveal quite different behavior. The article by R. Boyd and D. Gauthier which follows, summarizes recent research on pulse propagation effects in resonant material systems. The research has led to the rather intriguing discovery that pulses can propagate with negligible distortion through such systems with velocities that can be either very much smaller or very much greater than the velocity of light in vacuum, depending on the experimental conditions. Many of the results are made possible by the use of nonlinear optical techniques such as electromagnetically induced transparency. The consistency of these effects with causality and the possibility of their uses in applications are also discussed. The concluding article, by A. Torre, is concerned with the fractional Fourier transform and some of its applications in optics. The basic properties of such transforms are explained and their uses in Fourier optics and in wave propagation in free space and in graded-index media are discussed.

Preface

vii

It is clear that the articles in this volume cover a broad range of subjects, some of which are likely to be of interest to many scientists concerned with optical theory or with optical devices. Emil Wolf Department of Physics and Astronomy and The Institute of Optics University of Rochester Rochester, NY 14627, USA February 2002

E. Wolf, Progress in Optics 43 © 2002 Elsevier Science B. V. All rights reserved

Chapter 1

Active optics in modern large optical telescopes by

Lothar Noethe European Southern Observatory, Karl-Schwarzschild-Str 2, 85748 Garching, Germany

Contents

Page § 1.

Introduction

3

§ 2.

Principles of active optics

4

§ 3.

Relationship between active-optics components and parameters

13

§ 4.

Wavefront sensing

14

§ 5.

Minimum elastic energy modes

21

§ 6.

Support of large mirrors

26

§ 7.

Alignment

34

§8.

Modification of the telescope optical configuration

38

§ 9.

Active-optics design for the NTT, the VLT and the Keck telescope

40

§ 10.

Practical experience with active optics at the NTT, the VLT and the Keck telescope

54

§11.

Existing active telescopes

65

§ 12.

Outlook

66

Acknowledgements

68

References

68

§ 1. Introduction From the point of view of optics, the purpose of a telescope is to produce maximum light concentration of large fluxes in a star image, and, in general, maximum resolution, which is equivalent to optimum image quality. This requires that the optical configuration of the telescope be always close to an optimum state. The optimum state is defined with respect to the environment in which the telescope is operated. In space it is the diffraction image of the telescope and on the ground the image which can be obtained with a large optically perfect telescope in the presence of atmospheric disturbances, the so-called seeing disc. Deviations from this optimum state, due to wavefront aberrations generated by the optics of the telescope, are unavoidable. But, the telescope is still defined as diffraction-limited or seeing-limited if the degradation of the image is smaller than accepted limits. The criterion for diffraction-limited performance is that the ratio of the intensity of the real image at its center to the intensity of the diffraction image at its center, the so-called Strehl ratio, be larger than 0.8. This is achieved if the root mean square (rms) o^ of the wavefront aberrations is less than A/14, where A is the wavelength of the observed radiation. For a ground-based operation where the atmospheric effects are not corrected, the telescope can be defined as seeing-limited if the equivalent ratio of the intensity at the center of the real image to the one at the center of the optimum image, the so-called central intensity ratio (CIR) (Dierickx [1992]), is also greater than 0.8. Whereas, for small wavefront aberrations, the Strehl ratio depends on the square of the rms of the wavefront error, the CIR depends on the square of the rms o^ of the slopes of the wavefront error, and also on the current seeing, expressed as the fiill width at half maximum G of the seeing disk:

CIR = 1 - 2 . 8 9 ( 1 ) ' ,

(1.1)

where O depends on the wavelength A of the light and is proportional to A~^^^. The goal of the design of a telescope is therefore to limit the wavefront aberrations to amounts which will guarantee a diffraction- or seeing-limited performance. In old passive telescopes this was attempted by using special constructional design features. With the increase in size this proved to be no

4

Active optics in modern large optical telescopes

[1, § 2

longer sufficient (indeed, significant extrapolation beyond 5 m was possible neither technically nor costwise), but with the introduction of active elements, which can correct the aberrations during operation in a systematic way, the goals can nowadays be achieved also for very large telescopes. Such ground-based telescopes with the goal of a seeing-limited performance will be called active, those with the goal of diffraction-limited performance adaptive. In space, the goal of active optics would be a diffraction-limited performance. This article will only deal with active optics, which by definition does not include the correction of pointing errors, that is, guiding and tracking. § 2 gives an overview of the principles of active optics. § 3 introduces the relationships between the various components and parameters of an active-optics system with special emphasis on telescopes with a monolithic primary mirror. § 4 describes the properties and design of one type of wavefront analyser customised for an active-optics system. § 5 summarizes the major characteristics of the elastic modes of a meniscus mirror, which are of central importance for the control of a thin monolithic mirror, and § 6 deals with the theory of the support of such mirrors. § 7 shows how the alignment can be controlled by active optics, and § 8 the possibilities of changing the optical configuration and the plate scale of a telescope. § 9 describes the designs of the active-optics systems of the New Technology Telescope (NTT) and the Very Large Telescope (VLT), both of the European Southern Observatory, and the Keck Telescope, and § 10 summarizes some practical experience with these active-optics systems. § 11 gives a short overview of existing telescopes working with active optics, and § 12 presents an outlook for the implementation of active optics into ftiture telescopes with even larger mirror diameters and more than two optical components. Most of the review deals with two-mirror telescopes with altazimuth mountings and strong emphasis is put on the systems aspects. Earlier reviews have been given by Ray [1991], Hubin and Noethe [1993] and Wilson [1996], the latter also with a detailed presentation of the historical developments and an extensive list of references. More details about active optics with thin meniscus mirrors are given by Noethe [2001]. § 2. Principles of active optics 2.1. Error sources Since the design of a telescope is strongly based on the avoidance of wavefront aberrations, we discuss first the possible error sources, shown and classified according to their frequency bandpasses, in fig. 1.

1, § 2] 1—:

Principles of active optics Active Optics

^ ....

~

1

1

Adaptive Optics •- -

Atmosphere Wind Local air Telescope movements Tube temperature Mirror temperature

o

dc

Optical manufacturing 1 -2

10

1

1

10

10

-/

1

0

10

/

1

1

10

10

Hz

Fig. 1. Bandpasses of sources of wavefront aberrations in optical telescopes.

7. Optical manufacturing. These errors are constant in time. During the polishing phase the mirrors can usually not be tested together as one system. But, alone, neither of the two mirrors produces a sharp image, in particular not with an incoming spherical wavefront generated by a small pinhole. Therefore, interferometric testing is only possible with so-called null lenses which generate wavefronts which are identical to the required shapes of the mirrors. Predominantly rotationally symmetric errors in the manufacturing of these null lenses can then lead to severe errors in the shape of the mirrors in the form of spherical aberration. However, testing of null lenses is nowadays possible and, independently, the spherical aberration of the combined system can be measured in the manufacturing plant with the pentaprism test (Wetthauer and Brodhun [1920]). 2. Mirror temperature. Owing to their huge inertia and the ineffective heat exchange with the air, large telescope mirrors follow temperature variations only slowly, that is, the mirrors filter out all but the lowest temporal frequencies of the air temperature variations. Nevertheless, the day to night changes of the air temperature result in temperature changes of the mirrors of possibly a few degrees. Unless an extremely low expansion glass is used, this is sufficient for a noticeable change of the focus position and other aberrations. 3. Tube temperature. Owing to its much smaller mass and therefore lower inertia, and because of a faster heat exchange due to radiative cooling, the changes of the tube temperature are much faster and larger than the ones of the primary mirror. Again, as in the case of the change of the mirror temperature, the main and possibly only significant effect is a change of the focus position.

6

Active optics in modern large optical telescopes

[1? § 2

4. Telescope movements. Any movement of the telescope tube, for example a change of the zenith angle in telescopes with an altazimuth mounting, will change to some extent the alignment of the telescope and the forces acting on the primary mirror, both effects generating wavefront aberrations. While small telescopes can be intrinsically sufficiently rigid for these effects not to play a role, large telescopes with diameters of the primary mirrors of more than, say, two meters are always noticeably affected by elastic deformations unless they are actively controlled. 5. Local air. Local air is defined here as the air inside the telescope enclosure and the air in the ground layer in the vicinity of the telescope enclosure. The local air conditions in the enclosure can be influenced by the design of the enclosure, avoidance of heat sources and active devices to maintain small temperature differences between various parts of the telescope and the ambient air (Racine, Salmon, Cowley and Sovka [1991]). 6. Wind. Wind generates both movements and elastic deformations of the telescope structure, especially of the telescope tube, as well as elastic deformations of the primary mirrors if these are sufficiently thin. Inside enclosures the peak of the energy spectrum is at approximately 2 Hz. 7. Free atmosphere. The effects of the free atmosphere above the ground layer on the image quality are predominantly generated by a layer at an altitude of approximately 10 km. The frequency range is very large, ranging from approximately 0.03 Hz to 1000 Hz. The natural fi-equency for splitting the errors into two groups is the approximate lower frequency limit of the errors generated by the free atmosphere. Wavefront aberrations generated in the free atmosphere, especially at high altitude, are strongly dependent on the field angle, that is, they are anisoplanatic. With integrations times larger than 30 seconds the wavefront aberrations due to the free atmosphere are effectively integrated out and the remaining aberrations are then independent of the field angle, that is, isoplanatic. This important condition allows that the information about the wavefront aberrations obtained with a star anywhere in the field can be used to correct the images over the whole field. The lower frequency range up to the limit of 0.03 Hz includes the first four sources completely, and sources five to seven partially. Systems which systematically attempt to correct these telescope errors during operation leaving only the errors generated by the free atmosphere, and therefore achieving a seeing-limited performance will be called active-optics systems, those which are predominantly designed to correct the aberrations generated by the free atmosphere and achieve diffiaction-limited performance will be called adaptive optics systems. The latter work at much higher frequencies and are not the subject of this chapter.

1, § 2]

Principles of active optics

2.2. Classification of active telescopes

Up to the 1980s all telescopes were passive in the sense that after the initial setup the optical configuration was, apart from focusing, never or very rarely, and then only manually, modified. Active telescopes, on the other hand, are capable of modifying the optical configuration systematically even during operation, based on data obtained with measurements with the final, completely installed system. They can be classified according to the type of control loops and the type of correction strategies and capabilities. 2.2.1. Control loops From a design point of view, the major differences between a passive and an active telescope are the time periods for the stability requirements of the system defining the optics on the one hand, and the role of absolute versus differential requirements on the other hand. To illustrate this point, consider first the design of dipassive telescope with two mirrors. The optical configuration is fully defined by the shape of the primary mirror and the relative positions of the two mirrors. One therefore has to find support systems for both mirrors which maintain the shape and the relative positions independently of the telescope attitude for time periods of hours. The positions are mainly influenced by deformations of the telescope tube and the shape of the primary mirror by deformations of its cell. For large telescopes neither structural component can be built with sufficient stiffness since this would require deformations of the telescope tube of only a few micrometers and deformations of the primary mirror cell of less than the wavelength of light. But the variations of relative positions can be reduced by the use of Serrurier struts, which, despite the deformation of the telescope tube, make the support structures of both mirrors move in parallel when the telescope attitude is changed. The deformations of the primary mirror can be minimised by decoupling the primary mirror fi-om the deformations of its cell by using astatic supports, which can be either mechanical levers (Lassell [1842]) or hydraulic or pneumatic devices interconnected in three groups (Yoder [1986]). All these apply forces which are independent of the distance between the mirror and its cell. Clearly, both of these design features will only guarantee the stability of the force setting, that is, the application of the correct forces for any zenith angle, and the stability of the relative positions to a certain degree. Any force errors will generate deformations of the primary mirror which are inversely proportional

8

Active optics in modern large optical telescopes

[1, § 2

to its stiffness. The specifications for the tolerable wavefront aberrations will therefore define the minimum stiffness of the primary mirror and, up to diameters of approximately two meters, with the help of the scaling laws (6.1)-(6.5) for thin mirrors given in § 6.2, also its minimum thickness. For diameters of more than two meters, the mirrors become prohibitively thick. In addition, because of the influence of shearing stresses in thick mirrors, they are more flexible than suggested by the scaling laws mentioned above. The required stiffness can therefore not be achieved by simply increasing the thickness of the mirrors. The diameters of monolithic primary mirrors of passive telescopes capable of a seeing-limited or even a diffraction-limited performance are consequently limited to the order of two meters. In addition, the telescope should ideally be made of materials which do not deform under temperature variations, and, for the mirrors, guarantee a stable shape over long periods of time. The main effect of the temperature variations would be defocus, due both to a change of the length of the tube and a deformation of the mirrors. For the mirrors, the material which fijlfills both requirements is low expansion glass. But defocusing as a result of the contraction or expansion of the generally metallic structure cannot be avoided. Active telescopes, on the other hand, do not need the stability of the forces or positions to be maintained over long periods of time. Instead, forces and positions can be changed depending on the knowledge of the passively generated deformations. This is a much easier requirement than the passive stability over time periods of hours and allows the use of less rigid elements, in particular a less rigid and therefore thinner primary mirror. The additional important question is whether these modifications are carried out in open or closed loop. Open-loop changes require the knowledge and predictability of the optimum absolute forces and positions for all sky positions. A condition for this predictability is that the system be free of significant friction and therefore hysteresis effects. It should also be capable of setting these absolute forces and positions with the required accuracy over time periods of hours. On the other hand, pure closed-loop operations require the stability of the forces and positions only for small time periods between two measurements of the wavefront analyser. High accuracy is then predominantly required for differential force and position settings, which can be done much more accurately than absolute settings. As a consequence, the requirements for the stability and predictability of the deformations of the optomechanical elements can, compared with open-loop operations, be fiirther reduced. Since the number of free design parameters is much larger in active telescopes and, at least for a closed-loop operation, the system also needs a wavefront

1, § 2]

Principles of active optics

9

analyser adapted to the mechanics of the telescopes, the design of an active telescope is more complex than the one of a passive telescope. Clearly, from the considerations above, the goal should be a closed-loop active-optics operation based on information from the image-forming wauefront in the exit pupil. Nevertheless, open-loop or mixed open- and closed-loop operations are also feasible. In both cases the fiill active-optics system consists of wavefront sensors, which either online or offline measure the wavefront errors, and mechanical parts performing the corrections. 2.2.2. Correction strategy A complete and perfect correction would, in principle, require the capability of moving all elements in all necessary degrees of freedom and correcting the shapes of all optical components. The free positioning would also enable a perfect alignment with the axis of the adapter. Such a complete correction would require measuring devices to determine the shapes and relative positions of all components. For the shapes this could be individual devices for each component, and for the alignment, devices for the relative orientation of two neighboring components. In practice, a sufficient set of such devices is not always available. The alternative is to measure the combined wavefront aberrations generated by the deformations and misalignments of all components. This can be done and is only really possible by using the light from a star. The aberrations generated by the individual elements and the misalignments then have to be deduced from the total wavefront error. If this is not possible, the correction may be incomplete. On the other hand, if the errors cannot be attributed to individual elements, a correction by a subset of the elements may be sufficient; for example, the correction of the deformations in a two-mirror telescope with two monolithic mirrors may be accomplished by deformations of the primary mirror alone. The two extreme types of active telescopes are therefore, on the one hand, those which require the control and correction of the shapes of individual components and, on the other hand, those operating as a system, where one component can also correct errors introduced by other components. An example of the first kind is a telescope with a segmented primary mirror with comparatively large individual rigid segments and a monolithic movable secondary mirror. The errors introduced by the primary mirror, that is, the phasing and the alignment of the segments, are very different from the errors introduced by elastic deformations or the figuring of the secondary mirror and can therefore not compensate each other. As a consequence, the optical surfaces

10

Active optics in modern large optical telescopes

[1, § 2

of both elements have to be controlled individually. An example of the latter kind is a telescope with a flexible monolithic primary mirror and also a movable monolithic secondary mirror. Here, the nature of the errors is similar and one element can correct errors introduced by the other one. The elastic and figuring errors of both mirrors are usually corrected by the primary mirror since it is, first, more flexible, second, often defined as the pupil of the telescope, and, third, equipped with a large number of supports anyway. The correction of errors mainly introduced by incorrect positioning of the elements, that is, defocus and third-order coma, has to be done by appropriate movements of the optical elements in both types of active telescopes. For the type and support of flexible monolithic mirrors there are several options. The traditional type is a comparatively thick mirror with 2i force-based support, which is basically passive and astatic, with an additional capability of changing the forces differentially. Such a system is ideally suited for a pure closed-loop operation with time periods between consecutive corrections of the order of minutes, and, possibly with a reduced quality, also for a pure open-loop operation. With active optics position supports also become feasible. Since these are ftindamentally non-astatic they require more frequent correction and therefore, if the times between corrections are smaller than the minimum integration times for the wavefront sensing, usually a mixture of an open-loop and a closed-loop operation. An important advantage of active telescopes is the freedom to relax the requirements for the figuring of all optical elements, since some low spatial fi'equency aberrations can be corrected by the active-optics system. This gives the manufacturer the opportunity to concentrate on minimising the high spatial frequency aberrations. For very thin mirrors the shape of the mirror is, in a sense, only defined by the support forces. During the polishing process these cannot be controlled to the accuracy required for a perfect shape. The mirror therefore only functions together with the active-optics system and its shape is defined by that system only.

2.3. Modal control concept and choice of set of modes Most error sources generate wavefront aberrations which can be well described by certain sets of mathematical fianctions. Since, in many cases, a small number of these functions is sufficient to describe a wavefront aberration, a modal concept for the analysis and the correction of the wavefront errors is essential for an efficient, practical system. Which set of ftinctions is used depends on

1, § 2]

Principles of active optics

11

the dominant error sources and on the type of telescope. The choice is mainly between purely optical functions like the Zemike polynomials and vibration modes (Creedon and Lindgren [1970]) based on elastic properties of a flexible element, usually the primary mirror. A general requirement is that the set of functions should be complete with all functions mutually orthogonal. Although only a very limited number of functions is used in practice, the completeness guarantees that, in principle, any arbitrary wavefront aberration can be well approximated. The orthogonality ensures that the values obtained for the coefficients of certain functions do not depend on other functions used in the analysis. Another important feature is the thinking in terms of Fourier modes, which means that different rotational symmetries are considered separately. The wavefi*ont errors generated by misalignments are defocus, third-order coma and some field-dependent functions, all expressible as simple polynomials. The most commonly used complete set of orthogonal polynomials over the full or annular pupil are the Zemike or annular Zemike polynomials. The errors generated by deformations of thin monolithic mirrors, on the other hand, are best described by elastic vibration modes. These are functions with the property that the ratio of the elastic energy to the rms of the deflection is minimised. Both the Zemike polynomials and the elastic modes are also complete and orthogonal within each individual rotational symmetry.

2.4. Examples of active telescopes Most modem large telescopes with diameters of the primary mirror of more than two meters rely in some way on active optics. The prototype of an active telescope with a practical system approach (Wilson [1978], Wilson, Franza and Noethe [1987]) is the New Technology Telescope of ESO. It is a RitcheyChretien telescope with a meniscus primary mirror with a diameter of 3.5 m and a thickness of 241 mm. It possesses Sermrier stmts and astatic mechanical levers for the support of the primary mirror. The active elements are a motorized secondary mirror with the capability to move in axial direction and to rotate around its center of curvature, and movable counterweights in the supports of the primary mirror. This allows for a correction of defocus, third-order coma and a few of the lowest order modes of the primary mirror. The principle of active optics as used in the NTT is shown in fig. 2. Since the telescope still has the passive design features and, for its diameter a fairly conservative thickness, corrections are only necessary every few minutes.

12

Active optics in modern large optical telescopes

[1,§2

Fig. 2. Principle of active optics in telescopes with a thin meniscus primary mirror.

The telescope can therefore be operated in closed loop. The additional features of its successor, the ESO Very Large Telescope (VLT), a Ritchey-Chretien telescope with a meniscus primary mirror with a diameter of 8.2 m and a thickness of 175 mm, are a motorised control of the secondary mirror in six degrees and also of the primary mirror in five degrees of freedom. Because of its much lower rigidity due to the larger diameter of eight meters and the reduced thickness of 175 mm, corrections are necessary every minute, despite the use of the usual passive design features. This correction rate still allows a pure closedloop operation. The 10 m Keck Telescope is a Ritchey-Chretien design with a primary mirror consisting of 36 hexagonal segments, each 1.8 m across with a thickness of 75 mm and three position actuators. The telescope optics including the segments of the primary mirror is aligned approximately once per month based on data obtained from the wavefront in the exit pupil generated by a star. Afterwards the shape of the primary mirror, that is, the relative positions of its segments, is maintained by an internal closed loop based on piston measurements at intersegment edges, whereas the position of the secondary mirror is controlled in open loop (Wizinowich, Mast, Nelson and DiVittorio [1994]).

1, § 3]

Relationship between AO parameters

13

§ 3. Relationship between active-optics components and parameters If the active-optics corrections are done on a system level, the active-optics system is not a feature added to the telescope system, but rather an integral part of it, and for many design parameters the capability to do corrections is even the driver. Figure 3 shows the dependencies between various parameters and components for a telescope with a thin meniscus mirror. The first column contains ftindamental parameters which are independent of the particular design, like atmospheric effects, the safety of the mirrors under exceptional conditions like earthquakes or failures of the support systems. Also the light-gathering power, defined by the diameter of the primary mirror, and the optical quality are fixed initial parameters. The optical quality is, for active telescopes, conveniently defined by two separate specifications for the high and low spatial fi-equency wavefront aberrations (abbreviated 'Spec, low/high SF' in fig. 3). The parameters in the second column, that is, the decision to operate in either closed or open loop and the tolerable wind speed at the primary mirror, which is determined by the design of the enclosure, can be either input parameters or the result of the system analysis. The third column contains

Atmospheric effects [Safety aspects^ I Spec, high SF I Diameter Ml

I Spec, low SF Accuracy

Accuracy wavefront analysis

Wavefront analyser ^d Sampling Optical parameters

Fig. 3. Dependencies between the specifications and the parameters used in an active-optics design.

14

Active optics in modern large optical telescopes

[U § 4

intermediate parameters which Hnk most of the input parameters with the parameters in the fourth column, which define the properties of the mechanical and optical components of the active-optics system. Arrows from a parameter A to another parameter B mean that B depends either directly on A, as for example the stiffness of Ml on its diameter, or that a requirement relating to B depends on A, as for example the density of supports on the number of active modes, which are defined as the modes corrected by active optics. Lines with arrows at both ends indicate that the connected parameters can influence each other. It is then obvious from fig. 3 that limitations on mechanical parameters like the achievable accuracy of the force setting can have impacts on parameters like the allowable wind speed at Ml or the decision to operate in open or closed loop. The dependencies will be explained in detail in the following sections. The following example will show how the diagram should be read. The required accuracy of the force setting under the primary mirror is defined by the specification for the low spatial frequency aberration and by the stiffness of the primary mirror, which determines how easily these lowest modes can be generated. The minimum stiffness itself is defined by the requirement to reduce the effects of wind pressure variations to the level given by the specification for the low spatial frequency errors of the wavefront.

§ 4. Wavefront sensing 4.1. General considerations In particular for telescopes which operate in closed loop, the wavefi'ont analyser is an essential and critical part of the active-optics system. In general, it is much easier to obtain the wavefront information from devices exploiting the pupil information than fi-om measurements of the characteristics of the image. The two most widely used methods are the Shack-Hartmann method (Shack and Piatt [1971]) and curvature sensing (Roddier and Roddier [1991]). A Shack-Hartmann device, which is shown in fig. 4, measures the local tilts of the wavefront of a star somewhere in the field. A mask at the focus of the telescope prevents the light from other nearby stars entering the sensor. The telescope pupil is imaged on to an array of small lenslets, each producing in its focal plane a spot on a detector. The shift of the spot generated with light from a star compared with the position of the spot generated with a point reference source placed in the focus of the telescope is proportional to the average local tilt of the wavefront over the subaperture sampled by a single lenslet. The curvature sensing method

15

Wavefront sensing

l.§4]

Telescope focus

Collimator

Shack-Hartmann grid

Detector

Fig. 4. Shack-Hartmann optics in a telescope.

measures the intensity variations, that is, the Laplacian of the wavefront, and the shape of the edges, that is, the first derivatives of the wavefront, in defocussed intrafocal and extrafocal images. Both methods work, in the end, with similar accuracy. The wavefront sensor has to be adapted to the type of the telescope and the type of operation of the active-optics system, in particular the correction strategy. One important criterion is that the measured coefficients of the modes are not dependent on the particular number of modes. This requires that the modes fitted to the measured data be orthogonal over the area of the pupil. The independence of the results for individual modes gives, for example, the freedom to correct, depending on the results, only a certain subset without the need to do another analysis with only the modes contained in this subset. Another criterion is the question whether the rms of the wavefront error or the slopes of the wavefront error should be minimised. The first choice would be the optimum for a system aiming for diffraction-limited performance, the second for a system aiming for seeing-limited performance. For a system working with Zernike polynomials the first choice requires a conversion of tilt data from the Shack-Hartmann device into wavefront data and a subsequent fit of the orthogonal Zernike or, for annular pupils, annular Zernike polynomials, whereas the second choice requires a direct fit of Zernike-type polynomials, whose derivatives are orthogonal over the pupil, to the tilt data (Braat [1987]). A system working with elastic vibration modes of the primary mirror should fit ftmctions to wavefront data, since the elastic vibration modes, but not their derivatives, are orthogonal over the area of the mirror.

16

Active optics in modern large optical telescopes

[1, § 4

Furthermore, the wavefront sensor has to fulfill a number of requirements imposed by the environment and the specification for the required accuracy, given usually in terms of tolerable low spatial frequency wavefront errors. In the rest of this section we will concentrate on the Shack-Hartmann method.

4.2. Calculation of the wavefront coefficients The calculation of the coefficients is done in five steps. 1. Computation of local tilt values and indexing of the spots. The centroids of the Shack-Hartmann patterns obtained with the reference and the star light are computed. A problem may be to find the reference spot corresponding to a certain star spot. One possibility would be to mark certain lenslets by reducing their transmission. Another possibility is to use the irregularities of the lenslet array to find the relative shift between the two patterns for which certain combinations of local distances give the best correlation. The second method works well for well-corrected systems and grids with sufficient distortions. In practice, with highly regular grids available nowadays, the errors introduced by making a wrong correspondence are irrelevant for a first correction of strongly aberrated wavefronts. After this initial correction the pattern is so regular, that a well-designed telescope with good pointing and tracking will almost always place the star spots close to the corresponding reference spots. 2. Computation of the center of the pupil. The center of the pupil can be calculated as the simple weighted average of the positions of the reference spots of all double spots. Other more complicated algorithms may give a higher accuracy. The major goal, apart from finding the proper center of the patterns is to disregard distorted spots at the edges belonging to subapertures which are not fully inside the pupil. 3. Interpolation of tilt data to regular positions in the pupil. In general, the Shack-Hartmann spot pattern is neither symmetric nor fixed with respect to the pupil. Each fit of a set of modes to the data involves the computation of the values of all modes at the relative locations of all spots in the pupil. The alternative is to interpolate the data to a fixed regular grid and calculate in advance the values of the modes only once for the regular grid positions in the pupil. The interpolation is done by fitting a two dimensional polynomial to the data of the surrounding spots. For a 23 by 23 pattern the optimum is the use of a second-order polynomial taking into account all spots within a distance from the regular spot position of 20% of the radius of the full pattern.

1, § 4]

Wavefront sensing

17

4. Conversion of tilt data into wavefront data. The conversion of a shift of a centroid ^ccd on the detector, which is usually a charge-coupled device (CCD), into a slope of the wavefront is given by

^ = —L_4l^

Mn

dp

^ ^

2MelA^l^l "'•

Here w is the wavefront error, p is the normalised radial coordinate in the exit pupil, fo\ the focal length of the collimator, d\ the diameter, f the focal length and N\ =f\/d\ the f-number of a lenslet, and iVtei the f-number of the telescope. The tilt data are then integrated to wavefront data. This is done by integrating, for a square grid, along the «rc rows and «rc columns, stopping, if necessary, at the edge of the central hole with its «hoie missing rows or columns, and starting a new integration at the other side. If the vector field was curl-free, for all spots the two values obtained with the integration along the corresponding column or the integration along the corresponding row, would, with a proper choice of the integration constants, be identical. But with the noise added by the measurement, this is not the case. Since the approximate number of 0.15n^^ intersections is, for all practical grids, much larger than the number of 2(«rc + «hoie) integration constants, the optimum choice of the integration constants can be obtained with a least-squares fit. 5. Fit of chosen functions to the wavefront data. The next step is a straightforward least-squares fit of the chosen set of functions to the wavefront data on the regular grid. With a fixed set of functions the fit is a multiplication with a precalculated matrix. This yields the coefficients of the fitted modes and, in addition, the residual rms Oresid of the wavefront aberration after subtraction of the fitted modes. 6. Subtraction of field aberrations. Since the wavefront analysis is usually done in the field of the telescope, but the active-optics corrections require the coefficients at the center of the field, the contributions from the field aberrations have to be subtracted. In aligned systems these are rotationally symmetric, but in misaligned systems the patterns are more complicated as described in § 7. An accurate subtraction of the field effects therefore requires information on the actual misalignment of the telescope.

4.3. Definition of Shack-Hartmann parameters The focal length/;oi of the collimator is chosen such that the image of the pupil on the Shack-Hartmann grid and therefore also the spot pattern fits, with some

18

Active optics in modern large optical telescopes

[1» § 4

margin, on the detector. This leaves then only two adjustable parameters, namely the number of lenslets sampling the pupil and the f-number of the lenslets. • Sampling of the wavefront. The sampling is determined by two requirements (see fig. 3). First, it should be sufficient to guarantee accurate measurements of the coefficients of all fitted modes. For this, the major error sources are an inaccurate determination of the center of the Shack-Hartmann pattern, the averaging of the tilts over subapertures and the aliasing generated by the finite sampling. The error due to the first source is of the order of 2.5% for a 10 by 10 sampling, with the error being approximately inversely proportional to the sampling «rc in one direction. If the wavefront errors are expanded in Zernike polynomials, the latter two sources lead only to crosstalk into the next lower term in the same rotational symmetry. This crosstalk is of the order of e^, where e is the ratio of the diameter of the subaperture to the diameter of the pupil. Second, to guarantee for a closed-loop operation a full sky coverage with field sizes of the order of lOOarcmin^ available in most telescopes, the sampling should be sufficiently coarse, that is, the corresponding subapertures in the pupil should be large enough to gather, with the chosen integration time, enough light from stars of magnitude 13. Measurements with two wavefront analysers in different positions in the field have shown that only with integration times of 30 seconds or more the differences due to effects of the free atmosphere at high altitude are effectively integrated out. Measurements with these integration times are therefore effectively isoplanatic and 30 seconds is the minimum time between active-optics corrections in a closed-loop operation. With 30 seconds integration time sufficient maximum pixel values are, at least for seeing values up to l.Sarcsec, guaranteed with subapertures with diameters of approximately 400 mm. • f-number of the lenslets. This parameter is determined by the requirement that a wavefront analysis can be done with high accuracy under all relevant external conditions. The major external parameter is the atmospheric seeing. The image analysis should ftinction both under excellent seeing conditions with an expected minimum value O ^ 0.2arcsec and bad seeing conditions with seeing values up to at least 0 ^ 1.5arcsec. Above these values the tolerable errors, which would still guarantee a seeing-limited performance, are so large that a seeing-limited performance can also be achieved with open-loop operations (see § 10.3.3). This leads to four conditions for the f-number of the Shack-Hartmann lenslets (Noethe [2001]). 7. Minimum spot size larger than 1.5 times the pixel size. For an accurate centroiding the spot diameter has to be at least 1.5 times as large as the pixel

1, § 4]

Wavefront sensing

19

size Jp. The minimum spot size is generated by the reference Hght or possibly by star Hght under optimum seeing conditions and is given by the diameter of the Airy disk of the lenslets. This leads to the following condition for the f-number A^i of the lenslets. Nx > 1 . 2 5 ^ .

(4.2)

|im

2. Avoidance of swamping. In order not to overlap with the neighboring spots, the diameters of the Shack-Hartmann spots should be less than 0.7 times the lenslet diameter. For the spots generated by the reference light this leads to the condition Nx ^ 0 . 6 — . (4.3) |im In bad seeing conditions with an assumed worst seeing of 6)w the condition is TV, < 0 . 3 5 J , ^ - ^ ,

(4.4)

where du is the outer diameter of the primary mirror. 3. Maximization of sensitivity to transverse aberration. The measuring accuracy of the Shack-Hartmann sensor is mainly limited by the centroiding errors. The generated wavefront error is proportional to the centroiding error with an rms Ocen, the f-number A^i of the lenslets and, roughly, to the square root of the number of modes used in the analysis. This leads to the following condition for A^,: ^1 ^ 0 . 2 ^ ^ ^ y^i;;;^,

(4.5)

^wf, max

where Owf,max is the rms of the maximum tolerable wavefront error allocated to the wavefront analysis. Even with comparatively simple centroiding methods the centroiding error is of the order of only 5% of the pixel size. If the maximum pixel value is constant, the centroiding error does not depend on the spot size, but is only a function of the pixel size.

4.4. Wavefront analysers for segmented mirror telescopes For telescopes with segmented mirrors, the wavefront analyser should be capable of detecting the deformations of individual segments, relative tilt and piston

20

Active optics in modern large optical telescopes

[1, § 4

errors of individual segments, errors introduced by misalignments between mirrors, and possibly also errors introduced by deformations of monolithic mirrors. A special feature of two- or multiple-mirror telescopes with at least one segmented mirror is a possible degeneracy or near-degeneracy between certain modes generated by the segmented mirror on the one hand and by a monolithic mirror or misalignments of mirrors on the other hand. For example, defocus can be generated approximately by a misalignment of the primary mirror segments with the same change of the relative angle between all pairs of segments. It is also generated exactly by an axial movement of the secondary mirror. The two aberrations can compensate each other to first approximation. The difference would then be defocus errors of individual segments. A wavefi-ont analyser therefore has to be able to distinguish between these two effects. All these functions, most of them based on the Shack-Hartmann principle, have been realised in the Phasing Camera System (PCS) of the Keck telescope, which can operate in four modes (Chanan, Nelson, Mast, Wizinowich and Schaefer [1994]). The so-called passive tilt mode, where the light from each segment is collected into one spot per segment, can measure the tilt errors of the segments. The fine screen mode, where each of the 36 segments is sampled in 13 places, can measure the segment tilts, but also the defocus and decentering coma aberrations of the telescope optics, generated by a despace of the secondary mirror. Global defocus and coma introduce, over each subaperture corresponding to one segment, local defocus and astigmatism, respectively. The axial error in the position of the secondary mirror can then be calculated and corrected from the average defocus, and the tilt or decenter from the distribution of astigmatism over the subapertures. Both of these modes do not use common Shack-Hartmann lenslets, but rather a combination of prisms and a convex lens in the case of the passive tilt mode (Chanan, Mast and Nelson [1988]), and a combination of a mask, a defocusing lens and an objective with a focal length about five times smaller than the one of the defocusing lens in the case of the fine screen mode. The ultra fine screen mode samples just one segment with 217 close-packed hexagonal Shack-Hartmann lenslets. Finally, the segment phase mode (Chanan, Troy and Ohara [2000]) deduces the relative heights of adjacent segments using a physical optics generalization of the Shack-Hartmann test, with starlight from apertures with diameters of 120 mm centered at the intersegment edges. An alternative method, called phase discontinuity sensing (PDS), operates on the difference between intrafocal and extrafocal images and utilizes the light from the entire segments (Chanan, Troy and Sirko [1999]). It does not utilize the PCS hardware, but is installed in one of the infrared instruments. The physical optics Shack-Hartmann method utilizes two algorithms. The narrowband algorithm (Chanan, Ohara and Troy 42000])

1, § 5]

Minimum elastic energy modes

21

is based on the diffraction pattern obtained with quasi-monochromatic light. The pattern is a periodic function of the relative displacement of the adjacent segments. The capture range, which is the maximum difference between the heights for which the algorithm can be applied, is of the order of 15% of the wavelength A of the light. For A ^ 800 nm the accuracy is of the order of 6nm. The broadband algorithm takes the effects of the finite bandwidth into account. Both the capture range and the accuracy are roughly inversely proportional to the bandwidth of the light. For A ^ 800 nm and a bandwidth of 200 nm, the capture range is 1 |im and the accuracy 30 nm. Since both algorithms exploit interference effects, the coherence of the light over the subaperture should not significantly be degraded by atmospheric effects. This is guaranteed if the diameter of the subaperture is smaller than the atmospheric coherence length TQ for the wavelength used for the measurement. Under this condition, the results of the relative height measurements are largely independent of the current seeing. The method using the differences between the intrafocal and extrafocal images works at wavelengths of 3310 nm with a bandwidth of 63 nm. The capture range is 400 nm and the accuracy 40 nm. Piston errors start to limit the image quality if the atmospheric coherence length ro for the observed wavelength A approaches the dimensions of the individual segments (Chanan, Troy, Dekens, Michaels, Nelson, Mast and Kirkman [1998]). Since ro scales with the wavelength as X^^^, phasing becomes increasingly important for observations at longer wavelengths. For segments with diameters of 1.8 m, as in the Keck telescope, phasing is effectively irrelevant for observations with visible light, but at a wavelength of 5 \im and an rms piston error of 500 nm, the central intensity is reduced by approximately 60%. At the Keck telescope the phasing tolerances are set to ^ 100 nm for normal observing. However, for observations also using adaptive optics to correct the atmospheric disturbances or for telescopes in space, the tolerances should be much tighter.

§ 5. Minimum elastic energy modes The minimum-energy modes can be defined in the following way (Noethe [1991]). Each rotational symmetry m will be considered separately. Let Tm^o be the set of all functions of rotational symmetry m defined over the area of the mirror. The lowest mode e^nj is the one taken from the set Tni,o which minimizes the ratio F of the total elastic energy J of the mode to the rms A of its deflection perpendicular to the surface. Let ^„,, i be the set of all ftinctions of Tm, 0 which are orthogonal to e^,, i. The second mode 6^, i is the one taken fi-om

22

Active optics in modern large optical telescopes

[U § 5

!Fm^ 1 which mimimizes the ratio F. For an arbitrary / let J^mj- i he the set of all functions of !Fni,o which are orthogonal to all functions e^^^ i, •., ^w,/-1• Then, the /th mode Cn^^ / is the one taken from J^^u i - \ which mimimizes the ratio T. The actual construction of the minimum-energy modes requires the solution of the variational equation 5 ( J - | ^ ) = 0,

(5.1)

where | is a free parameter which can be interpreted as the energy per unit of the rms of the deflection. The use of variational principles leads, together with the assumptions of a thin shallow spherical shell, to a fourth-order differential equation, which can be transformed into two second-order differential equations for each rotational symmetry. Since the fixed points only define the position of the mirror in space and have no impact on its shape, the appropriate boundary conditions are the ones for free inner and outer edges. The solutions of the differential equations form, within each rotational symmetry m, a complete set {e\ of orthogonal fianctions, the elastic modes e^^j. The order of a mode within each rotational symmetry is denoted by the index /. If the eigenvalues ^ni, i are expressed as ^mj = {hycolj,

(5.2)

where y is the mass density of the mirror, h its thickness, and o;^,, / is interpreted as the circular frequency of a vibration mode with the order / within the rotational symmetry m, the differential equations are identical to the equations describing vibrations of a thin shallow shell under the assumption that in-plane inertial effects are neglected. The eigenvalues $^,,/, which can be shown to be proportional to the elastic energies of the modes, are therefore proportional to the squares of the eigenfrequencies of the corresponding vibration modes. For geometrically similar mirrors of the same material, the eigenfi'equencies scale with h/d^. Figure 5 shows the eigenfrequencies of the elastic modes of the VLT primary mirror with a diameter of 8.2 m, a thickness of 175 mm and a radius of curvature of 28.8 m as a ftinction of their order in a log-log plot. Two features of elastic modes are very useful in the context of active optics. First, the eigenfrequencies increase rapidly both with the symmetries m and with the orders /. Within each rotational symmetry m the increase in the log-log plot is approximately linear, with the symmetry 2 having the largest slope of approximately 2, that is, the eigenfrequencies are roughly proportional to f-. The lowest modes of the symmetries 0 to 3 show, for the lowest order, deviations from the linear behavior.

Minimum elastic energy modes

l.§5]

loglo(f'm,i/Hz)

(a)

23

(b)

: A

Number of ^;ubtrac•led lowest modes

Fig. 5. (a) Eigenfrequencies of the elastic modes of the VLT primary mirror for the lowest nine rotational symmetries m and lowest six orders / within each rotational symmetry, (b) Logarithm of the fraction O^/o^ of the deflection generated by a random pressure field left after subtraction a certain number of lowest modes.

For the rotational symmetries 0 and 1, the relative increase is due to membrane stresses induced by the thin shell. In a log-log plot of the eigenfrequencies against the rotational symmetry m the increases are, for w ^ 2, also linear, with a largest slope of approximately 2 for the lowest order one. In this order the eigenfrequencies are therefore proportional to m^. It is obvious from fig. 5a that the lowest elastic mode ^2, i of rotational symmetry 2 is by far the softest and therefore most easily excitable deformation. Its control is, therefore, together with defocus and decentering coma, which are generated by misalignments, the most important and demanding task of active optics. Because of the fast increase of the stiffness of the modes with the order and the rotational symmetry, any given set of forces or any given pressure field will generate significant deflections only in the lowest modes. If Op is the rms of the deflections generated by random white noise pressure fields, fig. 5b shows the ratio Od/Op, where o^ is the rms of the residual deflection after the subtraction of a given number of elastic modes with the lowest eigenfrequencies. A subtraction of the softest mode ei, \ alone reduces the rms of the deflection to 40%, and a subtraction of the softest five modes to 10%. Second, a pressure field, which is proportional to an elastic mode, will, since the mode is an eigenfunction of the underlying differential equation, generate a deflection with exactly the same functional dependence. The coefficient of the deflection is then inversely proportional to the eigenvalue, that is, the elastic energy, of this mode. This feature can be exploited to calculate the deflections generated by arbitrary pressure fields or sets of forces. The pressure fields

24

Active optics in modern large optical telescopes

[1,§5

Rotational symmetry 2 Elastic modes Zernike polynomials

Fig. 6. Lowest three eigenmodes of rotational symmetry 2 (dashed lines) and their corresponding annular Zernike polynomials (solid lines) as functions of the normalised radial coordinate p.

are directly expanded in terms of the elastic modes, whereas the forces are described as delta functions and then expanded. The total deflection is obtained by summing up the deflections in the individual modes, which are obtained by multiplying the expansion coefficients of the pressure field by factors inversely proportional to the elastic energies Sw,/ of the modes. Zernike polynomials z^^, and elastic modes e^,, are very similar in the respect that, in each rotational symmetry m, the number of nodes of the radial fiinction is defined by the order / of the mode. For rotational symmetries larger than 1 the Zernike polynomials z^j correspond to the elastic modes Cmj, but for rotational symmetries 0 and 1, where the lowest Zernike polynomials piston and tilt represent fiill body motions, the elastic modes Cnj^ / correspond to the Zernike polynomials z^jj+i. The major difference between the two sets of fiinctions is that the elastic modes are effectively linear near the outer edge but show stronger variations near the inner edge than the Zernike polynomials. The consequence is that particularly higher order elastic modes cannot be well approximated by a small number of annular Zernike polynomials. Figure 6 shows the first three annular Zernike polynomials and elastic modes of rotational symmetry 2. The residual errors after fitting to the elastic mode of order / the lowest / annular Zernike polynomials are, in fractions of the rms of the elastic modes, 0.05 for / = 1, 0.35 for / = 2, and 0.62 for / = 3. To push the residual fraction below 0.05 for the modes / = 2 and / = 3 one needs to fit 4 and 6 annular Zernike polynomials, respectively. Nevertheless, at least the lowest elastic modes are, if regarded as vectors in function space, effectively parallel to their corresponding Zernike polynomials. Examples of such pairs are Zernike defocus ZQ, 2 and the first elastic mode ^0,1 of rotational symmetry 0, Zernike third-order coma zi 2

1, § 5]

Minimum elastic energy modes

25

and the first elastic mode ei i of rotational symmetry 1, and Zemike thirdorder astigmatism Z2, i and the first elastic mode ^2, i of rotational symmetry 2. In general, members of such pairs should not be fitted simultaneously to a wavefi*ont. The relative difference between the lowest mode ^2, i of rotational symmetry 2 and the equivalent Zemike third-order astigmatism (y/6p^coslq)) is only of the order of 5%. But the forces to generate third-order astigmatism with an accuracy similar to the one achievable for the corresponding elastic mode 62,1 are significantly larger. The elastic mode ^2,1 of the primary mirror of the VLT can be generated, excluding print-through effects, with a relative accuracy of 0.00003 with maximum forces of F^ax = 1.68 N for a coefficient of lOOOnm. The accuracy, with which Zernike astigmatism can be generated, and the required forces depend on the number of elastic modes used for the approximation. With two modes the accuracy is 0.012 with F^ax = 4.2N and with six modes 0.0024 with Fmax = 13.6N. This shows again the advantage of working with elastic modes rather than Zemike polynomials in the active-optics corrections of elastically induced errors. Since alignment errors generate to first order nearly pure field-independent Zernike defocus and third-order coma, the Zemike polynomials zo,2 and zi,2 should be included and, consequently, the corresponding elastic modes eo, 1 and e\j excluded from the set of fitted modes. The two Zemike modes will not exactly be orthogonal to the higher elastic modes within their rotational symmetry, but this is in practice not a significant effect. The sets of functions used in active optics then contain Zemike defocus and third-order coma and, if monolithic mirrors are used, some of the elastic modes with the lowest energies. The number of elastic modes which will be considered depends mainly on the forces which are required to correct these modes. The fact that these forces increase much faster with the spatial frequencies of the modes than the coefficients of these modes generated by noise effects in the wavefront analyser, puts a natural limit on the number of modes which can be corrected. All modes which are actually corrected during the active-optics process will from now on be called active modes. The number of active modes can be defined in the following way. One can assume that the forces applied by passive actuators are accurate to approximately 5% of the nominal load over the range of usable zenith angles. Further, the rms of the wavefront error introduced by any mode should be well below the diffraction limit. Therefore, one should correct all modes which are generated with coefficients of more than, say, 15 nm by random force errors evenly distributed in a range of ± 5 % of the nominal load of each support.

26

Active optics in modern large optical telescopes

[U § 6

These coefficients, which will be inversely proportional to the square of the eigenfrequencies of the corresponding modes, can be calculated by the method mentioned earlier in this section, from several runs with independent sets of random forces.

§ 6. Support of large mirrors 6.1. System dependencies The properties of the supports of large monolithic mirrors, in particular of the primary mirrors, of large active telescopes are related to the basic requirements in a complex way. In fig. 3 the mechanical parameters, for which requirements are to be deduced from the input parameters, are shown in the upper six boxes in the fourth column. Friction is the only limitation for the predictability of the system. Clearly, it also has an influence on the stability. The latter depends on the general type of the support system, the astaticities of its components and the stiffness of the primary mirror. The major safety requirement is the need to keep the stress levels, in particular at the support points, well below the critical values. These depend on the material of the mirror, and the values of the generated stress primarily on the thickness of the mirror and the type of the support system, in particular the nature of the fixed points. The generated high spatial frequency aberrations, the so-called print-through, clearly depend on the specific weight and the elasticity module of the mirror material, on the thickness of Ml and the density of supports. But it can also be influenced by the general type of the support system, for example if part of the weight of the mirror is supported by a continuous pressure field at the back surface as realised in the support of the primary mirrors of the Gemini telescopes (Stepp and Huang [1994]). The stiffness of Ml, a central parameter for the activeoptics design, is a function of the diameter of Ml, its thickness and its elasticity module. Another intermediate parameter, the number of active modes, depends, as described at the end of §5, on the stiffness of Ml and the tolerable low spatial frequency errors. The required accuracy of the force setting depends on the stiffness of Ml, that is, predominantly on the stiffness of the softest elastic mode ^2, i, and on the tolerated low spatial frequency aberrations, again dominated by the mode ^2, i. The range of active forces depends on the stiffnesses of the active modes, and since the accuracy of a load cell is usually inversely proportional to its range, the active range is also directly related to the accuracy of the force settings.

1, § 6]

Support of large mirrors

27

6.2. Scaling laws for thin monolithic mirrors For a comparison of menisci with different diameters du and thicknesses h, and of their support systems with n^ individual supports, the following scaling laws can be used. They are given for the wavefront errors w and the corresponding slope errors t generated by deformations of the mirror. • Pressure field applied to the meniscus. If the pressure fields as fianctions of the normalized radii of the menisci are identical, the scaling law is given by

4

w oc -h^^ ,

' - #h^

(6.1)

• Sag under own weight. The sag between support points under the meniscus' own weight obeys the scaling law

""^^i^ ^°^A-

(6.2)

h^ni h'^nj These scaling laws can readily be derived from those for the pressure fields by noting that the forces applied by the meniscus' own weight are proportional to its thickness h. The factor l/nj ensures that for a constant thickness the sag stays the same if the number of the supports per area, which is proportional to dl^Ms, remains constant. • Single discrete force. For a single discrete differential force AF applied to the mirror, the scaling law is given by woc^AF,

tcx^AF

(6.3)

• Set of supports applying random force errors proportional to the nominal loads. If a force error is proportional to the nominal load of the support, one has AF (X d^Ji/n^. Furthermore, if «s supports are applying forces with random errors the expression for the effect of a single force has to be multiplied by y ^ . Together one then obtains from eq. (6.3) w o c f ^ ,

^ocf-^.

(6.4)

The design of the support system has to meet both the specifications for the high and the low spatial firequency aberrations. The former are dominated by the sag between the support points described by the scaling law (6.2) and the latter

28

Active optics in modern large optical telescopes

[!> § 6

by the deformation in the shape of the mode ^2,1 generated by random support forces. Since the achievable accuracy of the force setting will be proportional to the total force range, one can apply the scaling law (6.4), if the nominal load is understood as the force range. If, for different mirrors and their supports, the high spatial frequency errors are taken to be identical, the number of supports scales with n^ oc dl^/h for wavefront errors and n^ oc dj^^/h for slope errors. The low spatial frequency errors then scale with ^3

--^'

J2.25

^^Jf-

(6-5)

This scaling law shows that the requirements for the accuracy of the force setting increase strongly with the flexibility of the mirror. For example, if the thickness of the mirror and the density of the supports are kept constant, the required accuracy of the force setting as a fraction of the total range is inversely proportional to c/^ if the wavefront errors are to remain constant, and to J^^^ if the slope errors are to remain constant.

6.3. Types of supports for thin monolithic mirrors For thin monolithic mirrors there are three fundamental choices for the type of the support system. 1. Force- or position-based systems. While for passive telescopes force-based systems are the only option, for active telescopes both types are feasible. The force option is currently still preferred, since it allows a certain decoupling of the mirror from the mirror cell and therefore a pure closed-loop operation. For large mirrors a position-based support would, owing to the fast deformations of the mirror cell, also require open-loop corrections. 2. For force-based systems: combination of passive and active supports or purely active supports. The combination is the best and often the only solution for a pure closed-loop system, since the passive part, supporting the weight of the mirror, can be designed as an astatic system, which guarantees the required stability over sufficiently long time periods. Purely active supports usually have a level of non-astaticity which requires also open-loop corrections, although possibly not as frequently as with a position-based system. 3. Mechanical levers or, at least for the passive part, hydraulic or pneumatic supports. Mechanical levers add considerable additional weight and require real fixed points, which, in certain emergency cases, may have to support the ftill weight of the mirror. Astatic hydraulic or pneumatic systems can work

1, § 6]

Support of large mirrors

29

with supports connected in sectors and therefore virtual fixed points. Unwanted overloads are therefore distributed over several supports. This generates, in case of failure, much smaller stresses than a support with real fixed points and is, for very flexible mirrors, the safer and therefore preferred solution. Which of the above mentioned options is chosen depends on the maximum tolerable stress levels and the required stability of the optical configuration of the telescope system. With glass still the traditionally used, although not necessarily optimum material, the maximum stress level plays an important role. The choice of a system with real fixed points may then require a comparatively thick primary mirror, whereas a system with virtual fixed points and therefore a better distribution of the loads in exceptional circumstances may allow the use of a much thinner mirror. In the latter case a lower limit for the thickness of the mirror is defined by the stiffness required to limit deformations by wind buffeting to values defined by the specification for the effects of wind buffeting expressed in terms of low spatial fi-equency aberrations. This can be partially ameliorated by coupling the mirror for high temporal fi-equencies to its, in general, stiffer mirror cell, for example by using a tunable mirror support with optionally six fixed points at high temporal frequencies as described in §6.4.3 (Stepp [1993]). To be able to remove the mirror easily from its cell, for example for realuminization, it would be an advantage to have only push supports. While this is not possible for the optimum solutions for the lateral supports presented in § 6.5, it can be realised for the axial supports. The only restriction will be a limitation for the maximum zenith angle 0z, max at which the telescope optics can be corrected with the active-optics system. The reason is that the largest required negative correction force Fcorr has to be smaller than the remaining gravity load which varies with the cosine of the zenith angle. If FG,O is the nominal gravity load at zenith angle zero, one gets 0z,max = arccos (Fcorr/^co)- For larger zenith angles than 0z,max the mirror would, at a given support point, loose the contact with the support. One goal of the active-optics design should therefore also be to minimize the required range of the active forces.

6.4. Axial support of thin meniscus mirrors 6.4.1. Basic support geometry For the distribution of the axial supports one can choose between two basic geometries. One would be a regular geometry with hexagonal symmetry where neighboring supports form equal lateral triangles. This would be the most

30

Active optics in modern large optical telescopes

[1? § 6

effective solution in terms of the required number of supports, but the symmetry is not compatible with the circular shape of the mirror. The other choice implies discrete supports on circular rings. Over most of the area the support geometry is then irregular, but near the edges the deformations are more regular than those generated by the hexagonal support. The usual choice is the second option, also because analytical methods are available at least for the optimization of the ring radii. 6.4.2. Minimization ofwavefront aberrations The theory for the analytical optimization of the radii of the support rings for thin plates has been developed by Couder [1931] and that for thin shallow shells by Schwesinger [1988]. Both calculate first the deflections for a support on a single continuous concentric ring. The total deflection is then a superposition of the deflections generated by n rings, multiplied by the appropriate load fractions. Since the dependencies of the deflections on the radii are not linear, optimizations can only be done by trial and error methods. The final result of the optimization depends also on two other parameters which are, in addition to the radii, considered variable, namely the load fi-actions, and an overall deformation in the form of a paraboloid introduced by Schwesinger [1988], which can easily be corrected by an axial movement of the secondary mirror. Compared with the results which are obtained under the condition that all support forces are identical, that is, that the load fractions are fixed and no defocus is allowed, the rms of the sag between the supports can be reduced by approximately 30% if the additional degrees of freedom of the load fractions and, more important, the defocus are used for the optimization. The reason for the strong effect of the defocus component is that the deflections near the inner and outer edges are nearly linear and, if the support forces generate an overall shape similar to a parabola, a fitted parabola can intersect the deflection curve twice both between the inner edge and the inner ring and the outer edge and the outer ring. 6.4.3. Effects of fixed points Any basically astatic axial system needs three fixed points for the definition of the position of the mirror in space. These can be either real, as in the case of astatic mechanical lever supports, or virtual, as in the case of hydraulic or pneumatic supports, where all supports in each of the three sectors are interconnected. Since the volume of the fluid or gas is constant in each sector, the barycenter of the supports will stay constant. If the positions of the virtual fixed points are defined

1, § 6]

Support of large mirrors

31

as these barycenters, the two types of fixed points can mathematically be treated in the same way. In the case of the real fixed points, they usually replace, on one of the rings, three of the astatic or active supports at angular separations of 120"". The question now arises, whether modes of a given rotational symmetry m can be corrected with a given number of supports n^ on one ring without exciting appreciable deformations in other rotational symmetries. Let us assume that the force changes at the actuators on one ring follow the rotational symmetry m. The reaction forces on the fixed points due to changes of the actuator forces can easily be calculated from the conditions of the equilibrium of the forces and the two moments around two orthogonal axes perpendicular to the axis of the mirror. It can then be shown (Noethe [2001]) that the sums of the applied forces and the reaction forces on each support on the ring do not follow the rotational symmetry m any more, if the rotational symmetries of the applied forces are 0, 1, «s - 1, Ws or «s + 1 • For the rotational symmetries 0 and 1 the reaction forces can be made 0, if more than one ring is used and, in addition, for w = 0, the sum of load fractions on the rings is 0 or, for /w = 1, the sum of the products of the load fractions and the corresponding ring radii is 0. The largest rotational symmetry correctable with the axial support system is then n^ - 2, where n^ is the smallest number of supports on any of the rings. Another effect of the fixed points is that the correction of modes with all symmetries different from multiples of three lead to additional tilt. The coefficient of the tilt is roughly equal to the coefficient of the corrected mode. In practice, it is very small and anyway quickly removed by the autoguider. An interesting consideration, first suggested by the Gemini project (Stepp [1993]), is the use of 6 fixed points to couple the mirror to the, in general, stiffer mirror cell. This allows the reduction of wind buffeting effects on the primary mirror. Of course, the mirror should be coupled to its cell only for high temporal frequencies. For low temporal frequencies it has to be decoupled to facilitate active-optics corrections and, if intended by the design, to guarantee a basically astatic support with three fixed points. This can be achieved by splitting each of the three sectors in a hydraulic support system with interconnected supports into two smaller sectors and connecting the halves by a tunable valve. A straightforward calculation (Noethe [2001]) shows that only modes with rotational symmetries m = 6i-\

or

m = 6/

or

m = 6/ + 1,

/ = 0,1,2, ...

(6.6)

are compatible with a 6 sector support, that is, are decoupled from the mirror cell. For all other rotational symmetries, in particular the rotational symmetry 2 with

32

Active optics in modern large optical telescopes

[1, § 6

the softest and therefore most easily excitable first mode, the mirror is coupled to the cell and deformations in the form of these modes can therefore be reduced. 6.4.4. Effect of support geometry on mode correction Not only the fixed point reactions, but also the number of supports alone on any of the rings limits the correctability of certain modes. Let m be the rotational symmetry followed by the active forces on one support ring, 7} the offset angle, «s the number of supports on the ring, and the set S be defined by S = {p: p = j • ns, j = 0,1,2, ...}. The wavefi'ont aberration generated in an arbitrary rotational symmetry m is given by (Noethe [2001]): [ wa, ni, m( p) COS mq) COS mi}, w + m G S and H'b,w,w(p)COS(rficp + ml}), m-\-m e S and m-m y^ni,m(p,(P)= { ^c,m,m(p) COS (w(jP - mi}), m + m ^ S and m-m 0, w + w ^ S and m-m

m-m e S, ^ S, (6.7) e S, ^ S,

where Wa^m^jnip), Wb,w,w(p) and Wc,m,m(p) describe the dependencies on the radial coordinate p. The first three cases represent the combinations of the rotational symmetry m of the forces and the number «s of the equidistant supports on one ring which generate the requested deformation in the rotational symmetry rn = m or crosstalk into other rotational symmetries m ^ m. For example, a force pattern with a rotational symmetry m = 4 on a ring with ns = 9 supports will generate the required wavefront deformation Wc,4,4(p)cos4()p, but also an unwanted crosstalk of the form Wb,4,5(p) cos 5q). The same two wavefront aberrations are generated by a force pattern with the same maximum force but with a rotational symmetry m = 5, since the forces with rotational symmetries mi and m2 on a ring with m\ + m2 supports are identical. Most significant are couplings into the mode ^2,1. A support with, say, 9 supports on one of the rings will generate crosstalk into this mode if a mode with the symmetry 7 is corrected. 6.5. Lateral support of thin meniscus mirrors Lateral support systems are usually passive and should fulfill the following two requirements. First, they should not, for any inclination of the mirror, generate wavefi'ont aberrations which require significant active correction forces from the axial support system, since this would increase the range of active forces and therefore reduce the maximum usable zenith angle as described at the end of § 6.3. Second, the mirror should be supported at the outer edge only. Fortunately,

1, § 6]

Support of large mirrors

33

a type of lateral support with these characteristics exists. The analytical theory has been developed by Schwesinger [1988, 1991]. Instead of discrete forces it considers initially force densities at the edges with the three components/^ in radial, fi in tangential, and y^ in axial direction. In Fourier terms this implies that any force densities which follow a given rotational symmetry generate deformations in only this symmetry The only force densities which support the weight of the mirror are those with the rotational symmetry 1. A lateral support system should therefore only contain force densities of rotational symmetry 1. The lateral support is greatly simplified for telescopes with altazimuth mountings. In this case the directions of the forces with respect to a coordinate system which is fixed to the mirror are constant. Only the moduli depend on the inclination of the mirror cell. If the mirror is neither too steep nor too thin, it can be laterally supported at the outer rim under its center of gravity. But for steep and thin mirrors this is not the case and axial forces at the outer edge have to be used to balance the moment. The modulus of the axial force density y^, which is proportional to sin (p, where cp is the azimuth angle starting from the direction parallel to the altitude axis, is then defined by the weight of the mirror, its diameter and the distance between the plane of the supports and the center of gravity of the mirror. The radial force densities f^ must always be proportional to sin cp and the tangential force densities fx to cos q). The only free parameter is then the fi-action P of the weight supported by the tangential force density, with the remaining fraction 1 - fi supported by the radial force density. Schwesinger [1988, 1991] has derived analytical formulae for the dependence of the radial function of the deflection with the rotational symmetry 1 on the ratio ^. The deflection may contain third-order coma, which can be corrected by a movement of the secondary mirror. The residual wavefront error after fitting and subtracting third-order coma should therefore be the merit fiinction for the optimization with the ratio /?. These wavefront errors are, for an optimum choice of /?, in practice so small that a possible ftarther reduction with additional supports at the inner edge is not necessary (Schwesinger [1994]). Schwesinger's theory for the rotational symmetry 1 can be extended to all other rotational symmetries (Noethe [2001]). Similar to the method for axial forces described in § 5, this offers a fast and efficient alternative to the use of finiteelement methods for calculating the effects of general lateral forces on the mirror figure. Arbitrary continuous force densities and also discrete forces can be split and expanded in infinite series of continuous force densities/^,/ and^^ along the edges in all rotational symmetries m, that is, force densities being proportional to smmq) and cosmq). A force density proportional to sinmq) [cosmq] will

34

Active optics in modern large optical telescopes

[1, § 7

only generate deformations which are also proportional to sin mq) [cos mq)]. An overall deformation is then simply the sum of the deformations in all rotational symmetries. Since, as in the case of the elastic modes for axial deformations, the deflections decrease, for the same moduli of the force densities, rapidly with the rotational symmetry, the consideration of the lowest symmetries will be sufficient to calculate the overall deformations. If the lateral supports are combined with the axial supports as in the Subaru telescope (lye [1991]), the actual locations of the application of the forces have to be in the neutral surface to avoid unwanted moments. For solid monolithic mirrors this requires the drilling of additional bores. For mirrors with a honeycomb structure it may be the natural and best solution. 6.6. Segmented mirrors Although it is not a compulsory requirement, one goal of a segmented mirror design is that the shapes of individual segments do not need active corrections during the operation of the telescope. With diameters as large as 2 m they require passive, astatic supports as, for example, multi-stage whiffle trees which apply both axial and lateral forces. The deflections as functions of the number of supports per segment area and thickness follow the scaling laws for monolithic mirrors given in § 6.2. An optimization of the distribution of supports is usually done with finite-element calculations. To correct figuring errors in a d.c. mode, static devices like warping harnesses can be installed at the back surface of the segments. If each segment is intrinsically stable, the major problem is the alignment of the «seg segments both in piston and tilt. Each segment therefore needs three actuators capable of changing the axial positions of the three fixed points. The support of a segmented mirror as a whole is therefore position-based and requires frequent corrections, owing to the normally strong flexure of the cell with a change of the zenith angle. The alignment and control of a segmented mirror is discussed in § 7.2.

§ 7. Alignment 7.1. Alignment of a two-mirror telescope In a perfectly aligned two-mirror telescope the axes of the primary mirror, the secondary mirror and the rotator are congruent. This ideal case can, particularly

1, § 7]

Alignment

35

with large telescopes, only be achieved as an approximation. In particular, even if the alignment is sufficiently good for a certain zenith angle, the mechanical deformations of the telescope structure may generate misalignments at other zenith angles. A complete alignment of the telescope can be done in three steps. • Initial alignment with auxiliary equipment. Using autocollimation and finite focusing the axes of M2 and the rotator of a large telescope like the VLT can be aligned to an accuracy of approximately 3 arcsec for the angles between the axes and less than 1 mm for a shift of the vertex of M2 with respect to the axis of the rotator. But the position of Ml and therefore the angle between the axes of Ml and M2 and the shift between the vertex of M2 with respect to the axis of Ml are only defined within the mechanical tolerances of the Ml support, which are much larger than the accuracy of the alignment achieved for the relative alignment of the axes of the rotator and M2. The consequence will, in general, be a large amount of decentering coma. • Correction of decentering coma. The decentering coma generated by the misalignment between the axes of Ml and M2 can be measured by the wavefront analyser, and be corrected by a rotation of the secondary mirror around its center of curvature, by a full body movement of Ml, or by a combination of both. After this operation the telescope may still be a schiefspiegler (Wilson [1996]) in which the axes of Ml and M2 are not aligned, but intersect at the socalled coma-free point. This is a point around which the secondary mirror can be rotated without changing the value of field-independent decentering coma. • Alignment of the axes of Ml and M2. The residual misalignment can be determined from a mapping of the pattern of third-order astigmatism, that is from measurements of this coefficient at a few field positions. A complete correction can be done by rotating either the secondary mirror or the primary mirror or both around the coma-free point. An overview of aberrations in misaligned telescopes can be found in Wilson [1996] and of the ahgnment of telescopes in chapter 2 of Wilson [1999]. A general theory of low-order field aberrations of decentered optical systems has been given by Shack and Thompson [1980]. In particular it has been shown that the general field dependence of third-order astigmatism can be described by a binodal pattern, known as ovals of Cassini. Only for special cases such as a centered system do the two nodes coincide and the field dependence reduces to the well-known rotationally symmetric pattern with a quadratic dependence on the distance to the field center. These general geometrical properties have been used by McLeod [1996], starting from equations by Schroeder [1987], for the alignment of an aplanatic two-mirror telescope. McLeod showed that the

36

Active optics in modem large optical telescopes

[1, § 7

components Z4 and Z5 of third-order astigmatism of a two-mirror telescope with the stop at the primary mirror for a field angle 0 with components 0^ and (py are given by Z4 = Bo {(1)1 - 0l) + 5 , (0.a, - 0,a,) +^2 (a? - « ' ) ,

(7.1)

Zs = 2Bo(l),0y + ^1 (^.r^v + 0v«v) + IBia^a,.

{12)

Bo is the coefficient of field astigmatism for a centered telescope, whereas ^i and B2 only appear in decentered systems. Numerical values for Bo, B\ and Bi were obtained by using general formulae for field astigmatism of individual mirrors and adding the effects of the two mirrors. The values for a^ and a^ could then be obtained from measurements of Z4 and Z5 in the field of the telescope. Explicit expressions for the third-order astigmatism parameters BQ, B\ and B2 as fianctions of fundamental design parameters and optical properties of the total telescope and of the position of the stop along the optical axis give more insight into the characteristics of field aberrations of two-mirror telescopes. They have been derived for centered two-mirror telescopes by Wilson [1996] and for decentered ones by Noethe and Guisard [2000]. In a decentered system one also has to take into account the definition of the field center. The normal definition is the direction parallel to the axis of Ml, projected towards the sky. But in a decentered system the image of an object in this field center is not in the center of the adapter, where the instruments are located and which is therefore the practical field center. If the field astigmatism is calculated with respect to this practical field center, the structure of the eqs. (7.1) and (7.2) remains the same, but the parameters Bo, Bi and B2 change and 0 denotes the field angle with respect to the center of the adapter (Noethe and Guisard [2000]). To align the axes of the two mirrors and to put the intersection of this axis with the focal plane to the center of the adapter, one has to reposition both mirrors. In principle, two wavefront analysers would be necessary and also sufficient for a closed-loop alignment of a two-mirror telescope. With only one wavefront analyser available, mappings have to be done at various zenith angles and the alignment can be controlled only in open loop.

7.2. Alignment of a segmented mirror The alignment procedure described in this section is the one used in the Keck telescope. It is assumed that the shapes of the segments are not affected by changes of the zenith angle. The control of the position of a segment is restricted

1, § 7]

Alignment

37

to three degrees of freedom, a piston coordinate parallel to the optical axis of the telescope, and two tilt components for rotations around two orthogonal axes perpendicular to the optical axis. The alignment of the segments is done in two steps. First, the tilts of the «seg segments are measured optically with the passive tilt or the fine screen mode of the phasing camera system (PCS) described in § 4.4. The required corrections of the segment tilts are done by appropriate differential movements of the 3«seg piston actuators. Second, the differences in height at midpoints of intersegment edges are measured optically by the segment phase mode of the PCS. The number of these sampling points is larger than the number of degrees of freedom, which is equal to the number of segments minus one. The optimum differential piston movements, which are the ones that minimize the rms of the differences in height of adjacent segment midpoints, are obtained by a least squares fit. The relative positions of the segments, and therefore the overall shape of the mirror, are then maintained not by optical measurements with the PCS, but by measurements with a specific set of capacitive position sensors located at intersegment boundaries. These position sensors, which are capable of measuring changes in relative heights at intersegment boundaries, are sensitive to both relative piston movements and relative tilts of adjacent pairs of segments, but a single sensor cannot distinguish between the two. With the knowledge of all sensor readings, however, the proper fractions of the readings due to the piston movements and tilts, and therefore also the overall shape of the mirror, can be uniquely reconstructed. The actual readings after an alignment described above are defined as target values for subsequent corrections. The number of the piston sensors must be at least as large, but is usually larger than the number of actuators, which is 3«seg- The differences between the reference and the actual readings are, via a least squares fit, converted into actuator movements. Any noise in the sensor readings will lead to errors in the relative tilt and piston values of the segments. These aberrations can conveniently be expanded in socalled normal modes (Troy, Chanan, Sirko and Leffert [1998]). Similar to the elastic modes, which are eigenvectors of a differential equation describing the elastic behavior of the mirror, the normal modes are orthogonal eigenvectors associated with a singular value decomposition of the control matrix connecting actuator movements to the larger number of sensor readings. Apart from the discontinuities due to the segmentation, the normal modes, in particular the lower order ones, approximate to Zemike polynomials. If random noise is assumed for the sensor readings, the average of the coefficient of a normal mode contained in the wavefront error decreases rapidly

38

Active optics in modern large optical telescopes

[1, § 8

with the order of the mode. The normal mode which can most easily be generated by the segmented primary mirror control system of the Keck telescope is a defocus mode, followed by a mode similar to third-order astigmatism. The defocus mode is produced by a constant offset to all piston sensors, since the corresponding changes of the actuator lengths will exactly follow a parabola. Compared with the equivalent case of the average content of modes in a wavefront generated by random pressure fields on a monolithic mirror, the decrease is, above all for the higher order modes, much weaker. For example, the normal mode similar to the Zemike polynomial of rotational symmetry 2 and order 3, that is, 7th-order astigmatism, is only ten times weaker than the strongest mode, whereas the corresponding ratio for the elastic modes is of the order of 200. This slower convergence is of importance, since the higher order modes generate stronger edge discontinuities. On the other hand, the rms of the edge discontinuities related to tilt errors with a given rms value are much smaller than expected from a random distribution of the tilt errors over the segment, since most of the tilt error is contained in the smooth modes with small edge discontinuities.

§ 8. Modification of the telescope optical configuration A defocus aberration can be introduced both by an axial movement of the secondary mirror and a deformation of the primary mirror. This feature can be used to control the plate scale of the telescope. The defocusing with the secondary mirror can also, together with an elastic deformation of the primary mirror, be used to maintain the optical quality of the telescope during a change of its optical configuration. 8.1. Control of the plate scale The focal length/2' of M2 is assumed to be constant. The plate scale is therefore only affected by changes df( of the focal length / / of Ml and dd\ of the distance d\ between Ml and M2. A change of the shape of the primary mirror in the defocus mode, described by the coefficient Cdef of the equivalent wavefront change p^, generates the following change df( of the focal length of Ml:

dfl = mlc^,u

(8.1)

where A^i is the f-number of the primary mirror. The dependencies of the variations db, of the back focal distance b between the pole of Ml and the image.

1, § 8]

Modification of telescope optical configurations

39

and bf, of the focal length/' of the telescope, on the variations of// and d\ are given by Wilson [1996]: db={ml + \) ddx -m\dfl,

^f='^[fl^d,-{n^d,)dfl),

(8.2)

(8.3)

where W2 is the magnification of the secondary mirror. The two conditions for a control of the plate scale in the telescope are the amount of the change df and the requirement that the back focal distance b remain unchanged, i.e. 6Z? = 0. These two conditions can be fulfilled by variations of the two parameters / / and d\. From eq. (8.2) one then gets

dfl='^5d,.

(8.4)

Introducing this into eq. (8.3) and solving for dd\ one gets

mi-l-

di/f{

The accuracy of the control of the plate scale is limited by the accuracy of the force setting under Ml and the axial positioning of M2, that is, by the individual contributions from dfl and dd\ to df, and by the noise in the wavefront measurements. 8.2. Modification of the optical configuration If a two-mirror telescope has both Nasmyth and Cassegrain foci, it may not be possible to find a convenient design which places both foci at the same distance from the secondary mirror. Switching from one focus to the other therefore requires refocusing. In a classical Cassegrain design this will generate fieldindependent third-order spherical aberration, which was initially not present, and in a Ritchey-Chretien design in addition field-dependent third-order coma. Of those, the spherical aberration can be removed by a deformation of the primary mirror, that is, a change of its conic constant. Changing the shape of Ml by a function proportional to p^ requires comparatively strong forces, since p^ has strong curvature near the outer edge

40

Active optics in modern large optical telescopes

[1, § 9

contrary to elastic modes with effectively no curvature near the outer edge. The curvature of p^ near the outer edge can be greatly reduced by adding an appropriate amount of defocus which will be compensated by an additional axial movement of M2. This new deformation can be better approximated by elastic modes and can therefore be generated with much smaller forces.

§ 9. Active-optics design for the NTT, the VLT and the Keclc telescope 9.1. General requirements and specifications The NTT and VLT are examples of active two-mirror telescopes with monolithic meniscus mirrors. The NTT with a mechanical diameter of its primary mirror of 3.58 m was the first telescope with active optics as an integral part of its design. Nevertheless, since it was the first attempt to build an active telescope, one conservative requirement was that it could, with a reduced optical quality, also fiinction in a fiilly passive mode. The VLT with a diameter of its primary mirror of 8.2 m was envisaged to fianction only in the active mode, since its primary mirror is about 40 times as flexible. The designs of the active-optics systems of these two telescopes can serve as typical examples for two-mirror telescopes of the four and eight meter class with monolithic primary mirrors. The specifications for the NTT were given in terms of the diameter d%Q of the circle containing 80% of the geometrical energy. For a Gaussian point spread fiinction one has Jgo ~ 1-56^ ~ 2.54(7/ and for an atmosphericseeing point spread fiinction ^go ~ \.9d, where Q is the fiiU width at half maximum. The specifications were then Jgo = 0.15arcsec for the active and ^80 ^ 0.40arcsec for the passive mode. The figure for the active mode can be split into ^80 ^ 0. lOarcsec for the high and also d^^ = 0. lOarcsec for the low spatial frequency aberrations. The specifications for the VLT were given in terms of the central intensity ratio CIR, namely CIR = 0.8 for a seeing of 0.4 arcsec. The relevant contributions for the design of the active-optics system were CIR = 0.992 or, according to eq. (1.1), an rms of the wavefront slopes Ot = 0.021 arcsec for the high spatial frequencies of Ml, CIR = 0.979 or Ot = 0.034 arcsec for the activeoptics control errors of the wavefront, including both the wavefront analysis and the corrections, and CIR = 0.97 or o^ = 0.041 arcsec for the effects of wind pressure variations on Ml. Since the major aberrations generated by wind and active-optics control errors are low spatial frequency aberrations, the CIR figures for these error sources can be converted into approximate nns values Ow of wavefi-ont aberrations dominated by the mode ^2,1- CIR = 0.99 is then equivalent

1, § 9]

Active-optics design for NTT, VLT and Keck

41

to Ow = 140 nm and CIR = 0.98 to Ow = 200 nm. Finally, all possibly occurring stresses in the mirrors had to be well below the critical values for glass ceramics. With these specifications all parameters in the first column of fig. 3, which form the basis of the active-optics design, were defined. Two other parameters, which are in principle fi-ee in fig. 3, were also defined in advance. First, the active-optics systems of both telescopes were required to work fiilly in closed loop with wavefront analyser integration times of at least 30 seconds and a fiill sky coverage, and, second, the substrate of both primary mirrors was a glass ceramic. The VLT had the additional requirement that it should work both with the Nasmyth foci and a Cassegrain focus with the consequences described in § 8.2. Furthermore, because of the strong impact of temperature differences between the mirrors, the air in the enclosure and the outside air on the image quality, the VLT was required to control these differences within narrow limits instead of relying on natural ventilation only as in the case of the NTT. The specifications for the Keck telescope were given in terms of ^go- The error budget for the total telescope was 0.41 arcsec, with 0.24arcsec for the segment figure being the largest contribution. The total active-optics error budget was split into a contribution of 0.084 arcsec from zenith distance independent and of 0.058 arcsec from zenith distance dependent errors (Cohen, Mast and Nelson [1994]).

9.2. Active-optics design of the NTT 9.2.1. Thickness of Ml, type of support system and set of active modes The main driver for the thickness of Ml was the requirement, that the telescope could, although with a reduced optical quality, be operated also in a passive mode. Measurements at the equatorially mounted ESO 3.6 m telescope showed that the ^go values due to low spatial frequency elastic aberrations were of the order of 0.5 arcsec largely independent of the sky position (Wilson [1999]). Since the design of the Ml support of a telescope with an altazimuth mounting like the NTT was significantly easier, it was estimated that the NTT could, passively, achieve the same performance with a mirror of approximately half the thickness, which was then finally defined as 241mm. Another way of justifying this thickness of Ml is the following. The average coefficient of the dominant low spatial frequency mode ^2, i generated with random forces in the range of =blN would be 5.5nm, with maximum values of the order of 15nm. Random force errors in the range of ± 5 % of the nominal forces

42

Active optics in modern large optical telescopes

[1,§9

Table 1 Eigenfrequencies of the lowest elastic modes of the NTT and the VLT Symmetry

Order

Eigenfrequencies NTT

Symmetry

Order

VLT

Eigenfrequencies NTT

VLT

2

115

16

3

2

1131

160

3

273

38

1

2

1229

176

0

192

42

7

1

1383

192

4

479

66

4

2

1577

221

1

434

68

8

1

1779

246

732

102

2

3

1749

246

737

107

0

3

2050

272

5 2

2

0

2

852

119

5

2

2077

289

6

1

1034

143

3

3

2366

331

of approximately 760 N would then generate on average coefficients of ^2,1 of the order of 210 nm, which is equivalent to an rms of the slope errors of Ox ^ 0.082 arcsec and Jgo ~ 0.20arcsec. If equal tolerances were also given to the defocus and decentering coma errors, one would with a quadratic sum just fulfill the specification for Jgo = 0.4 arcsec for the passive mode. With such a thickness the stresses, which arise if the mirror is unintentionally supported by three points only, are well below the tolerable limit. This then allowed the use of a conventional support with astatic levers and consequently three real fixed points. To define the set of active modes, one can apply the procedure described at the end of §5. With the rms of 210nm for the average coefficient of ^2,1 generated with random forces in the range of ± 5 % of the nominal load, the frequency limit for the modes to be considered is V2,i 7210/15 ^ 430 Hz and the data in table 1 show that for the NTT the modes up to 64j should be corrected. But then, since the elastic mode eo, 1 is replaced by defocus, there would be no possibility to correct rotationally symmetric aberrations other than defocus. Because of the importance of spherical aberration, the elastic mode eo, 2 has to be added to the set of active modes. The chosen force range of the active actuators of ±30% of the nominal load was large enough to allow also the use of the equivalent Zemike modes instead of the more efficient elastic modes. Spherical aberration can be generated with much smaller forces by combining it with a defocus deformation of Ml.

1, § 9]

Active-optics design for NTT, VLT and Keck

43

This defocus can then easily be compensated by an appropriate axial movement of M2. For the NTT the best combination is p^ - 3.6p^. 9.2.2. Axial support of Ml The axial support of the primary mirror of the NTT consists of four rings with 9, 15, 24 and 30 supports. This distribution gives an rms Ow of the high spatial frequency wavefront aberrations of approximately 7 nm and an rms of the slope error of the wavefront of Ox ~ 0.02arcsec, well below the specification of Ox = 0.04arcsec, which is equivalent to dso = 0.1 arcsec. The chosen density of supports was also sufficient to generate all active modes with high accuracy. The largest error in terms of the rms of the relative difference between the requested and the actually generated shapes is, not considering the effects of the printthrough, of the order of only 2% for the second mode eo, 2 of rotational symmetry 0. For the other modes the relative errors are of the order of 0.1% or smaller. The modification with respect to a passive support with astatic mechanical levers were motorised counterweights which could change the support force by approximately ±30% of the gravity load on the support. Since the gravity loads are proportional to the cosine of the zenith angle, the correction of errors which are independent of the zenith angle like polishing errors would have required different positions of the counterweights for different zenith angles. For this reason additional springs were introduced which could apply correction forces independently of the zenith angle. The springs had no motorised control and could only be adjusted manually. With the comparatively large thickness, wind buffeting on Ml was no problem. In addition, the telescope optics was very stable over time periods of one minute and could therefore be operated in closed loop. The force-setting accuracy to achieve wavefront errors of Ow < 50 nm is of the order of ±ION. To be sure that the error is within the limit 95%) of the time and not only on average, the force-setting accuracy should be three times better, that is, ±3 N. 9.2.3. Lateral support of Ml Despite the relatively low f-number of 2.2 of the primary mirror, the plane perpendicular to the axis of Ml through the center of gravity intersects the outer rim. Ml could therefore be supported laterally under its center of gravity with all lateral forces in a plane perpendicular to the axis of M1. Figure 7a shows the dependence of the surface deflection along a central vertical line

Active optics in modern large optical telescopes

44 1

1

1

I

1

1

1

' 1 '

d e f l e c t i o n (nm)

, 1 , , , 1' NTT

100 ~

/5

" 0.00

" 0.8 X

/ J\ // ///

- ^

-

i" 1 1.1 Hi

" 0.3--^^ . 0 . 5 ^

[1,§9

^ l//'l --^ ^ - ^

^

- 1.0 \ \ ^^

\ s ^ _- - - -^/ / - (a) —^ -,,,!,,, 1 11 1 1,

-

My

-

100 —

p ' ,

1 , ,

1 1 "

Fig. 7. (a) Deflections of the primary mirror of the NTT as functions of the normaUsed radius p for various fractions (3 of the weight supported by the tangential forces, (b) Lateral forces for ^ = 0.5.

(Schwesinger [1988]) on the ratio /?. Apparently, the dependence on ft is not critical and for )S = 0.5 the deflection approximates to third-order coma, which can be corrected by a movement of the secondary mirror. The rms of the residual wavefront error is then approximately 20 nm. The choice of ^ = 0.5 is convenient, since for equidistant positions of the lateral supports the forces are all identical and parallel to the direction of the gravity vector, as shown in fig. 7b. With 24 supports the forces are of the order of 2500 N and the stresses well below the critical values. 9.2.4, Position control of M2 The control of defocus and decentering coma requires an accurate positioning of the secondary mirror. To reach an accuracy of Ot ~ 0.02arcsec for both modes, one needs an accuracy of the axial movement of M2 of approximately 2 ^m for the correction of defocus, and an accuracy of the rotation around the center of curvature of approximately 3 arcsec for the correction of decentering coma. The restricted number of motorised degrees of fi-eedom of the movements of M2 do not allow a motorised correction of a misalignment, which would require a rotation around the coma-free point. This can, however, be done by a combination of a mechanical adjustment of the M2 cell and a rotation of M2 around its center of curvature. 9.2.5. Wavefront analyser The wavefront analyser is a Shack-Hartmann device with a rectangular 25 by 25 lenslet array with lenslets of 1mm side length and a f-number of 170. To

1, § 9]

Active-optics design for NTT, VLT and Keck

45

fit the pattern on the CCD array with a side length of 11 mm, optics with a reduction factor of msh = 0.36 had to be used. In the conditions (4.2), (4.4) and (4.5) in § 4.3 the left hand sides then all have to be replaced by the product mshN\. With the chosen parameters these conditions are all fijlfilled. The size of 150 mm by 150 mm of a subaperture on the primary mirror corresponding to one lenslet may be too small to find a sufficiently bright guide star in the field for an arbitrary sky position. But the size of the subapertures could be increased to 350 mm by 350 mm, since a sampling of 10 by 10 would easily be sufficient for an accurate measurement of the small number of active modes.

9.3. Active-optics design of the VLT 9.3.1. Thickness of Ml, type of support system and set of active modes For an 8 m mirror as thin as the one of the VLT the stresses generated by an accidental support on three hard fixed points would have been dangerous. Since a basic passive support was required for a pure closed-loop operation, a hydraulic support system with all supports connected in each of the three sectors was chosen as the passive part of the axial support system. To avoid pressure differences due to gravity in inclined positions, it was designed as a two-chamber system (Schneermann, Cui, Enard, Noethe and Postema [1990]). The active part has electromechanical actuators which work in series with the passive support and therefore add the correction forces to the passive ones. The lower limit of the thickness of the VLT was partially defined by wind buffeting considerations. With expected wind pressure variations of 1 N/m^, the rms of the wavefi-ont aberrations could be limited to 150nm with a mirror thickness of approximately 175 mm. The wind pressure variations could have been reduced further by reducing the wind flow in the enclosure, but this could have generated local seeing effects due to insufficient flushing of temperature inhomogeneities created inside the enclosure. According to fig. 3 the definition of the thickness defined the stiffness and therefore also the rest of the activeoptics parameters. The set of active modes is defined by the procedure described at the end of § 5. For the VLT the average coefficient of ^2, i for random forces in the range of it 1 N is 85 nm. With random force errors of 5% of the nominal load of 1500N the expected average coefficient of ^2, i is then 6375 nm. The frequency limit for the active modes to be considered is therefore V2, i \/6375/15 ^ 330 Hz. The data in table 1 show that for the VLT the modes up to e^, i should be corrected.

46

Active optics in modern large optical telescopes

[1, § 9

As discussed in § 2.3 the modes eo, \ and e\j are replaced by the corresponding Zernike polynomials for defocus and third-order coma. 9.3.2. Axial support of Ml Support density. Since three is the highest order in the set of active modes, six rings are sufficient to generate these modes with the required accuracy. A uniform distribution of supports on the rings together with the requirement that the number of supports on each ring is a multiple of three then leads to a total number of 150 supports with 9, 15, 21, 27, 36 and 42 supports on the six rings. As a result of the scaling law (6.2), the rms o^ of sag of the mirror between its supports under its own weight would have been approximately ten times higher than the one at the NTT. Since the distances between the supports are larger than in the NTT, the rms Ox of the slopes of the wavefront would have been only six times higher, that is, Ot ^ 0.15 arcsec. For a seeing of G = 0.4 arcsec this would have given a central intensity ratio of CIR^ 0.6, far below the specification of CIR = 0.992 for the high spatial frequency aberrations generated by the printthrough. To reach the CIR specification, which is equivalent to Ot ^ 0.02 arcsec, the primary mirror would have required approximately 400 supports, which would have added significant complexity and cost. Instead, the specification could be reached by replacing each of the single-point supports by tripods (Schneermann, Cui, Enard, Noethe and Postema [1990]). According to §§ 6.4.3 and 6.4.4, with 9 supports on the inner ring a correction of modes with rotational symmetries 7 and 8 is not possible without generating crosstalk. Indeed, corrections of the modes ey, i and ^g, i with coefficients of lOOOnm generate 470 nm of ^2, i and 1053 nm of ^ u , respectively, since the symmetries of the force distributions and the crosstalk mode add up to 9, the number of supports on the first ring. In addition, generating lOOOnm of ^g, i also produces 2217 nm of ^2, i and smaller amounts of other aberrations because of reaction forces on the three virtual fixed points. These two modes should therefore not be corrected permanently in closed loop, but only once after a preset to a new sky position. The crosstalk to lower order modes is then removed by subsequent corrections. Accuracy of the force setting. To achieve on average an accuracy of 30 nm rms for the softest mode ^2, i generated by random force errors, the force setting accuracy has to be of the order of ±0.4 N. To obtain this accuracy 95% of the times would require an accuracy of ±0.1 N. Furthermore, to have some margin for this important and delicate part of the active-optics system, the value finally chosen was ±0.05 N.

1, §9]

Active-optics design for NTT, VLT and Keck

47

Fig. 8. Differences between required functions and those generated by using five elastic modes. Solid line, pure third-order spherical aberration (p^); dashed line, third-order spherical aberration combined with defocus (p^ - 4. Ip^).

Force range. The force range was primarily driven by three contributions. First, the required switch from the Nasmyth to the Cassegrain configuration needs active forces in the range of -180 N to +470 N. Figure 8 shows the residual wavefront errors Wresid for attempts to generate either pure third-order spherical aberration (p'*) or third-order spherical aberration combined with an optimised defocus component (p^ - 4. Ip^). The latter gives, as discussed in § 8.2, a residual rms of the wavefront error 4.5 times smaller and also with approximately 45% smaller maximum forces. The forces can be further reduced by using less than the maximum five elastic modes for the correction. For example, using only three elastic modes reduces the maximum forces to 173N but increases the residual rms from 46 nm to 80 nm. The second major contribution of ±120 N are the forces given to the optical manufacturer for the correction of low spatial frequency aberrations in form of the active modes which were not removed during the figuring process of Ml and M2. The third contribution are forces foreseen for corrections of aberrations introduced by the support system and, possibly, by local air effects. The total range of active forces was then defined as -500 N to +800 N. Astaticity and friction, A closed-loop operation requires a stability of the optical configuration equivalent to an rms of the wavefront errors of o^ < 50nm over time periods of approximately one minute. The major sources are the nonastaticity of the active electromechanical actuators and friction effects both in the lateral supports and the passive part of the axial supports. The limits for friction were entered into the specifications for the supports. The astaticity of the active actuators is directly related to the spring constant D^ of the springs in the electromechanical actuators. According to finite-element calculations, for

48

Active optics in modern large optical telescopes

[1» § 9

a change of the zenith angle of 90"", the deformation of the mirror cell with a rotational symmetry 2 due to its own weight and due to deformations of the centerpiece are of the order of d^ ^ ±350 |im at the outer edge of the cell. The rate of change depends on the position in the sky to which the telescope is pointing. During one minute the maximum deformation is, at the site of the VLT, (ic, minute = ±0.00343 • dc = ±1.2 p-m. Owing to the non-astaticity of the active supports with a spring constant D^ the deformations of the cell generate force changes of ±fi?c, minute A over one minute. These forces will predominantly generate a deformation of the mirror in the form of the first elastic mode ^2, i of rotational symmetry 2. If the forces have, over the area of the mirror, roughly the functional dependence of this mode, the coefficients of ^2, i can be calculated by dividing the maximum forces at the outer edge by the maximum calibration force Fmax on the outer ring needed to generate a specified amount of this mode. If (^2, i,max is the tolcrablc upper limit for the change of the coefficient of ^2,1 over one minute, the condition for the spring constants of the active supports is given by A

^

^2,l,maxF,.ax

^ 0 . 0 0 3 4 3 Je, minute

^^. ^

^

With Fmax ~ 1.7N/|im and 02, i^ax = 50nm one obtains A ^ 0.07N/(im. Coupling to the mirror cell. The condition (6.6) shows that out of the six elastic modes with the lowest eigenfrequencies the modes ei, 1, ^3,1 and ^4,1 are non-six-sector modes, that is they cannot be generated on a support with six fixed points. This is, of course, only strictly true if the support system is infinitely rigid. Otherwise the stiffnesses of the mirror, of the passive hydraulic support system and of the mirror cell have to be properly combined (Noethe [2001]). One then gets for each mode e^ui a ratio ri^^j of the deformations on a six-sector support to the ones on an astatic three-sector support. For six-sector modes like the rotationally symmetric modes the ratio is one. With respect to deformations in the form of the lowest elastic mode ^2,1 the mirror cell of the VLT is approximately five times stiffer than the primary mirror. Together with the stiffness of the passive support this gives a ratio of r/2,1 = 0.33. For the second softest mode ^3,1 one gets r/3,1 = 0.70, whereas the third mode eo, 1 is a six-sector mode with r/o, 1 = 10. Since the first three modes account for a large fraction of the deformation under wind pressure, a six-sector support reduces the wavefront aberration by approximately 50%. If the valves between the two halves of each sector are fiilly closed, the filtering effect of the six-sector support on the non-six-sector modes applies to all temporal frequencies. But the valves between the two subsectors of each

Active-optics design for NTT, VLT and Keck

§9]

49

of the three sectors must be partially open to enable slow active corrections of all modes. In this context one can define a damping frequency v^ as the inverse of the relaxation time t^, that is, the time after which an instantaneously applied pressure difference between the two subsectors drops to 1/e. To assure that 90% of an active-optics correction is done after 10 s, the conditions for the relaxation times and the damping frequencies are ^r ^ 4s and Vd ^ 0.25Hz, respectively. Measurements of wind pressure variations on a 3.5 m dummy mirror in the NTT enclosure have shown that the maximum of the power spectrum inside the dome is atfi*equenciesof approximately 2 Hz (Hortmanns and Noethe [1995]). Under the assumption that the mirror can instantaneously follow these pressure variations, a six-sector support with a relaxation time of 4 s will reduce the deformations for most of the relevant frequencies. Calculations with spectra obtained with pressure sensors on a dummy mirror inside the NTT enclosure have shown that the reduction is at least of the order of 40%. On the other hand, the coupling to the mirror cell over 4 s will generate wavefront errors in the mirror due to the flexure of the mirror cell, but these are only of the order of 17 nm.

9.3.3. Lateral support of Ml The VLT primary mirror cannot be supported in the plane of the center of gravity. Therefore, one needs axial forces around the edge to balance the moment generated by supporting the mirror at the center of the outer rim. The deflections obtained with the standard VLT boundary conditions are shown in fig. 9a (Schwesinger [1991]). The rms values o^ of the deflections and Od,resid of the deflections after subtracting third-order coma are shown in table 2. 1

deflect on (nm)

1

VLT

1

1

1

1

/

0.7450\

200

• ^0

_0.7500

N

,0.7529

^

^ ^. 0.7560

^\

•

•

-

^^0.7600

/

•7\^ / \ \ -

200

: (a) 1

1

1 1 1

0.2

-

P ^ 1

1

1

0.4

1

1

1

1

I

0.6

Fig. 9. (a) Deflections of the primary mirror of the VLT as functions of the normaUsed radius p for various fractions P of the weight supported by the tangential forces, (b) Lateral forces in the plane perpendicular to the axis of Ml with equidistant support points, (c) Lateral forces in the plane perpendicular to the axis of Ml with identical moduli.

50

Active optics in modern large optical telescopes

{\, ^ 9

Table 2 rms values o^ without and o^ resid ^ith subtraction of third-order coma of the deflections generated by the lateral support of the VLT with fractions /3 of the weight supported by tangential forces p

(7d (nm)

CTj, resid (nm)

P

^d (n^^)

^d, resid (nm)

0.7450

124.4

19.2

0.7560

49.2

6.1

0.7500

46.5

12.3

0.7600

111.5

7.0

0.7529

8.7

8.7

As in the case of the NTT, some of the deflections are very similar to thirdorder coma. But, contrary to the NTT, the deformations depend strongly on the ratio /?. It is therefore necessary to choose a ratio P ^ 0.15 to reduce the deformations to acceptable levels. The lateral forces projected onto the plane perpendicular to the axis of Ml, that is, the vectorial sum of the radial and tangential components only, are shown in fig. 9b. Unfortunately, the strong difference between the fractions of the weight supported by radial and tangential forces leads, with an equidistant distribution of the lateral supports as shown in fig. 9b, to three times larger lateral forces and therefore significantly larger stresses near the altitude axis than at angles of 90*^ from the altitude axis. The requirements to have not more than 64 lateral supports and to limit the lateral forces to 4000 N required a redistribution of the lateral supports. The new positions (jp, of the supports / were chosen such that the integrals of the force densities between q)i - (3/ and cpi + 5/ = (fi+\ - 6/ +1, where (3, is the identical distance between both the lower and upper integration bounds and cpi, gave identical total lateral forces. The resulting components in a plane perpendicular to the optical axis are shown in fig. 9c. 9.3.4. Position control of Ml and M2 If the specification for the low spatial frequency errors of 0[ = 0.034 arcsec is statistically split into three contributions, namely from the elastic deformation of Ml, from defocus and from decentering coma, an rms slope error of Ot ^ 0.02 arcsec could be allocated to each. For defocus and decentering coma this would require setting accuracies with rms values of a^ ~ 1.2 |im for movements in axial direction and (Jrot.coc ~ 14 arcsec for a rotation around the center of curvature, respectively. The specifications for the mechanical units were much tighter, namely o^ ~ 0.5 |im and (Jrot,coc ~ 0.3 arcsec. With these accuracies, which are also achieved in practice, the rms of the wavefront errors

1, § 9]

Active-optics design for NTT, VLT and Keck

51

are Ow,def ~ 35 nm and Ow,coma ~ 3nm. The control of defocus is therefore much harder than the one of coma and also of the shape of the primary mirror. If the mechanical specifications are fiilfilled and the three contributions are added up quadratically, the rms of the wavefront error from the low spatial frequency aberrations is of the order of 50 nm. Contrary to the secondary mirror of the NTT, the M2 of the VLT can be moved in all degrees of freedom, which also allows a motorised control of the alignment of the axes of Ml and M2. Furthermore, with the capability of a motorised control of the position also of Ml in five degrees of freedom, the telescope can be aligned such that the optical axis goes through the center of the adapter. 9.3.5. Wavefront analyser The Shack-Hartmann analyser of the VLT has a 20 by 20 lenslet array with lenslets with a side length of 0.5 mm and a f-number of 45. The lenslets therefore sample subapertures on Ml with a side length of 400 mm. The pattern fits on a CCD with a side length of 11 mm without the use of a reduction optics. With a pixel size of 23 |i,m all requirements listed in § 4.3 are then fiilfilled for a specified limit of 20 nm for the rms of the wavefront error generated by the noise of the wavefront analyser.

9.4. Active-optics design of the Keck telescope Each of the 36 segments of the primary mirror is supported by three 12-point whiffletrees. Low spatial frequency aberrations in the shape of an individual segment, mainly due to the manufacturing process, can manually be corrected by a warping harness. Each harness consists of 30 leaf springs, which apply moments about pivots of the whiffletree (Mast and Nelson [1990]). Through the use of the springs, the axial support forces can be adjusted at each of the support points, subject to the equilibrium conditions that the net forces and moments on the mirror be zero. The applied forces are also independent of the inclination of the segment. The optimum 30 pivot moments are calculated with a least squares fit of the deformations introduced by individual springs to the overall deformations of the segment, taking into account several hardware constraints. For the positioning of a segment in three degrees of freedom, each whiffletree is attached to a displacement actuator.

52

Active optics in modern large optical telescopes

[U § 9

The integration times in the phase camera system (PCS) described in § 4.4 are all of the order of 30 seconds, using stars of magnitude 9 in the passive tilt mode and of magnitude 4 to 5 in the other three modes. The segment phase mode uses 78 of the 84 segment edge midpoints. The 6 points closest to the center are omitted since the corresponding intersegment edges are partially obscured by the telescope tertiary tower. The diameter of the subapertures, centered at intersegment edges, of 120 mm is always smaller than the atmospheric coherence length for infrared wavelengths A > 2 jim. A complete alignment of the telescope optics is then done in three steps. First, the fine screen mode is used to measure and correct the defocus and decentering coma aberrations introduced by a despace of the secondary mirror as described in § 4.4. Without this step, these aberrations would be corrected by a then non-perfect alignment of the segments of the primary mirror. Second, either the fine screen mode or the passive tilt mode are used to stack the images of the 36 segments, that is to correct errors in the tilts of the segments. Finally, 78 relative piston errors of the segments are measured with the segment phasing mode. The appropriate piston movements to correct these errors are obtained from a least squares fit of the 36 axial movements to the 78 available data with the constraint of a zero mean movement. A full alignment takes approximately one hour. The need for bright stars prevents a full sky coverage, and the long time required for an alignment limits active-optics corrections based on data obtained with star light to open-loop control. For a change of the zenith angle of 90'' the primary mirror cell deforms primarily in the defocus mode by 170 |im rms, which is equivalent, in the worst sky position, to a change of 30 nm rms over one second. Since the support of the mirror as a whole is position based, the positions of the segments have to be adjusted at least once per second. Active-optics corrections therefore have to be done in open loop or in a combination of open and closed loop. An openloop control based on measurements after alignments at different zenith angles is probably not feasible, since the predictability to an accuracy of the order of, say 30 nm, for an overall deformation of 600 jim is not achievable, above all due to certainly existing hysteresis in the deformation of the mirror cell. The activeoptics system of the Keck telescope therefore works in two hierarchical levels. A lower level controls the shape of the primary mirror by an internal closed loop, based on internal measurements of the relative positions of the mirror, and an upper level controls the residual deformations of the primary mirror and the alignment of the primary and secondary mirrors in continuous open and periodic closed loop based on measurements with star light. For the lower level control, capacitive devices measure the changes in the relative height of adjacent

1, § 9]

Active-optics design for NTT, VLT and Keck

53

segments in the direction normal to the surfaces at intersegment boundaries. Two sensors are located at every intersegment edge close to the end of the edges. After an alignment the readings of the, in total, 168 sensors are stored as reference values. During operation the actual readings of the sensors have to be kept as close as possible to the reference values. The required movements of the 108 actuators, maintaining the average tilt and piston of all segments, are calculated from the 168 differences of the sensor readings via a least squares fit. The corrections are done twice per second. The quality of the correction depends, apart from the noise in the actuators, predominantly on the characteristics and noise of the sensors. The dependence of the sensor readings on the inclination generates mirror deformations in the defocus mode. But these dependencies can be accurately calibrated. The unavoidable random noise in the readings will only introduce random errors in the shape of the mirror. Without any other systematic error sources, the shape of the mirror would be stable, and the mirror could be regarded as a passive element without the need for correcting the shape in the upper level active-optics loop. But systematic error sources exist in the form of drifts of the sensor readings and other unknown effects. Whether the corrections of the ensuing wavefront aberrations can be done in open or closed loop depends the predictability and stability of the errors. On the one hand, the unknown effects may be predictable, for example from measurements of the deformations as functions of the zenith distance. They are then correctable in open loop, which in practice would be equivalent to a change of the reference values of the sensor readings as functions of the zenith angle. On the other hand, the drift of the readings is usually not predictable and requires closed-loop corrections, that is, a new alignment based on measurements with the PCS. The upper active-optics level therefore consists on the one hand of continuous open-loop corrections of the primary mirror and also of the alignment of the secondary mirror, and on the other hand of closed-loop realignments of the primary mirror segments at longer time intervals, typically of the order of one month. The active-optics systems in the Keck telescopes with their segmented primary mirrors and the NTT and VLT telescopes with their thin meniscus mirrors with force based supports are in principle similar, if the role of the basic astatic support of a thin meniscus mirror is seen as equivalent to the lower level closed-loop control of a segmented mirror. Both attempt to provide, at least to first approximation, a stable shape of the primary mirror independent of the inclination of the telescope. Whereas in the Keck telescope the residual errors, as well as the alignment of the two mirrors, are corrected in continuous open and sporadic closed loop, in the NTT and VLT this is done in closed loop.

54

Active optics in modern large optical telescopes

[1, § 10

§ 10. Practical experience with active optics at the NTT, the VLT and the Keck telescope 10.1. Intrinsic accuracy of the wauefront analysis The intrinsic quality of the wavefront analysis depends strongly on the centroiding accuracy and therefore on the number of photons in the brightest pixel of any of the Shack-Hartmann spots. With light levels of maximum pixel values of the order of a third of the saturation level of the CCD the error in the coefficient of the mode ^2,1 due to a finite flux is of the order of lOnm (Noethe [2001]). An upper limit for the intrinsic errors of the full analysis can be deduced from simultaneous wavefront measurements with two wavefront analysers. With integration times of 30 seconds, the rms of the variation of the differences between the coefficients of the elastic mode ^2.1, measured by the two analysers, was of the order of 40nm (Noethe [2001]). If all these variations were generated by intrinsic centroiding errors and not by residual anisoplanatic effects, which certainly exist, the rms of random centroiding errors would be of the order of 3% as can be seen from fig. l i b (§ 10.3.2). Especially for the coefficients of the lowest-order modes the effects of the intrinsic errors are negligible compared with the variations introduced by the air, even for integration times of 30 seconds, as described in § 10.3.2. In the Keck telescope the relative piston wavefront values of adjacent segments can be measured with an accuracy of 50 nm in the broadband and 12nm in the narrowband mode. The accuracy of the tip-tilt measurements of the segments is of the order of c/go ~ 0.03arcsec. The uncertainties in the measurements of errors due to segment deformations with the ultra-fine mode are of the order of c/go ~ 0.065 arcsec, or 20-25 nm rms for the lowest Zemike modes.

10.2. Actiue-optics operation at NTT and VLT 10.2.1. NTT The NTT suffered from spherical aberration which was caused by incorrect polishing of the primary mirror due to an error in the assembly of the null lens. In terms of third-order spherical aberration the wavefront error was of the order of 3500 nmp"^. This error alone generated a point spread ftinction with d^o ^ 0.7 arcsec exceeding the specification of (igo = 0.4 arcsec for an operation in the passive mode. Without the use of the active-optics system the primary mirror would have had to be repolished.

1, § 10]

Practical experience with active optics at NTT, VLT and Keck

55

A correction required forces of 420 N with the cahbration forces calculated by Schwesinger [1988], and 240 N with a calibration using the two lowest elastic modes of symmetry 0. The force range for corrections with the mechanical levers for a zenith distance 0z is approximately ±0.3 • 8OOcos0zN. A correction of spherical aberration with the force adjustments of the levers would therefore have been possible near the zenith only, with little reserves left for the correction of other aberrations. Instead of using the adjustable counterweights, the bulk of the error is therefore corrected with the springs which supply correction forces independently of the zenith angle. But another problem caused by the strong correction forces remains. Since the axial support system of the NTT is a pure push system, a negative correction force Fcorr at a given support cannot be higher than the gravitational load FQ at this support. Since the maximum negative active forces for the correction of spherical aberration alone are of the order of -200 N, the maximum usable zenith angle ^z.max defined in §6.3 is at most of the order of 75^ With a thickness of 241 mm of the primary mirror the NTT can, under average seeing condition, be operated with a few corrections per night. But, under good seeing conditions of, say, 6) = 0.5 arcsec the active-optics system should operate in closed loop. It has been shown that the optical quality of the NTT can then reach the specification of t/go = 0.15 arcsec for an operation in the active mode (Wilson, Franza, Noethe and Andreoni [1991]). 10.2.2. VLT According to the scaling laws for the wavefront and slope errors in eq. (6.4), the flexibility of the primary mirror of the VLT exceeds the one of the NTT by factors of 37 and 16, respectively. Therefore, the VLT has to be operated in the active mode all the time, even under bad seeing conditions. In principle, the corrections could be done in open or in closed loop. Since the closedloop corrections work well with an extremely low failure rate, initial open-loop corrections are done only after presets to new sky positions. Just after the installation of the telescope, a single manual intervention may be necessary to reduce the wavefront aberrations to levels which allow analyses with the wavefront analyser and therefore automatic corrections. The reason is, that without any correction forces, that is, when the mirror is supported by the passive hydraulic system alone, the transverse aberrations in the focal plane may be so strong that the Shack-Hartmann pattern is heavily distorted. Consequently, a significant number of the spots may be vignetted by the mask in the Shack-

56

Active optics in modern large optical telescopes

[1, § 10

Hartmann sensor. It is then necessary to remove manually, for an arbitrary zenith angle in a trial and error mode, the bulk of the two largest aberrations, namely third-order coma and the lowest elastic mode of rotational symmetry 2. The coefficients of these modes can be estimated from defocused images. This may take an hour, after which the transverse aberrations are sufficiently small to be analysed automatically. Such a manual intervention is therefore only necessary once after the installation of the telescope. Accurate coefficients of the active modes for the initial open-loop correction after presets will then, for all zenith angles, be obtained with the wavefront analyser, and stored as a look-up table in the database. After a preset to a new position in the sky the images are, without a correction, visibly deformed. Although the corresponding wavefront errors, which are dominated by the mode ^2, i, would not cause any problems for the automatic wavefront analysis, a first correction is always done in open loop based on the look-up table mentioned above. Afterwards continuous closed-loop corrections will be started. Since the maximum pixel values depend on the magnitude of the guide star, its color and the current seeing, the actual integration time is adapted to reach for the brightest pixels a level of at least 50% of the saturation level. If the CCD saturates with integration times of 30 seconds, exposures with shorter integration times are averaged. If the maximum pixel counts are too low, the integration time may be increased up to 60 s. Stars with magnitudes of the order of 12 to 13, which already guarantee a fiall sky coverage, are ideal, although stars with magnitude as faint as 15 may, depending on the color, be usable. In addition the primary mirror is kept in a fixed position with respect to the Ml cell by changing the oil volumes in the axial and lateral hydraulic sectors of the Ml support. Approximately 12 000 wavefront analyses and corrections are done on each telescope per month. All relevant data, in particular the coefficients of the modes and the residual rms aresid, are logged for further off-line processing. Apart from the correction of the optics these measurements are also an important maintenance tool, since they can detect errors in the telescope optics which may, because of the strong influence of the atmosphere, not be easily visible in the final image. An important feature of the VLT is the control of the temperature of the primary mirror by a cold plate under its back surface and of the air inside the enclosure also during the day by a ventilation system (Cullum and Spyromilio [2000]). Both temperatures are set to the outside temperature expected at the beginning of the night and, during the night, the mirror temperature is equilibrated to the normally falling temperature of the ambient air with the cold

1, § 10]

Practical experience with active optics at NTT, VLT and Keck

57

plate. The temperature differences are most of the time within a narrow band of ±1", for which the effects of dome and mirror seeing on the image quality are insignificant (Guisard, Noethe and Spyromilio [2000]).

10.3. Closed- and open-loop performance of the VLT 10.3.1. Purity of modes generated during correction An important criterion for the functioning of the corrections is the purity with which the active modes can be generated. This can be checked by generating large wavefront errors in a single mode and measuring the generated coefficients of all modes. If at all, crosstalk will mostly occur into lower modes of the same symmetry, and most important, into the softest mode ei, i. Several measurements have to be averaged to distinguish real crosstalk from the normal variations of the coefficients generated by the air as described in § 10.3.2. The strong crosstalk of 47% from the mode ey i into the mode ^2,1 mentioned in §9.3.2 could be verified. Other crosstalk of the order of 20% exists from some modes of higher order into lower order modes of the same rotational symmetry or into the softest mode ^2,1, that is,fi*om^4,2 into e^^ \ and ^2,1, from ^2,3 into ^2,2 and ^2,1, and from ^0,3 into ^0,2 and eo, i- Since the coefficients of these higher order modes are always small, the crosstalk is not significant. For the rest of the active modes the crosstalk into other modes is smaller than 10% and therefore also negligible. 10.3.2. Wavefront variations without corrections To measure the evolution of the wavefront errors primarily as a function of the zenith angle, wavefront measurements have been done, without performing any corrections, following a star going through a position close to the zenith. The 7-component of the coefficient of the elastic mode ^2,1 obtained during such a drift measurement, which started at a position near the zenith, is shown in fig. 10a. Its evolution can clearly be separated into a smooth low temporal frequency variation representing elastic effects, and high temporal frequency variations representing primarily atmospheric effects, as will be shown later on. The low temporal frequency behavior is obtained by fitting a sixth-order polynomial, indicated by the dashed line in fig. 10a. The difference between the measured data and the fitted curve is shown in fig. 10b. The average of the residual rms Oresid during this measurement was approximately lOOnm. From these data one can calculate rms values Oeia of the low and Ohf of the high temporal frequency variations of the mode ^2,1, and similarly of all other

58

Active optics in modern large optical telescopes 1

nm

1

1

,

,

[1, §10

_

mode Cg 1 y - c o m p o n e n t 6000

~ 4000

-

2000

-

0

- f

(a)

-

zenith angle ,

,

1

,

:

Fig. 10. (a) ^-Component of the coefificient of the elastic mode ^2, l ^s a function of the zenith angle. The dashed line is a best fit of a sixth order polynomial, (b) Residual variations after the subtraction of the fitted polynomial. 1 '

-

1

1 2 3 ~ 4 5 6 7 ~ 8 ~

- \ / \

W \ W

• -A \

A

-

'

'

63,1 •^0.2

e^i Z,,2 ^5,, ^2.2 ^0.2

9 10 11 12 13 14 15 16

-

ee, ^3.2 e,2 e,^ e,3 e^, 6^3 e„,3

\ /'* \

-

-

_ -

>*
• log

io(^

1

,

,

,

A.

la / n m )

log ,o(500000/(i^

: (a)

150

100

-

A

_

nm

1

,

1

_

sec)^) 1

i

1

10 mode 15

i

"

, , ,,

1 2 3 4 5 6 7 8

~~ A

-t -\ \ •

- U' \

1 ' -

^31

~

•^0,2

-

^4,,

-

Zl,2

-

^5,1

~

^2,2

^

^0,2

•

\ /' \/ 4

50

~ - •.

9 10 11 12 13 14 15 16

'

1 '

~

63.2

-

e,,2

-

-

e,,i

-

-

64,2

-

-

^8,1

_

~

^2,3

_

"

^0,3

^hf

A ^Kolm A

• ^cenl

^

~

A -A

^

""""•—^#L '^

, , ,,

~

1 1

,

'

1 ,

10 mode 15

Fig. 11. (a) Correlations between inverse stiffness of modes and measured rms values of the changes of the coefficients, (b) Correlations between expected rms values of the variations of the coeflhcients due to centroiding errors ((Tcent) or Kolmogoroff turbulence ((JKolm) ^^^ the rms o^f of the measured high frequency non-elastic variations.

measured modes. These rms values a^a and Ohf are displayed as solid circles in figs. 11a and lib, respectively. Because of the large variation of the figures for (Tela for different modes, the data in fig. 11a have been plotted on a logarithmic scale. The figures for OQ\^ should be compared with parameters which describe the expected elastic overall variation of the coefficients with the zenith angle, and those of Ohf with the expected measuring noise due to centroiding and to high temporal frequency variations generated by the air.

1, § 10]

Practical experience with active optics at NTT, VLT and Keck

59

Elastic low temporal frequency variations. According to eq. (5.2) the energy of a certain mode for a unit rms deformation, and therefore also its stiffness are proportional to the square of the eigenfrequency of the mode. As a consequence, the coefficient of any mode contained in a deformation generated by random forces will be inversely proportional to the stiffness of this mode. In fig. 11a the inverse figures of the squares of the eigenfrequencies are represented by triangles. The arbitrary scale factor of 50000 mentioned in the plot has been chosen such that, with the exception of defocus and coma, the triangles fall close to the corresponding circles, representing the measured elastic rms variations Oeia. The measured coefficients of ^3,1 and ^4,1 are smaller than expected, whereas the coefficients of Zemike defocus and coma are much larger than expected from the elastic properties of the primary mirror. These comparatively large variations of defocus and coma are due to the deformations of the structure rather than elastic deformations of the primary mirror. Coma, in addition, could also be generated by systematic effects in the lateral support system. The proportionality between the variations of the coefficients of the modes and the inverse of their stiffnesses indicates that the deformations are not generated by systematic effects, but rather by random force errors. The coefficient of ^2,1 changes by approximately 7000 nm for a change of the zenith angle of 45"". Assuming that the change is equally generated by random-force errors in the axial and the lateral support, the random-force errors in the axial support would be in the range of ±60 N, which is approximately 4% of the nominal axial load on a single support at zenith. Non-elastic high temporal frequency variations. The coefficients acent of the considered modes expected in a spot diagram generated by random centroiding errors with an rms of 2% of the pixel size are represented by squares in fig. lib. Not only the moduli, but also the relative values of the coefficients expected from centroiding errors are very different from the corresponding measured figures Ohf, represented by the circles. In particular the comparatively large values for the low order coefficients indicate that, at least for the lowest, say, eleven modes, the measurements are not limited by the accuracy of the wavefront analysis. Instead, the relative ratios agree nearly perfectly with the ones expected from fully developed Kolmogoroff turbulence (Noll [1976]). This indicates that the high temporal frequency variations of the wavefronts averaged over 30 seconds are predominantly generated by the air. For an assumed atmospheric coherence length or Fried parameter of TQ ~ 500 mm, the expected coefficients, represented by triangles in fig. lib, are nearly identical to the measured coefficients.

60

Active optics in modern large optical telescopes

[1, § 10

10.3.3. Open-loop performance The performance of the telescope achievable with open-loop active-optics corrections can be measured by presetting the telescope to zenith distances in the range from 5"" to 65'' in steps of IS'' several times down and up. At each position the calibrated forces for the specific zenith angle are applied and the wavefront aberration is measured. The two important recorded figures are, at each zenith angle, the differences Achi between the average values of the coefficients measured after a preset from a previously higher or lower zenith angle, and the rms values o^c of the scatter of the coefficients around the average values. Not considering the results for defocus, which are, in addition to elastic effects, strongly influenced by temperature variations, the only significant differences Achi are of the order of 600 nm for ^2,1, 90 nm for ^2,1 and 15nm for ^4,1. The ratios follow the ratios of the corresponding stiffnesses quite well. Similarly, the most important scatter comes from ^2,1 with o^^ ~ 300 nm, which has to be compared with the scatter ofo ^ 150 nm introduced by the atmosphere. The additional errors arising from an open-loop operation therefore generate, given a good calibration of the active forces with the zenith angle and in the case of defocus with the relevant temperatures, slope errors of the wavefront with o ^ 0.05arcsec. This is equivalent to a diameter d^o ^ 0.1 arcsec for the diameter of the circle containing 50% of the geometrical energy, which would have to be added quadratically to the FWHM of the image obtained with an optimum closed-loop correction. An open-loop operation of the VLT is therefore, with only a minor reduction of the image quality, feasible. It would still be mandatory to make frequent wavefront analyses to detect any changes in the optomechanical system invalidating the look-up table used for the open-loop corrections. 10.3.4. Closed-loop performance Closed-loop measurements showed that, on average, the coefficients of the modes are proportional to the residual rms (Jresid of the wavefront error after the subtraction of the contributions from the active modes (Guisard, Noethe and Spyromilio [2000]). The constants of proportionality are therefore a measure of the average content of the modes in the measured wavefront. Figure 12 shows the rms values of the coefficients measured in closed-loop operation. To be able to compare the data with the ones measured without corrections, where during the drift measurement the residual rms aresid was on average of the order of lOOnm (see § 10.3.2), the values Oci plotted in fig. 12 also correspond to Oresid ~ lOOnm. Apparently, the correlation with the rms aKoim of the coefficients expected from

§ 10]

Practical experience with active optics at NTT, VLT and Keck 1

nm 250

-\ 200 - \ \

1

1

r

f

150

- 1 A

100

r

1 ' '' ' 1'n 1 1 - ^z.x 9 2 - ^3,1 1 0 3 ~ ^0.2 11 4 - e^ J 1 2 5 - Zl,2 1 3 6 - ^5,1 1 4 7 " ^2,2 1 5 8 ~ ^0,2 1 6

1

^

61

\

- ee,, -

^3,2

-

- e^g : - e^'j - e^'g -_ -

63,

-

-

62,3

_

-

e,,3

-

V \ \

A a^^,^ \

t

_

• A

50

•

', n

1

0

5

1

1

1

1

1

10

1

mode 15

Fig. 12. Correlations between expected rms values of the variations of the coefficients due to Kolmogoroff turbulence (crj<;^olm) ^^^ the rms 0^ of the coefficients in closed-loop operation.

pure Kolmogoroff-type turbulence, represented with an appropriate scaling by the triangles, is as good as in fig. 1 lb. The only difference between the two plots is that the coefficientsfi*omthe closed-loop operation are, on average, 50% larger than the ones measured without corrections. Therefore, not only the wavefront errors generated by the air itself, but also the wavefront errors generated by the telescope optics follow the Kolmogoroff statistics. This indicates that the errors in the telescope optics are mainly introduced through the corrections, based on the measured wavefront errors generated predominantly by the air. Other mechanical error sources are effectively negligible, since they would generate predominantly the lowest mode ei, i. The coefficients of this mode would then, compared with the ones of the other modes, be larger than expected from Kolmogoroff statistics. As the correlation between successive data in fig. 10b is only of the order of 25%, the high temporal frequency variations of the coefficients are effectively random. A correction based on these data will therefore increase the rms of the wavefront error by \/2, but will not change the ratio of the coefficients. This is exactly what can be seen from a comparison of the plots in fig. l i b and fig. 12. Summing up quadratically the values of all the coefficients in fig. 1 lb, the rms of the total low spatial frequency wavefront errors generated by the telescope optics itself is on average of the order of 250 nm. Whereas the wavefront error due to the air cannot be eliminated with active optics, the error in the telescope optics can be avoided by using filtered data for the correction. The filter should eliminate the high temporal frequency variations due to the air, and the active

62

Active optics in modern large optical telescopes

[1, § 10

optics corrections would then follow the dashed line in fig. 10a. Apart fi*om the alignment, the quality of the telescope optics would then be limited by the accuracies of the force and position settings, and according to the specifications for the correction devices the remaining wavefront aberrations would have rms values of the order of 50 nm. The telescope optics itself can therefore approach the diffraction limit even with the measuring limitations imposed by the atmosphere. As a consequence, the rms wavefront errors measured in closedloop operations should be reduced by a factor of approximately 1/A/2, and should then be identical to the ones of the high temporal frequency aberrations shown in fig. lib. The importance of a filter to remove the quickly varying local air effects from the closed-loop corrections is obvious also from three other considerations. First, a comparison of the two plots of fig. 11 shows that for all modes above mode 7, i.e. ^2,2, the rms values of the high temporal frequency variations are larger than the rms of the smooth variations from small to large zenith angles. For these modes the corrections are therefore much larger than the elastic variations during the time between two corrections. It would even be sufficient to correct these modes only once after a preset to a new sky position. Second, the maximum correction forces generated by the variations of the coefficients due to local air effects are of the order of 30 N, if all modes except ^7,1 and eg, 1 are corrected. Although the support system can apparently, despite the large force changes, accurately correct even the softest mode ei, \, such correction with strong and frequently changing forces should be avoided. If only the modes ^2,1, ^3,1 and ^4,1 were corrected, the maximum forces would be of the order of 2.5 N, and if the corrections of all modes were based on filtered data, the maximum forces could be further reduced to approximately 0.3 N. Third, if the corrections are based on unfiltered data, the optical quality of the telescope optics itself due to low spatial frequency errors is approximately given by the quadratic sum of the rms values of the slopes corresponding to the coefficients given in fig. lib. The sum is equal to Ox ^ 0.06arcsec, which is about twice as large as the specified value for the active-optics control errors of 0.034 arcsec given in §9.1. The best image obtained so far with the VLT with a long exposure time of 5 minutes had a FWHM of 0.25 arcsec, during which time five closed-loop active-optics corrections were done, and the best image with a short exposure time of a few seconds had a FWHM of 0.18 arcsec. The latter figure is certainly close to the ultimate limit achievable with ground-based active telescopes. To approach the much smaller diffraction limit of 0.015 arcsec FWHM one has to use, in addition, adaptive optics to correct the fast aberrations introduced by the atmosphere.

1, § 10]

Practical experience with active optics at NTT, VLT and Keck

63

10.4. Alignment of the VLT According to eqs. (7.1) and (7.2) a misalignment also adds linear terms to the field dependence of third-order astigmatism. These are, since the wavefi-ont is measured with the guide star in the field, interpreted as field-independent astigmatism, and, with an active-optics correction incorrectly introduced at the center. The error at the center should be significantly smaller than the error of (Jt = 0.034 arcsec allocated to the active-optics control, say Ox = O.Olarcsec, which is roughly equivalent to a coefficient of 180nm of third-order astigmatism p^ cos2(/P. From eqs. (7.1) and (7.2) with the value of 5i for the VLT and a field angle of 13 arcmin the tolerable misalignment angle a is then of the order of 30 arcsec. With just one wavefront analyser the misalignment angle can only be detected by measuring the astigmatism at a minimum of two positions in the field. Noise is introduced by two effects. First, inaccuracies in the measurement of the coefficient of coma give rise to movements of the secondary mirror which modify the true misalignment angle with every correction. Second, inaccuracies in the measurement of the coefficient of astigmatism lead to errors in the misalignment angle calculated from eqs. (7.1) and (7.2). Together, they generate under average conditions, errors for a of the order of 50 arcsec. Simultaneous measurements with two wavefront analysers in the center and in the field reduce the errors in the coefficients of astigmatism, and consequently the error in a to about 30 arcsec. Under the assumption that the misalignment does not show strong hysteresis, the angle a can be determined as a fiinction of the zenith angle and be corrected in open loop. Otherwise, two permanently installed wavefront analysers in the field are required for a closed-loop control of the alignment. 10.5. Plate scale control If the plate scale were measured in a closed-loop operation every minute with integration times of one minute, its variations could be split into two contributions, similarly as for the variations of the coefficients shown in § 10.3.2. On the one hand, smooth systematic variations of the plate scale as a function of zenith angle similar to the dashed line in fig. 10a will be generated by deformations of Ml in the form of the first mode of rotational symmetry 0, which is similar to defocus, and a correction of this defocus by an axial movement of M2. If this smooth variation can be measured as a function of the zenith distance, it can be corrected in open loop. On the other hand, fast defocus changes introduced by the air have optically the same effect as a change of the

64

Active optics in modern large optical telescopes

[1, § 10

curvature of Ml and will, with a closed-loop operation, also be corrected by an axial movement of the secondary mirror, generating fast, random variations of the plate scale. Relative variations of the plate scale of approximately 10~^ at the VLT, measured with integration times of the order of one minute, can quantitatively be explained by defocus variations due to the air. By filtering the measured defocus data as described in § 10.3.4 these variations can be reduced by a factor of \/2. But for long exposures the random variations of the plate scale are also, to some extent, averaged out.

10.6. Active-optics operation and performance of the Keck telescope At the beginning of each night and possibly also a few other times at night, a stacking procedure is used to correct defocus and decentering coma errors introduced by a misalignment of M2. By adding a constant to all position sensor readings, the Ml defocus mode is generated, resulting in a uniform spread of the images formed by the individual segments. From the difference between the expected and measured positions of these images the overall focus and decentering coma error is calculated and corrected by a corresponding movement of M2. With a subtraction of the previously added constant in the position sensors, the images are then again superimposed. This procedure, which is somewhat similar to the PCS passive tilt mode, can also not distinguish between wavefront aberrations introduced by Ml segment misalignments and M2 positioning. In particular, since identical offsets at all position sensor may occur with much higher probability than expected from random errors in the sensors, substantial amounts of the Ml defocus mode can be generated. As these are compensated by the corresponding axial movements of M2 to first order only, an appreciable amount of the residuals mentioned in §4.4 may accumulate over time. Similar residuals, but usually in much smaller amounts, may occur in the decentering-coma mode. This stacking procedure may also be used to remove other segment tilt errors. The tilts cannot be corrected without changing the heights at the centers of intersegment edges. Additional piston changes of the segments are then introduced, which minimize the rms of the change of the relative heights at these locations. The revised sensor readings are then stored as the new reference values for the position sensors. A correction with the stacking procedure takes approximately 10 minutes. To remove in particular the errors accumulated due to the near-degeneracy of the Ml focus mode with global focus, a realignment of the whole telescope

1, § 11]

Existing active telescopes

65

optics, which takes about one hour, is done approximately once per month. The position of the secondary mirror is controlled in open loop, whereas the shape of the primary mirror is controlled with an internal closed loop running at 2 Hz. The combined effect of the wavefront errors attributed to the active-optics control system, which are independent of the zenith distance, is ^go ~ 0.28arcsec. This is dominated by the long-term drift in the readings of the piston sensors. Both the sensor noise and the actuator noise are of the order of 5 nm rms, which, with random errors, would lead to an image blur with Jgo = 0.027 arcsec. The errors in the alignment of the individual segments are Jgo ~ 0.10 arcsec in tilt and approximately 30 nm rms in piston. The distribution of the tilt errors over the segments is not entirely random. The ratios of the coefficients of the normal modes in the expansion of the wavefront error follow quite well the ratios of the eigenvalues of the control matrix described in § 7.2. The absolute values of the coefficients immediately after a mirror alignment are approximately three times larger than expected from the random sensor noise, and increase by another factor of two after 11 days without a realignment. The rms of the edge discontinuities is 76 nm, with contributions of 24 nm from tilt errors, 23 nm from phasing errors and, the largest figure, 59 nm from segment aberrations. The zenith-distance-dependent aberrations are, with the internal closed-loop control of the primary mirror and accurate calibrations of the dependencies of the piston sensor readings on the inclination, reduced to Jgo ~ 0.5 arcsec at a zenith distance of 55"". The deformations in non-focus-modes are, as ftjinctions of the zenith angle, quite predictable. With a correction of these aberrations in open loop, the image blur due to zenith-dependent active-optics control errors is anticipated to be of the order of ^80 ~ 0.03 arcsec.

§ 11. Existing active telescopes Apart from the telescopes NTT, VLT and Keck already mentioned, other modern large telescopes also rely on active-optics corrections. The largest group of these telescopes are the ones with solid thin meniscus primary mirrors. The active optics of the 2.5 m Nordic Optical Telescope (NOT) is operated in pure open loop. The 3.5 m Galileo telescope (TNG) is similar in design to the NTT. In the 8.2 m Subaru telescope each of the 264 supports supplies active axial and passive lateral forces (lye [1991]). To avoid unwanted moments from the lateral forces, the application points of the support forces have to be in the neutral surface of the mirror, which required the drilling of 264 bores into the solid meniscus blank. The two 8 m Gemini telescope are

66

Active optics in modern large optical telescopes

[1, § 12

equipped with a three stage axial support system (Stepp and Huang [1994]). The passive astatic part consists of a continuous pneumatic support under the back surface supporting 75% of the weight, and a hydraulic support like the one in the VLT supporting the remaining 25%. The third stage has active electromechanical actuators as in the VLT. Telescopes with structured primary mirrors like the 3.5m WIYN, the 6.5m Magellan and the 2 x 8m LBT (Columbus) telescopes with honeycomb mirrors have the advantage that the primary mirrors are, for the same weight and diameter, stififer than the thin meniscus mirrors, making them more resistant to wind buffeting deformations. But with diameters as large as 8 m also these mirrors require active-optics control for the fine tuning of the optics. Each actuator applies both axial and lateral forces in the neutral surface of the mirror. The increased stiffness has the disadvantage of reducing the dynamic range of active-optics corrections and corresponding reduction of the low spatial frequency tolerance relaxation in fabrication. The primary mirror of the Hubble Space Telescope was equipped with a figure correction system, consisting of 24 actuators. These could correct, for example, up to half a micron of astigmatism, but the range was far too small to correct the spherical aberration caused by incorrect polishing (see the remark in Mast and Nelson [1990]). Further details about existing active telescopes are given by Wilson [1999].

§ 12. Outlook The extension of the active-optics principles for two-mirror telescopes to telescopes with three or more elements and also with combinations of segmented and monolithic mirrors does not require new principles and techniques. For telescopes with more than two elements additional information about the misalignment can be obtained from measurements of the field dependence of Zernike polynomials with rotational symmetries larger than two. In general, the field dependencies of the aberrations are given by generalizations of the ovals of Cassini for third-order astigmatism, that is, with normally some other number of nodes than 2. Usually, the coefficients of the higher order polynomials in the field decrease rapidly with the order. In current large Ritchey-Chretien two-mirror telescopes the only significant field aberration is third-order astigmatism, but in larger multi-element telescopes higher order Zernike field aberrations may also be detectable. If not, it would be sufficient to have an alignment which only corrects the field dependence of third-order

1, § 12]

Outlook

67

astigmatism, although it is in principle not perfect. If they can be measured, an alignment has to be done by moving more than only two elements. The number of wavefront analysers has to equal at least the highest number of nodes occurring in the field patterns of any of the polynomials used for the alignment analysis. Consequently, two-mirror telescopes should also be equipped with two wavefront analysers in the field to be able to do closed-loop alignment corrections. Future extremely large telescopes may have more than one flexible monolithic mirror (Dierickx, Delabre and Noethe [2000]) and therefore the capability of correcting also field effects. Segmented mirror technology will become increasingly important for the fiiture generation of very large telescopes. For these, the development of active-optics control systems, which can operate in closed loop at time intervals of the order of one minute, will be essential. Telescopes with somewhat different applications of active optics include the Chinese LAMOST project (Su, Cui, Wang and Yao [1998]) and the hexapod telescope of the University of Bochum (see the overview given by Wilson [1999]). LAMOST is a 4m meridian-type Schmidt telescope with a thin reflecting aspheric corrector plate, where the optimum shape of this plate depends on the zenith distance. The required deformations as a function of the zenith distance could be introduced by active forces at the back surface of the corrector plate. In the 1.5 m hexapod telescope the primary mirror is effectively the thin front plate of the mirror cell. The support is position based and the mirror behaves like mirrors used in adaptive optics, in the sense that the actuation of one support generates a local deformation of the mirror. Active optics is particularly suited to telescopes in space. Since even with the disturbing effects of the atmosphere on the wavefront analysis, diffractionlimited performance of the telescope optics itself can be achieved on the ground for telescopes with mirror diameters of eight meters, it should be much easier to realise this goal in space. Low expansion glasses are still the favorite substrate for large mirrors. In the case of monolithic mirrors the advantage of the stability of the shape at different temperatures is, at least with a closed-loop active-optics system, irrelevant. Other substrates, especially aluminium, may be better suited for large blanks. The advantages are a higher thermal conductivity and therefore a faster equilibration with the ambient temperature, a lower safety risk, and possibly a reduced cost for the blank production. Long-term instabilities of the shape would most likely be in the low spatial frequency modes and could easily be corrected by active optics.

68

Active optics in modern large optical telescopes

[1

Acknowledgements The author would Uke to thank G. Chanan, S. Guisard, J. Spyromilio and R.N. Wilson for carefully reading the manuscript and for suggestions.

References Braat, J., 1987, J. Opt. Soc. Am. A 4(4), 643. Chanan, G., T. Mast and J. Nelson, 1988, in: Proc. ESO Conf. on Very Large Telescopes and their Instrumentation, Garching, Germany, ed. M.-H. Ulrich, Vol. I, p. 421. Chanan, G., J. Nelson, T. Mast, P. Wizinowich and B. Schaefer, 1994, in: Proc. Conf. on Instrumentation in Astronomy VIII, SPIE Proc. 2198, 1139. Chanan, G., C. Ohara and M. Troy, 2000, Appl. Opt. 39(25), 4706. Chanan, G., M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast and D. Kirkman, 1998, Appl. Opt. 37(1), 140. Chanan, G., M. Troy and C. Ohara, 2000, in: Proc. Conf. on Optical Design, Materials, and Maintenance, SPIE Proc. 4003, 188. Chanan, G., M. Troy and E. Sirko, 1999, Appl. Opt. 38(4), 704. Cohen, R., T. Mast and J. Nelson, 1994, in: Proc. Conf on Advanced Technology Optical Telescopes V, SPIE Proc. 2199, 105. Couder, A., 1931, Bull. Astron., 2me Serie, Tome VII, Fasc. VI, pp. 266, 275. Creedon, J.F., and A.G. Lindgren, 1970, Automatica 6, 643. Cullum, M., and J. Spyromilio, 2000, in: Proc. Conf on Telescope Structures, Enclosures, Controls, Assembly/IntegrationA^alidation, and Commissioning, SPIE Proc. 4004, 194. Dierickx, Ph., 1992, J. Mod. Opt. 39(3), 569. Dierickx, Ph., B. Delabre and L. Noethe, 2000, in: Proc. Conf on Optical Design, Materials, and Maintenance, SPIE Proc. 4003, 203. Guisard, S., L. Noethe and J. Spyromilio, 2000, in: Proc. Conf on Optical Design, Materials, and Maintenance, SPIE Proc. 4003, 154. Hortmanns, M., and L. Noethe, 1995, in: Proc. 9th Conf on Wind Engineering, New Delhi, India, p. 469. Hubin, N., and L. Noethe, 1993, Science 262, 1390. lye, M., 1991, Technical Report 2 (Japanese National Large Telescope, Japan). Lassell, W., 1842, Mem. R. Astron. Soc. XII, 265. Mast, T, and J. Nelson, 1990, in: Proc. Conf on Advanced Technology Optical Telescopes IV, SPIE Proc. 1236, 670. McLeod, B.A., 1996, Proc. Astron. Soc. Pac. 108, 217. Noethe, L., 1991, J. Mod. Opt. 38(6), 1043. Noethe, L., 2001, Active optics in large telescopes with thin meniscus primary mirrors, Habilitationsschrift D 83 (Technische Universitat Berlin). Noethe, L., and S. Guisard, 2000, Astron. Astrophys. Suppl. Ser. 144, 157. Noll, J.N., 1976, J. Opt. Soc. Am. 66, 3. Racine, R., D. Salmon, D Cowley and J. Sovka, 1991, Proc. Astron. Soc. Pac. 103, 1020. Ray, FB., 1991, in: Proc. Conf. on Analysis of Optical Structures, ed. DC. O'Shea, SPIE Proc. 1532, 188. Roddier, F, and C. Roddier, 1991, Appl. Opt. 30(11), 1325.

1]

References

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Schneermann, M., X. Cui, D. Enard, L. Noethe and H. Postema, 1990, in: Proc. Conf. on Advanced Technology Optical Telescopes IV, SPIE Proc. 1236, 920. Schroeder, D.J., 1987, Astronomical Optics (Academic Press, San Diego, CA). Schwesinger, G., 1988, J. Mod. Opt. 35(7), 1117. Schwesinger, G., 1991, J. Mod. Opt. 38(8), 1507. Schwesinger, G., 1994, Appl. Opt. 33(7), 1198. Shack, R.V, and B.C. Piatt, 1971, J. Opt. Soc. Am. 61, 656. Shack, R.V, and K. Thompson, 1980, in: Proc. Conf. on Optical Alignment, SPIE Proc. 251, 146. Stepp, L., 1993, Gemini report TN-O-G0002 (Gemini Observatory, Tucson, AZ). Stepp, L., and E. Huang, 1994, in: Proc. Conf. on Advanced Technology Optical Telescopes V, SPIE Proc. 2199, 223. Su, D., X. Cui, Y. Wang and Z. Yao, 1998, in: Proc. Conf on Advanced Technology Optical/IR Telescopes, SPIE Proc. 3352, 76. Troy, M., G. Chanan, E. Sirko and E. LefiFert, 1998, in: Proc. Conf on Advanced Technology Optical/IR Telescopes VI, SPIE Proc. 3352, 307. Wetthauer, A., and E. Brodhun, 1920, Z. Instrumentenkd. 40, 96. Wilson, R.N., 1978, in: Proc. ESO Conf on Optical Telescopes of the Future, eds E Pacini, W. Richter and R.N. Wilson (ESO, Geneva) p. 99. Wilson, R.N., 1996, Reflecting Telescope Optics I (Springer, Berlin). Wilson, R.N., 1999, Reflecting Telescope Optics II (Springer, Berlin). Wilson, R.N., K Franza and L. Noethe, 1987, J. Mod. Opt. 34, 485. Wilson, R.N., R Franza, L. Noethe and G. Andreoni, 1991, J. Mod. Opt. 38(2), 219. Wizinowich, P., T. Mast, J. Nelson and M. DiVittorio, 1994, in: Proc. Conf on Advanced Technology Optical Telescopes V, SPIE Proc. 2199, 94. Yoder, PR., 1986, Opto-Mechanical Systems Design (Marcel Dekker, New York).

E. Wolf, Progress in Optics 43 © 2002 Elsevier Science B. V. All rights reserved

Chapter 2

Variational methods in nonlinear fiber optics and related fields by

Boris A. Malomed Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

71

Contents

Page § 1. Introduction § 2.

73

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

87

§ 3. Variational approximation for the inverse scattering transform

.

123

§4.

Internal dynamics of vector (two-component) solitons

127

§ 5.

Spatially nonuniform fibers and dispersion management . . . .

140

§ 6.

Solitons in dual-core optical

164

§ 7.

Bragg-grating (gap) solitons

174

§ 8.

Stable beams in a layered focusing-defocusing Kerr medium . .

182

§ 9.

Conclusion

186

fibers

Acronyms adopted in the text

186

Acknowledgements

187

References

188

72

§ 1. Introduction 1.1. The nonlinear Schrodinger equation and simplest optical solitons The mathematical basis of nonhnear optics is Maxwell's system of equations governing propagation of electromagnetic waves in a material medium, combined with relations accounting for the nonlinear response of the medium to the electromagnetic field (Newell and Moloney [1992]). In most cases, application of well-known asymptotic methods makes it possible to derive simplified partial differential equations (PDEs) governing the spatial and/or temporal evolution of essential field modes in the medium. A typical and most important example of the thus derived asymptotic PDE is the nonlinear Schrodinger (NLS) equation, which governs the propagation of an electromagnetic wave in a glass fiber, or the spatial evolution of the electromagnetic field in a planar waveguide. In the case of a single-mode fiber, i.e., one permitting the propagation of a single electromagnetic-wave mode, the electric component of the field with a fixed polarization is taken in the form f (z, t) = u(z, T) Voir) exp(i^o^ - icoot),

(1)

where z, r and t are, respectively, the propagation distance along the fiber, the radial coordinate in the transverse plane, and time; the frequency COQ and wavenumber ko of the carrier wave obey a linear dispersion relation for the fiber, k = k(a)), and Vo(r) describes the transverse structure of the propagating mode [the physical field is given by the real part of the complex expression (1)]. The dispersion relation determines the carrier's group velocity Fgr = l/k\ dispersion coefficient D = -k", and "reduced time" T = t - z/Fgr, where the prime stands for the derivative d/do; taken at co = WQ. Both D > 0 and D < 0 are possible, being referred to as, respectively, anomalous and normal dispersion. The NLS equation for the slowly varying amplitude w(z, r) of the modulated wave (1), derived from the Maxwell equations in the absence of dissipation, is (Agrawal [1995]) \U, + \DUJT + Y\U\^U = ^.

(2) 73

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Variational methods in nonlinear fiber optics and related

fields

[2, § 1

Here, the nonlinearity coefficient is 7=-;—,

(3)

where «2, c and A^ff are, respectively, the Kerr coefficient, the hght velocity in vacuum, and the fiber's effective cross-sectional area. Usually, y is scaled out of eq. (2) by means of an obvious transformation. The dispersion coefficient D can also be scaled out, provided that it is constant. However, in many important applications which will be considered in detail below in § 5, D is a function of the propagation coordinate z. The dispersion D can readily be made variable (modulated), as it is contributed to by the material dispersion of the silica glass and the geometric dispersion of the fiber waveguide. These two contributions may nearly cancel each other near the zero-dispersion point, so relatively small variations in the fiber's cross-section area, while only slightly affecting y, can strongly change the small residual dispersion coefficient. Thus, a continuous variation of the cross-section in the process of drawing the fiber from glass melt gives rise to dispersion-decreasing fibers (see §5.1). Uniform fibers can be fabricated with different constant values of D, making it possible to build a long dispersion-compensated optical link by periodically alternating pieces with anomalous and normal dispersion. This is a basis for the dispersionmanagement (DM) technique, which finds important applications in transmitting signals through fiber-optic links in linear (Lin, Kogelnik and Cohen [1980]) and nonlinear regimes, see § 5.4. The same NLS equation finds another well-known application to nonlinear optics, describing the spatial distribution of the stationary electromagnetic field in a planar waveguide (film). In that case, the electric field with fixed polarization is taken as £(z,x, t) = u(z,x) Vo{y) exp(iA:oz - icoot)

(4)

(cf eq. 1), where x and y are transverse (relative to the propagation distance z) coordinates, directed, respectively, along the film and perpendicular to it, and the ftinction Vo(y) accounts for the transverse structure of the propagating mode. Note that, unlike the case of propagation in a fiber, the slowly varying amplitude u from eq. (4) is a function of the transverse coordinate x, rather than the temporal variable r. The NLS equation governing the spatial evolution of w(z, x) in the lossless waveguide can be derived, after rescalings, in the form (see details in the book by Hasegawa and Kodama [1995]) iw- + ^w.o- + \u\^u = U(x) w,

(5)

where, as in eq. (2), the cubic term is generated by the Kerr effect (a nonlinear correction to the effective refractive index in the material medium), while the

2, § 1]

Introduction

75

second-derivative term, unlike that in eq. (1), accounts for the spatial diffraction of the field, rather than temporal dispersion. The term on the right-hand side (rhs) of eq. (5) takes into regard possible modulation of the waveguide in the transverse direction, which gives rise to an effective real potential U{x). Note that the positive sign in front of the nonlinear term in eq. (5) assumes that the Kerr nonlinearity is self-focusing (corresponding to a positive nonlinear correction to the effective refractive index), which is the case in most optical media, including silica glass. In the opposite case of a self-defocusing Kerr nonlinearity, which occurs in semiconductor waveguides (see, e.g., the paper by Michaelis, Peschel and Lederer [1997] and references therein), eq. (5) takes the form iw- + \uxx - \u\^u = U(x) u. The NLS equation with constant coefficients is one of the basic equations of modem mathematical physics. This equation finds numerous applications, not only in optics, but also in plasma physics, hydrodynamics, etc. Its most fundamental property is exact integrability by means of the inverse scattering transform (1ST), which is based on a representation of the constant-coefficient NLS equation as a compatibility condition for two systems of auxiliary linear equations (see books by Zakharov, Manakov, Novikov and Pitaevskii [1980], Ablowitz and Segur [1981] and Newell [1985], and some details in §3 below). The exact integrability makes it possible to produce a vast class of exact solutions to the NLS equation, the simplest and most fundamental one being a soliton (solitary wave), 7]

f

Tl

\

Wsoi = -7= sech ^ ( r - To) exp(^ir/^z + i0o),

(6)

where r] is an arbitrary amplitude of the soliton, which also determines its temporal width ~ y/D/r] and propagation constant (wavenumber shift) \rf-, and To and 00 are arbitrary real constants. The NLS equation (2) is invariant with respect to the Galilean transformations, which allows one to generate a family of walking solitons (this term was introduced by Tomer, Mazilu and Mihalache [1996]) out of the "quiescent" one (6): T]

ft)

\

r

/

c \

c

Msoi = - ^ s e c h l - ^ ( r - c z - r o ) j e x p U i ('J^ ^ ;o ) ^ + ' 5 ' ' + '<*' (7) where c is a real walk parameter. Physically, c represents a shift of the central frequency in the soliton's Fourier transform, which gives rise to a velocity shift via the fiber's dispersion.

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Variational methods in nonlinear fiber optics and related

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[2, § 1

As concerns the propagation of a soliton in an optical fiber, the most important length scale is the soliton period ZQ: it is the propagation distance over which the soliton's phase changes by ^;r (Agrawal [1995]), so that ^0 = ^ .

(8)

As will be explained in different sections of this review, an essential transformation of a strongly perturbed soliton requires a propagation distance z > ZQ. The soliton solution (7) is characterized by its area, energy and momentum,

S = f

\u(T)\dT = JtJ-,

(9) /D

/ +00 CX)

/

|M(r)|'dT=^—r/,

1 uu^,dT= -^crj

7

(10)

(11)

oc

(the factor j in the definition of E is introduced in order to simplify notation below). The energy and momentum, which are defined for an arbitrary field configuration by means of the integral expressions in eqs. (10) and (11), are dynamical invariants (integrals of motion) of eq. (2), while the area is not a dynamical invariant. Due to the fact that the NLS equation is exactly integrable by means of 1ST, the energy and momentum are but the two first items in an infinite set of dynamical invariants conserved by the NLS equation. The third invariant is the Hamiltonian of the NLS equation, {D\Ur\'-YW\')dT,

(12)

OO

while higher-order invariants do not have a straightforward physical interpretation (Zakharov, Manakov, Novikov and Pitaevskii [1980]). Solitons in various optical media have attracted a great deal of attention, first of all, as objects for fundamental research. In glass fibers, temporal solitons, predicted by eq. (2), were first observed by Mollenauer, Stolen and Gordon [1980], and the first observations of spatial solitons in planar waveguides of various types, predicted by eq. (5), were reported by Maneuf and Reynaud [1988] and Aitchison, Weiner, Silberberg, Oliver, Jackel, Leaird, Vogel and Smith [1990]. Besides being a physical object of fundamental interest, solitons in fibers

2, § 1]

Introduction

11

also have a great potential for application to optical communications, as a basis for the so-called return-to-zero format of data transmission, in which a soliton carries a single bit of information. Detailed descriptions of this topic can be found in books by Agrawal [1997] and lannone, Matera, Mecozzi and Settembre [1998]. 1.2. Introduction of variational methods 1.2.1. Models without losses In a real physical situation, it is necessary to deal with perturbed (deformed) pulses whose area is different from that given by eq. (9). In other words, for real pulses the initial relation between their amplitude and width may strongly deviate from that for the ideal soliton, even if the functional form of the pulse is still close to sech. The evolution of such a perturbed soliton is a problem of great practical importance. Formally, it can be solved exactly by means of 1ST, but the exact solution is really usable only at an asymptotic stage of the evolution (at z —» oo), which makes it necessary to develop an approximation that yields a sufficiently accurate explicit result for all values of z. The corresponding variational approximation (VA) for solitons in optical fibers was introduced in the cornerstone paper by Anderson [1983], following the pattern of VA for solitons in other physical media (chiefly, plasmas), which had been developed earlier by Bondeson, Lisak and Anderson [1979]. The VA technique for optical solitons was fiirther developed in an important paper by Anderson, Lisak and Reichel [1988a]. These works became the basis for the rapid development of analytical methods in nonlinear optics based on VA. The approximation begins with postulating an ansatz, i.e., a trial analytical form of the field configuration sought for (in most cases, the configuration is a solitary wave). In the case of the NLS equation (2), a commonly adopted ansatz approximating a perturbed soliton is Wansatz(^, ^) = ^ s e c h f - j exp(i0 + iZ?r^).

(13)

The functional form of the ansatz is fixed as concerns its r-dependence, while it contains several free parameters, for instance the real amplitude A, width a, phase 0, and the so-called chirp b in the case of the ansatz (13). The free parameters are allowed to be fiinctions of the evolutional variable, which is z in the case of eq. (2). Equations governing the evolution of these parameters in z can be derived in a natural way, provided that the underlying equation(s) (e.g., eq. 2) can be

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Variational methods in nonlinear fiber optics and related

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[2, § 1

derived by means of the standard variational procedure, equating to zero the variational derivative bS/bu* of the corresponding action functional S{u,u*} (the asterisk stands for complex conjugation). The variational representation is usually available for conservative models [including those with an explicit coordinate dependence, e.g., the above-mentioned case when the dispersion coefficient D in eq. (2) is a function of z]. Only in some special cases do dissipative models also admit a natural variational representation, see below. The action is represented in the form S = J Ldz, where z is realized as the evolutional variable, and L is a Lagrangian, which is represented in its own integral form, = / Cdrdr,

(14)

where £ is a Lagrangian density, that must be real, and r is the vector set of transverse coordinates (implying the possibility to consider spatiotemporal evolution of fields in two- and three-dimensional dispersive nonlinear media). If the transverse coordinates are present, the ansatz must be a definite fiinction of both r and r. For the NLS equation (2), the Lagrangian density is C = ^i(w*w_- - uu*)-^D\ur\^ + ^y\u\\

(15)

and dr is dropped in eq. (14). Generally, in the case of a system of NLS-like equations for complex variables u„(z, r, r), the density is C = C [un, w*, ( w j _ - , « ) - , ( w „ ) , , « ) , , Vw,„ Vul), where V is the gradient with respect to the transverse coordinates. In this case, the equations following from the variational principle, bS/bu* = 0, take the form d

dC

d

dC

dz d[{K)_]^ dT d[{K)y

dC d{VK)

# = 0 .

(16)

The Lagrangian representation of the nonlinear wave equations is related to their Hamiltonian representation, which, for a broad class of equations of the NLS type, is

(«a=-ig,

(17)

where the fiinctional //{w, w*} is the Hamiltonian. In particular, for the NLS equation (2) proper, it is given by the expression (12).

2, § 1]

Introduction

79

In order to apply VA to a given problem, one should insert the adopted ansatz into the expression (14) and calculate the integral in an analytical form. The necessity to perform the integration analytically imposes conditions on the choice of the ansatz: on one hand, it must not be too primitive, in order to have a chance to accurately approximate basic features of the pulse, and, on the other hand, it must not be too complex, otherwise VA will be intractable. If the ansatz contains a set of free parameters pj{z) [for instance, p\ = A, P2 = a, p3 = b, p4 ^ 0 in the case of the ansatz (13)], the calculation of the integral (14) after the substitution of the ansatz yields an effective Lagmngian, Zeff, which is a function of pj and their derivatives dpj/dz = pj (the derivatives appear because of the presence of the z-derivatives in the Lagrangian density). For example, the effective Lagrangian obtained by substitution of the ansatz (13) into the NLS Lagrangian corresponding to the density (15) can be easily calculated analytically: ^NLS) ^ _2^2^^, _ ^ ^ 2 ^ 3 ^ _ 1 ^ _ ^DA'a'b' ett ^ 6 3 ^ 3

+ I yA'a. ^

(18)

It is noteworthy that only the z-derivatives of the phase parameters 0 and b appear in this expression, and 0 itself does not appear at all. The effective Lagrangian gives rise to a set of variational equations for the variables pj(z).

^^d{p;)

dp.

which can then be solved by means of analytical or numerical methods. In particular, the system of variational equations generated by the effective Lagrangian (18) for the NLS soliton will be considered in detail in §2. First of all, one should find fixed points (FPs) of the ordinary differential equation (ODE) system (19), dpj/dz = 0, which correspond to a stationary soliton of the underlying model. Next, stability of the fixed points against small perturbations can be analyzed, linearizing eqs. (19) near the FP solutions, which should predict whether the soliton is expected to be dynamically stable. Full dynamical solutions to eqs. (19) (rather than linearization around the fixed points), that correspond to a strong perturbation of the solitons, may also be of interest from the viewpoint of the underlying model. Thus, the essence of VA is approximating an unknown field configuration by an appropriate ansatz, whose free parameters evolve in z according to the system of ODEs (19). It is necessary to stress that there is no direct formal

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Variational methods in nonlinear fiber optics and related

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[2, § 1

relation between the underlying PDEs (for instance, the NLS quation) and the system of variational equations (19). Thus, VA is always based, to a large extent, on physical intuition rather than on rigorous mathematical arguments, and the relevance of the application of a particular variational ansatz to a given problem can only be checked a posteriori by comparison of the results with direct numerical simulations of the underlying PDEs. Comparison with direct simulations is especially necessary if one is dealing with the stability problem: while the shape of a static soliton may be readily mimicked by a reasonably chosen ansatz, the approximation can miss a specific perturbation mode leading to an instability of the soliton; moreover, VA can sometimes introduce a false instability that the soliton in fact does not have, see § 7.1 below. Despite its drawbacks, VA turns out to be a very efficient technique, as it is, as a matter of fact, the only consistent approximation producing analytical or semi-analytical results for complex dynamical models. As for the necessity to verify the validity of the results against direct simulations, this does not devaluate VA, since it is frequently sufficient to perform the comparison at a few different values of the problem's control parameters. If the comparison at several benchmark points corroborates the applicability of VA, then its (semi-)analytical predictions are reliable enough to describe solitons in broad parametric regions.

1.2.2. Generalization to models with losses and gain or drive 1.2.2.1. Models with intrinsic gain. A physically important and relatively simple generalization of the NLS equation is that which includes losses and amplification. In the general case, it can be written in the form lUz + \DUTJ + Y\U\^U = ia(z) w,

(20)

where the coefficient a(z) includes a constant negative part -ao accounting for fiber losses, and an array of 6-fianctions accounting for the action of strongly localized amplifiers. Thus, in the typical case. a(z) = -ao + g ^

Hz - Zan),

(21)

where g > 0 is the gain provided by an individual amplifier, and z« is the amplification spacing. In the general case, with an arbitrary density a(z) of the

2, § 1]

Introduction

81

distributed losses and gain, the term on the right-hand side of eq. (20) can be eUminated by means of a transformation w(z, r) = exp U

a{z) dz j • u{z, r),

(22)

which converts eq. (20) into the NLS equation (2) for the field u{z, r) with a variable nonlinear coefficient, iu. + ^DurT-^YQxpil 2

a(z)dz

•|w|^w = 0

(23)

Jo /O

(Bullough, Fordy and Manakov [1982]). An advantage of this transformed equation is that, unlike the underlying equation (20), it admits a variational representation with the same structure of the Lagrangian density as in eq. (15), y being replaced by 7(z) = 7 e x p ( 22 / /a , .a-( ,zd) dzz)) .

(24)

Then, ansdtze^ of the usual type, e.g., eq. (13), may be used to approximate the field £;(z, r). 1.2.2.2. Models with an external drive. Another type of models describe systems in which dissipation is compensated not by the intrinsic gain, but rather by an external drive. The first model of this type was introduced by Kaup and Newell [1978]: iut + \uxx + \u\^u = -\au + e exp(-ia;0,

(25)

where a > 0 is a dissipation constant, and e and o) are the amplitude and frequency of the AC drive applied to the system (this equation is written in "non-optical" notation, as it is less relevant to optics than to other applications). By means of an obvious transformation, u{x, t) = u(x, t) Qxp{-'\(jot),

(26)

eq. (25) can be cast into a more convenient time-independent form, \Vt + ^t^xY + (^ + \u\^) u = -\au + 6.

(27)

Finally, the dissipative term may be removed from eq. (27) by means of the same transformation (22) as above, leading to an equation representable in the ' The word ansdtze is plural for ansatz (which is a synonym for a trial wave form in the variational approximation).

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[2, § 1

Lagrangian form, which opens the way to apply VA to it. In particular, driving and stabilization of a cnoidal wave, i.e., as a matter of fact, a periodic array of NLS solitons, was considered, following this way, by Friedland [1998]. Another possibility is to drive solitons parametrically, as described by the following version of the perturbed NLS equation (see, e.g., a paper by Barashenkov, Bogdan and Korobov [1991], where VA was used), iut + |wxY + \u\^u = -iau + ew* exp(-2ia;0,

(28)

the asterisk standing for the complex conjugation. The same transformation (26) as above casts eq. (28) into a time-independent form. iw./

+ ^Wvv + (w + |w|^) u = -iau + 6w*.

(29)

Note that the last term on the rhs of eq. (29) can be derived from an extra term in the Lagrangian density, AC = \e w'+(w*)' . Therefore, subsequent application of the transformation (22) makes it possible to present eq. (29) in a fully Lagrangian form. 1.3. Comparison with other approximations Application of VA to optical solitons was not the first instance where this technique was used. Earlier, it was applied by Whitham [1974] to the cnoidal waves in the Korteweg-de Vries (KdV) equation (recall that these waves are periodic arrays of solitons). An exact solution for cnoidal waves in the KdV equation is known in terms of elliptic functions. However, an approximation is necessary when considering a case where parameters of the cnoidal wave are initially subjected to a long-wave modulation. In that case, the ansatz is based on the exact solution, whose arbitrary constant parameters are allowed to be slowly varying functions of the coordinate and time. Upon substituting the ansatz into the corresponding Lagrangian, one can explicitly perform the integration over the rapid variables, arriving at an effective Lagrangian for the slowly varying parameters. Then, the effective Lagrangian yields a system of so-called Whitham's equations (which are also PDEs, but essentially simpler than the underlying KdV equation) governing the slow evolution. The Whitham equations can be used for analysis of various dynamical processes involving the cnoidal waves, e.g., decay of an initial configuration in the form of a step (see chapter 4 in the book by Zakharov, Manakov, Novikov and Pitaevskii [1980]).

2, § 1]

Introduction

83

As concerns solitary waves proper in models different from those occurring in optics, VA was applied in a systematic way by Gorshkov, Ostrovsky and Pelinovsky [1974] and Gorshkov and Ostrovsky [1981]. Models studied in those works were similar to the KdV equation (but nonintegrable). A typical problem was interaction between far-separated solitons. Using the Lagrangian representation of the underlying model, an effective potential of the interaction between solitons was derived. Mathematical models for solitons in plasmas are sometimes similar to those in nonlinear optics. In a systematic way, the application of VA to plasma solitons was developed by Bondeson, Lisak and Anderson [1979]. In that work, a generalization of VA allowing to incorporate effects produced by dissipative terms, that cannot be directly derived from the Lagrangian representation, was put forward too. It should be stressed that when one is dealing with slightly perturbed solitons (for instance, in the case of interactions between far-separated ones), the use of VA is quite legitimate but not necessary. Instead, one may use direct perturbative methods. The most powerful among such methods is based on 1ST, provided that the underlying PDE is a perturbed version of an integrable equation. This is indeed the case for many problems in nonlinear optics, when the model is described by a perturbed NLS equation. The IST-based perturbation theory was first elaborated by Kaup [1976] (see also a paper by Kaup and Newell [1978]) and, independently, by Karpman, Maslov, and Solov'ev (see an early review by Karpman [1979] and a later important paper by Karpman and Solov'ev [1981], in which the interaction between NLS solitons was treated as a perturbation). Many results obtained by means of the perturbation theory based on 1ST were collected in a review by Kivshar and Malomed [1989a]. Second-order perturbation effects for the solitons in optical fibers may be taken into regard to improve the accuracy of this technique; this was systematically investigated by Kaup [1991]. As a matter of fact, VA belongs to a class of nonrigorous approximate methods whose objective is to reduce complex dynamics described by PDEs to a relatively simple system of a few ODEs. All these methods aim to "project" the full dynamics onto a finite-mode space, or, in other words, truncate a system with infinitely many degrees of freedom to a finite-dimensional one. This general procedure is often called Galerkin truncation (its mathematically rigorous description can be found in a book by Blanchard and Briining [1992]). It applies not only to conservative systems which admit the Lagrangian representation, but also to dissipative and mixed conservative-dissipative ones. In some cases typically, slightly above a threshold of an instability that gives rise to formation

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[2, § 1

of nontrivial patterns - the truncation of dissipative or mixed systems can be performed in a consistent way, using a corresponding small parameter {overcriticality). Examples are the derivation, by Malomed and Nepomnyashchy [1990] in the ID case, and by Zaks, Nepomnyashchy and Malomed [1996] in the 2D case, of a finite-dimensional dynamical system to approximate the pattern formation in the complex cubic Ginzburg-Landau equation with periodic boundary conditions just above the threshold of the modulational instability of a finite-amplitude spatially uniform state. However, in most cases no small parameter is available, and the Galerkin truncation is, as a matter of fact, based solely on intuition. A specific version of the truncation is the method of integral momenta, when the underlying PDE is replaced by several relations obtained, after substituting an adopted ansatz for the approximate solution, by multiplication of the equation by certain weight functions and integration of the resultant expression over the temporal and/or transverse spatial variables. The momenta method in its various forms has been used widely in various problems of nonlinear optics, e.g., by Caglioti, Trillo, Wabnitz, Crossignani and DiPorto [1990], Romagnoli, Trillo and Wabnitz [1992] and Maimistov [1993] for the study of soliton dynamics in dual-core fibers, by Akhmediev and Soto-Crespo [1994] for the description of soliton dynamics in a bimodal birefringent fiber, and by Turitsyn, Schaefer and Mezentsev [1998] and Belanger and Pare [1999] in the study of pulse propagation in dispersion-managed fiber links. A similar method was employed by Barashenkov, Smimov and Alexeeva [1998] and Barashenkov and Zemlyanaya [1999] to consider bound states of solitons in the driven NLS equations (25) and (28). The VA technique does not have a rigorous justification either. Nevertheless, it is essentially less arbitrary than other truncation-based approximations, as it is based on the variational principle, which is known to be the most fundamental one unifying various physical models. In this connection, it is relevant to mention that VA for linear physical systems (unlike nonlinear ones which are the subject of the present review) has been developed long ago under the name of the Rayleigh-Ritz optimization procedure, reviewed by Gerjuoy, Rau and Spruch [1983], that has well-known applications, e.g., to finding stationary wave ftinctions in quantum mechanics (Landau and Lifshitz [1977]). It is relevant to mention that essentially the same method was used by Barashenkov, Bogdan and Korobov [1991] to analyze the stability, in terms of the corresponding eigenmodes, of a soliton in the parametrically driven NLS equation (28), and by Barashenkov, Gocheva, Makhankov and Puzynin [1989] in their consideration of the stability of dark solitons. A rigorous

2, § 1]

Introduction

85

mathematical account of the Rayleigh-Ritz procedure is given in the book by Blanchard and Briining [1992].

1.4. Objective of this review There is a huge number of papers using VA in various problems of nonlinear optics and in other areas of "nonlinear physics". The present review, being limited in size, is necessarily limited in scope too. It does not aim to give a comprehensive review of all applications of VA to optics, nor does it give references to all relevant publications. Instead, the objective is to collect most important examples of the application of variational methods to solitons in optical fibers, and a few examples concerning solitons in other optical media (chiefly, in planar waveguides), which can be used as paradigms for many other applications. The review is focused on solitons (this term is realized in a loose mathematical sense, i.e., it does not imply integrability of the underlying models), as they are the most natural objects for the application of variational methods, and the absolute majority of results have been obtained for solitons. Fibers are selected as the main medium to be considered in this review, as in this field variational methods have been developed better than in any other, and fibers are most important for applications. In §2, the consideration will start with the most fundamental case of a single soliton in a uniform nonlinear optical fiber. Then, at the end of § 2 and in subsequent sections, more complex models will be introduced and considered, increasing the number of solitons, or the number of equations, or considering nonuniform optical media. In several cases, which are fundamentally important for applications, the presentation is not limited solely to results which can be obtained by means of VA, but a more comprehensive account of the problem as a whole is given; examples are bound states of solitons (§2.3.2), and generation of solitons of different types by a pulse passing a point where the local dispersion changes sign from normal to anomalous (§5.2). Three large topics belonging to the field of nonlinear optics are not included in this review. These are systems with quadratic (x^^O nonlinearities (second-harmonic-generating media), spatiotemporal solitons ("light bullets"), and discrete systems. The first topic has recently been reviewed in a systematic way by Etrich, Lederer, Malomed, T. Peschel and U. Peschel [2000]. That review includes, inter alia, 3. thorough account of the application of VA to x^^^ systems. Additionally, variational methods for x^^^ models were the main subject of another (more special) recent review by Malomed [2000].

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In particular, as concerns "light bullets", a large part of the theoretical analysis, and the only experimental observations of the spatiotemporal solitons reported thus far (by Liu, Qian and Wise [1999b] and Liu, Beckwitt and Wise [2000]), pertain to x^^^ media. The theoretical description of x^^^ spatiotemporal solitons relies heavily upon VA (Malomed, Drummond, He, Berntson, Anderson and Lisak [1997]), and this was included in the above-mentioned recent reviews. Variational techniques prove to be very useful also for consideration of multidimensional solitons in media with different nonlinearities, such as cubicquintic (Quiroga-Teixeiro and Michinel [1997], Desyatnikov, Maimistov and Malomed [2000]). In fact, a review of spatiotemporal solitons seems to be necessary, but it cannot be given in the present article due to length limitations. As for discrete systems, this is a large field which calls for a separate review. Variational methods are fi-equently used in this field too (see, e.g., a paper by Malomed and Weinstein [1996]), but their technical implementation is quite different fi*om what is considered in the present article. Lastly, it is necessary to mention that variational techniques, similar to those developed in nonlinear optics, find applications to the description of soliton-like objects in other physical systems. An important example is the Bose-Einstein condensate, i.e., a cloud of ultracold atoms obeying the Bose quantum statistics and held together in a trap. The corresponding model is based on the Gross-Pitaeuskii equation, which, as a matter of fact, is the three-dimensional NLS equation with an external potential representing the trap. The cubic term in the Gross-Pitaevskii equation has, in most cases, a sign corresponding to repulsive interaction between atoms in the condensate, although it may sometimes be attractive, then making the condensate prone to collapse. VA for the Bose-Einstein condensates with both repulsive and attractive interactions was developed by Dodd [1996], Perez-Garcia, Michinel, Cirac, Lewenstein and Zoller [1997], and Perez-Garcia, Konotop and GarciaRipoll [2000]. Another noteworthy example of the application of an "optical-like" VA to nonoptical systems is the description of intrinsic vibrations of an (effectively) onedimensional soliton in the Zakharou system, which is a fundamental model of the interaction between electron (Langmuir) and ion-acoustic waves in plasmas. As was demonstrated by Malomed, Anderson, Lisak, Quiroga-Teixeiro and Stenflo [1997], VA reduces the internal dynamics of this soliton to a Hamiltonian system with two degrees of freedom, which, in particular, may give rise to dynamical chaos.

2, § 2]

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§ 2. Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide 2.1. A soliton in an optical fiber 2.1.1. Anderson approximation fi^r a nonstationary NLS soliton The application of VA to nonlinear optics was initiated by Anderson [1983] when he considered the evolution of a strongly perturbed NLS soliton governed by eq. (2). In that pioneering work, a Gaussian ansatz for the soliton was used. While this type of approximation is very useful in the case of dispersion management (see § 5), the most appropriate ansatz for a soliton in a uniform optical fiber is the hyperbolic-secant-based one (13). In fact, the variational equations derived by Anderson [1983] on the basis of the Gaussian ansatz are very close to those which will be displayed below for the ansatz (13). The effective Lagrangian for this ansatz is given by the expression (18). The corresponding system of variational equations (19) was first derived by Anderson, Lisak and Reichel [1988a]. After some transformations, the equations can be conveniently cast into the following form, which is also valid in the important case when the dispersion coefficient D in eq. (2) is a fiinction of z (Malomed [1993]): d (A'a) = 0, dz 1 da b=

(30) (31)

IDaTz'

d dz \DAz)

(32)

da

UMa)^^,{

E =A \ V«^

(33)

«/'

and a separate equation for the phase 0, rl/A

7r2

.

/ r\h

\

1

First of all, eq. (30) implies the existence of the dynamical invariant E = A^a. The conservation of this quantity is a straightforward manifestation of the conservation of the energy (10) in the NLS equation. Indeed, the substitution of the ansatz (13) into the definition of the energy yields A^a.

Variational methods in nonlinear fiber optics and related fields

[2, §2

U(a)

Fig. 1. Shape of the effective potential (33) for D = \, E ^ 4JT~ (the large value of E serves to emphasize the characteristic shape of the potential).

An essential remark concerning the formal properties of VA is that one may replace the combination A^a everywhere in the effective Lagrangian (18) by constant E, which is not subject to the variation, and then perform the variation (after this, the phase-evolution equation (34) is derived by the variation in E). The resultant equations have exactly the same form as above. This feature makes it possible to simplify the derivation of the variational equations. Equation (31) shows that the intrinsic chirp of the soliton is generated by its deformation (change of width). This equation also explains why the chirp must be included into any self-consistent ansatz: otherwise, intrinsic evolution of the soliton, the study of which is the basic objective of VA, cannot be described. Equations (32) and (33) demonstrate that the evolution of the soliton's width can be represented, in closed form, as the motion of a Newtonian particle with mass D~' and coordinate a(z) in a potential well UQij{a), the shape of which is shown in fig. 1, while the propagation distance z plays the role of time. In fact, as stressed by Abdullaev and Caputo [1998], the effective potential ^eff(<^) is exactly the same as in the classical Kepler problem (see a book by Landau and Lifshitz [1975]). Note that when the dispersion coefficient Z) is a function of z, both the particle's mass and the potential depend explicitly on "time". There is an equilibrium position «eq = ^

(35)

at the bottom of the potential well (33) (if D = const). Comparison with expression (6) shows that the ansatz (13) with a = ^eq coincides exactly with the

2, § 2]

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89

unperturbed soliton solution, i.e., VA correctly reproduces the exact stationary soliton. The Hamiltonian corresponding to eq. (32) is / / = {\/2D){da/dzf + Uefr(a), and, as demonstrated by Anderson, Lisak and Reichel [1988b], it can be obtained in exactly this form by the substitution of ansatz (13) into the Hamiltonian (12) of the underlying NLS equation. lfD = const, the Hamiltonian is a dynamical invariant of eq. (32). Dynamical trajectories with / / < 0 and H > 0 correspond to the motion of a particle which, respectively, is trapped in the potential well or escapes to infinity. Oscillations of the trapped particle correspond to internal vibrations of a soliton-like pulse which is initially chirped and/or has a relation between its amplitude and width different from that for the exact soliton solution. Near the equilibrium position (35), oscillations with a small amplitude af^ have the form a(z) = ^eq + «i sin(^o^ + 6),

KQ = 2^-^^.

JTD

(36)

Here, 8 is an arbitrary constant, and the spatial period of the small oscillations, Zosc = 2JT/KO = Ji^D/(yE)^, is not much different from the soliton period (8) of the unperturbed soliton, which, in the notation used in eq. (36), is ZQ = JtD/(yEy (in fact, 4zo is more appropriate for the comparison with ZQSC, as ZQ proper corresponds, by definition, to the change of the soliton's internal phase by ^Ji, rather than 2JT). Exact results for eq. (32) are available too. In particular, an expression for the spatial frequency K of anharmonic oscillations of the trapped particle can be found in papers by Afanasjev, Malomed, Chu and Islam [1998] and Abdullaev and Caputo [1998]. It takes a compact form in terms of the Hamiltonian H,

and attains the maximum value KQ given by eq. (36) sX H = -2(yE/jT)^D~\ that corresponds to the bottom of the potential well. Exact solutions to the effective equation of motion (32) with the potential (33) can be represented in a parametric form, using known results for the above-mentioned Kepler problem (Abdullaev and Caputo [1998]). Setting D = y = 1, the exact solutions describing oscillations of the particle trapped in the potential well are 2E a= —r—-(l-eocos^),

fc

= $-eosinS,

(38)

where K is the frequency (37), CQ = y/\ - Ji^\H\/2E^ plays the role of the eccentricity in the Kepler problem, and £ is an auxiliary dynamical variable (the parameter).

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Persistent internal vibrations of a perturbed NLS soliton can be easily observed in direct simulations of eq. (2); see, e.g., detailed numerical results in the paper by Kath and Smyth [1995]. In fact, the explanation of the soliton's vibrations as oscillations of the trapped particle in terms of the ansatz was the first result of VA explaining a nontrivial dynamical behavior of the perturbed soliton; note that, while this result is obtained by means of VA in quite a simple way, it is not straightforward to predict the vibrations of perturbed solitons by means of the rigorous 1ST formalism. The regime of motion with / / > 0, corresponding to a(z) -^ oo, implies unlimited spreading out of the pulse, i.e., as a matter of fact, its decay into radiation. Thus, VA can indirectly predict transformation of the pulse into radiation, although the ansatz does not take into regard radiation degrees of freedom. The separatrix / / = 0 is a border between the pulses that are predicted to self-trap into soliton-like states and those which decay completely. For an initial unchirped pulse (13) with b = Q and arbitrary values of the amplitude ^o and width a^, the soliton content can be found in an exact form, in terms of 1ST, from a solution to the corresponding ZS equations (Satsuma and Yajima [1974]). An important exact result is that the pulse produces a soliton, in the limit z ^ cxo, provided that (39) On the other hand, the condition / / < 0, which is necessary for the formation of a soliton-like pulse in terms of VA, yields, for the same initial configuration. ^0^0

> \/D/2Y.

(40)

Comparing this to the exact result (39), one concludes that VA underestimates the soliton's stability, in terms of the soliton-formation threshold, by a factor \/2. An empirical modification of the variational technique, which can remedy this shortcoming, was proposed by Anderson, Lisak and Reichel [1988a]. 2.1.2. Solitons in extended versions of the NLS equation It is well known that, even if an optical fiber has no losses, the NLS equation for very narrow solitons (roughly speaking, with temporal width < 1 ps and, accordingly, with high power) should be modified against its classical integrable version (2). Additional terms take into regard the third-order dispersion (TOD)

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

91

with a corresponding coefficient D and, sometimes, a higher-order (quintic) correction to the Kerr nonUnearity, with a coefficient y: iuy + \DUTJ + y| w|^w = XDUTTT + y| w|^w.

(41)

If only the quintic term is added, the corresponding cubic-quintic (CQ) NLS equation can be rescaled into a normalized form, iu, + \ujj + |w|^w - \u\\ = 0.

(42)

Equation (42) is not integrable, but it has an exact single-soliton solution (Kh.I. Pushkarov, Pushkarov and Tomov [1979]; see also D.I. Pushkarov and Tanev [1996]), u - e'^^ • ^ ^ W 1 + y^l-^A:coshr2v^r)

(43)

where k is the propagation constant, taking values 0 < A: < ^ . As the quintic term in eq. (42) corresponds to an extra term -^|w|^ in the Lagrangian density for the NLS equation, VA can be developed to describe internal vibrations of a perturbed soliton in the CQ equation, as was done by Kumar, Sarkar and Ghatak [1986] [they also took into regard a dissipative term, eliminating it by means of the transformation (22)] and De Angelis [1994] on the basis of a Gaussian ansatz. In this connection, it is relevant to mention that, for the NLS equation with a general nonlinear term \u\^'u, where q is an arbitrary positive number. Cooper, Shepard, Lucheroni and Sodano [1993] developed VA based on a super-Gaussian ansatz, assuming In

M(Z, r)

= A{z) exp {-\+ib{z))

W{z)

(44)

where W(z) and b(z) are real width and chirp variables, A(z) is a complex amplitude, and n is an appropriately chosen positive constant. This ansatz makes it possible to analyze not only regular dynamics of a perturbed soliton, but also spatiotemporal collapse of the pulse, i.e., formation of a singularity after a finite propagation distance, which takes place (in the one-dimensional case) if (7 > 4, see a review by Berge [1998] (VA for describing the collapse of three-dimensional pulses in the usual cubic NLS equation was elaborated by Desaix, Anderson and Lisak [1991]). A general super-Gaussian ansatz was also

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[2, § 2

used by Dimitrevski, Reimhult, Svensson, Ohgren, Anderson, Berntson, Lisak and Quiroga-Teixeiro [1998] to analyze dynamics of axisymmetric beams in a bulk medium with the CQ nonlinearity (which, in fact, amounts to considering the CQ NLS equation with two transverse coordinates). Lastly, it is relevant to mention that various forms of VA were also applied to construct spinning solitons, i.e., solitons with internal vorticity, in the two-dimensional (Wright, Lawrence, Torruellas and Stegeman [1996], Quiroga-Teixeiro and Michinel [1997]) and three-dimensional (Desyatnikov, Maimistov and Malomed [2000]) NLS equations with the CQ nonlinearity. The TOD term in eq. (41) can also be derived from an extra term in the Lagrangian density, viz., (i/2)D (ww*^^ - w*Wrrr), hence VA applies to this version of the NLS equation too. It is necessary to stress that, strictly speaking, the NLS equation with this additional term has no soliton solution, as any solitary pulse gradually decays into radiation, due to the form of the equation's linear spectrum (Wai, Chen and Lee [1990]). Nevertheless, if the TOD coefficient is small enough, the rate of radiative decay is exponentially small, and it then makes sense to consider evolution of a soliton in this equation. A VA-based approach to the problem was developed by Desaix, Anderson and Lisak [1990]. As the NLS equation upon addition of the TOD term loses its invariance with respect to a sign change of r, an appropriate ansatz should not be even in r. In the above-mentioned paper, the ansatz was taken as w(z, r) = A(z) sech(r - T(z)) X exp[-i(r - T(z)) Q{z) - iM{z)tanh(r - T(z)) + ib(z)(T -

T(z)fl (45) where the amplitude A(z) is complex, and all the other variational parameters are real, cf. eq. (13). Consideration of evolution equations for the variational parameters has demonstrated that the soliton shifts itself, in the frequency domain, deeper into the anomalous-dispersion region, so that the relative size of the TOD term becomes small, and the soliton becomes close to its ordinary NLS counterpart. This result is, generally, confirmed by numerical simulations reported by Wai, Menyuk, Chen and Lee [1987], although the simulations also demonstrate that a relatively small wave packet separates from the initial pulse and then drifts in the opposite direction, deeper into the normal-dispersion region, where it completely decays into radiation. 2.1.3. Radiative losses and damping of internal vibrations of a soliton The most essential limitation of VA is the fact that a simple ansatz, like that given by eq. (13), completely ignores radiation degrees of freedom of the field.

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

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In fact, as known from both the exact solution produced by 1ST and from numerical simulations, a perturbed soliton, while vibrating in accord with the VA prediction, is also emitting small-amplitude radiation waves, which gives rise to gradual decrease of the vibration amplitude. The exact result of 1ST is that, at z —> cxD, the pulse will shed a finite fraction of its energy as radiation, and will eventually assume the form of an exact soliton with a reduced value of the energy. A modification of the ansatz (13) that accounts for the radiation background around the soliton was proposed by Kath and Smyth [1995]: WansatzC^, ^) = [A sech(r/(
(46)

where g{z) is a real amplitude of the radiation background that is assumed to be uniform (r-independent) across the soliton; it was also assumed that \g\ <^ A, i.e., the background's amplitude is much smaller than that of the soliton. Of course, the substitution of the modified ansatz (46) into the Lagrangian density (15) and subsequent integration in the expression (14) for the fijll Lagrangian will give rise to a divergence as the term ~g does not vanish as |r| -^ oc. Therefore, the integration was confined to a finite interval \r\ < /; in particular, the net energy of the wave field is then A^a + \g^l. To select the parameter /, the condition was adopted that the (spatial) frequency of the small oscillations of the amplitude of the slightly perturbed soliton matches the frequency in eq. (36), which yields / = 3:T^/8«eq5 where ^eq is the equilibrium width (35), and it is implied that D = y = 1 in eq. (2). The variational equations derived by means of the ansatz (46) were further amended by adding, to an equation accounting for the energy conservation, an extra term that directly took into regard radiation losses, as calculated from a linearized equation for the radiation wave far from the soliton's body. The modified variational equations [which turn out to be much more complicated than the system of eqs. (30) through (34) produced by the Anderson approximation] were then solved numerically, showing not only persistent internal vibrations of a perturbed NLS soliton, but also gradual damping of the vibrations due to the emission of radiation. Comparison with direct numerical simulations of eq. (2) has demonstrated that this modified version of VA yields very good accuracy in the description of the soliton's dynamics. Direct comparison of the VA predictions for the internal vibrations of the NLS solitons with direct simulations of eq. (2) was also a subject of a work by Kuznetsov, Mikhailov and Shimokhin [1995]. In this paper, it was claimed that VA is essentially wrong, as the frequency of the small vibrations revealed

94

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[2, § 2

by extremely long simulations was quite different from the expression given by eq. (36). However, this conclusion was a result of an apparent misunderstanding: in fact, the numerical results presented in that work pertained to a very late stage of the evolution, when the emission of radiation by the vibrating pulse has actually ended, and the observed (extremely small) oscillations of the soliton's amplitude were not vibrations of the perturbed pulse, but simply beatings between the stationary soliton and a very small low-frequency component of the radiation wave which had not yet separated from the soliton. Of course, the beating frequency is different from that of the vibrations of a pulse consisting of the soliton and trapped radiation. 2.1.4. The soliton-compression problem and modified variational ansdtze The Anderson approximation, based on the simple ansatz (13) or its Gaussian counterpart used in the first paper by Anderson [1983], can be better adjusted to specific problems without adding radiative degrees of freedom. A particular problem important for applications is compression of pulses based on the socalled soliton effect, i.e., passing a stationary (fundamental) soliton to a fiber with a smaller value of the dispersion coefficient, where the pulse will be a higher-order soliton and will start to self-compress, developing internal chirp (an allied problem is the investigation of conditions for wavebreaking-free propagation of nonsoliton pulses in an optical fiber, which was earlier considered by Anderson, Desaix, Karlsson, Lisak and Quiroga-Teixeiro [1993]). For a given ratio N^ = D\/D2 of the dispersion coefficients D\ in the fiber in which the soliton was formed as a fundamental one and Di in the compressing fiber (or, in terms of the soliton effect, for a given order N of the initial A^-soliton), and for given energy of the soliton, the most important characteristic of the process is the optimum compression length L of the second fiber at which the narrowest chirp-free pulse is expected to come out. To minimize the number of arbitrary parameters, one can take eq. (2) with D = y = 1^ ^^^ consider compression of the initial A^-soliton pulse in which the width is set to be 1, uo= N sech r,

(47)

so that E = N^. Then, the optimum compression length should be found as a function of the single free dimensionless parameter E. This problem was considered in detail by Afanasjev, Malomed, Chu and Islam [1998], who compared, against direct numerical results, predictions provided by the traditional ansatz (13) and by a modified one, u = A sech(r/a) exp[i0 + ib tanh^(r/flf)],

(48)

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

95

4 6 Soliton Energy E Fig. 2. Optimum compression length for the initial A'^-soliton pulse (47) in a fiber governed by eq. (2) with D = y = 1 vs. energy £":(!) prediction based on the traditional ansatz (13); (2,3) predictions produced by the modified ansdtze (51) and (48); dots represent results of direct simulations, connected by an interpolating curve.

where b is the chirp parameter. The introduction of this ansatz was suggested by direct simulations, which demonstrated that the intrinsic phase structure of the compressed pulse was very different from the parabolic function assumed by eq. (13). Instead, the phase distribution is parabolic near the soliton's center, and saturates at a constant value far from the soliton's center, which is mimicked by the modified ansatz (48). VA based on the usual ansatz (13) predicts compression of the initial A^-soliton pulse (decreasing a{z)) up to the turning point z = n/K, where K is the spatial frequency of the soliton's vibrations given by eq. (37) with H = (2/jr^)(l - 2E). Thus, z = Jt/K is the optimum compression length as predicted by the usual ansatz. An explicit formula for the predicted optimum compression length is

L=

\n^E\/2E-\.

(49)

It has no meaning for £" < 1, as in this case the deformed soliton will be initially expanding, rather than compressing. VA also predicts the degree of compression.

a(z = L)

(50)

In fig. 2, the dependence L{E) as given by eq. (49) (curve 1) is displayed vs. the dependence obtained by direct simulations of the NLS equation with the initial conditions (47). The numerical results are represented by dots connected by an

96

Variational methods in nonlinear fiber optics and related

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[2, § 2

interpolating curve. The two curves are quite close at £ ^ 2, which corresponds to small compression degrees, but for larger E the actual optimum compression length is essentially smaller than the predicted value. For the case of relatively small compression rates, a very detailed comparison between direct numerical simulations of the A^-soliton compression and the corresponding predictions of the usual version of VA, based on the ansatz (13), has been given by QuirogaTeixeiro, Anderson, Berntson and Lisak [1995]. To improve VA in the case of large compression rates, the modified ansatz (48) was tried, along with the following one: w=^[sech(r/a)]'^*^e''^

(51)

with a real chirp parameter b and the phase b\n(sech{T/a)) which is growing linearly at |r| :^ a, i.e., it is sort of intermediate between the phases in eqs. (13) and (48). Note that this ansatz follows the pattern of the initial pulse configuration MQ = ^[sech(r/flr)]^ +'^, which is the most general one for which the ZS equations can be solved in an exact form (Maimistov and Sklyarov [1987], Grunbaum [1989]). The ansatz (51) gives rise to the effective Lagrangian (cf eq. 18) Eb^,

\Eb^

\ E

2E^

i^eff - - — ^ ' - T — - ^ - + ^ - ^

3"^~3?~^3T

(52)

and a set of evolutional equations that reduces to

^-i't --KM)Comparison with eqs. (32) and (33) shows that, although the expression for the chirp parameter is different from eq. (31), the evolution equation for the soliton's width keeps the same form of the Newton equation of motion for a particle with a coordinate a(z) in the Kepler-problem potential (33), the only difference being a change of the particle's mass from 1 to m^|^ = 9/jr^ ^ 0.912 (recall we now set D = y = 1). In terms of the plot L(E), this difference amounts to a simple rescaling: the original curve 1 in fig. 2 should be uniformly stretched in the horizontal direction by the factor (w^[^)~'^^ ^ 1.047, which gives rise to curve 2. This minor change renders the theoretical prediction slightly closer to the numerical data at £ < 2, but it does not remedy the major discrepancy at larger values of E.

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

97

25 r ^ 20

c ^ 10

0 Time < Fig. 3. Intensity distribution \u\^ in a pulse with energy £" = 5.5 at point z = 0.5, close to the optimum compression point, as predicted by the modified variational ansatz (48) (dashed curve) and as per direct simulations (solid curve).

The effective Lagrangian for the modified ansatz (48) is lEb , 3 a

32 Eb^ 105 a^

1E

2E^

3a^^ 3 a'

(54)

producing the evolution equations (cf eqs. 53, 53) b=

^aa\

'

32 / 1 ^ 3 5 ^ - ?

(55)

Thus, the evolution equation for a(z) can again be obtained from the effective potential (33), the corresponding effective mass being mf^ = 35/SJT^ ^ 0.443. Its drastic difference from m^J^, corresponding to the ansatz (51), and from m^^ = 1, corresponding to the traditional ansatz, is noteworthy. The change in the curve L(E) produced by the new value of the effective mass again amounts to stretching, this time by a factor (w^J^)'^^ ^ 1.502, yielding curve 3 in fig. 2. An immediate conclusion is that the new curve is much worse than the previous ones at £" ^ 2; in the range 2 ^ £ ^ 3.5 the numerical data fall between curves 1 and 3; and at £" ^ 4 the modified ansatz (48) definitely gives a better approximation. To directly illustrate the strong compression of the soliton, we display in fig. 3 its intensity profile |w(r)p at a point close to the optimum compression length, for £" = 5.5. As one sees, this profile is reasonably well approximated by the modified ansatz (48), while the traditional ansatz (13) predicts in this case a profile which is completely off the actual one.

98

Variational methods in nonlinear fiber optics and related fields 50 F

[2, §2

Wl

>> ^ 0 E

(O

I 30 I -g 20 0)

/^ A

/

\

1

I

IGF

Fig. 4. (a) Peak power and (b) width of a pulse with energy E = 5.5 vs. propagation distance. The sohd and dotted curves display, respectively, direct numerical results and analytical predictions produced by the modified variational ansatz (48). Two fiill compression-dilatation cycles are shown.

Further information about the accuracy (or inaccuracy) of the modified version of VA is given by the dependences of the pulse's peak intensity and width on the propagation distance, displayed in fig. A for E = 5.5. A general inference suggested by these plots is that, at this quite large degree of compression, the modified VA overestimates the peak intensity very close to the optimum compression point, but, otherwise, provides a reasonable analytical approximation, and is quite accurate in predicting the optimum compression length, which is most important for applications. In a recent work, Smyth [2000] has revisited detailed comparison of direct numerical simulations of the compression problem with results predicted by VA, adding the above-mentioned sophisticated version of VA worked out by Kath and Smyth [1995], which includes the small radiation background. A conclusion was that, while the modified ansatz (48) and the ansatz including the radiation predict the optimum compression length equally accurately for large values of the compression degree, the latter ansatz predicts the amplitude and width of the compressed pulse, and the phase distribution in it, essentially better. Lastly, it is relevant to mention the problem of soliton compression in conjugation with the action of localized or distributed amplification, which is described by the modified NLS equation (20). Detailed investigations performed

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

99

by Quiroga-Teixeiro, Anderson, Andrekson, Berntson and Lisak [1996] and by Chu, Malomed and Peng [1996] have demonstrated that VA based on the ansatz of the usual type (13), taking into account the effective variable nonlinear coefficient (24), provides for sufficiently accurate predictions for compression of the soliton in such a setting. 2.7.5. Compression of a soliton in a three-fiber configuration The pulse-compression technique described in the previous subsection does not make it possible to transform a given soliton into a compressed fundamental soliton corresponding to the smaller value of the dispersion coefficient. Instead, it produces a vibrating chirped pulse. The problem of compression of solitons without disturbing their fundamental character is of great interest. As follows from the general expression (6) for the soliton, its width can be presented in terms of the energy as ^ = VD/T] = D/yE; hence, if the fundamental soliton is compressed by lowering the dispersion coefficient from D\ to D2, without energy loss and at a constant value of the nonlinearity coefficient, the ideal compression factor is ^ ]

- ^.

(56)

One possibility to achieve nearly ideal compression is to use a dispersiondecreasing fiber with a gradually decreasing local dispersion coefficient, which is able to perform adiabatic compression of a sohton, as described below in §5.1. However, a much simpler possibility is to use the configuration proposed by Anderson, Lisak, Malomed and Quiroga-Teixeiro [1994], in which an intermediate fiber segment, with a value D of its dispersion coefficient taking some specially chosen value between the initial and final values D\ and D2, is inserted between the incoming and outgoing fibers. In terms of the standard VA, the incoming sohton corresponds to a particle resting at the bottom of the potential well (see fig. 1) corresponding to D = D\. In passing to the second fiber, and then to the third, the soliton jumps from one potential well into another, corresponding to a different value of D (it is assumed that the nonlinear coefficient is the same in all the fibers involved). The energy E, width a, and chirp b of the pulse must be continuous across the jump. According to eq. (31), the continuity of ^ implies that the combination D~^da/dz must keep its value, as does a, across the jump, while the derivative da/dz itself changes its value by a jump. Within the framework of this description, an ideal transformation of an incoming fundamental soliton, which was adjusted to the dispersion coefficient D = Di,

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fields

[2, § 2

U(a)

Fig. 5. Potential wells corresponding to three different values of the dispersion coefficient (here denoted a) for a fixed value of the soliton energy. The dashed trajectory demonstrates the possibility for ideal compression of the input soliton into an output soliton, keeping its fiindamental character.

into an outcoming fundamental soliton, adjusted to D = D2, is achieved if the value D of the dispersion in the intermediate fiber and its length Z* are selected in such a way that after the jump from the first potential well into the second the soliton performs exactly half a cycle of oscillations in the second well, hits its wall, and at this point jumps into the third potential well corresponding to D = D2, 2iS illustrated by fig. 5. An elementary calculation at constant energy yields D=

2D1D2 D,+D2'

(57)

(here, the nonlinear coefficient is 7 = 1). In this approximation, the same result is expected if the soliton passing the intermediate segment performs any odd number of half-cycles of the oscillations. This prediction was checked against direct simulations. To estimate the efficiency of the scheme, the soliton was passed through the intermediate segment with the value D taken as per eq. (57) with different values of its length. The soliton component in the energy of the output pulse was determined as corresponding to the discrete eigenvalue obtained from the numerical solution of the ZS equations for this pulse. Figure 6 shows the most essential numerical result, viz., the share of the input soliton's energy which is kept by the output soliton at different values of the dispersion ratio D\/D2, vs. the length of the intermediate segment measured in units of the length Z* predicted by eq. (57). The last curve, corresponding to D\/D2 = 10, includes two optimumcompression points corresponding to both one and three half-cycles of the

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

101

Fig. 6. Results of direct simulations for the energy of the compressed soliton, E^, normalized to the energy £"0 of the input soliton, vs. the length of the intermediate segment L normalized to the optimum-compression length L^ predicted by VA (eq. 57). The four curves displayed pertain to dispersion ratios D\/D2 = 2, 3, 5, 10.

oscillations. It is noteworthy that, although degradation of compression quality does occur in direct simulations with increasing dispersion ratio, the degradation is not catastrophic: even at a very large value of the dispersion contrast, D1/D2 = 10, as much as 84% of the input energy is kept in the soliton component of the output pulse (and the best result is achieved at the second optimumcompression point). It is interesting too that the actual value of the (first) optimum-compression length decreases with increasing of the dispersion ratio. In the same work by Anderson, Lisak, Malomed and Quiroga-Teixeiro [1994], a related problem was analyzed by means of VA, viz., "tunneling" of a soliton through a finite segment of a purely linear fiber inserted between two nonlinear ones. Predictions produced by VA for this problem (e.g., the critical length of the linear segment behind which the soliton gets completely destroyed) were compared to direct simulations, resulting in good agreement. The three-fiber compression scheme was tested in a real experiment by Bertilsson, Aakjer, Quiroga-Teixeiro, Andrekson and Hedekvist [1995]. For instance, an input ftindamental soliton with width 11 ps was successfully compressed to a ftindamental soliton with width 2.4 ps, when the soliton was passed from a fiber with D\ = 5ps/(kmnm) to one with Di = lps/(kmnm) through a 20 km-long intermediate segment with dispersion D = 1.7 ps/(km- nm), which is quite close to that predicted for this case by eq. (57). The compression

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fields

[2, § 2

factor achieved, W\/W2 ~ 4.6, is quite close to the ideal one, D\/D2 = 5, predicted by eq. (56). 2 J. 6. Resonant excitation of soliton internal vibrations by periodic amplification A realistic model of a fiber communication link should take into account losses, periodic amplification, and filtering, which makes it necessary to consider a perturbed NLS equation, iuz + \ujj + \u\^u = ia{z) u + ijiujj,

(58)

where we set Z) = y = 1, the term ~ fi accounts for the filtering (which is taken in the distributed approximation, averaging the discretely placed filters along the fiber link), and the function a{z) combines the uniformly distributed losses and periodic amplification as per eq. (21). Stationarity of the soliton transmission regime requires the mean net rate of attenuation and amplification for the soliton, averaged over long distance, to be zero. Neglecting filtering losses, as well as emission of radiation by the soliton, this condition amounts to setting gZa = ao in eq. (21). When the additional losses are taken into account, they must be compensated by a{z) having a residual positive mean value a. Therefore, in the general case it is natural to split the function a{z) into a mean value and a variable part a{z) with zero average value, a{z) = ~a + a\{z).

(59)

The dissipative term'-- a in eq. (58) can be converted into a variable coefficient in front of the nonlinear term by means of the transformation (22). In the present case, it is reasonable to apply this transformation only to the variable part of a{z), leaving the mean value a aside, which leads to the equation iv, + \vrr + Q^'^^'-^\v\^v ='\{au + jiu jj),

(60)

where A = f a\(z)dz. Periodic perturbation of a soliton obeying eq. (60), a physical origin of which is the periodic amplification of the soliton in a long fiber link, may get into a resonance with the free internal vibrations of a deformed soliton described above. This problem was considered, by means of VA, in a paper by

2, § 2]

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103

Malomed [1996]. The exact resonance takes place if the period of the small vibrations of a perturbed soliton, which is

with KQ given by eq. (36) (recall that now D = y = 1), is equal to the amplification spacing ZQ, see eq. (21). Proximity to the resonance is determined by a detuning parameter,

The result takes the form of the variational equations derived above, in which the energy is replaced by E{z) = E exp(2A(z)), and, additionally, the filtering term gives rise to an effective friction force that should be added to eq. (32), so that it becomes d'a

4 ri]

_ _E^^^^n E Q ^

dz2

^H^^^%2p^ 3jr3

(63)

' dz

(strictly speaking, the friction force takes this simple form only for smallamplitude oscillations near the bottom of the potential well, seefig.1). Besides, the relation between the chirp b and the varying width a changes against eq. (31): I da 4ai(z) 2a dz Jt^a^ [recall that a\(z) is the variable part of a defined in eq. (59)]. In eq. (63), z-periodic functions can be decomposed into Fourier series, and nonlinearities are to be expanded, assuming oscillations with a small amplitude near the bottom of the potential well. Keeping quadratic and cubic nonlinear terms in the latter expansion, it was demonstrated that the final equation can be mapped into the standard equation for a resonantly driven nonlinear oscillator, provided that the detuning (62) is small enough. Using well-known results for the latter equation (Landau and Lifshitz [1975]), the amplitude of established oscillations can be found, and their stability can be examined. In particular, a bistability region was found in the parametric space, where two different solutions for the driven internal vibrations of the soliton may exist, being

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fields

[2, § 2

simultaneously stable. The expression for the bistability region takes a simple form when the filtering is disregarded, ^ = 0:

-(If [recall that g is the gain parameter from eq. (21), and e is the detuning (62)]. The difference between two coexisting stable propagation modes of the soliton in the bistable range is in the size of the chirp: one mode is characterized by low chirp, while the other has relatively large chirp. Moreover, it was shown that a subharmonic resonance, which takes place when the period (61) of small vibrations of the perturbed soliton is close to Iza, also gives rise to a bistability. Thus, the soliton may propagate along the fiber link in the state of persistent internal vibrations, which are resonantly driven by the periodic amplification. 2.2. A spatial soliton in a periodically inhomogeneous planar waveguide 2.2.1. A stationary soliton A peculiarity of physically relevant problems for spatial solitons is that they may interact with an effective external potential, as per eq. (5). For the simplest case, with a potential of parabolic shape, VA was applied to the corresponding spatial soliton by Michinel [1995], who used a Gaussian ansatz including a degree of freedom accounting for a possible shift of the soliton off the waveguide's center. In particular, it was demonstrated that this ansatz generated decoupled evolution equations for the internal vibrations of the soliton, and for oscillations of its center about the center of the waveguide. A model with great potential for applications to photonics introduces a periodically inhomogeneous nonlinear waveguide that may be a basis for a switchable multichannel system guiding light signals. The basic version of this model postulates a simple sinusoidal spatial modulation of the waveguide, so that eq. (5) takes the form iw- + ^WxY + e cos(^x) • u + \u\^u = 0,

(65)

where L = Ijz/q and e are the period and amplitude of the modulation; using the invariance of eq. (65), it is possible to set L = 1, i.e., q = 2jt, which will be assumed below. As a matter of fact, the same equation (65) also describes a planar array of densely packed nonlinear waveguides, a medium in which actual experiments with the spatial solitons have been performed (Eisenberg,

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

105

Silberberg, Morandotti, Boyd and Aitchison [1998]). Indeed, a chain of coupledmode equations for the array reduces, in the dense-packing approximation, to the NLS equation in which the residual discreteness manifests itself in the form of an effective harmonic Peierls-Nabarro potential (Kivshar and Campbell [1993]), i.e., exactly eq. (65). This model was analyzed in detail by means of a combined analytical (VAbased) and direct numerical methods by Malomed, Wang, Chu and Peng [1999]. The first objective of the analysis was to find stationary one-soliton solutions of the form u{x, z) = exp(i^z) U(x)

(66)

with a real propagation constant k and real U{x). The solution describes a solitary beam trapped in a trough (one of the channels induced by the periodic spatial modulation). Substitution of eq. (66) into eq. (65) leads to an ODE, ^ W' + [e cos(2jrx) -k]U + U^ =0,

(67)

which can be derived from the Lagrangian [(U'f + {2k~e cos(2;rx)) U~ - U^] dr.

(68)

The solution is approximated by a simple ansatz, U = A sech(^x). Placing the center of the soliton at x = 0, one assumes e > 0 in eq. (65), then x = 0 is a local potential minimum for the soliton. Substituting the ansatz into the Lagrangian and performing the integration, the variation in A and r] leads to ^2 o^2 [^^^ cosh(jrVry) - 3^sinh(jr^/^/)] r - iJi'e^ ^ -, ^ ^ = 2k,

^'-\

r/^ + 6A: -

?7sinh(:TV?7)

(69) (70)

Equations (69) and (70) have exactly one solution at any e > 0 and any A: > 0. In particular, the asymptotic form of the solution for very small and very large k is A^ = if]^ = Ik. Comparison of the VA prediction for the soliton shape with numerical solutions of eq. (67) is presented in fig. 7. Note that, at small k, the width of the soliton is essentially larger than the modulation period. This explains the wavy

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Variational methods in nonlinear fiber optics and related fields

/fi\

U

0.6

numerical /t

[2, §2

(a) ^^analytical

l\

0.4

0.2

0 - 8 - 6 - 4 - 2

0

2

X

u

numerical

/

\

2

analytical

x/

^^^^ -

(b)

p

-

1

0

1

2

X

Fig. 7. Comparison between the one-soliton solutions to eq. (67) at f = 1 with (a) k = 0.2 and (b) k = 5.0, obtained numerically (solid curves) and by means of VA (dashed curves).

shape of the soHton in fig. 7a with k = 0.2. Of course, this feature is not included in the simple ansatz adopted above, which explains some disagreement between VA and the numerical results at small A:: at A: = 0.2, the amplitude predicted by eq. (70) differs by less than 2% from the numerical value U{x = 0) = 0.622. At larger k, the soliton becomes narrower, and it is then very close to the shape predicted by VA, see fig. 7b. 2.2.2. Soliton stability and the Vakhitou-Kolokolou criterion Numerical simulations of the fiall PDE (65), using an ansatz with the width and amplitude (69) and (70) predicted by VA as an initial configuration, have demonstrated that, at all values of e and k, the initial configuration gives rise to stable solitons. Actually, VA makes it possible to predict the stability by means of a criterion proposed by Vakhitov and Kolokolov [1973] (the VK criterion). According to this criterion, one should calculate the power of the solitary beam, F = J ^ |wp dx, which is thus obtained as a function of e and k. The VK cri-

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

107

6r

4[

numerical

3[

Fig. 8. Solitary-beam power F vs. propagation constant k for the one-soliton state in model (65): shape found numerically (solid curve) and shape predicted by VA (dashed curve).

terion states that a necessary (but, generally, not sufficient) condition for the stability of the soliton is dF/dk > 0. A typical example of the dependence F{k) evaluated on the basis of both numerical and variational solutions is displayed in fig. 8, which clearly shows that the numerical and variational results are fairly close, both showing that the slope dF/dk is positive everywhere. An issue crucially important for the use of this model as a multichannel system is the existence and stability of two-soliton states, with the solitary beams trapped in two adjacent channels. The two-soliton state can be destabilized by the mutual attraction of the two beams, which can lead to their merging into one beam. Malomed, Wang, Chu and Peng [1999] had found a stability region for twosoliton states by means of direct simulations. 2.2.3. Switching a soliton between adjacent channels A more sophisticated problem that was also considered by Malomed, Wang, Chu and Peng [1999] is to model controllable switching of the soliton from a given trough into an empty adjacent one (the principal possibility of switching spatial solitons was demonstrated experimentally by Shalaby and Barthelemy [1991]). To this end, one may assume that a laser beam launched in the direction transverse to the planar waveguide is focused on a small spot with coordinates (x = xo, z = 0) somewhere between the two troughs [0 < xo < 1; recall q = 2JT in eq. (65)]. Through cross-phase modulation (XPM), the bright spot gives rise to an attraction center, which is described by an additional localized perturbation added to eq. (65): iuz + ^Uxx + e cos(2JTx) • w + I w| u = -j.1 b{x - Xo) 6(^) • w,

(71)

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[2, § 2

^ being proportional to the intensity of the transverse beam. The attracting spot has a chance to throw the soUton over the dividing potential barrier into the adjacent trough. To analyze this possibility, the change of the soliton induced by the perturbation concentrated at the spot can be found in an exact form. Indeed, representing the soliton solution as w(x,z) = a(z,x)exp(i0(x,z)) with real amplitude a and phase 0, it is straightforward to see that the spot does not introduce any instantaneous change of the amplitude, while the change of the phase is A0(x, z) = 0(x, z = +0) - (pix, z = -0) = jLi b(x - xo).

(72)

Further analysis can be carried out by means of perturbation theory, treating both e and jn as small parameters, and the soliton as a particle. The unperturbed NLS soliton should be taken in the general "walking" form (7), which, in the present case, corresponds to a(x, z) = y/lk sech f V2k(x - cz - ^) j , where k is the propagation constant introduced in eq. (66), the small "velocity" c is, in fact, a ramp of the solitary beam in the (x, z) plane, and ^ is the coordinate of the beam center at z = 0. With regard to the definition (11) of the momentum of the "walking" soliton, it may be interpreted as a particle with the following momentum, kinetic energy, and mass: P = Mc,

^k.n = ^ ,

M-lVlk.

(73)

We consider the situation in which the beam at z < 0 was trapped in the given channel (trough), so that it has c = ^ = 0. As follows from the general expression (11) for the momentum, the instantaneous phase change (72) gives rise to a jump of the momentum from 0 to a value that can be found in exact form: n^-yz

+OC

/

a^(x) A0\x) dx = lii / DC

a\x) b\x-xo)

dx = -2f.ia{x^) a'ix^).

^-DC

(74) The substitution of the unperturbed soliton form, \/2k sechf \/2kx j , into eq. (74) yields an explicit result for P. Thus, the localized perturbation plays the role of a sudden push that lends the particle a kinetic energy, which can be found at the first order of perturbation theory, using eqs. (73) and (74), £kin = lii\2kf^ sinh^ (V2^^vo) sech^(y/2kxo] . (75)

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

109

The interaction of the unperturbed soHton with the periodically modulated refractive index is described by an effective periodic potential W{^), which is generated by the corresponding part of the Lagrangian (68), W{^) = -e

COS(2JTX)

• a^(x) dx =

J-oc

— ; = ^ cos(2;r^). sinh (jt/Vik)

(76)

According to eq. (76), the height of the potential barrier separating two adjacent troughs is

f ' . .

AW^

ill)

sim^i I JT/y/lkj The soliton set in "motion" (physically, given the ramp c) by the sudden push will pass the separating barrier and get into the adjacent trough if £'kin > A ^ . Substitution of equations (75) and (77) into this inequality shows that the attracting spot created at the point XQ is able to switch the solitary beam into the adjacent channel if its strength /.i^ exceeds a threshold value 2JJ^2^

cosh^f\/2^xoj

(2^)^^^ sinh ( j T / v ^ ) sinh^(v^xo) In particular, fi^^^, considered as a function of XQ, takes a minimum value at the point where cosh^ (\/2^JCo) ^ | • In the framework of the lowest approximation of the perturbation theory, the soliton kicked out from the trough where it was originally trapped will not be trapped by the adjacent trough, but will keep moving farther. However, radiative losses not taken into account in the lowest approximation are likely to help trapping the soliton. Direct simulations demonstrate that radiative losses take place indeed, and the soliton can be trapped by the adjacent trough after having been pushed by the spot (Malomed, Wang, Chu and Peng [1999]). 2.3. Interactions and bound states of solitons 2.3.1. Potential of interaction between two far-separated solitons 2.3.1.1. General analysis. The variational methods can also be quite efficiently used for the description of multi-soliton complexes, the simplest and most important example of which is a pair of far-separated solitons. In the case of

110

[2, §2

Variational methods in nonlinear fiber optics and related fields

the unperturbed NLS equation, the interaction force between two distant soUtons was calculated analytically by Karpman and Solov'ev [1981] on the basis of the perturbation theory for a single soliton, which treated the overlapping between one soliton and a vanishing tail of the other as a small perturbation (similar work was done by Gordon [1983]). Essentially the same results were obtained by Anderson and Lisak [1986a] by means of VA, postulating an ansatz in the form of a linear superposition of two solitons. The interaction force between solitons, predicted by Karpman and Solov'ev [1981], was directly measured by Mitschke and Mollenauer [1987] in an experiment with solitons in an optical fiber. Interactions between spatial solitons are also amenable to direct experimental studies, as first demonstrated by Reynaud and Barthelemy [1990] and Aitchison, Weiner, Silberberg, Leaird, Oliver, Jackel and Smith [1991]. Following these ideas, it is natural to consider two far-separated solitons as particles, describing their interaction in terms of the corresponding effective potential. It will be shown below, following Malomed [1998a], that VA makes it possible to find the effective interaction potential in a very general and fairly simple analytical form. For two far-separated solitons, the wave field is assumed to be a superposition of their individual fields u\ and W2, (79)

w(z, r) = U\(Z,T)-\- U2(Z, T).

Note, however, that a weak "tail" of one soliton can be essentially distorted where it overlaps with the "body" of the other soliton. The general analysis outlined below does not neglect this distortion. The configuration with two solitons to be considered here is defined so that the center of the first soliton is set at r = 0, and that of the second is at T = -T, where T is a large separation between the solitons. The interaction potential is, with the minus sign, part of the Lagrangian produced by the overlapping of each soliton with the small tail belonging to the other. Substituting the superposition (79) into the Lagrangian, one arrives in the first approximation at the following general expression for the potential: 8£ ^in

d_ dz

6w*

+ C.C. } + {l

dC

• w; +

9w*

("|)r

d^

^2}, (80)

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

111

where c.c. stands for the complex conjugate expression, the integral is taken over a vicinity of the first soliton where the tail of the other one is small, and {1 ?=^ 2} stands for a symmetric contribution from a vicinity of the second soliton. The presence ofd/dz in one of the terms of the integrand implies that the z-derivative was transferred, in that term, from the multiplier u^ as per integration by parts with respect to z, which is implied because the Lagrangian L = J^ C dt should be further inserted into the action, f^^ Ldz. If integration by parts (with respect to r) is applied to the last term in the integrand in eq. (80), one arrives at the following integral expression:

/[(

•M^dr,

(81)

which is exactly equal to zero, as the one-soliton solution (for the first soliton) is obtained from the Lagrangian exactly in the form stating that the expression in the square brackets in eq. (81) is zero. Therefore, the only nonzero contribution to the interaction potential in its general form (80) comes from the integration limits when integrating by parts the last term in the integrand in eq. (80): ^int = \D{z) [{ux\ u\ + C.c] 1^:"^^ + { 1 ^ 2 } .

(82)

As the integral in eq. (80) is to be taken over a vicinity of the first soliton, the lower integration limit TQ is realized here as some value of r such that /y ' < To < r .

(83)

where if]~^ is the width of the soliton (see below), and T is the large separation between the solitons defined above. The condition ^/"^
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fields

[2, § 2

the gain/loss term as in eq. (20), and also a filtering term, so that the linearized form of the perturbed NLS equation becomes (cf eq. 97 considered below) iw- + ^D(z) UTT = iOt(z) U + i/5(z) Urr-

(84)

Here, fi is the filtering coefficient, and the most general case is considered, in which D, a and ji may all be periodically modulated, in order to take into account, respectively, possible dispersion management, periodic alternation of the losses and gain, and discrete allocation of filters in a real lumped model (as opposite to the simplified distributed model, which assumes the filtering to be uniformly "smeared" along the fiber link). Accordingly, these coefficients may assume the forms 0 ^ > . D . ^ . aU-.^B.^yW. « z , ^ / j . ^ , ,85, dz dz dz where overbars indicates average values, and the terms represented by the derivatives account for the purely variable parts with zero mean values (in particular, A(z) is called accumulated dispersion, which is defined so that it does not include a contribution from the average part of the dispersion coefficient). A solution to the linearized equation (84) for an exponentially decaying tail may be sought for as wtaii(z, T) = Aexp[-0(z) + ii/;(z)-(r] + \X)\r\] , (86) where A, rj and x are real constants. In fact, A and rj, the latter constant determining a characteristic soliton's width ^ r]~\ may only be found by matching the tail (86) to the soliton solution of the full (nonlinear) perturbed NLS equation, so in the solution (86) they figure as arbitrary real constants, while x must be found along with 0(z) and xp(z). The final form of the solution is simplified due to the fact that the dissipative terms on the rhs of eq. (84) may be treated as small perturbations. The solution must satisfy the condition that the function (p{z) in (86) may oscillate in z, but may neither decay nor grow systematically, as the interaction between established (quasi-stationary) solitons is considered. This condition yields, at first order of perturbation theory, X = ^ ,

(87)

and then one finds 0(z) = rjxAiz) - {A{z) + ii^B{z)\ ,

i/;(z) = \ r]' [Dz + A(z)] + iPo,

(88)

where A(z), A(z) and B(z) are the oscillating functions defined in eq. (85), and i/^o is an arbitrary real constant.

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

113

2.3.1.3. Interaction potential for two solitons in an optical communication link. The solution (86) for the tails of both solitons u\ and 1/2 can be inserted into the general expression (82) for the interaction potential (80). Because the condition for equilibrium between gain and losses selects the same parameters A and rj for all solitons in the system, it is sufficient to consider the case of interaction between identical solitons. One can then immediately check that the contribution from the outer integration limit with respect to both solitons, i.e. at r = +oc, vanishes, while the contribution from the intermediate limit T = -TQ has mutually cancelling exponential factors Qxp{±r]To) produced by the expressions (86) for the tails of u\ and U2, so that the effective potential does not depend on the arbitrary value of TQ:

UUT, AV^) = -2A^ r]D{z) exp(-20(z) - rjT) cos(xT) cos(Ai/;),

(89)

/.(2) where Aip = % - t/^o^^ is the phase difference between the two solitons. A remarkable feature of this expression is that the potential does not decay monotonically with the separation T between the solitons, but shows oscillations accounted for by the multiplier cos(xT), as seen in fig. 9. As first shown by Malomed [1991b], this opens the way to the existence of stationary bound states of the two solitons at values of T corresponding to extrema of the potential (89),

Fig. 9. Schematic form of the potential of the interaction (89) between two solitons for A0 = jr. The point T = T\ corresponds to the first bound state generated by the potential, see eq. (90).

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Variational methods in nonlinear fiber optics and related

fields

[2, § 2

i.e., cot(xT) = -X/T]- The first bound state corresponds to the smallest possible separation between the solitons, which is 71

— + tan , \r] X 2

2X

(90)

(see fig. 9), where it is taken into account that |x/^| ^ 1 as per eq. (87). However, the stability of those bound states is a tricky problem, which will be considered in the next subsection. It should be noted that the general expression (82) for the interaction potential was obtained for a model that has the Lagrangian representation (81). In fact, the analysis outlined above remains completely correct in the presence of loss and gain, as they can be easily included in the Lagrangian by means of the transformation based on eqs. (22) and (24). Contrary to this, the use of the potential is not quite consistent in the presence of filtering, for which a simple Lagrangian representation is not available. Nevertheless, the concept of the effective potential may be employed in the case when filtering acts as a small perturbation. Note also that the oscillations in the shape of the potential (89), which represent its most nontrivial feature, demand the presence of loss and gain, as is obvious from eqs. (90) and (87), but not necessarily the presence of filtering. The general approach to the calculation of an effective interaction potential between two far-separated solitons outlined above was generalized by Malomed [1998c] for two- and three-dimensional solitons. As well as in the ID case, the integral contribution to the potential can be eliminated by means of integration by parts, which, in the multidimensional situation, takes the form of the Gauss theorem, that reduces the potential to a contribution from the corresponding surface term, taken along circumferences (in the 2D case) or spheres (in the 3D case) of a large radius p surrounding each soliton. The radius p is chosen so that ro <^ p <^ R (cf eq. 83), where TQ is the size of the solitons, and R is the separation between them. Then, the surface term can be easily calculated in a general explicit form, as both solitons are represented in it by their asymptotic tails. The final expression for the interaction potential, which is (cf eq. 89) t/int ~ R~^^~^^^^ exp(-R/ro)cos(Aijj) (in the purely conservative model), where Z) = 2 or 3 is the dimension, does not depend on the intermediate radius po, similar to the fact that expression (89) does not contain the intermediate time scale TQ. A similar result was obtained by Malomed, Maimistov and Desyatnikov [1999] for the interaction of multidimensional solitons belonging to different components in a two-component model.

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

115

2.3.1.4. Generalization for dissipative systems. The concept of the effective interaction potential generated by the overlapping of the tail of each soliton with the body of the other can also be applied to purely dissipative systems that admit representation in the gradient (pseudo-Hamiltonian) form [cf. the Hamiltonian representation (17)]:

bA

r^

W/ = - ^ ,

A=

A(w,w*,w,-,w*)cbc,

(91)

where the variational derivative acts on the Lyapunov functional A with a real density A. An obvious consequence of eq. (91) is dA/d^ < 0, hence the dynamical evolution described by eq. (91) drives the system to a state with a minimum value of the Lyapunov functional. A physically interesting example of the gradient system is a parametrically driven Ginzburg-Landau (GL) equation, which was introduced by Coullet, Lega and Pomeau [1991] [cf. the parametrically driven damped NLS equation (29)]: Ut = u- \u\^u + Uxx + yw*,

(92)

that can be derived from the Lyapunov functional with the density ^=

n J -oo

i\uA^-\u\^

+ \\u\'-{[u^

^

+ {un''])dx,

(93)

^

7 being a real parameter. Equation (92) has an exact solution in the form of a Block domain wall (BDW), alias kink, Rew = OyJ\ + y t a n h f y 2 7 x j ,

Imw = opi^l - 3y sechf y ^ x j ,

(94) where a = ± 1 and 11 = ±\ are two independent polarities of the kink. BDW is stable in its existence interval 0 < y < | , which gives rise to a natural problem of the interaction between two BDWs with a large separation L between them. The interaction is accounted for by a part in the Lyapunov functional (93) that is generated by overlapping between the two kinks. This part of the functional will be referred to as a pseudopotential of the interaction. Linearizing the integrand in the integral expression for the pseudopotential with respect to the fields representing weak tails of the kinks, and taking into account the fact that an isolated BDW is an exact solution to eq. (92), which is generated by the Lyapunov functional as per eq. (91), it was shown by Malomed

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Variational methods in nonlinear fiber optics and related

fields

[2, § 2

and Nepomnyashchy [1994] that applying integration by parts reduces the pseudopotential Uint to a general expression similar to eq. (82), UM=

[ K ) V " 2 + C . C . ] | ; ; ; " ^ ^ + {1 ^ 2 } ,

(95)

where XQ is an intermediate point similar to fo in eq. (82). The substitution of straightforward asymptotic expressions for the tails of the two BDWs (94) into the general expression (95) yields an effective interaction pseudopotential that, as well as its counterpart (89) in the conservative model, does not depend on the choice of the intermediate point:

UUL) = 16^1 f(l - 3y)AiA2 ^xpf-^^l)

- ^

^xpf-l^^l)

(96)

(recall that Ai,2 are intrinsic polarities of the two kinks). In expression (96), it is taken into account that 0\a2 = -1 for two adjacent kinks. This result is most interesting in the case when (1 - 3y) is small: then the pseudopotential (96) combines attraction at L < LQ ^ A / | | l n ( l - 3 y ) | and repulsion at Z > LQ. Therefore, one may conclude that a periodic array of BDWs in an infinite system, or in a long one subject to periodic boundary conditions, is stable if the array spacing exceeds LQ (Malomed and Nepomnyashchy [1994]). 2.3.2. Full analysis of bound states of solitons in a realistic model of an optical communication link For a fiall description of the interactions between solitons (in particular, for the analysis of the stability of their bound states), it is necessary to consider the interactions by means of direct perturbation theory, rather than limiting the analysis to finding the effective interaction potential. The model takes into account, as above, losses, gain, and filtering, but in the distributed approximation, so that the accordingly perturbed NLS equation actually takes the form of the complex Ginzburg-Landau (GL) equation with constant coefficients. In the GL model, soliton-like pulses have, in accordance with eq. (86), tails which decay exponentially with oscillations, in contrast to the monotonically decaying tails of the NLS soliton (6). In the simplest GL equation with a cubic nonlinearity, solitary pulses are obviously unstable, as the zero solution, i.e., the soliton's background, is unstable in that equation due to the presence of the linear gain. Therefore, interactions between solitons and their bound states can be studied in a consistent way, as was

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

117

done by Afanasjev, Malomed and Chu [1997] in the framework of the cubicquintic (CQ) GL equation, which combines linear loss, cubic gain, and quintic loss: \UZ + \UTT + \U\ U =-\au+'\PujT-^'\e\u\

u-\r\u\

u.

(97)

Here, we set D = y = 1, and the positive parameters a, ^, e and F account for, respectively, linear losses, spectral filtering, nonlinear gain, and stabilizing higher-order nonlinear losses (a model of this type was first introduced by Petviashvili and Sergeev [1984]). The linear and quintic losses provide for the linear stability of the zero solution and for the global stability of the model, respectively. The nonlinear gain, accounted for by the term ~€ in eq. (97), can be produced, in a fiber-optic communication link, by a combination of the usual linear amplifiers with nonlinear saturable absorbers, see, e.g., the book by Hasegawa and Kodama [1995]. As demonstrated first in an appendix to the paper by Malomed [1987], in the case when the gain and dissipation terms in eq. (97) are small perturbations, which is relevant for the application to optical fibers, the CQ model gives rise to two different stationary soliton-like pulses which are close to the NLS soliton: u

= r]SQch[rj(T-T)] exp[i (^r/^z + 0)] ,

rj^ = (lery^

[5(2e -P)±

V25(2e - PY - 480arl ,

(98) (99)

where T and 0 are arbitrary constants. The upper and lower signs in eq. (99) correspond, respectively, to stable and unstable pulses. Besides selecting the definite value of the soliton's amplitude, which is arbitrary in the case of the NLS soliton proper, the small dissipative perturbations in eq. (97) also cause the asymptotic form of the soliton far from its center to be oscillating (cf eq. 86), u^2rjQxp(-r]\T\+ix\T\),

(100)

where x = <^V~^ + Pv [cf eq. (87); the small parameter x is absent in the zeroorder approximation (98)]. To consider the interaction between pulses with equal amplitudes r/, it is convenient to define the normalized propagation distance x = 2y/2r]^z, the separation between the pulses, r = rj{T\ - T2), and the phase difference between

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Variational methods in nonlinear fiber optics and related

fields

[2, § 2

them, yj = ipi - \lJ2. Considering overiapping between the soUtons as another small perturbation, one can derive a system of evolution equations for r and ijJ: (fr Vl dr —T + -Vi5 ^ + e"" [cos(br) + b sin(br)] cos t/^ = 0, ax^ 3 djc

(101)

where the four original control parameters combine into three final ones: ft

A= ^,^25{2e-l3f-4S0ar,

b= ~^^f^

.

(103)

Notice that cos(Z?r) and sm(br) in eqs. (101) and (102) are induced by the oscillations in the soliton's tail as per eq. (100). Equations (101) and (102) may be regarded as equations of motion for a mechanical system with two degrees of freedom, r and yj, in the presence of friction, in the potential U(r, \p) = -e"' cos(br) cos ip, which has a set of local extrema at Z?ro = tan"^ Z)+^:^(1+2«),

xp^^jtm

(104)

(cf. eq. 90), where n = 0,1,2,..., and m = ±l,rb2,.... Normally, points of the potential minimum would be stable fixed points (FPs) of the underlying dynamical system and, thus, they would produce stable bound states of the two pulses. However, a pecuHarity of the system (101) and (102) is that, while the effective mass corresponding to the coordinate r is +1, the mass corresponding to 1/; is - 1 . The presence of the negative effective mass drastically changes the stability of the FPs: all the local extrema (104) are saddles. It is easy to find a pair of eigenvalues that determine the character of the saddle FP (so that small perturbations around the static solution are growing as exp((Tz)):

Ox2 = ±bJ—-Q-'\

(105)

Due to the assumed smallness of the parameters on the rhs of eq. (97), the coordinate ro of the FP given by eq. (104) is large, hence the eigenvalues (105) are exponentially small. Notice that, in the framework of the fourth-order system (101) and (102), the FP must have four eigenvalues. Two others that are missing

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

119

in (105) are negative, and they are not exponentially small, i.e., they correspond to quickly decaying (stable) small perturbations around the FP. Besides the saddles (104), eqs. (101) and (102) also have another set of FP's, brQ = \n{\+2n\

i/^o = ^^(1+2m).

(106)

Comparing FP's (104) and (106), one concludes that, for the same value of «, they have nearly equal separation r between the bound pulses, but the relative phase \l) differs by \n. Stability analysis of the FP's (106) reveals that they have two relatively large negative eigenvalues corresponding to rapidly decaying perturbations [as well as in the case of FP (104)], and two exponentially small complex eigenvalues

(T|,2 = ± i f e J ^ e - ' " + ^ {~\

L/2p+^x\

e^^'";

(107)

hence, the FPs (106) are unstable spirals. Thus, we obtain two types of unstable bound states in the CQ GL model: depending on the phase difference between the pulses, their bound state are unstable as the saddle or as the spiral. Exactly this was observed in numerical experiments performed at non-small values of the perturbation parameters (and with normal dispersion, i.e., with the opposite sign in front of the dispersion term) by Afanasjev and Akhmediev [1996]. Therefore, one may conjecture that the above analytical results should plausibly remain valid even when the perturbation theory cannot be applied. Returning to perturbative analysis, one can notice a very important difference of eq. (107) from eq. (105). Namely, for the same «, i.e., nearly the same ro, the real part of the eigenvalue (107), accounting for the instability of the spiral, is proportional to the square of the exponentially small factor exp(-ro), while in the case of the saddle the growth rate of the instability is linear in this factor. Thus, the instability of the spiral is extremely weak. Nevertheless, it is an issue of fundamental interest to explore the result of a developing instability at extremely large propagation distances. To this end, it is necessary to take into account that the fourth-order system (101) and (102) implies relatively quick decay of the perturbations corresponding to the above-mentioned relatively large (non-exponential) stable eigenvalues, and very slow evolution of perturbations corresponding to the exponentially small eigenvalues (105) and (107). In this connection, a natural simplification of the full system is to project it onto the two-dimensional space of the slow modes, eliminating two rapidly decaying ones.

120

[2, §2

Variational methods in nonlinear fiber optics and related fields

Technically, this implies treating the second derivatives in eqs. (101) and (102) as small perturbations. In the zeroth approximation, one simply omits the second derivatives, so that eqs. (101) and (102) reduce to dr dx dij)

-^ dx

(108)

e ' [cos(br) + b sm(br)] cos \p,

V2I3 V2

(109)

= —r-Q ' cos{br) sin ip. A

Within the framework of this system, FP (104) remains the saddle, while (106) is neutrally stable. At the next step, one restores the second-derivative term by means of the identity ^ = ^ ( ^ ), and similarly for r, making use of the expressions (109) and (108) for di/^/dx and dr/dx. To perform the second differentiation, one can use eqs. (108) and (109) once again. Retaining essential corrections to eqs. (108) and (109) produced by this procedure, one arrives at the system dr dx

y/213

e '' [cos(br) + b sm(br)] cos V^ + ; ^ e ' cos{br) sin^ xp

dip V2 —— = -r-Q ' cos(Z?r)sin ip dx A be

Vix^p

0,

(110)

(111)

[cos(Z7r) + b sm(br)] sm{br) sm(2\py

It is straightforward to verify that the reduced system (110) and (111) has exactly the same FPs (104) and (106) as the full system (101) and (102), with the eigenvalues given by the same expressions (105) and (107). However, unlike the complicated full system, it is very easy to understand the general character of the dynamical trajectories on the phase plane of the reduced system. Indeed, one can check that the saddles (104) are connected by a rectangular grid of dynamical trajectories of the form r = ro, \p = ip(x), and r = r(x), ip = xpo, where ro and xpo are the values of r and ip at FPs (104). These trajectories are stable and unstable separatrices of the saddles, and they are exact solutions to both eqs. (101,102) and (110,111). From this fact, and from the knowledge of the eigenvalues of the FPs, there follows a phase portrait of the reduced system shown in fig. 10. Looking at the figure, one concludes that the spirals, except for those corresponding to « == 0 in eq. (106), asymptotically approach, atx ^ co, infiniteperiod limit cycles coinciding with an elementary cell of the separatrix grid. The spirals corresponding to « = 0, i.e., to the bound states with the smallest possible

2, § 2]

Dynamics of solitons in a single-mode nonlinear optical fiber or waveguide

121

27T

•K^S^-

6r

-2TT 7T :

37T

2 \

2

Fig. 10. Phase portrait of the reduced dynamical system (108) and (109), showing the long-distance evolution of the bound state of two solitons in the cubic-quintic Ginzburg-Landau model.

separation between the pulses, formally also asymptotically tend to a similar cycle, that, however, passes through r = 0, which implies a collision between the two pulses. The latter event is not described by the above approximation. It is natural to expect that the colliding pulses will undergo fusion into a single pulse. These conclusions have been checked by direct simulations of the full fourdimensional system (101) and (102), with the conclusion that the full system always produces resuhs virtually identical to those obtained from the reduced system, even if the perturbation parameters are not really small (Afanasjev, Malomed and Chu [1997]). Thus, a general conclusion is that the spiral-type bound states of the pulses in the CQ GL model are either practically stable in the usual sense, if one may neglect the exponentially weak instability, or stable as dynamical states corresponding to the limit cycle. 2.4. Dark and "symbiotic" solitons This review focuses on "bright" solitons, which are represented by solutions vanishing at infinity. It is, nevertheless, necessary to mention another class of solutions, in the form of dark solitons (DSs), which look like a hole in a uniform continuous-wave (CW) background. As is well known, the DS solution exists and is stable in the usual NLS equation (2) in the case of normal dispersion {D < 0): Wdark == a exp(ie(2^ + i0o) tanh f .

a{T-TQ)

(112)

122

Variational methods in nonlinear fiber optics and related

fields

[2, § 2

where a is the ampHtude of the background supporting DS, and 0o and TQ are arbitrary phase and position constants. DSs were first observed in a nonhnear optical fiber by Krokel, Halas, GiuHani and Grischkowsky [1988]. The description of the dynamics of perturbed DSs is essentially complicated by the presence of the fixed-amplitude background. In any perturbation-theory approach, separation of the internal DS degrees of freedom and the background is a crucial issue (Kivshar and Yang [1994], Kivshar and Krolikowski [1995]; see also a review by Kivshar and Luther-Davies [1998]). Moreover, a perturbed dark soliton can easily generate waves propagating on top of the background. Due to strong radiative losses generated by these waves, the dark soliton actually has no effective "quasimode" of its internal vibrations, contrary to the Anderson quasimode of the bright soliton described above. An appropriately modified version of VA for DS was developed by Kivshar and Krolikowski [1995]; it also includes the possibility of having nonlinearities in the corresponding NLS equation different from the cubic nonlinearity in eq. (2), and other perturbation terms. The technique was applied to various problems, notably the interaction (in fact, repulsion) between two DSs. VA for moving DSs was developed, with an emphasis on the stability problem, by Barashenkov and Panova [1993]. A necessary stability criterion for moving DSs, obtained in that work in a fairly simple form, states that the properly defined momentum of the DS must be a decreasing function of its velocity; this was later rigorously proved by Barashenkov [1996] by means of a Lyapunov functional (i.e., an integral functional with the property that it may only decrease as a result of the evolution of the fields) which can be introduced in this problem. Another known model that gives rise to DS is a system of NLS equations coupled by the nonlinear cross-phase-modulation terms, that will be considered (for bright solitons) in detail below in § 4. If the two equations correspond to two carrier waves with different wavelengths copropagating in a fiber, it is quite possible to encounter a case where one carrier wave (typically, that with the larger wavelength) has anomalous dispersion, while the other wave has normal dispersion. Then, it is interesting to consider a bound state consisting of a bright soliton in one subsystem and a DS in the other. Detailed analysis shows that the only possible bound state of this type has the bright-soliton component in the «orwa/-dispersion wave, and the DS component in the anomalousdispersion wave (Trillo, Wabnitz, Wright and Stegeman [1988], Afanasjev, Dianov and Serkin [1989], Wang and Yang [1990]). Because such soliton components, obviously, cannot exist in isolation, the bound state was given the name "symbiotic soliton" by Lisak, Hook and Anderson [1990], who studied it in detail by means of an accordingly modified version of VA. Note, however,

2, § 3]

Variational approximation for the inverse scattering transform

123

that symbiotic solitons are always unstable, as the CW background, which is necessary to support its DS component in the anomalous-dispersion subsystem, cannot be stable.

§ 3. Variational approximation for the inverse scattering transform As mentioned in the introduction, the NLS equation, which is the most important model for nonlinear optical fibers and waveguides, is amenable to an exact solution by means of 1ST. In the context of this method, the first step is to find scattering data corresponding to a given initial pulse w(r), solving the ZS (Zakharov and Shabat [1971]) linear equations for the two-component complex Jost function {\l)^^\x), ^)^^\x)), iV;[^^ + At/;^^^ + w*(r)V^^^^ = 0 , \\l)f^ - Xip^^^ + u(r)\l)^^^ = 0,

(113) (114)

where the asterisk indicates complex conjugation, and A is the spectral parameter which takes values in the upper complex half-plane. The most important characteristic of the pulse, viz., its soliton content, is determined by discrete eigenvalues A„, each giving rise to a soliton (7) with amplitude r] = 2 Im(A„) and velocity shift c = 4Re(A„), provided that D = y = 1 in eq. (2). Note that, as was discovered by Ablowitz, Kaup, Newell and Segur [1973, 1974] (see also books by Ablowitz and Segur [1981] and Newell [1985]), the ZS equations are used in application of 1ST not only to the NLS equation, but also to other integrable equations, including, in particular, those describing selfinduced transparency in an optical medium filled with two-level atoms (Ablowitz, Kaup and Newell [1974], Kaup [1977]). Thus, it is quite important to develop methods for solving ZS equations. There are very few field configurations for which the ZS equations can be solved exactly. These include a rectangular box without chirp (Manakov [1973b]), and a pulse of the form wo(r) = A [sech((7r)]'^'^' with real a and (.i (Maimistov and Sklyarov [1987], Griinbaum [1989]). In other cases, numerical methods had to be used (Bofifeto and Osborne [1992]); in some cases, a WKB(Wentzel-Kramers-Brillouin)-like analytical approximation can be applied to the ZS equations (Lewis [1985]). In particular, an important problem which requires numerical calculation is the influence of chirp on the soliton content of pulses, as the increase of the chirp gives rise to bifurcations, generating new solitons and then pushing solitons apart by lending them opposite velocities (Hmurcik

124

Variational methods in nonlinear fiber optics and related

fields

[2, § 3

and Kaup [1979], Kaup and Scacca [1980]). The problem can be solved in the simplest way for a rectangular box with the chirp accounted for by the phase function 0(r) = Z)| r|, with b = const., so that all of the initial chirp is concentrated at the soliton's center (Kaup, El-Reedy and Malomed [1994]). On the other hand, the ZS equations have a natural variational representation, which was used by Kaup and Malomed [1995] to develop VA for a semianalytical calculation of the discrete eigenvalues A, an account of which is given below. Independently, essentially the same was done by Desaix, Anderson and Lisak [1994] and by Desaix, Anderson, Lisak and Quiroga-Teixeiro [1996] (in the latter work, rectangular-box, sech^, and Gaussian initial pulses were considered). Multiplying eq. (113) by ip^^^ and integrating the result over dr, one can obtain the following representation for the spectral parameter: A=-^,

(115) dr,

A^= /

i/;^'V^''dr.

(116) (117)

Varying expression (115) with respect to \p^^^ and i/^^^^ produces equations (113) and (114), i.e., eqs. (115)-(117) give an effective Lagrangian for the ZS equations. It is noteworthy that this Lagrangian exactly coincides with the eigenvalue sought for; a similar fact is well known in quantum mechanics, where the linear Schrodinger equation can be obtained by varying a functional which is the energy eigenvalue (Landau and Lifshitz [1977]). However, an essential difference from quantum mechanics is that the functional (115) is not real. Note that the terms in expression (116) containing r-derivatives cancel mutually if i/^^^^ is proportional to ip^^\ Thus, variational ansdtze for these components should be functionally different. For example, if one uses Gaussian trial functions, it is necessary to allow the two Gaussians to have shifted centers, so that the ratio i/^^^Vi/;^^^ would be r-dependent. As a first example, one can take a rectangular pulse with an internal phase jump A0 = 2e:

fo

u{T)= I I ^exp(iesgnr)

if

|r|>l,

if

|r| < 1.

(118)

2, § 3]

Variational approximation for the inverse scattering transform

125

Here A is the real amplitude, and the width of the pulse can always be scaled to be 2, as implied in eq. (118). The simplest possible ansatz for the Jost functions corresponding to this pulse is ' 2exp(-;U(T IP^^\T)=

<

1))

T+l ^0

^(2) = ,

if

r > 1,

if if

|r|
'0 5(1 - T)

if if

r > 1, |r| < 1,

^2 Bexp(p.(T + 1))

if

r<-l.

(119)

(120)

The integrals (116) and (117) calculated with this ansatz are L = 2iB-^lA [B^ (e"'' + 7e'') - (7e'' + e"'')] ,

N = p;

(121)

note that they do not depend on jj.. Substitution of expressions (121) into eq. (115) and varying the only nontrivial parameter B yields B = ±i. Inserting this back into eq. (115), one obtains the eigenvalues Im A = - § ± 2 ^ cose,

ReA = T i ^ s i n e .

(122)

Since only the eigenvalue with ImA > 0 is meaningful, eq. (122) shows that, with increasing area ^S = 2^ of the pulse (118), the soliton appears, with an infinitesimal amplitude and finite velocity c = - | tan e, at the threshold value ^thr = Z . 2 cose

(123)

In the case e = 0, when the ZS equations for the pulse (118) have an exact solution, eq. (123) yields the threshold area | , while the exact result is JT/2 (Manakov [1973b]). Thus, the present crude approximation, using a single variational parameter, gives an error < 5%. This approximation fails to predict additional solitons which appear with a further increase of the area, the total number of solitons produced by the rectangular box with e = 0 being [(2S - JtyijT] + 1 (Manakov [1973b]). Equation (123) predicts that Sthr diverges in the limit 2e = Jt, when the pulse (118) turns into a combination of two pulses with opposite signs. For this case, an improved ansatz with an additional free parameter yields the

126

Variational methods in nonlinear fiber optics and related fields

[2, §3

result that the two-box configuration may produce only a pair of solitons with equal amplitudes and opposite velocities, provided that A exceeds a threshold value 3/2^^^ VA may also be applied for predicting the soliton content in the practically important case of a chirped Gaussian pulse, Uo(T) = BQXp[-{o^

(124)

+ il3)T^],

with real B, fi and o. A natural form for the Jost-function ansatz is also Gaussian, V;^") = 5 , e x p [-(W,j^ + 2l;„T)] ,

n = 1,2,

(125)

where the variational parameters B,,, W„ and t,n may be complex, provided that RcWn > 0. The subsequent calculation of the effective Lagrangian and variation can be done analytically, but the expressions are cumbersome. Final results are presented in fig. 11 as plots of the soliton amplitude rj vs. the area of the initial pulse at different fixed values of the initial chirp p (in all cases shown,

Fig. 11. Amplitude of the single soliton generated by an initial Gaussian chirped pulse (124) vs. its effective area A = B/ {y/rio) for a = 1 and different values of the initial chirp, /3 = 0,10,20,30,40,50.

2, § 4]

Internal dynamics of vector (two-component) soHtons

127

exactly one soliton is generated). These curves are almost the same as those obtained numerically for the same problem by Kaup and Scacca [1980]. Note that the plots clearly show the increase of the area necessary for the formation of the soliton with an increase of the initial chirp.

§ 4. Internal dynamics of vector (two-component) solitons 4.1. General description An important aspect of nonlinear fiber optics is copropagation of two or several modes in one fiber. Although standard fibers are designed to admit propagation of a single mode determined by its transverse structure (Agrawal [1995]), the polarization of light makes the fiber bimodal. Another very important possibility is to launch different modes carried by different wavelengths, which is the basis of the wavelength-division multiplexing (WDM) technique. WDM is the cornerstone of the present-day development of optical telecommunications, as it allows to create many parallel channels in a single fiber core. In the latter case, the most essential dynamical process is collision between two solitons belonging to different channels. In the application to optical telecommunications, quasi-random collisions in a multi-channel system are very detrimental, as they generate random walk (temporal jitter) of the solitons, which interferes with data transmission. VA can be naturally applied to the collision problem, reducing it to interaction of particles, as was shown for a two-channel system in an early work by Anderson and Lisak [1986b], and for multisoliton states in multi-channel systems by Ablowitz, Biondini, Chakravarty and Home [1998]. A related problem appears when two copropagating waves have their frequencies on opposite sides of the fiber's zero-dispersion point, so that one wave has normal dispersion, and the other propagates at anomalous dispersion. While only the latter wave can carry bright solitons, the normal-dispersion channel can be used to launch a periodic structure (which, loosely, may be realized as a periodic array of dark solitons). This support structure in the normal-dispersion channel induces, through the XPM interaction, an effective periodic potential in the soliton-carrying anomalous-dispersion channel (Shipulin, Onishchukov and Malomed [1997], Malomed and Shipulin [1999]). The periodic potential may be quite useftil, helping to stabilize the temporal position of solitons against the jitter (random walk) induced by interaction of a soliton with optical noise in the fiber. The suppression of the jitter by the periodic support structure has

128

Variational methods in nonlinear fiber optics and related

fields

[2, § 4

been considered using a combination of variational and numerical methods. In particular, the soliton dynamics reduce to an equation of motion for a particle in a periodic potential under the action of a random driving force, which, in turn, gives rise to the corresponding Fokker-Planck equation. From the standpoint of applications of VA, the case of a bimodal system corresponding to two polarizations in a nonlinear fiber is more interesting, as it allows one to consider different types of vector two-component solitons and their internal vibrations. Vector solitons in bimodal systems are considered below. A basic model of a bimodal nonlinear fiber includes two nonlinearly coupled NLS equations (see a detailed derivation by Menyuk [1987] and a book by Agrawal [1995]) for the amplitudes u(z,t) and u(z,t) of the electromagnetic waves in two linearly polarized modes, iu. + icur-\-ku + \utt + (|w|^ + ||t;|^) w+ jt;^w* = lu,

(126)

where the asterisk stands for complex conjugation, and the terms ~ ± c and ±A: take into account, respectively, the group-velocity and phase-velocity birefringence due to deviation of the fiber's cross-section from the ideal circular shape (usually, the group-velocity birefringence is much weaker than its phase-velocity counterpart). The nonlinear cross-coupling terms preceded by the coefficient | are insensitive to the phase difference between the u and u fields, and represent the cross-phase modulation (XPM) between the two polarizations, induced by the Kerr effect. The phase-sensitive terms preceded by the coefficient \ account for another manifestation of the Kerr effect in a multimode system. Viz., four-wave mixing. Lastly, the linear-coupling terms (with real A) on the rhs of the equations take into regard possible twist of the fiber, which causes linear mixing between the two linear polarizations (Trillo, Wabnitz, Banyai, Finlayson, Seaton, Stegeman and Stolen [1989]). Equations (126) and (127) can be derived from the Lagrangian density L= \

[(M*W-

+ v^'u- + icw*Wj- -

XCV^'UT)

+ C.C]

+ fc(M^-H2)-i(kl' + kP) + Hl"r + H')

(128)

hence VA can be applied here. The dynamics of two-component solitons governed by eqs. (126) and (127) are rather complicated if four-wave mixing is taken into account, as the phase difference between the components will play

2, § 4]

Internal dynamics of vector (two-component) solitons

129

the role of an additional degree of freedom, along with the widths of the two components and the separation between their centers, see below. This problem was analyzed by means of VA, neglecting the group-velocity birefringence and twist-induced linear coupling, in the works of Muraki and Kath [1989, 1991] and Anderson, Kivshar and Lisak [1991], and a generalization for a case when the birefringence coefficient varies randomly along the fiber was developed by Ueda and Kath [1992]. A similar analysis based on VA which, however, took into account the group-velocity birefringence and linear coupling between the polarizations was independently developed by Malomed [1991a]. In all these works, the variational ansatz was based on sech, and results were presented in the form of normal forms of various modes of intrinsic vibrations of twocomponent solitons. As demonstrated by Malomed [1991a], in the presence of the group-velocity birefringence, VA also makes it possible to explain another interesting dynamical phenomenon: internal degrees of freedom of the twocomponent soliton get coupled to the motion of its center of mass, so that the internal vibrations generate periodic oscillations of the velocity at which the soliton propagates (this effect had been known from direct numerical simulations of eqs. (126) and (127) reported by Trillo, Wabnitz, Wright and Stegeman [1989] and Wright, Stegeman and Wabnitz [1989]). However, in a realistic situation the fiber birefringence is so strong that the corresponding length of beatings between two polarization components is much smaller than any propagation length relevant to the evolution of solitons, hence both the four-wave-mixing nonlinear and twist-induced linear phase-dependent coupling are negligible. Thus, the most fiandamental model is based on the simplified version of eqs. (126) and (127), iu, + \utt + {\u\^ ^+3 \\u\^)u = 0 , 2^tt ^ \\^\ \v, + \vtt + {\v\^-^\\u\^)v

= 0,

(129) (130)

in which the birefringence terms are dropped because they may be readily eliminated in the absence of the phase-dependent couplings. The dynamics of two-component solitons in this simplified model will be considered below, following works by Ueda and Kath [1990], Kaup, Malomed and Tasgal [1993], and Malomed and Tasgal [1998]. Notice that the XPM coefficient | in eqs. (129) and (130) is relatively close to its special value 1, at which the two coupled NLS equations constitute a model integrable by means of 1ST, as was found by Manakov [1973a]. The proximity to the Manakov system can be used to develop a perturbative approach based on a

130

Variational methods in nonlinear fiber optics and related fields

[2, §4

combination of the Lagrangian and Hamiltonian representations of the equations (Malomed [1991a]). Before proceeding to detailed consideration of vector solitons, it is relevant to mention another problem which appears in the model of a bimodal fiber, in the case when the dispersion is normal, and NLS equations for two waves with opposite circular polarizations are used. In this case, the XPM coefficient in the coupled NLS equations is 2 (the same as for the interaction between waves at different wavelengths) rather than | , see eqs. (129) and (130). While bright solitons are impossible with normal dispersion, it was demonstrated by Malomed [1994a] that another nonlinear structure occurs in this case, viz., an optical domain wall, separating two temporal domains filled with waves having opposite circular polarizations. Variational methods are quite useftil in the study of these domain walls and interactions between them. 4.2. Solitons in a bimodal birefringent fiber 4.2.1. Ansatz and stationary states A general ansatz for vector solitons must make it possible to describe solitons with different widths of their two components, as well as independent vibrations of the two widths. Because a product of hyperbolic secants with different widths cannot be integrated in analytical form, the only option is to adopt a Gaussianbased approximation, as was first done by Kaup, Malomed and Tasgal [1993]. The ansatz for vector solitons generated by eqs. (129) and (130) is

M(Z,

t) = All e x p

1

(t-yu

2 V Wii

u(z, t) = Au exp

1

[t-yc

2 V Wc

exp{i [(pii + bii{t -yii) + Cuit -yuf] ] ,

exp{i [(0, + b,(t -y,) + c,{t - 7 . ) ' ] } ,

(131)

(132)

where, as usual, all the free parameters are real and are assumed to be fianctions of z. The ansatz accommodates a pulse with arbitrary amplitudes {An^Ai), widths {Wii, Wa), and central positions {yu,yu)- Each component is also allowed to have arbitrary phase {au^ac), central frequency (bi,,bu), and frequency chirp (Cii^Cu). Note that the ansatz admits splitting of the vector soliton into single-component ones. The VA technique yields a set of equation for the twelve parameters of the ansatz. The set includes four dynamical invariants and six evolutional

2, § 4]

Internal dynamics of vector (two-component) solitons

131

equations which are written below as three second-order equations. There are also equations for the phases 0u and 01^ which involve other variables but do not themselves influence anything else, so they are not displayed below The dynamical invariants are the energies in the two modes and their net momentum, oo

/

rz.

^^At^^AlWu,

(133)

E, = { r\v\'dt=^AlW„ OO

(134)

•

1

/ - (w/M* - uu^ + UfU* - uu^)dt = —(Euyu-^Evyu) = -(E^bu + EM,

.^^^.

and the Hamiltonian of eqs. (129) and (130), which is not needed in an explicit form. It is convenient to define the soliton's polarization angle 6, by tan^ 6 = E./E^. The equations of motion for the widths W^^^; and relative position y = yu -yv of the components of the vector soliton are

w -^E.W.{Wl

+ fv^)

-^/^•w^wl

+w^)

d2

d^ dz2^

-3/2 /,

ly"- \

d az

I

y^

(138)

= -^(E„.£.)(^,^^.y^^exp(-^ and c„ = {2W„r\d/dzW,„ c, = {2W,)-\d/Az)W,. Fixed points (FPs) of eqs. (136)-(138) correspond to steady-state vector solitons with y = c,, = c^ = 0. The corresponding stationary values of the widths were first found, by means of VA, in a paper by Kaup, Malomed and Tasgal

132

[2, §4

Variational methods in nonlinear fiber optics and related fields

[1993] (independently, the same family of general stationary solitons was found in a purely numerical form by Haelterman, Sheppard and Snyder [1993]): W„ =

(139)

W, =

(140)

where the width ratio r = Wu/W^ is determined by 3/2 4

" \ .

"

1

A

(141)

3 \\+r^

Thus, the energies of the two modes Eu and E^; are free parameters of the stationary vector soliton, which determine its amplitudes Au and A^ and widths Wu and Wij according to the above equations. While the relative frequency {bu - bu) is zero at the fixed point, the mean frequency {bu + bu)/2 can take any constant value, a nonzero one adding a net momentum to the soliton, making it "walking". Table 1 summarizes the VA predictions for the parameters of the stationary vector soliton in a range of polarization angles. The values of the widths in the table refer to either the Gaussian ansatz with net energy E = Ei,-\- E^ = VJT/2, or (see below) a hybridGaussian-sech ansatz with E = I. Note that the negativeness of the Hamiltonian is a necessary existence condition for the soliton (if the Hamiltonian is positive, the pulse will decay into radiation). The stationary widths for other values of the energies can be obtained from table 1: in either ansatz, the widths scale as the reciprocal of the net energy, so, to obtain the widths for arbitrary net energy £, the values in table 1 should be multiplied by Table 1 Widths of the two components predicted for the stationary vector soliton by the variational approximation based on the Gaussian ansatz with total energy E = \/JT/2 or the sech ansatz with total energy E = 1, for different values of the vector soliton's polarization angle 6

en

Polarization 0

5

10

15

20

25

30

40

45

1.0000

1.0038

1.0149

1.0330

1.0571

1.0860

1.1175

1.1774

1.2000

n/a

1.1881

1.1930

1.2003

1.2089

1.2168

1.2220

1.2150

1.2000

2, § 4]

Internal dynamics of vector (two-component) solitons

133

{y/n/2/E) or (l/fi"), for the Gaussian or sech ansatz, respectively. For polarization angles > 45°, one should take the complement of the angle and interchange u and V. 4.2.2. A hybrid ansatz In the exactly solvable cases, t; = 0 or w = t;, when the coupled system degenerates into a single NLS equation, and in the Manakov system, exact vector-soliton solutions have |w| = A^^"^^ sech(r/^^^^'^), with width and amplitude related to those predicted by the Gaussian ansatz in a simple way: p^sech ^

/ ^ ^Gauss^

^sech ^

/ _ J _ ^Gauss^

(^42)

provided that the energy of the soliton, E = (^sech^2^sech^ j^ equal to that of the Gaussian ansatz. This relation suggests that making the same adjustment in thQfinalresults produced by the Gaussian-based VA, i.e., replacing the Gaussian by the sech pulse with the parameters rescaled according to eq. (142), may help to get the approximate vector-soliton shape closer to the true pulse shape. To estimate the importance of the adjustment, one can note that, in the solvable cases, the standard FWHM width predicted by the Gaussian approximation differs from that of the exact sech soliton with the same energy by a factor (i/2cosh"^ ViyVjTlnl] ^ 0.845. Thus, to improve the accuracy of VA, one may take a solution to eqs. (136)(138), which govern the evolution of the parameters of the Gaussian ansatz, and insert the solution not into the Gaussian ansatz but rather into M(z, 0 = ^ r ' sechf ^ " j exp[i(0„ + b.,{t -y„) + c„(t -y,.f)] v(z,t)

= 4 - S e c h ( ' ^ ' ) e x p [ i ( 0 , + M ' - J . ) + c.(r->'.)')],

,

(143) (144)

with the widths and amplitudes rescaled according to eq. (142). Comparison with numerical simulations clearly shows advantages offered by the hybrid ansatz. It was observed that, starting with the initial conditions corresponding to the FP (139)-(141), predicted by the Gaussian-based VA, more than 99% of the initial energy is ultimately retained by the soliton, with the exact size of the radiative losses slightly depending on the soliton's polarization angle 6. So, by this measure - the share of the net energy going into the soliton the predictions of the Gaussian VA are very good. However, in terms of the

134

Variational methods in nonlinear fiber optics and related fields

[2, §4

soliton widths 1.25

solfton widths

Fig. 12. (a) Evolution of the vector soliton widths produced by simulations of eqs. (129) and (130), starting from the initial condition predicted by the usual Gaussian approximation for the vector soliton with polarization ^ = 30° and energy E = VJT/I. (b) Same, starting from the initial conditions predicted by the hybrid Gaussian-sech approximation, based on eqs. (142)-(144). The larger and smaller widths pertain to the less energetic and more energetic components of the vector soliton, respectively.

2, § 4]

Internal dynamics of vector (two-component) solitons

135

eventual widths of the vector soHtons, the agreement of the same version of VA with the numerical results is worse, with a relative error of about \. To illustrate, fig. 12a shows the numerically simulated evolution of the pulse widths, starting from the fixed point of the usual Gaussian approximation, with total energy E = \fl7i and polarization Q = 30°. The hybrid approximation, based on eqs. (142)-(144), yields much more accurate predictions for the widths of the stationary states than the usual Gaussian VA: even in the worst case the relative error is below 1%, while in most cases the accuracy is even higher than that. The radiative energy shed by the evolving vector soliton starting from the hybrid ansatz was too small to measure, being much less that 1%, which is another drastic improvement offered by the hybrid VA model. Figure 12b illustrates this, showing the evolution of the widths in the simulations, starting from the stationary solution as predicted by the hybrid model with total energy E = y/lJt and polarization 6 = 30°. 4.2.3. Intrinsic vibrations of a vector soliton For small oscillations around the stationary vector soliton, linearization of eqs. (136)-(138) shows that small vibrations of the separation;^ between the two components decouple from vibrations of the widths W^ and W^. Two distinct eigenmodes of the width vibrations can be identified: one "in-phase", with both widths oscillating synchronously, the other "out-of-phase", with the two widths oscillating with a phase shift Jt (Kaup, Malomed and Tasgal [1993]). For the case 0 = 45°, when the energy of the vector soliton is equally divided between its two components, the y- and in-phase width oscillation eigenmodes were first identified by Ueda and Kath [1990]. In this case, the eigenfrequencies of the separation (y-), in-phase-width, and out-of-phase-width vibration modes, calculated by means of the sech ansatz, are 2/72

{k;''\k^^'\K'^'')

= (0.50,0.69,0.99) • ^ .

(145)

The same eigenfrequencies were found from direct simulations by Malomed and Tasgal [1998] to be ( C " ^ , ^ " ' , ^ ? ) ^(0.53,0.54,0.56)

.

(146)

Comparison with eq. (145) shows a large error of the VA-based prediction for the in-phase-width mode, and a very large error for the out-of-phase-width mode.

136

Variational methods in nonlinear fiber optics and related

fields

[2, § 4

The stability and instability of different oscillation modes of the perturbed vector soliton can be predicted by considering the unperturbed one as a nonlinear structure that protects itself from decay into radiation by placing its eigenfrequency in a spectral gap in which radiation modes do not exist. In other words, the unperturbed soliton is, essentially, a gap soliton, as defined in the review by de Sterke and Sipe [1994]. In particular, for the case 0 = 45°, the gaps in the spectra of the spatial frequencies k for u- and t;-components are identical, and they can easily be found in exact form from eqs. (129) and (130) linearized around the stationary soliton, without resorting to VA: \k\ < 0.5454

.

(147)

Jt

If the mode's eigenfrequency lies inside the continuous spectrum (outside the band gap), the oscillation mode couples to the radiation and is therefore subject to decay Comparing the eigenfrequencies (146) with the spectral gap (147), one concludes that the predicted frequency of the separation oscillations ky belongs to the gap, and the frequency of the in-phase-width oscillations is located virtually exactly at the edge of the gap. In contrast with these, the frequency of the out-of-phase-width oscillations lies deep inside the continuous spectrum. These conclusions suggest that the oscillations of the separation between the two components of the vector soliton should be the stablest eigenmode, while the out-of-phase width oscillations should be unstable. The system of ODEs (136)-(138) produced by VA also predicts the possibility of dynamical chaos if the vector-soliton's internal vibrations have sufficiently large amplitude. However, in the corresponding PDE simulations, the degree of freedom corresponding to the out-of-phase-width oscillations dies out quickly, which actually prevents the appearance of chaos (Malomed and Tasgal [1998]). It is relevant to mention work by Yang [1997a] (see also Yang [1997b]), who studied the vibrations of vector solitons and emission of radiation from them via a different method, based on numerical algorithms for determining the true form of the small vibrations. This work provides a considerable advance in detailed understanding of the dynamics of small vibrations in vector solitons; in particular, the out-of-phase mode of the width vibrations was identified there with a combination of radiation modes, which accords with the results outlined above. Vector solitons with more than two components can also be considered in a model of several waves carried by different frequencies and interacting via XPM. By means of a combination of variational and numerical methods, a threecomponent vector soliton of this type was considered by Tran, Sammut and Samir [1994].

2, § 4]

Internal dynamics of vector (two-component) solitons

137

4.3. Resonant splitting of a vector soliton in a bimodal fiber with periodically modulated birefi^ingence It is quite easy to fabricate a bimodal optical fiber with periodically modulated birefi-ingence, which gives rise to a model with a group-velocity difference between two polarization modes that changes sign periodically. This suggests a possibility of a resonance between the separation mode of internal vibration of the vector soliton, considered above, and the periodic modulation of the birefringence. This model was introduced by Malomed and Smyth [1994]. It is based on coupled equations (cf. eqs. 126 and 127) iu,-\-ic(z)ur + ^Utt-\-(\u\^-^B\u\^)u

=0,

(148)

w,-ic(z)ur-\-\urt

= 0,

(149)

+ (\u\^^B\u\')u

where physically relevant values of the XPM coefficient are 5 = | and B = 2 corresponding, as explained above, to linear and circular polarizations. The group-velocity difference between the polarizations is assumed to be modulated as c(z)-^sin(^z).

(150)

In what follows, we set A: = 1, which can always be achieved by obvious rescaling. In order to derive equations for internal oscillations of the vector soliton, the following ansatz is adopted (cf eqs. 131 and 132):

u = A(z)scchf^^^)

cxp[i0,(z) + iQ{z)iT-yiz))^ib(z){T-y(z)f]

, (151)

u = A{z)scchf^^)

exp[i(l>2(z)-iQ(z)(T^y(z)) + ib(z)iT+y(z)f] . (152)

Straightforward VA-based calculations lead to the following equations of motion for the separation y(z) between the centers of the two components and their common width W{z): g=2S/:r-^F'(|)+ecosz,

^{'

(153)

w

w \w

(154)

138

Variational methods in nonlinear fiber optics and related

fields

[2, § 4

where K = A^W is the dynamical invariant which represents the conserved energy in each polarization (the notation K for energy is used instead of E in this section), and F(x) = (xcoshx - sinhx)/sinh^x. At £ = 0, the system of equations (153) and (154) has a fixed point (FP) at j^ = 0, ^ = (1 + B)~^K~\ which corresponds to the stationary vector soliton. Linearizing the equations in a vicinity of the FP, one readilyfindstwo eigenfrequencies of small oscillations in the absence of the periodic modulation: ql =f^B{\+BfK\

(155)

ql =±(\+BfK\

(156)

where the subscript indicates the type of the corresponding eigenmode. Several different types of resonance between the internal vibrations of the vector soliton and the periodic modulation of the birefringence are possible. The simplest (fundamental) resonance is expected for the value of the soliton's energy at which K-'^ = ^^B(\^B)\

(157)

when, according to eq. (155), the eigenfrequency qy coincides with the modulation wave number (which is 1 in the notation adopted). A second-order resonance may take place at

K~^ = ( L ± ^ ,

(158)

JT

when qw "= 2 according to eq. (156). Indeed, eq. (153) shows that in this case the variable y is driven at the frequency 1, and, in turn, it resonantly drives the variable W through eq. (154) at the frequency 2. In order to realize how the resonances predicted by considering small internal vibrations of the vector soliton manifest themselves, the system of equations (153)-(154) was simulated numerically. It was found that, with increasing modulation amplitude e, the driven vibrations of the vector soliton become more and more chaotic and, finally, the vector soliton is split into two singlecomponent ones, which corresponds to 7 ^ oo at z -^ oc in terms of eqs. (153) and (154), at a certain critical value ^cr- Figure 13 shows an example of the evolution of the separation y(z), finally resulting in splitting, in the case when e slightly exceeds ^crA numerically found dependence of ^cr on energy K is shown (for B = 2, i.e., circular polarizations) infig.14. In this case, eqs. (157) and (158) predict the

2, § 4 ]

Internal dynamics of vector (two-component) solitons

139

Fig. 13. Example of the splitting of a vector soliton into two single-component solitons under the action of a periodically modulated birefringence, as predicted by simulations of eqs. (153) and (154) with B = ^ (i.e., for linear polarizations), at AT = 0.8 and e = 0.13 (slightly above the splitting threshold).

Fig. 14. Critical amplitude of birefringence modulation, e^x^ ^s. soliton energy K, as obtained from simulations of eqs. (153) and (154) with 5 = 2 (i.e., for circular polarizations).

fundamental and second resonances at ^ = 0.363 and A^ = 0.591, respectively. The plot in fig. 13 indeed has the deepest and second-deepest minima fairly close to these two points. The accuracy with which the positions of the minima are predicted by eqs. (157) and (158) is remarkable, as the analytical results were obtained fi-om the consideration of small oscillations, while the splitting implies indefinitely large amplitudes of the oscillations prior to the splitting.

140

Variational methods in nonlinear fiber optics and related

fields

[2, § 5

§ 5. Spatially nonuniform fibers and dispersion management 5.1. Dispersion-decreasing fibers As was explained in detail in §2.1.4, the problem of compression of a pulse in an optical fiber without disturbing the pulse's fundamental-soliton character is of great practical importance. If the original pulse is already sufficiently narrow in the temporal domain, and/or the fiber's dispersion is high enough, so that the soliton period (see eq. 8) is not too large, a natural idea is to pass the soliton through a long piece of fiber with a gradually decreasing dispersion coefficient (Kuehl [1988]). If the length of the piece essentially exceeds the soliton period, one may hope that the pulse will adiabatically follow the decreasing dispersion coefficient, while remaining the fijndamental soliton. This idea was realized in dispersion-decreasing fibers (DDF), in which the variable dispersion is created by tapering the fiber, i.e., gradually varying the diameter of its core. Experimentally, high-quality strong compression of fiandamental solitons by means of DDF has been demonstrated in a number of works, e.g., by Chernikov, Dianov, Richardson and Payne [1993]. DDF may find a particular application in improvement of the amplification of (sufficiently narrow) solitons in a long fiber-optic communication link, as proposed by Malomed [1994b]. A problem is that, as a matter of fact, a linear (erbium-doped) amplifier instantaneously multiplies the soliton temporal profile by an amplification factor, transforming the fundamental soliton into a "lump", that will later split into an amplified soliton proper and a noisy radiation component. However, the amplified pulse may be fed immediately into a fiber with a higher dispersion value, for which it will remain a fundamental soliton, and then DDF can adiabatically transform it into a fundamental soliton adjusted to the value of the dispersion in the system (bulk) fiber. 5.2. Formation of a soliton fi-om a pulse passing a zero-dispersion point An interesting realization of the situation considered above is when the dispersion is varied along the propagation length so that it changes from normal to anomalous. As proposed by Malomed [1993], in that case a pulse launched in the normal-dispersion part of the fiber may self-trap into a soliton after passing the zero-dispersion point (ZDP). The process can be analyzed by means of VA, using the general equations (32) and (33) with the variable D(z). The most essential prediction is that formation of a soliton is possible if the pulse's energy exceeds a certain threshold, which is proportional to the value of the slope dD/dz at ZDP.

Spatially nonuniform fibers and dispersion management

2, §5]

141

4[

;^ 2

Fig. 15. Comparison of results produced by direct simulations of the NLS equation (solid curves) with the variable dispersion coefficient passing through zero and simulations of the variational equation (32) (dashed curves) in the same case. For energy E = 4 and area M = Jt/2 of the initial pulse (159), we show (a) the evolution of the field amplitude |w| at the center of the pulse, r = 0, and (b) the temporal shape of the pulse, |w(r)|, at the point z = 1.

This process was simulated numerically by Clarke, Grimshaw and Malomed [2000], within the framework of eq. (2) with y = 1 and D(z) taken in the simplest form providing for the continuous passage through ZDP (at z = 0): D(z) = sgn(z) at \z\ > 1, and D(z) = z at \z\ < 1. In fact, the simulations commenced at z = - 1 with the initial pulse u(z =

-1,T)=

A sech(/?r).

(159)

Thus, the possible outcomes of the process are controlled by two positive parameters A and h introduced in eq. (159), i.e., by the energy and area of the initial pulse, which are E = lA^/h and M = nA/h according to eqs. (10) and (9) [in this section, the definition of energy does not include the factor ^ in front of the integral in eq. (10), and the symbol for the area is M ("mass" of the soliton) instead of S]. Comparison of direct PDE simulations with those of the variational dynamical equation (32) has demonstrated that the agreement between them is quite good for sufficiently narrow initial pulses, for which VA is expected to be applicable, see fig. 15 for an example. Results of many simulations are summarized in a diagram showing qualitatively different outcomes of the pulse evolution for different values of the initial area and energy (fig. 16). These outcomes may be: decay of the pulse into radiation, formation of a single fijndamental soliton, formation of a higher-order soliton {breather), and also formation of a pair of two separating fundamental solitons. A noticeable feature of the diagram is a

142 4

-

.

.

^

I

1

•

,

' /'—1—'

'

C

3 R t*q

//

2

B 5

1 •

/

; :

/ / /

0.0

0.^

\

: -

0 1.0

1.5

2.0

Fig. 16. Chart showing different outcomes of the evolution of a pulse (159) passing from normal to anomalous dispersion, for different values of the pulse's initial area M = JtA/h and energy E = lA^/h. Symbols: R, decay into radiation; S, formation of a single fundamental soliton; B, formation of a breather; C, formation of a pair of separating fundamental solitons.

virtually direct transition from the single-soliton state to the pair of separating solitons (regions S and C in fig. 16), although, in theory, the transition may only occur via an intermediate breather state. Plausibly, the intermediate layer is so thin that it cannot be seen in the computer-generated diagram. 5.3. Fibers with periodically modulated dispersion 5.3.1. Variational analysis The fact that VA predicts persistent internal vibrations of a perturbed soliton, described in the exact parametric form by eq. (38) or in the approximation of small oscillations by eq. (36), suggests a possibility of resonances between these vibrations and a periodic modulation of the local dispersion coefficient along the fiber. This problem was considered first by Malomed, Parker and Smyth [1993], who assumed the simplest sinusoidal form of the modulation, D{z) = 1 + fsinz,

(160)

where it is implied that the period of the modulation may always be made equal to 2JZ by means of a rescaling of the NLS equation (2). Possible nonlinear

2, § 5]

Spatially nonuniform fibers and dispersion management

143

resonances were studied analytically, assuming f
(161)

The sequence of different dynamical regimes observed with increasing 8 is illustrated by a set of typical plots in fig. 17, which pertain to the case KQ = 2, i.e., E = y/jT, when eq. (36) predicts the second-order resonance. When the modulation amplitude e is small, the rate of direct emission of radiation by the soliton can be calculated by means of perturbation theory, as done by Abdullaev, Caputo and Flytzanis [1994]; the role of radiation loss in the destruction of the soliton is considered in detail in the next subsection.

144

Variational methods in nonlinear fiber optics and related fields

[2, §5

0.725r

0

10

20

30

40

50

60

70

80

90

IOC

0

20

40

60

80

100

120

140

160

180

200

Fig. 17. Typical solutions to eq. (32), with the effective potential (33) and D{z) taken as per eq. (160). The energy of the soliton is £ = y/Jr, which corresponds to the frequency A:O = 2 of small free vibrations (36) of the soliton near its equilibrium shape. The modulation amplitude is (a) e = 0.01, (b) e = 0.05, and (c) e = 0.25, the latter value being slightly larger than the critical amplitude which gives rise to destruction of the soliton.

2, §5]

Spatially nonuniform fibers and dispersion management

145

^E' • D D D aaa D a D D aSa D D D " •aa

a aaa a aaa BBa a a a a aaS B-a a a a 0D aaa pa DmDD aaa a a aB aaa aSB •aa a a a OB D D D D aaa a aBB a a aa a a BO! m a n aaa a aBB n Ia aa a a BB! m a g aaa a BBa D Ia aa a a D D D n aaa a BBa a a ' aa a a mmOm aaa a BBa n a aa a n agDD a a a B-a a a aa a a -DBD a a a BB moma a a GBB Baa a a aa a a a a aBB Baa a a aaaa BB IBB • B a a a a B B B aaa n a aSaa aaa a Dm aaa BB aaa aaaa aaa aaaaaa SBB B a a a BBB BBO aaa •Baaaa IBB B 9 B S IBB B B B B BBBBBBl iBBBBBaaa B B B B B B BBBBBBI IBBBBBBBa B B B B0.5 BBBB

^8

Fig. 18. Phase diagram in the parametric plane {£,£-) of the NLS equation with the local dispersion modulated as per eq. (160). The solid and open rectangles correspond, respectively, to stable and splitting solitons.

5 J.2. Comparison with direct simulations Grimshaw, He and Malomed [1996] compared the VA predictions for a soliton in a fiber with sinusoidally modulated dispersion with results of direct simulations of the NLS equation (2) with y = 1 and D{z) taken as per eq. (160). Results of systematic simulations are summarized in fig. 18. Two gross feature of this diagram roughly comply with the predictions of VA. Firstly, destruction of the soliton may take place if the modulation amplitude exceeds a critical value, which varies, essentially, within the interval 0.15 < ^cr < 0.20, that should be compared to the interval (161) predicted by VA. Secondly, destruction of the soliton actually takes place, for e not too large, if the initial squared soliton energy E^ exceeds a minimum value E^j^ varying between 1.8 and 2.0, which may be compared with the value E^ = ^JT that, according to eq. (36), gives rise to the fiindamental resonance between small vibrations of the perturbed soliton and the periodic modulation (160). The most essential qualitative difference between the assumptions on which VA was based and numerical results is that the fiindamental mode of the soliton destruction under the action of the sinusoidally modulated dispersion is not decay into radiation, but splitting of the soliton into two apparently stable secondary solitons, which is also accompanied by emission of a considerable amount of

146

Variational methods in nonlinear fiber optics and related

fields

[2, § 5

Fig. 19. Typical example of the splitting of a fundamental soliton into two secondary ones in the NLS equation with sinusoidally modulated dispersion, observed at f = 0.3 and E^ = 2.9, cf fig. 18.

radiation. A typical example of this splitting in displayed in fig. 19. Obviously, the ansatz (13) does not admit any splitting; nevertheless, VA predicts the basic characteristics of the destruction of the soliton qualitatively and semiquantitatively correctly, even if the actual destruction mode is dififerent from that implied by VA. Detailed inspection of the numerical results presented by Grimshaw, He and Malomed [1996] shows that, prior to splitting, the soliton performs a number of irregular vibrations, which resembles the picture produced by VA, see fig. 17c. In accordance with that picture, the vibrational stage preceding the destruction of the soliton is quite long if the splitting takes place at e slightly exceeding fcrThe soliton stability diagram for the sinusoidally modulated model, displayed in fig. 18, has a number of other interesting features, such as a "stability isthmus" and general restabilization of the soliton at large e [note that for ^ > 1, the local dispersion becomes sign-changing according to eq. (160)]. However, these features are found too far outside the domain of applicability of VA. Very interesting additional results concerning the comparison between VA and direct simulations in the above model were obtained by Abdullaev and Caputo [1998]. They have also found that the destruction of the soliton takes place via splitting into two secondary ones, and demonstrated that agreement between VA and direct simulations is fair as long as the frequency KQ of the small vibrations (see eq. 36) remains smaller than the modulation spatial frequency (equal to 1 in the present notation). At KQ > 1, intensive emission of radiation takes place (even without complete destruction of the soliton), which, naturally, strongly deteriorates the agreement with VA, that completely disregards the radiation component of the field. These conclusions are illustrated by figs. 20 through 22,

2, § 5 ]

Spatially nonuniform fibers and dispersion management

147

1

0.9

'^ !A

f\

n

M U i MM M 0.8

3•i

^

1 ;

n

1..;1

\^

0.7 100

200

X

Fig. 20. Comparison between oscillations of the soliton width as predicted by VA (dashed curve) and found by Abdullaev and Caputo [1998] from direct simulations of the NLS equation (solid curve) with sinusoidally modulated dispersion at KQ = \ and e = 0A.

Mi' CO

ri

2

¥^¥f r^ 100

200

X

Fig. 21. Same as fig. 20, but for f = 0.6. Both VA and direct simulations predict destruction of the soliton in this case.

CO

2

Fig. 22. Same as fig. 20, but for KQ = 1 and e = 0.2. In this case, VA predicts decay of the soliton, but in direct simulations it remains stable, as the internal vibrations predicted by VA are strongly damped by radiation losses.

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Variational methods in nonlinear fiber optics and related

fields

[2, § 5

which compare the analytical and direct numerical results for different values ofKo and e. Note that the destruction of the soliton in the case shown in fig. 21 actually proceeds via splitting. Another important numerical finding reported by Abdullaev and Caputo [1998] is that, in cases when the variational and direct numerical results are generally close, a more subtle (and quite natural) effect of radiation losses is strong suppression of higher harmonics in the soliton's internal vibrations predicted by VA.

5.4. Dispersion management In application to real optical telecommunications, the concept of variable sign-changing dispersion has gained great popularity under the name of dispersion management (DM). For long fiber-optic links, however, the use of fibers with harmonically modulated dispersion, as in eq. (160), is impractical. A much simpler possibility, which is DM proper, is to build a long link composed of periodically alternating segments with positive (normal) and negative (anomalous) dispersion, so that the path-average dispersion (PAD) is close to zero. It is necessary to stress that this concept, in the form of periodic dispersion compensation, has been known for a long time, and has been implemented in existing telecommunication networks, in application to the linear regime of optical signal transmission (Lin, Kogelnik and Cohen [1980]). However, a great deal of interest in the propagation of optical solitons in dispersion-compensated links has arisen not long ago, starting with works by Smith, Knox, Doran, Blow and Bennion [1996] and others (in particular, Knox, Forysiak and Doran [1995], Suzuki, Morita, Edagawa, Yamamoto, Taga and Akiba [1995], Nakazawa and Kubota [1995], Gabitov and Turitsyn [1996]). VA is a natural technique for the analysis of DM schemes; therefore it was used in numerous works (see papers by Bemtson, Anderson, Lisak, QuirogaTeixeiro and Karlsson [1996], Gabitov, Shapiro and Turitsyn [1997], Matsumoto [1997], Malomed [1997], Turitsyn [1997], Lakoba, Yang, Kaup and Malomed [1998], Turitsyn, Gabitov, Laedke, Mezentsev, Musher, Shapiro, Schafer and Spatschek [1998], Kutz, Holmes, Evangelides and Gordon [1998], Bemtson, Doran, Forysiak and Nijhof [1998], and Turitsyn, Aceves, Jones, Zhamitsky and Mezentsev [1999]). Very recently, the approach proposed originally by Kath and Smyth [1995] in order to incorporate the radiative component of the field into VA for the usual NLS equation was generalized by Yang and Kath [2001] for the case of DM. A common feature of different forms of VA developed for DM models is that they are based on the Gaussian (rather than sech) ansatz, as

2, § 5]

Spatially nonuniform fibers and dispersion management

149

the Gaussian provides for an exact solution to the linear Schrodinger equation in the dispersion-compensated model, see below, and is therefore the most natural basis for VA. When PAD is close to zero, it may be necessary to take into account thirdorder dispersion (TOD). The VA technique for a DM system including TOD was worked out by Hizanidis, Efremidis, Malomed, Nistazakis and Frantzeskakis [1998] (the TOD coefficient was assumed to be constant). Comparison with direct simulations has demonstrated that VA makes it possible to take TOD into account in quite an accurate form. It is relevant to mention that description of DM solitons may be based not directly on the corresponding NLS equation in the temporal domain, but rather on its integral counterpart in the frequency domain, as shown by Ablowitz and Biondini [1998] (see also a work by Pare, Roy, Lesage, Mathieu and Belanger [1999]). In relation to this, an interesting version of VA for the DM model was proposed by Pare and Belanger [2000]: using the fact that the above-mentioned integral equation can be derived from its own Lagrangian, VA can be applied to this equation. In fact, contrary to the usual approach, this implies approximating the DM soliton by means of an ansatz (for which the Gaussian form was adopted) not continuously along the fiber link, but only at junctions between the DM cells. It was demonstrated that results produced by this version of VA are in fairly good agreement with direct simulations. The version of VA for DM pulses which is presented below follows, chiefly, the works by Lakoba, Yang, Kaup and Malomed [1998] and Malomed and Berntson [2002]. The NLS equation governing pulse propagation in the DM transmission line is iuz + \d(Z)uTT + \u\^u = 0,

(162)

where 8(Z) is the local piecewise-constant dispersion coefficient, so that f5i,0 ^ L\
Imap,

which is repeated periodically, Imap being the DM period. The most interesting case is then the strong-DM regime, corresponding to the situation with ^map < yPo and Tp < l^iZil ^ I52I2I, where PQ and Tp are the peak power and width of the pulse. In terms of rescaled variables, z = Z/Lmap, r = T/JL\L2\d\

- (52|/lmap5 eq. (162) takes the form

iu,-\^{z)Urr+{-\li^UrT

+ \u\^u) = 0,

(164)

150

Variational methods in nonlinear fiber optics and related fields

[2, §5

d\L\ + 62L2 "1,1215,-62!'

(165)

with

A

m

fZ), = s g n ( D | - D 2 ) Z , | , £»2 = sgn(Z)2-£>,)L2,

0

(166)

L\ < z < 1

(which is repeated with period 1), where /3o is PAD and Z-1,2 •- Lij/Lm^p, so that the coefficients are subject to normalizations £>,L,+£)2i2 = 0,

\DM

(167)

= \D2L2\ = \.

In the strong-DM case, PAD and nonlinearity are much weaker than the local dispersion, hence the expression in brackets in eq. (164) is a small perturbation. A well-known exact solution of eq. (164) in zero-order approximation, when the perturbation is omitted, is the Gaussian pulse Uo =

Po exp l+2i(4(z)/r2) '•[

(T^+liAiz))

(168)

+ i0

Here PQ and TQ are, respectively, the peak power and minimum width of the pulse over one DM period, A(z) = AQ - /Q )8(Z') dz' is accumulated dispersion (from which a contribution produced by PAD is subtracted), and ^o and
Mo = ci(z) exp

W\z)

+ ic(z) r^ + i0

(169)

then its amplitude a(z), width W(z) and chirp c(z) are expressed in terms of the parameters introduced in eq. (168) as follows:

^4^AA{zf a(z) =

\^2i(A(z)/T^y

W{z) =

c(z) = To

lA{z) T0^ + 4Z\(Z)2

(170) The parameter x^, which is dimensionless in view of the normalization conditions (167), plays a dominant role below; in works on this topic, another constant is frequently used, called DM strength, 1.443 S = (171) (Bemtson, Doran, Forysiak and Nijhof [1998]; the factor 1.443 appears due to the use of the FWHM definition of the width of the Gaussian pulse).

2, § 5]

Spatially nonuniform fibers and dispersion management

151

The exact solution (168) of the linear-DM model is used as the variational ansatz for the nonlinear model. The application of VA yields the following evolution equations for the parameters of the ansatz: dro

/;zEroA(z)

jfV-fe,^Kf')-^a. dz

'^

,,,3,

2^W\z)

where, as usual, the soliton's energy PQTQ = E is the dynamical invariant. In fact, E plays the role of a small parameter measuring the relative weakness of the nonlinearity in comparison with the local dispersion. An issue of fundamental interest is to find conditions allowing for the stationary propagation of the pulse, i.e., a dynamical regime in which the parameters IQ and ^o return to their initial values afi:er passing one DM period. Because, as seen from eqs. (172) and (173), changes of TQ and A^ within one period are small ~ (l3o,E), in first approximation one may insert unperturbed values of TQ and AQ into the rhs of eqs. (172) and (173), and demand that (recall the DM period is 1 in the present notation) f ^ ^ , = f ^ ^ , = 0. (174) Jo dz Jo dz This yields the stationary-propagation conditions for the Gaussian pulse in an explicit form: Ao = - \ ,

I3O =

^A'T'O

(175)

The meaning of the condition AQ = -^ is quite simple: it requires the pulse to have zero chirp at the midpoint of each fiber segment. The second of conditions (175) predicts that the DM soliton propagates steadily at anomalous PAD, ft < 0, provided that TQ > [TQ)^^^ 0.30, at ft = 0 if ri = (r^)^^, and at normal PAD, ft > 0, if (r^)^^^ ^ 0.148 < T^ < (r^)^^. The latter case is quite interesting, as the classical NLS soliton cannot exist at normal dispersion. Further analysis of eq. (175) shows that, in this case, the solution exists in a limited interval of the normal-PAD values, 0^ ^ ^ (-) \

^ 0.0127.

(176)

/ max

Inside this interval, eq. (175) yields two different values of the minimum width TQ for a given value of ft/fi", while in the anomalous-PAD region, TQ is a uniquely

152

Variational methods in nonlinear fiber optics and related fields 1 0.9 0.7

I

0.6

Q. •D Q) .N

E

-

/

0.8 0

[2,

bo = -0.2

-

/

0.5 0.4

y

0.3 0.2 0.1

^^^,H
/

/

T^

-

\ bo = 0.02

bo = - 0 . 0 2 ^ /

bo = 0

,

m^

0 3

4

5

6

,

, 10

Map strength, S Fig. 23. Peak power of the stationary DM soliton vs. map strength at different values of the pathaverage dispersion ^ (= bo), as predicted by the variational approximation based on eq. (175). Asterisks mark particular cases for which the corresponding model with random DM was investigated in detail, see § 5.5.

defined function of PQ/E. It can be concluded that the DM sohton corresponding to the larger value of TQ is stable, while that corresponding to smaller TQ is unstable, The border between the stable and unstable solitons corresponds to Po/E = (A)/^)max (see eq. 176), and it is located at T^ = {T^)^^^ ^ 0.148 [i.e., all stable and unstable solitons have, respectively, TQ > {TQ)^.^^ and TQ < {T^O)^-^JThe results concerning the stability of these two solitons were reproduced in a mathematically rigorous form by Pelinovsky [2000]. Translating TQ into the standard DM-strength parameter S according to eq. (171), one concludes that VA predicts the following: • stable DM solitons at anomalous PAD ifS< Scr ~ 4.79; • stable DM solitons at zero PAD ifS = S^r ~ 4.79; • stable DM solitons at normal PAD if 4.79 < 5' < S^ax ^ 9 . 7 5 ; • no stable DM soliton if S > S^ax ~ 9.75. The normalized power of the DM soliton, which is P = 4 - 1.12Po (the factor 1.12 is the ratio of the FWHM widths for the sech-shaped and Gaussian pulses) vs. the DM strength at different fixed values of PAD, ft, is shown, as predicted by eq. (175), in fig. 23. A counterpart of the same dependence, obtained by Berntson, Doran, Forysiak and Nijhof [1998] from direct simulations of eq. (164), is displayed in fig. 24. In fig. 23 the curves are shown only in the region S < 9.75, where the solitons are expected to be stable. The curves in fig. 24 corresponding to normal PAD (ft > 0) terminate at points where the DM soliton becomes unstable. Comparison of figs. 23 and 24 shows that VA yields acceptable results for

2, §5]

Spatially nonuniform fibers and dispersion management

153

(D

O Q. (D N

E

10

12

20

Map strength, S Fig. 24. Counterpart of fig. 23, obtained by direct numerical simulations of eq. (164). Asterisks mark particular cases for which the corresponding model with random DM was investigated in detail.

relatively small values of the soliton power, where the above approximation, which treated the nonlinearity as a weak perturbation, should be relevant. In particular, the VA-predicted value Scr ~ 4.79 is different from, but reasonably close to, the critical DM strength Scr ~ 4 which was found from direct simulations for the small-power case. With increasing power, the numerically found Scr grows. It is also noteworthy that the value 5'max ~ 9.75, predicted by VA as the stability limit for DM solitons, is indeed close to the result of direct simulations for small powers, see fig. 24. 5.5. Random dispersion management Existing terrestrial optical telecommunication webs are patchwork systems, which include links with very different lengths (Agrawal [1997]). This circumstance of practical importance suggests to consider random DM. A short account of recent results obtained for random DM by Malomed and Berntson [2002] by means of VA is given here, the most salient feature being a sharp difference between robust and unstable soliton propagation regimes. A random-DM model of a different type was considered by Abdullaev and Baizakov [2000] (see also work by Abdullaev, Bronski and Papanicolaou [1999]), where the local values of the dispersion, rather than the fiber-segment lengths, were distributed randomly. In these works, the above-mentioned drastic difference between stable and unstable regimes of soliton transmission was not reported. The basic equation and normalizations are the same as in the previous section, i.e., they are given by eqs. (164) and (167). In the case of random DM, the normalizations must be applied to mean values of the random lengths. Limiting

154

Variational methods in nonlinear fiber optics and related fields

[2, §5

the consideration to the case when the mean lengths of the segments with anomalous and normal dispersion are equal, L\ = I2, eqs. (167) yield L\2 = j and \D\2\ = 2. To comply with the former condition, one may assume that the random lengths L\2 are distributed uniformly in the interval 0.1 < L < 0.9. The minimum length 0.1 is introduced because, in reality, the length can neither be very large (say, larger than 200 km) nor be very small (shorter than 20 km). The same ansatz (168) and variational equations (172 and 173) may be used with the randomly distributed lengths. As explained above, the change of the soliton's parameters, TQ -^ TQ + 6TO, AQ -^ AQ + bAo, within one DM cell is small. Therefore, the evolution of the pulse passing many cells is approximated by smoothed differential equations, dto/dz = 8TO/ ( I * " ' + Z , 2 " J and dAo/dz = bAo/ f Z,',"* +12"*) (here n is the cell's number), which take the following form, dro _

"57

V2ET^

8 ( i 2 + i , ) [ y ^ 4 + 442

^T^ + 4iA> +

2L2-2L0' (177)

yJr* + 4{Ao + 2L2f\ dAo = -/^ dz V2ETt

2Zin

8(Z,2+ii)

sjrt + ^Al

2iAo +

2L2'2L,)

yj4 + A{A^ + 2L2-2L^f

^ T 4 + 4(zio +2^2)2

(178)

- i In \1(AQ + 2L2 - 21,) + ^ r 4 + 4(/io + 2 L 2 - 2 I i ) 2 j + In ( 2(4o + 2Z.2) + ^j4+A{AQ + 2L2y The most essential characteristic of the pulse propagation at given values of /3b and E is the cell-average pulse's width, W

' ^

j'

W{z)diz.

(179)

JceW

Simulations of eqs. (177) and (178) reveal that there are two drastically different dynamical regimes. If the soliton's energy is sufficiently small (hence

2. §5]

Spatially nonuniform fibers and dispersion management

E=3.6

a 8 ^

4

^

2

155

0

100

200

300

400

500

600

700

800

900

1000

Distance of propagation, unit cells Fig. 25. Evolution of the soliton's cell-average pulse width (normalized to its initial value) in the random-dispersion-management model with zero PAD. The mean values (solid curve) and standard deviations (dashed curves) are produced by averaging over 200 different realizations of the randomlength set. The propagation distance is given in units of the average DM cell length. The bottom and top plots correspond to the DM solitons with low energy £" = 0.1 and high energy £" = 3.6, respectively.

the approximation outlined in the previous section is relevant) and PAD is anomalous or zero, i.e., ft ^ 0 (especially, if ft = 0), the pulse performs random vibrations but remains, in fact, fairly stable over long propagation distances. When the energy is larger, as well as when PAD is normal, ft > 0, the pulse demonstrates fast degradation. Typical examples of the propagation are displayed in fig. 25 for the zeroPAD case, which is the best in terms of the soliton stability. Simulations of eqs. (177) and (178) have been performed with 200 different realizations of the random-length set, chosen so that L^^"* = L^-^"* (equal lenghts of the anomalousand normal-dispersion segments inside each DM cell). Figure 25 shows the evolution of (^(z)), i.e., the mean value of the width (179) averaged over the 200 random realizations, along with the corresponding normal deviations from the mean value. The figure demonstrates that some systematic slow evolution takes place on top of the random vibrations, which are eliminated by averaging over 200 realizations. Systematic degradation (broadening) of the soliton takes place too, but it is extremely slow if the energy is small. In the case shown in the bottom part of fig. 25, the pulse survives with very little degradation in propagation over more than 1000 average cell lengths (in fact, as long as the simulations could be run). It is not difficult to understand this: in the limit of zero power, i.e., in the linear random-DM model, an exact solution for the pulse is available in essentially the same form as given above for periodic DM,

156

Variational methods in nonlinear fiber optics and related

fields

[2, § 5

see eq. (168). If PAD is exactly zero, this exact solution predicts no systematic broadening of the pulse. If the soliton's energy is larger, further simulations of eqs. (177) and (178) show that, after having passed a very large distance, the sluggish spreading out of the soliton suddenly ends in a blowup (complete decay into radiation). This seems to be qualitatively similar to what was predicted by VA for periodic sinusoidal modulation of the dispersion, see §5.3.1. and fig. 17c: a long span of chaotic but nevertheless quasi-stable vibrations is suddenly ended by rapid irreversible decay. In fact, the case ft = 0 is a point of sharp optimum: at any finite anomalous PAD, i.e., ft < 0, the degradation of the pulses is essentially faster, especially for those with larger energy, and at any small normal value of PAD, ft > 0, very rapid decay always takes place, virtually at all values of energy. Malomed and Berntson [2002] have also performed a comparison of the results predicted by VA with direct simulations of the same random-DM model. The direct numerical results prove to be quite similar to what was predicted by VA. In particular, the most stable propagation is again observed at zero PAD, the soliton's broadening is faster at nonzero anomalous PAD, and all solitons decay very quickly at nonzero normal PAD. The soliton's stability in the direct simulations drastically deteriorates with increasing energy, as also predicted by VA. Detailed comparison shows that, surprisingly, the direct simulations yield somewhat better results for the soliton's stability than VA: the actual broadening rate may be ^20% smaller than that predicted by VA. The slow long-scale oscillations, clearly seen in fig. 25, are less pronounced in the direct simulations. The sudden decay into radiation, predicted by VA after very long propagation, does not take place in the direct simulations; instead, the soliton eventually splits into two smaller ones, quite similar to what is observed in direct simulations of the model with periodically modulated dispersion, see fig. 19.

5.6. Interactions between dispersion-managed solitons 5.6.1. Collisions between solitons belonging to different channels in wavelength-diuision-multiplexed systems Wavelength-division multiplexing (WDM), i.e., creation of a large number of channels in the same fiber, carried by different wavelengths, is the most important direction in the development of optical telecommunications. In soliton-based systems, the most serious problem related to WDM is crosstalk due to collisions of pulses belonging to diflferent channels. Collisions are inevitable, as the

2, § 5]

Spatially nonuniform fibers and dispersion management

157

inherent dispersion of the fiber gives rise to different group velocities of the carrier waves in different channels. Very promising results are produced by a combination of WDM and DM, especially with respect to the suppression of collision-induced effects, as shown in simulations reported by Niculae, Forysiak, Gloag, Nijhof and Doran [1998]. Here, an account of VA-based analysis of collisions in the combined WDM/DM system will be given, following a work by Kaup, Malomed and Yang [1999]. The simplest two-channel system is described by the following equations (cf. eq. 164): i(Mz + CUT) + W, +

ID(Z)

lD(z)UrT^

UTT

+

DuUrr + y (\uf

D,UrT-^y(\u\^^2\uf^u^

+ 2 |w|^) u\ = 0,

= 0,

(180)

(181)

where c is the inverse group-velocity difference between the channels, D(z) is the main part of the dispersion (with zero average), which may be assumed the same in both channels, Du^^ are the values of PAD in the two channels, which are different in general, and the nonlinear terms represent, as usual, the self-phase modulation (SPM) and cross-phase modulation (XPM) effects. The analysis uses the same ansatz (168) for the solitons as above. However, in order to describe the dynamics of the interacting pulses, the ansatz may be taken in a more general form, which is obtained from eq. (168) by the Galilean boost, w(z, r) = uo(z, r - T(z)) exp(-ia;r + iV^(z)),

(182)

where co is an arbitrary frequency shift, and the corresponding position shift is generated by the following equation: ^=-co{D(z)

+ D,).

(183)

In the absence of interaction, the parameters of solitons in both channels are selected by the conditions (175). Since these conditions were obtained treating the SPM nonlinearity as a small perturbation, the XPM-induced interaction between solitons may also be considered as a perturbation in the Lagrangian of eqs. (180) and (181) (the Lagrangian representation of XPM-coupled equations

158

Variational methods in nonlinear fiber optics and related

fields

[2, § 5

was considered in § 4). This approach makes it possible to derive the following evolution equation for the soliton's frequency shift in the presence of XPM: ^^^y^PXc^f_c^r^\^ <^ [T^ + 4A\z)f' \ [r^ + 4AHz)]

(184)

where Pu is the peak power of the pulse in the t;-channel. In the same approximation, the evolution of the position is governed by the unperturbed equation (183). Equation (184) implies that the centers of the two solitons coincide at z = 0. In a two-channel model without DM, a dynamical equation similar to (184) was derived by Ablowitz, Biondini, Chakravarty and Home [1998]). However, there is a principal difference between the collision in the system with DM and that in the system with constant dispersion: as the coefficient D(z) in eq. (183) periodically changes sign, it is easy to see that, in the strong-DM regime, colliding pulses pass through each other many times before separating. It is necessary to distinguish between complete and incomplete collisions. In the former (generic) case, the solitons are far separated before the collision, while in the latter case, which takes place when the collision occurs close to the input point, the solitons overlap strongly at the beginning of the interaction. In either case, the most important result of the collision is a net shift of the soliton's frequency 6a;, which can be calculated as ba)= [ ^ ^dz, 4 dz

(185)

where dco/dz should be taken from eq. (184). The lower limit of the integration in expression (185) is finite in the case of the incomplete collision, while a complete collision corresponds to ZQ = -oo. The net frequency shift is very detrimental, as, through the dispersion, it gives rise to a change of the soliton's velocity. If the soliton picks up a "wrong" velocity, information carried by the soliton stream in the fiber-optic telecommunication link may be lost completely. An estimate of physical parameters for dense WDM arrangements, with a wavelength separation between channels of 6A < 1 nm (this is the case of paramount practical interest) shows that the group-velocity mismatch c may be regarded as a small parameter, hence the function cz varies slowly in comparison with the rapidly oscillating accumulated dispersion A(z). In this case, the integral (185) and similar integrals can be calculated in a fully analytical form, as shown by Kaup, Malomed and Yang [1999]. In particular, the net

2, § 5]

Spatially nonuniform fibers and dispersion management

159

frequency shift is zero for the complete colHsion, which shows the abiUty of DM to suppress colHsion-induced effects. In fact, the zero net shift is a result of the multiple character of the collision (see above): each elementary collision may generate a finite frequency shift, but they sum up to zero. Once the net frequency shift is zero, the collision is characterized by a net position shift, which is a detrimental effect too, but less dangerous than the frequency shift. The position shift can be found from eq. (183): 87=/ --dz J-oo dz

= eD J_^

z^dz+ / dz J_^

A(z)--dz, dz

(186)

where integration by parts was performed. Then, substituting the expression (184) for dw/dz, one can perform the integrations analytically, to obtain a very simple final result (for definiteness, it is written for the soliton in the w-subsystem): bT, = V2^D,P,^.

(187)

Note that this result contains a product of two small parameters, namely, PAD Du and the power P^; (the latter is small as it measures the nonlinearity in the system, and it was assumed from the very beginning that the nonlinearity is a small perturbation). The net frequency shift generated by the incomplete collision can be found similarly. In this case, the worst (largest) result is obtained for the configuration with the centers of the two solitons coinciding at the launching point z = 0:

(6ft)),,, = V2 g In (5 + V]T¥)

,

(188)

where S is the DM strength defined by eq. (171). These analytical results obtained by means of VA were compared with numerical simulations. First of all, simulations show that the net frequency shift induced by complete collisions is very small indeed (much smaller than in the case of incomplete collisions at the same values of the parameters). As for the position shift in the case of complete collision, the analytical prediction (187) is compared to numerical results in fig. 26, showing reasonable agreement. In the case of incomplete collisions, simulations yield a nonzero frequency shift, which was compared to the analytical prediction (188) by Kaup, Malomed and Yang [1999]. In this case, there is acceptable agreement again. However, in contrast to the case of complete collision (fig. 26), the difference between the

160

Variational methods in nonlinear fiber optics and related fields

[2, §5

solid: numerical

0.25 [

dashed: analytical 0.2 [ c0.15^

L o.n 0.05

0.4

0.6

0.8

1.2

1.4

Fig. 26. Analytically and numerically found position shift of the soliton induced by complete collision in the two-channel model described by eqs. (180) and (181) with L\ = 0.4, L2 = 0.6, D\ = ^, ^2 ^ ~f' c = 0.3, and peak powers of the colliding pulses Pn = P^, = 0.1.

analytically predicted and numerically found results (this time, the frequency shifts) decreases with increasing DM strength S. 5.6.2. Interactions between solitons inside one channel As demonstrated in the previous subsection, DM makes it possible to suppress the crosstalk between solitons in different channels of the WDM system. Thus, the most serious remaining limiting factor in the strong-DM regime is the interaction of pulses in the same channel, as demonstrated by Yu, Golovchenko, Pilipetskii and Menyuk [1997]. In fact, in this regime the intrachannel interactions turn out to be stronger than in the absence of DM, which suggests that it may be optimal to use a moderate-DM regime rather than strong DM. Analysis of the interactions inside one channel can be effectively performed by means of VA, although a particular variational technique which yields good agreement with direct simulations turns out to be rather cumbersome. Below, main results for the intrachannel interactions between solitons are presented, following the paper by Wald, Malomed and Lederer [1999]. Earlier, VA based on the Gaussian ansatz was applied to this problem by Georges [1998], Matsumoto [1998] and Malomed [1998a,b] (see also a related paper by Kumar, Wald, Lederer and Hasegawa [1998]). It was shown that this simple version of VA correctly describes the interactions in the strong-DM regime, despite the fact that the Gaussian ansatz approximates only the cores of the DM solitons adequately, but not their "tails", whose genuine form is exponential rather than Gaussian (Ablowitz and Biondini [1998]). The incorrect approximation for the tails is not significant in the case of strong-DM, as the

2, §5]

Spatially nonuniform fibers and dispersion management

161

huge periodic spreadings of the pulses lead to strong overlapping between them, involving their cores rather than tails. In the moderate-DM case, however, the tails play a dominant role in the interactions (Malomed [1998a]), and the simple Gaussian ansatz fails in this case. To describe the interaction of two separated pulses, one may substitute w(z, r) = u\{z,r) + U2(Z,T) into eq. (164), which describes the one-channel DM model, and split it, following Karpman and Solov'ev [1981], into separate NLS equations for the two pulses, treating the interaction between them as a small perturbation, r\

r\2

i-^-\ji{z)—-j+ \u„\^u„ = -ulul_„ + 2\u„\^Ui^,„ n = 1,2 dT^ dz

(189)

These equations can be derived from the Lagrangian density C = C\ + C2, where (cf eq. 15)

Cn =

+ lw.l'

'" dz

(190)

\Un\^ (w„W3_,,+ W>3_,;) ,

the last term accounting for the interaction. An ansatz which proves to be adequate for the description of the interactions between solitons is

U„(z,T)=A„(z)f(tn)QXp

f{tn) = exp -yJan{z)-Vtl

\0n{z)-'^1^n{z%

+

'\b,,{z)f-

t„ = q(T - T„(Z)),

dtl

(191) (192)

where real variational parameters are ^1,2, ^1,2, ^\2^ ^1,2, a\,2 and T\2, while q is an auxiliary constant that is not to be varied, see below. This ansatz combines a Gaussian-like core and exponentially decaying tails, as/(/,,) '^ exp(-|/,,|) as \tn\ -^ 00. The shape of the pulses is controlled by the parameters an{z)\ the larger «„, the more Gaussian-like the pulse is. Calculating the full Lagrangian L = J^^{C\ + £2)dr with the ansatz (191)-

Variational methods in nonlinear fiber optics and related fields

162

[2, §5

(192) in analytical form is not possible, therefore the resulting variational equations for the parameters of the ansatz are cast into the following form:

dz

f

fit„)fiti.„)

J -o

1 d^f sin(A0) dr. dt„ \fit„) dti

3/„cosA0^^_j^„^^__a

\/a„ + tl

(193) Here, E„ = Al J^^f\T,a„)dT = 2AJ,y/aKi{2y/a), with K^ the modified Bessel fiinction, are the energies of the two pulses, which are conserved separately in the approximation used here. Further, Alp = Kit\ -K2t2

&(«)

+

bi d^f

b, d^f

f{t2)dti

m)dt]'

ML, e , ( . ) . M o ^ ,

Q.ia^.'M^,

(194)

(195)

k{a) =

2^K,{2^),

hl'i-hli

The evolution equations (193) were solved numerically, and the results were compared to direct simulations of the interaction between two solitons in eq. (164) with DM taken in the simplest symmetric form (cf. eq. 166),

f/3,,

K^)

1^2, I A?

0 ^ z < \zu \z\


(196)

Z2 + yZi < Z ^ Zi +Z2,

and the launch point in the middle of one of the segments. In the simulations, the lengths of the DM segments in eq. (196) were z\ = zj = 0.1, and the initial FWHM width was fixed to be 21n( 1 + %/2] ?^ 1.763, i.e., the same as that of the pulse A sech r. To keep this fixed value of the width for an arbitrary initial

2, § 5]

Spatially nonuniform fibers and dispersion management

163

value «1,2(0) = ao of the parameter a{z) in the ansatz (191), (192), the constant q in eq. (191) was chosen so that lr» 9

q^ = : [ - ^ ^ ( l n 2 + 4 v ^ ) ^ 0.155 + 0 . 8 9 2 ^ ^

(197)

(recall that q was defined as a wowvariational auxiliary parameter). Note that, in this notation, the expression for the DM strength is

instead of eq. (171). First, eqs. (193) were applied to an isolated pulse. The objective was, solving numerically the first two equations of the system (193), to find the initial values ao and r/o = ^(O)exp(-y^) that provide for a stationary DM pulse, i.e., strictly periodic evolution of a(z) and b(z) [in other words, an analog of the conditions (175) obtained for the simple Gaussian ansatz (168)]. Because of the presence of two unknown initial values [with q taken as per eq. (197)], the simulations generate a whole set of values (ao, rjo) that give rise to a stationary pulse (in fact, the r/o thus found is nearly constant, while ao may vary within broad limits). This implies that an extra optimization condition may be imposed, in order to select a unique set of values that gives rise to the most accurate approximation for the DM soliton. The extra condition was the demand that not only the transmission of the isolated pulse, but also interactions between identical pulses must be correctly described by the fiill system of variational equations (193). To this end, the fiill system was solved numerically, and the collision distance predicted by this solution, i.e., the value of the propagation distance z at which a collision of two solitons has to take place, was compared to the actual collision distance obtained from direct numerical simulations of eq. (164). This procedure, repeated at many different values of parameters, has yielded an empirical result giving the value of ao, optimized against the description of the interactions, as a fianction of the DM strength S (see eq. 198), ao{S)=l.2 + 5.SS\

(199)

in the range S < 1.5 (i.e., for moderate DM). Detailed results reported by Wald, Malomed and Lederer [1999] show that the ansatz based on eqs. (191) and (192) and optimized as outlined above, although being somewhat cumbersome, generates a shape for an isolated DM soliton

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[2, §6

4 n

-2 H

1 ' ' 5

'

•

1

10

1 '

'

15

'

1 1 1 • 1

20

25

' ' I 30

distance z

Fig. 27. Positions of two interacting identical solitons in one dispersion-managed channel vs. the propagation distance z. Curves: prediction of the variational approximation based on the ansatz (191)-(192) with parameters optimized as described in the text; symbols: direct numerical simulations for various values of the DM strength S.

which is extremely close to the shape found numerically, and simultaneously provides for a very accurate description of the interaction, as illustrated by fig. 27. The figure compares the evolution of the temporal positions of two interacting identical DM solitons, as predicted by the present version of VA and as obtained from direct simulations of eq. (164).

§ 6. Solitons in dual-core optical fibers 6.1. Solitons in a basic model of the dual-core fiber A dual-core fiber (DCF), alias directional coupler, is a system of two parallel identical or different fibers with a gap between them on the order of the wavelength, so that light can linearly couple from one core into the other. DCF is a basis for design of optical switches (Trillo, Wabnitz, Wright and Stegeman [1988], Friberg, Weiner, Silberberg, Sfez and Smith [1988]). It can also be used for efficient compression of solitons by passing them into a fiber with a smaller value of the dispersion coefficient: as demonstrated by HatamiHanza, Chu, Malomed and Peng [1997], the highest-quality compression is achieved when two fibers with different dispersion coefficients are connected not by splicing, but rather when they form an asymmetric coupler. Nonlinear DCFs are a promising medium for the observation of new types of optical solitons, for the description of which VA is a natural technique, as it was

2, § 6]

SoUtons in dual-core optical

fibers

165

first shown by Pare and Florjanczyk [1990] and Maimistov [1991] (see also an independent work by Chu, Malomed and Peng [1993]). The applicability of VA to the DCF model and its limitations were discussed by Ankiewicz, Akhmediev, Peng and Chu [1993]; however, the version of VA considered in that work was not flexible enough. Results obtained by means of VA for solitons in DCF are presented in this section, following the work by Malomed, Skinner, Chu and Peng [1996]. DCF is described by a system of linearly coupled NLS equations, iu- + \uu H- \u\^u + Kv = 0 ,

(200)

w, + \vjr + \v\^v + Ku = 0 ,

(201)

where K is the coupling constant accounting for the light exchange between the two cores. These equations admit the usual variational representation, the linear coupling being accounted for by additional terms in the Lagrangian density, A£ = K{U''V + UV"). An ansatz for a soliton with a component in each core can be taken as u = A cos 6 sech (-)

exp [i(0 + i/^) + ibr^] ,

(202)

u = ^ sine sech ("-] exp [i(0-V^) + iZ?r^] .

(203)

New parameters, in comparison with the sech ansatz (13) for the singlecomponent soliton, are the angle 6 which measures the distribution of energy between the two cores, and the relative phase ip between them. Note that the ansatz (202)-(203) assumes that the centers of the two components of the soliton are stuck together. This implies that the linear coupling between the two cores is strong, which corresponds to a real physical situation. However, one may also consider a case when the linear coupling is a small perturbation, so that a two-component soliton is a weakly bound state of two individual NLS solitons belonging to the two cores, as it was done by Abdullaev, Abrarov and Darmanyan [1989] and Kivshar and Malomed [1989b], using straightforward perturbation theory (see also a paper by Cohen [1995], where the Hamiltonian formalism was used to analyze the stability of bound states of solitons in weakly coupled fibers). Switching of a soliton between the two cores was considered, on the basis of a fiill system of variational equations for the ansatz (202)-(203), by Uzunov, Muschall, Golles, Kivshar, Malomed and Lederer [1995] (independently, a less sophisticated version of VA was used by Doty, Haus, Oh and Fork

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Variational methods in nonlinear fiber optics and related

fields

[2, § 6

[1995] to analyze interactions of solitons in DCF; accurate numerical results for the interaction were reported by Peng, Malomed and Chu [1998]). The corresponding ODEs were solved numerically, and the results were compared against direct numerical solutions of eqs. (200) and (201), showing very good accordance over a broad parametric region. It was demonstrated by Smyth and Worthy [1997] that the description of the switching dynamics in DCF can be improved further if the radiation component of the wave field is incorporated into the ansatz, similarly to what was done by Kath and Smyth [1995] in the model of the single-core fiber. Here, consideration is focused on static solitons, which can be found fi"om the variational equations generated by the ansatz (202)-(203), in which all parameters except the phase 0 are assumed constant: sin(20)sin(2V^) = O, 3a a dz

(204)

cos(20) - K cot(20) cosilxp) = 0,

(205) (206)

1 - I sin^(2e) 6a^ 3a I

1 - I sin^(20)] + /csin(20)cos(2V^),

where E is the net energy of the soliton, E ~ ^ J^^ (I ^ P + I ^P) dr = A^a. Equation (204) requires either sin(20) = 0 or sin(2t/;) = 0. According to the ansatz (202)-(203), the former solution implies that all the energy resides in a single core, which contradicts eqs. (200) and (201), hence this solution is extraneous. The latter solution, sin(2i/;) = 0, implies that cos(2i/^) =^ ± 1 . According to the numerical findings of Soto-Crespo and Akhmediev [1993], the solutions corresponding to cos(2t/;) = - 1 , i.e., with a phase shift Ji between the two components, are almost everywhere unstable. Therefore, only the case cos(2i/;) = +1, corresponding to solitons with in-phase components, is considered below. Then, the width a is eliminated by means of eq. (206), and the remaining equation (206) for the energy-distribution angle 6 takes the form cos(20) I ^ sin(20) [l - \ sm\26)\

- 11 = 0.

(207)

A simple analysis reveals that, in the interval 0 <E^ < E^, where El=lV6K^5.5nK,

(208)

the only solution to eq. (207) is the symmetric one, with 9 = \ji, which implies equal energies in both components according to eqs. (202) and (203). When

Solitons in dual-core optical fibers

2, §6]

167

Fig. 28. Bifurcations between symmetric and asymmetric solitons in a dual-core nonlinear optical fiber. The solid and dashed branches correspond to stable and unstable solitons, respectively, and the thick dots indicate the bifurcation points.

the soliton's energy attains the value E\, asymmetric solutions emerge with cos(20) = ± l / \ / 3 . When E^ attains a slightly larger value, 6K,

(209)

a backward (subcritical) bifurcation occurs, which makes the symmetric solution with 6= \7i unstable. The corresponding bifurcation diagram is displayed in fig. 28. Note that the quantity cos(29), which is used as the vertical coordinate in the diagram, measures the asymmetry of the soliton because, as follows from eqs. (202) and (203), cos(20) :

£'(»H-£(2)'

(210)

where E^J"^ is the energy in theyth core. Even without detailed stability analysis, one can easily distinguish between stable and unstable branches in the diagram, using elementary theorems of bifurcation theory (see a book by looss and Joseph [1980]). Thus, VA predicts backward bifurcation at the soliton energy Ei = \/^K ^ 2A5\/K, whereas the known exact value is ^J\K ^ 23\\/K (Wright, Stegeman and Wabnitz [1989]), which illustrates the accuracy of VA. The bifurcation diagram produced by VA agrees with the numerical results (Akhmediev and Ankiewicz [1993], Soto-Crespo and Akhmediev [1993]) in showing that the region of bistability extends over a very narrow range of energies.

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Variational methods in nonlinear fiber optics and related fields

[2,

6.2. Fibers with a variable separation between the cores An interesting dynamical generalization of the static problem outlined above is the consideration of bifurcations of solitons in a model of DCF with a periodically modulated coupling constant, i.e., K = KQ + K\ cos(A:z) in eqs. (200) and (201). The periodic modulation may uncover hidden intrinsic resonances in the two-component DCF soliton, as was demonstrated, also by means of VA, in the work of Chu, Malomed, Peng and Skinner [1994]. A final result of the analysis is the prediction of a bifiarcation from a two-component soliton whose energy oscillates symmetrically between the two cores to a pair of mutually symmetric solitons with broken symmetry of the oscillations (fig. 29). A dual-core configuration of practical importance is di fused coupler, in which two far-separated fibers are bent so that they converge, reach a minimum separation at which light can couple between them, and then diverge again. The accordingly modulated coupling coefficient is, for instance, K{z) = KQ sech(/rz). In this case, a natural dynamical problem is to consider how the energy of a soliton launched into one core is split between the cores after the passage of the coupling region. The variational technique can be applied to this problem in a straightforward way and, as demonstrated by Skinner, Peng, Malomed and Chu [1995], it produces results which are very close to those obtained from direct simulations of the NLS equations coupled by the linear terms with the variable coefficient K(z). Note that the limiting case of the fused coupler is when the coupling may be assumed to be concentrated at a single point, K(z) = Kob(z). In this limit, the

Fig. 29. Schematic representation of the symmetry-breaking bifurcation of a sohton whose energy oscillates between the two cores in a dual-core fiber with the coupling constant periodically modulated along the propagation distance. Here, 6\ = ^ - ^jr is the asymmetry parameter (see eqs. 202 and 203), and the rest of the notation is as in the paper by Chu, Malomed, Peng and Skinner [1994]. The solid and dashed lines represent two asymmetric solitons existing past the bifiircation point; the unstable symmetric solution remaining beyond the bifurcation (cf fig. 28) is not shown.

2, § 6]

Solitons in dual-core optical

fibers

169

problem of the soliton passage though the fused coupler admits an exact solution, as shown by Chu, Kivshar, Malomed, Peng and Quiroga-Teixeiro [1995]. Another source of z-dependence of the coupling constant K in the DCF model may be small fluctuations of the separation between the cores, as K is very sensitive to the exact value of the separation. However, it was demonstrated by Mostofi, Malomed and Chu [1998] that the solitons in DCF are not critically sensitive to fluctuations of K, except for an extremely narrow vicinity of the bifurcation point. 6.3. Gap solitons in asymmetric dual-core fibers Asymmetric DCFs, consisting of two different cores, can be fabricated easily, and the properties of solitons in these DFCs may be quite different from those in the symmetric DCF. A general model for an asymmetric DCF is (cf eqs. 200, 201) \u- + qu+\ujj+

\u\^u + v

\u-_-6'{qv-^\vjr)^\v\^v

=0, +u -0,

(211) (212)

where the real parameter d accounts for the difference in dispersion coefficients in the cores, and the real q defines the phase-velocity mismatch between them; possible group-velocity terms, - iur and lUj, can be eliminated from the equations. The influence of the asymmetry between the cores on soliton bifurcations was considered in the above-mentioned paper by Malomed, Skinner, Chu and Peng [1996], and in more detail by Kaup, Lakoba and Malomed [1997]. In the latter work, an analytical approach based on VA showed good agreement with direct numerical results. A noteworthy feature of bifurcations in the asymmetric model is the possibility of hysteresis in a broad region (in the symmetric model, hysteresis is only possible in the narrow bistable region between the two bifurcation points, see fig. 28). The most interesting version of the asymmetric model is that with 5 > 0 in eq. (212), i.e., with opposite signs of the dispersion in the two cores, which was studied by means of VA and direct numerical methods by Kaup and Malomed [1998]. To understand the fundamental properties of solitons in this model, it is first of all necessary to analyze its linear spectrum. Substituting w,t; ~ exp (i^z - io;r) into the linearized equations (211) and (212) yields the dispersion relation k=\{d-\){o?-2q)

± ^ ^ ( ( 3 + l ) ' ( a ; 2 - 2 ^ ) ' + l.

(213)

Solitons may exist at values of the propagation constant k that belong to a gap in the spectrum (213), i.e., where both values of o)^ corresponding to a given k

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Variational methods in nonlinear fiber optics and related fields

[2,

10 r 5

2 k 0

-2 _

i

y^

1

T ^ \

\

i

J

-4

-6 - 4 - 3 - 2 - 1 0 1 2 3 4 (0

(b)

Fig. 30. Typical dispersion curves given by eq. (213) for a dual-core fiber with opposite dispersions in the cores, described by eqs. (211) and (212) with 6 = 1: (a) ^ = - 1 ; (b) q = +\.

are nonphysical (negative or complex). Moreover, if these values are complex, soliton tails decay with oscillations, rather than monotonically. In particular, it follows from eq. (213) that, in the subgap 0 ^ A:^ < 48/(\-^df, only solitons with oscillating decaying tails may exist. The existence of this type of soliton is an essential result (similar solitons were also found by Malomed [1995] in a model with equal dispersions in both cores, 8 = -\, and a groupvelocity mismatch between them, which is a special case not comprised in eqs. 211 and 212; as shown in that work, the solitons may form bound states, interacting through the oscillating tails). In the case of opposite dispersions in the two cores (8 > 0), the spectrum always contains a finite gap; typical results for negative and positive values of the mismatch q are shown in fig. 30. Once the gap has been found, stationary gap solitons residing in it are sought for as u{z,r) = U(T)Qxp(ikz), U(Z,T) = F(r)exp(i^z) with real U and V that obey ODEs

(q-k)U-^^U'' + U^ + V = 0, -(8q^k)V-^8V''+V^

+ U 0, (214)

2, § 6]

Solitons in dual-core optical

fibers

171

the prime standing for d/dr. Approximate solutions to eqs. (214) were constructed by means of VA, using the Gaussian ansatz ^ =^exp(-^),

^-«exp(-^).

(215)

The amplitudes A and B can be eliminated from the resulting system of variational algebraic equations, leading to the following equations for the widths a and b: [3 - 4(A: - q) a^] [36 + 4(A: + dq) b^]

,

lb _ ^2) (^2 ^ yiy^ ^ 32iabf {b" - 3fl2) (3^2

(216)

[2>-M.k-q)a^] {3a^-b^f [3d + A{k + dq)b'^]{ib'^-a^) _a^[6-

^217)

(3^2 + ^,2) + 4(^ + Sq) ^2 ^yi _ ^2)j

b^[{3b'^+a^)+A{k-q)a^{b'^-a^)] Equations (216) and (217) were solved numerically to find a and b as functions of the control parameters, 6 and q, and the propagation constant k. To present the results in a physically meaningful form, one should define, as usual, the energies of the two components of the soliton, +00

/»+00

/

|C/(T)p dt = VnA^a, OO

E,=

|F(r)p dt = sfjtB^b, J

-OO

(218) and the net energy E = E^ + E^. The dependence E(k) is particularly important as, according to the condition put forward by Vakhitov and Kolokolov [1973] (VK), a necessary condition for the stability of the soliton is dk/dE > 0. Detailed results presented in the above-mentioned paper by Kaup and Malomed [1998] show that the gap solitons exist indeed in some part of the available gap, and, in most cases, they are stable according to the VK criterion; however, another part of the gap remains empty (there are intervals of the propagation constant k inside the gap, in which no soliton can be found). A noteworthy property of the gap solitons is that (slightly) more than half of their net energy always resides in the normal-dispersion component u (i.e., Eij/E > ^, see eq. 218), despite the obvious fact that the normal-dispersion core cannot, by itself, support any (bright) soliton. Accordingly, a typical soliton

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Variational methods in nonlinear fiber optics and related fields

[2, § 6

2.0

Fig. 31. Numerically found (solid lines) gap-soliton solution to eq. (214) with oscillating decaying tails, in the case 6 = I, q = 0.2, and E = 2.734, displayed along with its variational counterpart (dashed curves).

predicted by VA (see fig. 31) has a narrower component with a larger amplitude in the anomalous core, and a broader component with a smaller amplitude in the normal one. As can be seenfi*omfig.31, VA in general correctly approximates the soliton's core, but the simple ansatz (215) does not take into account the fact that, as explained above, the soliton tails decay with oscillations. The contribution of the tails is also amenable for a conspicuous difference of the energy share Eij/E in the normal-dispersion core against the value predicted by VA for the same net energy E: for example, in the case shown in fig. 31, the predicted value is Eu/E = 0.585, while the numerically found one is EJE = 0.516 (but still larger than ^, as stressed above).

6.4. Two polarizations in the dual-core fiber A physically interesting extended model of DCF, that was developed by Lakoba, Kaup and Malomed [1997], takes into account the fact that light may have two polarizations in each core. The model (a bimodal dual-core fiber) is based on a system of four equations. i(Wl)z + ^(Wl)rr + (|W11^ + f |i^l |^) Wl + W2 = 0, K^\)z + ^(uOrr + (|t;i 1^ + | | w i 1^) Ui+U2 = 0, i(W2)r + ^(W2)rr + (1^2 |^ + ^\U2\^) ^2 + W, = 0, i(U2)z + ^(i^2)rr + {\U2\^ + f |W2|^) U2 ^ Ui = 0,

(219)

2, § 6]

Solitons in dual-core optical

fibers

173

where u and v refer to two linear polarizations (in the case of circular polarizations, the XPM coefficient | should be replaced by 2), the subscripts 1 and 2 label the cores, and the coupling coefficient between them is AT = 1. Four-component soliton solutions to eqs. (219) can be sought for by means of VA based on the Gaussian ansatz, wi,2(z, r) = Ax2 exp(i/?z -

WT^), (220)

^^1,2(2", r) = Bx2 exp(i^z - \b^T^), with arbitrary real propagation constants p and q. In the general case, the corresponding variational equations for the ansatz parameters An.Bn and a,Z?, which are sought for as functions of/? and q, are cumbersome. The equations admit both symmetric solutions, with A \ = A\ and B]= B\, and asymmetric ones, which are generated by symmetry-breaking bifurcations, similar to the model of DCF with a single polarization considered above. The existence regions of all the solutions in the (/?, q) plane, obtained from numerical solution of the algebraic variational equations, are displayed in fig. 32 for the most important case when the signs of the amplitudes A\2 and Bx2 inside each polarization coincide (other cases can also be considered, but they yield unstable solutions only). Outside the hatched area, there are only solutions with a single polarization (i.e., with either ux^ = 0 or wi,2 = 0), which amount to solutions considered in §6.1; in particular, at the dashed-dotted borders of the hatched area, asymmetric four-component solitons (designated by the symbol ASl in fig. 32) change over to the two-component asymmetric solitons of the single-component DCF model. Symmetric solitons exist inside the sector bordered by the solid lines. The biftircation which gives rise to the asymmetric solitons ASl and destabilizes the symmetric solitons occurs along the short dashed curve in the lower left part of the hatched area. There is an extra asymmetric soliton (denoted by AS2 in fig. 32) inside the area confined by the dashed curve. Thus, the total number of soliton solutions changes, as one crosses the bifiarcation curves in fig. 32 from left to right, from 1 to 3 to 5. However, the soliton AS2 is generated from the symmetric soliton by an additional symmetry-breaking bifurcation which takes place after the symmetric soliton has already been destabilized by the bifurcation which gives rise to the asymmetric soliton ASl, therefore the soliton AS2 is always unstable, while the primary asymmetric soliton ASl is, most plausibly, always stable. Further details about the stability of different solitons in this model can be found in a paper by Lakoba and Kaup [1997].

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[2,

Fig. 32. Regions of existence of the symmetric and two types of asymmetric (stable, ASl, and unstable, AS2) solitons in the plane {p,q) in the bimodal dual-core-fiber model (219). The symbols M = 0, t; = 0, and u = u refer to particular solutions with a single polarization and equal polarizations.

§ 7. Bragg-grating (gap) solitons 7.1. Instability of gap solitons In the systems described by the single or coupled NLS equations, the secondderivative terms account for the intrinsic material dispersion of the fiber or waveguide. Contrary to this, strong artificial dispersion can be induced by a Bragg grating (BG), i.e., a periodic modulation of the refractive index written along the fiber, the modulation period being half the wavelength of the light

2, § 7]

Bragg-grating (gap) solitons

175

signal. The model for a nonlinear optical fiber equipped with BG is based on the coupled equations (see the review by de Sterke and Sipe [1994]) \Ut + \Ujc + {o\u\^-^\u\^)u^v

=0,

it;,-i(;;, + (|M|^ + (T|t;|^)i; + w = 0 ,

(221) (222)

where u and v are the amplitudes of the right- and left-traveling waves, the linear coupling terms take into account resonant reflection of light on BG, and the cubic terms account for the usual SPM and XPM nonlinearities. In this context, the SPM coefficient takes the value o = \, while in the case a = 0 eqs. (221) and (222) constitute a massive Thirring model, which is exactly integrable by means of 1ST. The limiting case a ^ oc, when eqs. (221) and (222) take the form iwr+ iwY +|w|^M + i; = 0,

it;,-it;.v +|t;|^t; +w = 0,

(223)

has a different application to nonlinear optics: after making the replacements t -^ z and X -^ r/c, eqs. (223) describe a dual-core fiber with a group-velocity mismatch 2c between the cores, while their intrinsic dispersion is neglected, cf eqs. (211) and (212) (Malomed and Tasgal [1994]). Although the system of equations (221)-(222) with a ^ 0 is not integrable, it has a family of exact soliton solutions found by Aceves and Wabnitz [1989] and Christodoulides and Joseph [1989]. In particular, the expression for zero-velocity solitons is u = (l-\- a)~^^^(sin Q) sech(jc sin Q- {iQ)- exp(-i/cos 0 , i; = -(1 + a)"^^^(sin Q)sech(x sin Q + ^iQ) • exp{-itcos Q), where the parameter Q, which takes values 0 < Q < Jt, determines the soliton's width and amplitude. These solitons are frequently called gap solitons (GSs), as they exist inside the gap, w^ < 1, in the linear spectrum, o)^ = \ -\- k^, of the system (221)-(222). The exact zero-velocity GS solution to eqs. (223) is obtained from eqs. (224) by setting a = 0. A problem that may be considered by means of VA is internal vibrations of perturbed GSs. The analysis, developed by Malomed and Tasgal [1994], has produced an unexpected prediction - an intrinsic instability of a part of the family of GS solutions (224). At the time this resuh was published, it seemed to be an artifact generated by VA, and it was even regarded as a major failure of the variational technique, stimulating a sophisticated analysis of situations when VA may generate false soliton instabilities (Kaup and Lakoba [1996]; it was

176

Variational methods in nonlinear fiber optics and related

fields

[2, § 7

concluded that a spurious instability is possible, roughly speaking, in models in which the quadratic part of the Hamiltonian is not positive definite, which is the case for eqs. (221), (222), but not for the single or coupled NLS equations). Indeed, in the case a = 0, the solitons of the integrable massive Thirring model have no instability. However, rigorous results of direct investigation of the soliton stability in the general model (221)-(222) with a ^ 0, based on numerical solution of the corresponding linearized equations, which were later reported by Barashenkov, Pelinovsky and Zemlyanaya [1998] and De Rossi, Conti and Trillo [1998], have confirmed that a part of the GS family (224) is indeed unstable if (7 ^ 0. In fact, the border between stable and unstable solitons in the cases a = I and a = oo, which are relevant to nonlinear optics (see above), is close to that predicted by Malomed and Tasgal [1994] on the basis of VA, see details below. The variational ansatz for perturbed GS follows the pattern of the exact solution (224): w=/7,(l + (j)-'/2[sin(e + ^)] X sech [(x + t ) sin(^ + ^) - K ^ + q)] X

exp [-i {a,, + bu(x + £) + ^c,, sin(\Q) • (x + t)^)] ,

(225)

u=^ - / / , ( l + (j)-'/2[sin(e-^)] X

SQch[(x-Osm(Q-q)^^^{Q-q)]

X exp [-i {a, + b,(^ - C) + i c , s i n ( ^ 0 • (x - Cf)] , where r]u, rju, Q, q, Uu, a^, bi,, Z?^, c^, Q and t are variational parameters that may be functions of t. This ansatz lets one vary independently the central position, width, amplitude, phase, carrier frequency, and chirp of the u- and i;-components. Equations (221) and (222) can be derived from the Lagrangian L = f^ C dx with the density C = ^ [w*(a, + d,)u-

u{dt + d,) w* + u\dt - d,)u- u(dr - d,)u'^] (226)

+ ^(T(|w|" + |t;|^) + |i/|>|2 + w*t;4-wt;* [for the model (223), one should set a = 1 and drop the XPM term |wp|t;p in eq. (226)]. Substituting eqs. (225) into the Lagrangian, performing the integration, and varying with respect to the free parameters yield a cumbersome system of dynamical equations which have a fixed point (FP) corresponding to

111

Bmgg-grating (gap) solitons

2, § 7 ]

the unperturbed soliton (224): rj^ = rju = ^, Q = const. = Qo, Ui, = a^ = ^cosig, bu = b, = Cu = c, = q = ^ = 0. Linearization of the general variational equations about FP leads to a sixthorder system of equations for small internal vibrations of GS, which give rise to the corresponding eigenfrequencies. They take a simple form in the case Q^ <^ I, when GS (224) has a small amplitude and large width, its shape being close to that of the usual NLS soliton: COq =

±]

16(3+ ;r2) 45

80+192(1+ a)-> ^ 675

124 , 45

, (227)

± [2.14-(1.01 + 0.66(1 + ar^)Q'] , 48(1 + a ) - ' + 2 0 K' 135

(228)

± {2.09+ [0.61-0.84(1 + a)-^] Q^] , 4(l + (j)-i + l COQ-

;r2-12

(229)

±0.61g^ {1 - [0.51 - 0.40(1 + a)-'] Q^} (note that (jOg_ is much smaller than w}^ and COQ^, as Q^ <. I). The eigenmode corresponding to cOg is dominated by the oscillations of the dynamical parameters q/Q, (TJI - r]l) and {QU - «^) in the ansatz (225), while the oscillation amplitudes of other variables are smaller by a factor ~ Q. The O)Q+ and (DQ- eigenmodes are dominated by oscillations of the variables t and {QU + a^;), but in different ratios. With increasing Q, the shape of GS becomes essentially different from that of the NLS soliton, and it becomes unstable at some critical value Q^r- In fact, each of the three eigenfrequencies becomes unstable at some critical value of Q, as illustrated by fig. 33, which displays the eigenfrequencies vs. Q/n in the limit case corresponding to eqs. (223), i.e., a ^ oc (recall it is a model for a group-velocity-mismatched dual-core fiber). In this case, the smallest Q^v is generated by the eigenfrequency co,,, which becomes imaginary, giving rise to a monotonic (nonoscillatory) instability dXQ ^ QAn (see fig. 33a). This instability is indeed spurious, as it has no counterpart in the numerically exact results which were later reported for the same case by Barashenkov, Pelinovsky and Zemlyanaya [1998]. However, two other eigenfrequencies found by means of VA become complex (rather than imaginary) at ^ ^ O.SZJZ, giving rise to oscillatory instabilities. The onset of this instability, which is oscillatory too.

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Variational methods in nonlinear fiber optics and related fields

[2, §7

Fig. 33. Eigenfrequencies of internal vibrations of the gap soliton in the model (223) vs. the intrinsic soliton parameter Q (see eq. 224), as produced by VA. The real part of the eigenfrequencies is shown by the solid curve above the Q axis, and the absolute value of the dashed branches below the Q axis gives the instability growth rate |Ima>|, if any.

2, § 7]

Bragg-grating (gap) soli tons

179

was discovered by Barashenkov, Pelinovsky and Zemlyanaya [1998] at virtually the same point, Q ^ 0.53JT. Moreover, it is seen in figs. 33b,c that a secondary oscillatory instability sets in at a still larger value of Q, which also complies with the numerically exact results. Lastly, as concerns the spurious nonoscillatory instability generated by the eigenfi-equency cOq (fig. 33a), this artifact can probably be explained by the theory developed by Kaup and Lakoba [1996]; note that the maximum growth rate of the spurious instability is 6 times as small as that of the genuine instability, cf. figs. 33a and 33b, hence the spurious instability is not so important in practical terms. 7.2. Solitons in linearly coupled waveguides with Bragg gratings A natural generalization of the model for an optical fiber equipped with a Bragg grating (BG) is a system of two parallel-coupled cores with the grating written on both of them. As shown by Mak, Chu and Malomed [1998], this model gives rise to generalized gap solitons (GSs) with interesting dynamical properties. The model can be cast into the following normalized form [cf eqs. (221)-(222) for the single-core BG fiber and (200)-(201) for the dual-core fiber without BG]: '\u\t + iwu + (^|wi|^ + |t^i|^)wi +v\ + Aw2 = 0, \v\t-\v\y:-^{\\vx\^

+ |wi|^)(;i +wi +Xv2 = 0,

iw2r+ iw2.Y + (^|w2|^ + \v2\^)u2 + Vi + Awj = 0, W2t-W2x + C2\v2\^'^\u2\^)V2 + U2+Xux

= 0,

(230) (231) (232) (233)

where the usual ratio 1:2 between the SPM and XPM coefficients is implied, the BG-induced coefficient of the conversion between left- (wi,2) and right- {u\^2) traveling waves is normalized to be 1, and A is the coefficient of the linear coupling between the two cores. The same model can be realized as describing stationary field distributions in two parallel-coupled planar waveguides with BGs in the form of a system of parallel scores, in which case t and x play the roles of the propagation distance and transverse coordinate, the diffiaction in the waveguides being neglected. Zero-velocity solitons are sought for as wi,2 = exp(-ia;0 ^i,2(^),

^\a "= exp(-ia;0 ^i,2(-^),

(234)

where the reduction Fi 2 = ~^\2 ^^Y ^^ imposed [in fact, the exact GS solutions (224) in the single-core model are subject to the same reduction].

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Variational methods in nonlinear fiber optics and related

fields

[2, § 7

Substituting this into eqs. (230)-(233) leads to coupled ODE's (with the prime standing for d/djc), a ; ^ i + i ^ ; + f | w ^ i | 2 ^ i - ^ * + A^2 = 0 , coU2 + iU^^l\U2\^U2-U^+Wi

(235)

= 0.

(236)

A possible existence range for solitons in the (A, (o) plane can be found from the linear dispersion relation for eqs. (230)-(233). Looking for a linearized solution in the form wi,2,f12 ^ exp(i^x-iwt), one obtains

a)^ = X^-^l-^k^± iXVYTk^.

(237)

As was mentioned in the preceding subsection, the solitons can only exist in the gap of the linear spectrum, i.e., at values of w which cannot be obtained from eq. (237) at any real value of A:. At A = 0, when the two waveguides decouple, the gap is widest, -1 < a; < 1. At |A| = 1, the gap closes up, i.e., no soliton may exist at |A| > 1. To summarize, the soliton existence region is a part of the rectangle |A| < 1, |ft;| < 1. The stationary equations (235) and (236) can be derived from the Lagrangian with the density

c = co(Uiu; + U2U;) + ^ [u[u; - (u^yui + u^u^ -(U^yu2] + Id^il' + \U2\')-^,{uf + c/,*' + u^ +1/;^) +A(t/i^2* + ^1*^2). Then, the following ansatz is adopted for the complex soliton solution sought for: ^1,2 =^i,2sech(^jc) + L5i 2 sinh(|ax)sech^(^),

(238)

with real ^1,2, ^1,2 and ^. The corresponding effective Lagrangian is

I

Cdx

J-o = M"' [2(o{A] +Al)+\aj{B\+Bl)-\^{A,B,

+A2B2)

(239)

+ {A1 + A\) - 1.2857(S| + B\) + 1{A]B] + AJBl) -2(A\ + Al) + \{B] + B\) + 4kA\A2 + \XBxB2] (the numerical coefficient 1.2857 is given by some integral), which generates variational equations 3A^2,i - 3(1 - (jo)A\2 + 2A]2 + \AU2B\2

- M^i.2 = 0,

A52,, + |fl,.2 -3.8575^2 + f^[2'Si.2 -M^:.2 = 0, 2w{A] +Al)+l(o{B]+B\)

+ {A\+A\)-

\.2%51(B\+B^,)

+ \{A]B] + AJBJ) - 2{A] + A\) + \{B] + B\) + AkAxA2 + \XBxB2 = 0.

(240) (241)

2, §7]

Bragg-grating (gap) solitons

181

Fig. 34. Bifurcation diagram for zero-velocity solitons in the model of a dual-core nonlinear optical fiber with Bragg gratings written on both cores.

A general result, following both from numerical solution of eqs. (240)-(242) and from direct numerical solution of ODEs (235) and (236), is that a symmetric solution, with A\ = A\ and B\ = Bj, exists at all values of co and A inside the above-mentioned spectral gap, and it is the only soliton solution if the coupling constant A is large enough. However, below a critical value of A (which depends on co), the symmetric solution bifurcates, giving rise to three branches: one remains symmetric, while two new mutually symmetric branches represent nontrivial asymmetric solutions. The bifurcation can be conveniently displayed in terms of an effective asymmetry parameter

e

uL uL uL^ui:

(243)

where U\m and Uim are the amplitudes (maxima of the absolute values) of the fields U\2 in the two cores. A complete three-dimensional plot of the bifijrcation, i.e., 0 vs. 0) and A, is shown in fig. 34. At A = 0, when eqs. (235) and (236) decouple, the numerical solution matches the exact single-core solution (224), while the other core is empty. Generally, this symmetry-breaking biftircation is similar to that shown in fig. 28 for the dual-core nonlinear fiber without BG. However, unlike the (slightly) subcritical biftircation in fig. 28, both VA and direct numerical solutions show that the present biftircation is supercritical (alias a forward biftircation). The bifurcation diagram in fig. 34 was drawn using direct numerical results obtained from eqs. (235)-(236), but its variational counterpart is very close

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Variational methods in nonlinear fiber optics and related fields

[2, § 8

Fig. 35. Shapes of the larger component U\ of the asymmetric soliton in the dual-core Bragg-grating fiber at a> = 0.5 and A = 0.2. The upper and lower panels show Re U\(x) and Im U\(x). In each panel, the solid and dashed lines represent the numerical and variational results.

to it: the relative discrepancy between the VA-predicted and numerically exact values of A at which the bifurcation takes place for fixed o) is - 5 % . To illustrate the accuracy of VA, fig. 35 presents, for a typical case, a comparison between the shapes of the asymmetric soliton predicted by VA and obtained from direct numerical integration. A direct numerical test of the stability of symmetric and asymmetric solitons in the present model has yielded results exactly conforming to what should be expected on the basis of the general bifurcation theory (see a book by looss and Joseph [1980]): all the asymmetric solitons are stable whenever they exist, while all the symmetric solitons, whenever they coexist with the asymmetric ones, are unstable. However, beyond the bifurcation points, where the asymmetric solitons do not exist, all the symmetric ones are stable.

§ 8. Stable beams in a layered focusing-defocusmg Kerr medium It is well known that the standard NLS equation governing the spatial evolution of signals in bulk nonlinear optical media cannot support stable soliton-like cylindrical beams: if the nonlinearity is self-defocusing (SDF), any beam spreads out, while in the case of a self-focusing (SF) nonlinearity, a stationary-beam solution with a critical value of its power does exist (Chiao, Garmire and Townes

2, § 8]

Stable beams in a layered focusing-defocusing Kerr medium

183

[1964]; as a matter of fact, this was the first soliton considered in nonlinear optics), but it is unstable because of the possibility of wave collapse (see review by Berge [1998]). Recently, Berge, Mezentsev, Juul Rasmussen, Christiansen and Gaididei [2000] have demonstrated, by means of direct simulations, that the beam can be partly stabilized if the nonlinearity coefficient is subjected to weak spatial modulation along the propagation direction, so that the beam power (which is virtually constant, as radiative losses turn out to be negligible) effectively oscillates about the modulated critical value, sometimes being slightly larger and sometimes slightly smaller than it. As a result, it was observed that the beam could survive over a large propagation distance, although eventually it might be destroyed by the instability. Here, a model is considered in which the nonlinearity is subjected to a more radical modulation, so that SDF and SF layers alternate periodically. The model is based on the NLS equation iwz + I Viw + Y{Z)\U\^U = 0,

(244)

where the diffraction operator V^^ acts on the transverse coordinates x and y, and the nonlinearity coefficient y assumes positive and negative values Y± inside alternating layers with widths L±. While particular realizations of such a layered medium are not discussed here in detail, it is relevant to note that it has been demonstrated experimentally by Liu, Qian and Wise [1999a] that narrow layers with a large negative value of the effective Kerr coefficient can be created, using the cascading mechanism based on the quadratic nonlinearity. A novel result, obtained recently by Towers and Malomed [2002] by means of both VA and direct simulations, is that this type of nonlinear medium gives rise to completely stable beams, which is the subject of the present section. Axisymmetric spatial solitons are sought for in the form w(z, r, 6) = exp(i5'0) U{z, r),

(245)

where r and d are the polar coordinates in the transverse plane, the integer S is vorticity ("spin"), and the fianction U{z, r) obeys the PDE i^z + ^f^..

+ ^ ^ , - ^ ^ ' ) + 7 ( z ) | ^ | ' ^ = 0.

(246)

To apply VA to eq. (246), a natural ansatz is adopted, U = A(z)r' exp [ib(z)r' + i0(z)] s e c h ( ^ ] .

(247)

where b and W are the soliton's chirp and width. Skipping details of straightforward calculations, the following set of variational equations for the parameters

184

Variational methods in nonlinear fiber optics and related fields

[2, § 8

3

No Fixed Points

0.5 -1

-0.8

-0.6

-0.4

-0.2

Fig. 36. Parameter space of the variational model describing the cylindrical zero-vorticity beam in a layered focusing-defocusing medium. The fixed point is stable in the speckled area.

of the ansatz (247) can be derived. First, due to the conservation of energy E (actually, E is the power of the beam), there is a dynamical invariant A^W(z) 2{S+\)

.

const. = E,

(248)

which makes it possible to eliminate the amplitude A in favor of the width W. After that, there remains a second-order equation for W(z), 2/2

2/4

^,

~d^

w-

(249)

the chirp being expressed in terms of W(z) as b(z) = (2W) 'dW/dz, cf similar equations (32) and (31) for the usual ID soliton. The constants I\x4 are integrals resulting from VA; fovS = 0 (zero-spin beam), /1.2.4 ^ (1.352,0.398,0.295). For the piece-wise constant function y(x) defined above, eq. (249) can be integrated inside each interval where y is constant. The resuk is

d^j ^^ = ^^'

(250)

where V = W\ T = 8 [V/i - ( V / i ) / ] , H = 8/?, and h [which is the Hamiltonian of eq. (249) with 7 = const.] is an arbitrary integration constant. Within the interval 0 < z < Z+, the parameter F keeps a constant given value r+, then it assumes another constant value r_ in the interval

2, §8]

Stable beams in a layered fociising-defocusing Kerr medium

185

30,

30 ^

-

20 -

2.25

3^ ~ 10 .

0 > 20

^ ^ 5

1 5 ^ ^ ^

20

z'°

5 ^

-

Y

^K^^,

10

12

14

16

18

20

Fig. 37. Numerically simulated evolution initiated by the configuration (247) with 5" = 0, L = 1, and r = -1.3. The upper and lower plots show the evolution of the beam's peak amplitude and cross-section, respectively, vs. the propagation distance.

L+
L(F-\) Vo = ± AVLTIV-I-LF'

v^

V-\-LF VL+\

(251)

which make sense only for negative values F < -\/L. To investigate the stability of FP, one should find eigenvalues A of the Jacobian of the map, d (FQ, VQ)/ d [VQ, VQ). The FP is stable if both eigenvalues satisfy the condition |A| ^ 1. The resuks of this analysis are summarized in fig. 36. No

186

Variational methods in nonlinear fiber optics and related

fields

[2

FP exists beneath the curve L = -l/F. Above this curve, FP is stable inside a speckled band. Outside the band, FP is unstable. To test the VA-based analytical results against direct simulations, the underlying equation (244) was solved numerically, using the ansatz (247), whose parameters were taken at FP (251), as the initial configuration. A typical result is shown in fig. 37: after a short relaxation period, the initial beam reshapes into a nearly stationary stable one, which propagates with small residual oscillations. This seems to be the first example of a stable cylindrical beam in a medium with a Kerr nonlinearity. Simulations of the beams with nonzero vorticity S, see eq. (245), show that, unlike the 5* = 0 beam, they are all unstable.

§ 9. Conclusion The aim of this review was to demonstrate that a combination of the variational approximation with direct numerical simulations is the most natural and efficient approach to many problems in nonlinear optics and other areas of physics which are based on nonlinear PDEs. Although the technique has already been applied to a large number of systems, its potential is far from being exhausted. The outburst of activity in the field of dispersion management has been responsible for the recent renaissance of variational methods. Another fast developing topic which calls for the development of these methods at a higher level is the study of spatiotemporal pulses in multidimensional optical media. Beyond the limits of nonlinear optics, the variational approximation has recently often been used in studies of Bose-Einstein condensates. Thus, variational methods remain a powerful and universal tool in the arsenal of modem-day nonlinear science.

Acronyms adopted in the text ID

one-dimensional

2D

two-dimensional

BDW

Bloch domain wall

BG

Bragg grating

c.c.

complex conjugate (in equations)

CQ

cubic-quintic (equation)

CW

continuous wave

DCF

dual-core fiber

2]

Acknowledgements

DDF

dispersion-decreasing fiber

DM

dispersion management

DS

dark soliton

FP

fixed

187

point (of dynamical equations or map)

FWHM fill! width at half-maximum (of a solitary pulse) GL

Ginzburg-Landau (equation)

GS

gap soliton

1ST

inverse scattering transform

KdV

Korteweg-de Vries (equation)

NLS

nonlinear Schrodinger (equation, or soliton)

ODE

ordinary differential equation

PAD

path-average dispersion (in a dispersion-managed fiber-optic link)

PDE

partial differential equation

rhs

right-hand side (of an equation)

SDF

self-defocusing (nonlinearity)

SF

self-focusing (nonlinearity)

SPM

self-phase modulation

TOD

third-order dispersion

VA

variational approximation

VK

Vakhitov-Kolokolov (stability criterion for solitons)

WDM

wavelength-division multiplexing

XPM

cross-phase modulation

ZDP

zero-dispersion point

ZS

Zakharov-Shabat (equations)

Acknowledgements Outstanding contributions to the fields comprised by the review were made by D. Anderson, D.J. Kaup and M. Lisak. This review is, to a large extent, a result of my collaboration with them. It is also my pleasure to appreciate valuable collaborations and/or discussions with other colleagues working in the areas comprised by this review: FKh. Abdullaev, VV Afanasjev, J. Atai, Y. Band, I.V Barashenkov, A. Bemtson, A.V Buryak, J.G. Caputo, A.R. Champneys,

188

Variational methods in nonlinear fiber optics and related

fields

[2

RL. Chu, S.R. Clarke, RD. Drummond, C. Etrich, D.J. Frantzeskakis, B.V Gisin, R. Grimshaw, K. Hizanidis, R Kevrekidis, Yu.S. Kivshar, G. Kurizki, F. Lederer, T.I. Lakoba, W.C.K. Mak, D. Mazilu, D. Mihalache, A.I. Maimistov, U. Reschel, M. Quiroga-Teixeiro, G.D. Reng, A. Shipulin, I. Skinner, N.F. Smyth, M. Weinstein, F. Wise and J. Yang. Important contributions to results included into the review were made by my younger collaborators: L.-C. Crasovan, R.S. Tasgal, A. Desyatnikov, I. Towers and M. Wald. Special thanks are due to J. Refina, who suggested writing this review for Progress in Optics. Lastly, I appreciate valuable aid in the preparation of this article for printing from Jeroen Soutberg, of ISYS Rrepress Services.

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E. Wolf, Progress in Optics 43 © 2002 Elsevier Science B. V. All rights reserved

Chapter 3

Optical works of L.V. Lorenz by

Ole Keller Institute of Physics, Aalborg University, Pontoppidanstrcede 103, DK 9220 Aalborg 0st, Denmark

195

Contents

Page § 1.

Introduction

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§2.

Biography of Lorenz

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§ 3.

Aether vibrations in polarized light

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§ 4.

Surface optics: the first theory

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§ 5.

Lorenz begins to doubt the elastic light theory

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§ 6.

The phenomenological light theory of Lorenz

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§ 7.

The electrodynamic theory of Lorenz

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§8.

The discovery of the Lorenz-Lorentz relation

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§ 9.

Light scattering by molecules and a sphere

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§ 10.

Lorenz and the aether

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References

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§ 1. Introduction In 1867, the Danish physicist Ludvig Valentin Lorenz (1829-1891) pubHshed a paper (Lorenz [1867a]) in which he identified the vibrations of light with electrical currents. In the same year the paper was translated fi-om Danish into both German and English and appeared in the Annalen der Physik (Lorenz [1867b]) and Philosophical Magazine (Lorenz [1867c]). The paper marked the culmination of his efforts towards understanding the nature of light and uniting the natural forces, and it has turned out that the electrodynamic theory of Lorenz essentially is equivalent to that proposed by James Clerk Maxwell a few years earlier. In the annals of science the decisive step in establishing the theory of electrodynamics is rightly associated with the name of Maxwell, but it is interesting to reflect on the fact that this step was also taken, quite independently, by his Danish contemporary Lorenz. The path which Lorenz followed was quite different fi-om that of Maxwell although they both had to abandon the action-atdistance conception. Lorenz did this by introducing retarded potentials, and he identified the velocity of retardation with the velocity of light. In his amazing 1867 paper he established the inhomogeneous wave equations for what we call the vector and scalar potentials, and he showed that these potentials were linked by the relation most scientists nowadays incorrectly refer to as the Lorentz gauge condition. The Dutch physicist H.A. Lorentz was only fourteen years old when L.V Lorenz established his gauge condition between the potentials in order to satisfy the equation of continuity for the electrical charge. Lorenz had begun his inquiries into the nature of light in 1860, starting fi-om the elastic theory of light. In this theory the aether (ether) hypothesis played a crucial role. Within a few years Lorenz broke away from the elastic light theory, and step by step he was led, by 1867 if not before, to the conclusion (long before Maxwell) that there was no need for an aether. In a popular article in Danish, written in 1867 (Lorenz [1867d]) and bearing the title "Om lyset" (about light) he writes towards the end:

"Altogether it is a very unscientific approach to invent a substrate (the aether) when its existence does not manifest itself in an otherwise definite manner." 197

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[3, § 1

Only by the end of the nineteenth century was it generally understood that there was no room for an aether, a view which had accomplished itself in the mind of Lorenz many years before. In Lorenz' set of electrodynamic equations, the material side contains only (free) charges and associated current densities, therefore if one looks at his equations from a formal point of view they are identical to the microscopic Maxwell equations written in the relativistically invariant form. Although it certainly would be an exaggeration to assert that Lorenz himself understood the far-reaching consequences of his set of equations in this respect, the correct set of microscopic Maxwell equations in the manifestly covariant form was on the scene in 1867! For good reasons the Dutch physicist H.A. Lorentz is well-known to all physicists, but the name of L.V Lorenz is (best) known as the "other name" appearing in the Lorentz-Lorenz formula, relating the refractive index to the molecular polarizability. The works of Lorenz were well-known to his contemporaries, and there is no doubt that he was considered an eminent theoretical physicist among the most prominent scientists in Europe in the second half of the nineteenth century. Except for one article (Lorenz [1890]), which was published only in Danish, yet well-known outside his country, all his papers in optics (electrodynamics) appeared in the most esteemed international journals of his time (Annalen der Physik, Philosophical Magazine, and Journal fur die reine und angewandte Mathematik), and we know that many eminent physicists and mathematicians red his papers and were inspired by them. Lorenz' important memoirs were soon forgotten and his impact on the development of theoretical physics seems to have been insignificant despite of his splendid achievements. It is always difficult, not least in cases like that of Lorenz, to place a scientist on the proper shelf of skill, but then again, that is not terribly important. To be remembered, quoted, and understood in science we know that it is important to have personal contacts. Despite the fact that Lorenz throughout his life took up central issues in fundamental physics, he did not have much personal contact with physicists and mathematicians outside Denmark, and in his own country, though respected, very few people (if any) understood his theoretical work. For good or for bad he was a lonely thinker, but the fact that he was able to attain far-reaching goals in physics all by himself indicates that he should be ranked among the most famous figures of physics in the nineteenth century. In Denmark there has been a long tradition in research in that part of physics relating to optics. Thus, in 1669 Rasmus Bartholin [Erasmus Bartholinus] (16251698) discovered the double refraction of light by studying calc-spar crystals from Iceland. Shortly after, Ole Romer (1644-1710) from observations of the eclipses of Jupiter's moons discovered what he in Danish called "lysets toven"

3, § 1]

Introduction

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(the hesitation of light), i.e. the finite speed of light. The value he obtained for the speed of light was only ten percent lower than the correct value. In an electrodynamical experiment Hans Christian 0rsted (1777-1851) observed in 1820 that a current could affect a nearby compass needle, and this led him to the conclusion that magnetism and current were related phenomena. There is no evidence that 0rsted, who was a monumental figure in Denmark in the first half of the nineteenth century, and not only in the natural sciences, had any personal contact with Lorenz (0rsted died when Lorenz was a student at the Technical High School at Copenhagen), but Lorenz inherited from 0rsted as a guiding principle the idea of the unity of the forces of nature. In contrast to 0rsted, Lorenz was an eminent mathematician, and one may say that theoretical physics was bom in Denmark with Lorenz. Not only was he the father of theoretical physics, he also was alone in Denmark in his century, and thus there was no deep understanding of his achievements in his home country. The revolution which took place in physics in the first quarter of the twentieth century may have contributed to the fact that Lorenz was almost forgotten in Denmark, not least because Niels David Bohr (1885-1962) moved into the centre of the stage of physics in 1913 with his new theory for the spectral lines of hydrogen, a theory which definitely broke with classical physics and paved the way for the quantum theory which has been in the centre of ftindamental physics ever since. Lorenz carried out theoretical research not only in optics and electromagnetics, but also in thermal physics, in elasticity, and in molecular kinetics. Lorenz' most important contribution outside optics (electrodynamics) was in the field of metallic conductivity, where he established the so-called Lorenz law, according to which the ratio between the conductivities of heat {K) and electricity (a) is proportional to the absolute temperature {T). The quantity K/{OT) is known as the Lorenz number. Lorenz mastered both theory and experiment and he carried out many important experiments in optics, electricity, and heat physics. Thus, in the determination of specific electrical conductivities of metals Lorenz in 1873 invented an absolute method which won international acceptance and served to fix the international unit of resistance (the ohm) for many years. In a paper never published Lorenz developed a theory of currents in telephone cables, and he was the first to establish the basic differential equations of telegraphy. Lorenz also made contributions to mathematics, among others to the theory of prime numbers and to asymptotic formulas for the Bessel functions. The collected works of Lorenz were published in French in 1896-1904 under the title "Oeuvres Scientifiques de L. Lorenz" by the Danish mathematician H. Valentiner (publisher: Lehmann Copenhagen), who also added useful comments and corrections

the period and edited and Stage, (Valentiner

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[3, § 1

[1896-1904]). The collected works were reprinted in 1964 by Johnson Reprint Corporation, New York. In 1939, M. Pihl published the first comprehensive study (in German) of the works of Lorenz under the title "Der Physiker L. V. Lorenz eine kritische Untersuchung" (Pihl [1939]). The work of Pihl embraces all the physical studies of Lorenz and gives a fine and balanced account of Lorenz' achievements in physics. Since 1939 no major work on Lorenz has appeared. The role of Lorenz in the evolution of electrodynamics is beautifully described by L. Rosenfeld in a review entitled "The Velocity of Light and the Evolution of Electrodynamics" (Rosenfeld [1957]). Rosenfeld ranks the achievement of Lorenz very high. Thus, he ends his article with the sentence: "There is no more striking illustration of the tenseness of physical thought in that heroic period than Lorenz' and Maxwell's struggle to establish the electromagnetic theory of light." In a "Survey of Some Early Studies of the Scattering of Plane Waves by a Sphere", N.A. Logan [1965] emphasizes that the general theoretical result of the scattering theory published by Mie in 1908 (Mie [1908]) had been given already in 1890 by Lorenz, in a paper (Lorenz [1890]) which was well-known at least in Britain before the turn of the century. A brief account of Lorenz' scientific achievements was given thirty years ago by Pihl [1972], and Pihl [1973] also wrote briefly about Lorenz and his scientific works in Gillispie's "Dictionary of Scientific Biography". In this dictionary Lorenz' first name Ludvig is spelled incorrectly with a W instead of a V Ten years ago, H. Kragh [1991] presented a short but valuable description of Lorenz and his optical works in Applied Optics. Sixty years after the appearance of the comprehensive treatise of Pihl [1939], the time has come to remind scientists of the works of Lorenz, and this review presents a more detailed account of the theoretical optical works of L.V Lorenz than previous publications. § 3 describes Lorenz' first published work in optics, which deals with the bending of light at an opening in a screen, and it is shown that the so-called Rayleigh diffraction formulae of the second kind were discovered by Lorenz long before Rayleigh. In §4, it is described why the experiments of Jamin and the subsequent theoretical analysis of Lorenz, starting from the Fresnel formulae, can be considered as the experimental and theoretical birth of surface optics. Lorenz begins to doubt the elastic theory of light in 1861, and this phase in the development of his understanding of the nature of light is described in §5. The phenomenological light theory of Lorenz is given in § 6. Due to the fact that his wave equation governing light propagation in inhomogeneous but transparent media is identical to the one obtained from the macroscopic Maxwell equations, Lorenz was able to derive the boundary

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conditions for the electric and magnetic fields that we use today if charge and current densities are absent at the interface. Subsequently, his important accounts for the double refraction, optical activity and chromatic dispersion are presented. The great electrodynamic theory of Lorenz from 1867, which I have already mentioned, is discussed in § 7, and § 8 presents the discovery of the Lorentz-Lorenz (or maybe better Lorenz-Lorentz) relation. § 9 presents Lorenz' description of the interaction of light with point-like atoms surrounded by a sphere of influence, where the light velocity of vacuum is modified, and it is shown that even with our (quantum) theory of electrodynamics at hand one cannot deny that light behaves in a special manner in the near-field zone of atoms. The section ends with a description of Lorenz' ingenious, but unknown work on plane-wave light scattering from a spherical particle. Our survey of the optical works of Lorenz is concluded in § 10 with a brief account of the development which took place in his mind with regard to the aether hypothesis, viz., from belief in the existence of an aether in 1860 to his certainty of its nonexistence in 1867, if not before. My own interest in the theoretical works of Lorenz came about by the merest chance. In the spring of 1998 I submitted to the Physical Review A a paper which put forward a propagator description of the spatial confinement of quantized light emitted from an atom (Keller [1998]). At one point in the text I used the words "Lorentz gauge", and an anonymous referee pointed out that the historically correct designation was "Lorenz gauge" after the Danish physicist L.V Lorenz. Although apparently very few people know this, as a Dane I ought to have known it! I checked the referee's assertion, and found that he was right. Soon after, I started my studies on Lorenz, and when Emil Wolf visited me in Aalborg in the autumn of 1999, in connection with his nomination as Doctor Honoris Causa at my university, I had already planned to write about Lorenz. When I mentioned this to him, we soon realized that Progress in Optics would be the right place for putting the optical works of Lorenz back on the scene of physics.

§ 2. Biography of Lorenz Ludvig Valentin Lorenz (fig. 1) was born on 18 January 1829 in Helsingor (Elsinore), the Danish city with the castle where William Shakespeare laid the scene for his famous play "Hamlet". Lorenz' father Johann Gottfried Lorenz (1796-1849), who owned a bakery in Helsingor, was born in Strahlsund in Germany, and when he came to Denmark he married Charlotte Christine Scherfin (1797-1885). The Scherfin family was of French origin, descending from the

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"} Fig. 1. This photograph of Lorenz is identical to the one pubhshed in Oeuvres Scientifiques de L Lorenz, edited by H. Valentiner, Vols. 1, 2 (Lehman and Stage, Copenhagen, 1896-1904). In the signature below the photo, Lorenz has left out the initial V (for Valentin). In all his scientific articles Lorenz also wrote his name as L. Lorenz.

Huguenots, and Charlotte Christine's father came from Riigen in Germany. Lorenz was six years old when the family moved to Maribo, a small city on the island of Lolland, and here Lorenz spent the rest of his childhood.

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Lorenz received private lessons (his father gave him the first instructions in arithmetic and mathematics) until he was fourteen years old, and then came to the cathedral school (gymnasium) in Nykobing, where he took his preliminary higher education three years later. In an unpublished autobiographical sketch, probably dating from ~1877, Lorenz mentioned that an evening lecture in physics during the winter 1841-42 aroused his interest so much that he soon knew that he would make the study of mathematics and physics his calling. In 1846 he went to Copenhagen to enroll at the Polytechnic High School, founded by H.C. 0rsted in 1829. He attended a variety of courses both at the High School and at the University of Copenhagen, including Hebrew, mathematics, pathology, and geology. Apparently he was strongly influenced also by the philosophy of Soren Kierkegaard (1813-1855). In 1852 he graduated in chemistry, not because he liked this field in particular but rather, as he wrote, "because I believed to benefit most from my years of study by practical works in the laboratory". Lorenz never received a formal education in physics, maybe because physics education in Denmark was at a low level and was dominated by 0rsted's qualitative (nonmathematical) idea of physics, and he never had (took) a job as a chemist. Apparently, Lorenz preferred to earn his living by occasional teaching since this would allow him to study physics and mathematics by himself and in depth. In 1854 he received the Gold Medal of the University of Copenhagen for a prize essay on the geometrical properties of Fresnel wavefronts. Supported by the Government, the university and private grants he spent almost a year (1858-59) in Paris to improve his knowledge of theoretical physics, and here his interest particularly in the theory of elasticity, a field of physics (at that time) closely related to optics, grew, maybe because the possibilities of using advanced mathematics were good in this branch of physics. An examination paper on the theory of elasticity reviewed by G. Lame, J. Bertrand and the famous mathematician J. Liouville, formed the basis for his paper "Memoire sur la theorie de I'elasticite des corps homogenes a elasticite constantes" published in Crelles Journal (Lorenz [1861a]). It is in this paper that retarded potentials occur for the first time in the published literature. On 12 August 1862 Lorenz married Agathe Fogtmann (1831-1922) in Copenhagen, and in 1866 he became a member of the prestigious Royal Danish Academy for Sciences, a membership he earned because of his splendid work in optical theory. In the same year (1866) he obtained a permanent position as a physics teacher for army cadets at the Military High School (Academy) just outside Copenhagen. Here he also had at his disposal an excellent laboratory where over the years he carried out many important experimental studies, particularly in optics and thermal physics. Although the theoretical work was always closest to his heart, the laboratory

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at the Military High School gave him "a welcome opportunity to be occupied with practical physics" as he said. Lorenz stayed at the Military High School for 21 years, and despite the facts that his salary was rather low and that he had to struggle hard to finance his research, all in all he appears to have been satisfied. He was a man with no great demands to the material aspects of life, but recognition of his scientific achievements pleased him much. Apparently, he was appreciated in his daily life by people around him, and during his years at the Military High School he wrote several textbooks in elementary and intermediate physics. One of these, a textbook in optics with the Danish title "Laeren om Lyset" was published in 1876 and shortly after translated into German under the title "Die Lehre vom Licht" (Lorenz [1877]). It is a wonderful book which shows how gifted Lorenz also was when it came to communicating elementary physics in written form. Lorenz did not lack recognition in his age. Thus he received the title of professor in 1876, became a so-called etatsraad (titular councillar of state) in 1887, and in the same year was appointed Doctor Honoris Causa by the University of Uppsala. Despite his eminent qualifications he was unable to obtain a position at either the University of Copenhagen (the only one in Denmark in the nineteenth century) or the Polytechnic High School. A contributory cause may have been the circumstance that he had a critical attitude towards (the professional works of) other scientists at the university. In 1887, just four years before his death, the newly established Carlsberg Foundation offered him a position as "an independent researcher" for the rest of his life in recognition of his achievements in physics. He accepted the offer and in February 1887 wrote to the Carlsberg Foundation: "as a result of the economic independence and release ft-om any interrupting work I will be able to take an optimistic view of the fiiture and concentrate my efforts on larger problems to the best of my ability". In 1890, Lorenz' great work on the scattering of light from a sphere was published (Lorenz [1890]). Lorenz died suddenly on 9 June 1891 in Copenhagen and was buried there. A brief, but recommendable description of Lorenz' life has been given by Meyer [1938].

§ 3. Aether vibrations in polarized light 3.1. The unknown dualism between electric and magnetic fields By the end of the nineteenth century it had been known for a long time that the laws of (classical) mechanics are invariant under a Galilean transformation. The laws of electromagnetism were not invariant under this transformation, however.

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and this in itself seemed to indicate the existence of an aether (ether), and a preferred coordinate system in which the ether was at rest. A number of experiments carried out during the second half of the nineteenth century finally led to the abandonment of the ether hypothesis and the birth of the special theory of relativity in 1905. Before Einstein developed his theory on the basis of the postulates that (i) the laws of nature are identical in all reference frames moving with constant velocities relative to one another, and (ii) the speed of light is independent of the motion of the source, the Dutch physicist H.A. Lorentz in 1904 showed that the Maxwell equations in vacuum were invariant under a coordinate transformation now called the Lorentz transformation. His findings, together with the work of Poincare who showed that the transformation of charge and current densities could be made in such a way that the fiall set of Maxwell equations are form-invariant under the Lorentz transformation, in 1905 turned out to be a consequence of the two postulates of Einstein. The covariance of electrodynamics appears in an explicit manner via the second-rank, antisymmetric field-strength tensor F , the elements of which are given by F^^=^-^^

(3.1)

where {A} = (A,iq)/co) is the four-vector potential formed by the vector (A) and scalar (q)) potentials [CQ being the speed of light in vacuo], and {x} = (r,icot) is the four-space-time coordinate. In terms of the electric, E = (E„Ey,E,) = (^,,£2,^3), and magnetic, B = (Br,By,B,) = {BuBi.B^l field vectors, the 4x4 field-strength tensor (^, v = 1,2,3,4) has the explicit form /

0 B, -By -iE,/co\ -B, 0 B, -iEy/co By -B_, 0 -i£-/co \iEj,/co iEy/co iE-/co 0 /

(3.2)

The two inhomogeneous (microscopic) Maxwell equations are contained in the manifestly covariant equation

E ^dxv = M.

(3.3)

V

where J/^i is the ^uth component of the four-vector current density {J} = (J, icop), J = {Jx,Jv,Jz) being the three-vector current density and p the charge density.

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For pi = \, 2 and 3 one obtains the x, y, and z components of the Maxwell equation V x B = JLIOJ + c^'^dE/dt, and pi = A gives the equation V • E = p/Co. The two homogeneous Maxwell equations follow from

?^ + f^ + f^=0, OXy

OXx

(3.4)

OX/^i

where the three indices are different. If the three indices are equal to 1, 2, or 3 the Maxwell equation S/ B = 0 appears, and if one of the three indices equals 4, one finds the equation V x £" = -dB/dt. In 1860 the dualism between the electric and magnetic fields of light was unknown, and in his first scientific paper (Lorenz [1860a]), published in a Scandinavian journal under the Danish title "Om Retningen af ^therens Svingninger i det polariserede Lys", and shortly after (Lorenz [1860b]) translated into German and published in the Annalen der Physik under the title "Bestimmung der Schwingungsrichtung des Licht-athers durch die Polarisation des gebeugten Lichtes", Lorenz attempts to determine the direction of the "light vibrations" by analyzing the diffraction of light propagating through an opening in a plane opaque screen. In passing, it is interesting to note that Lorenz in all his articles writes his name as L. Lorenz, thus omitting the initial letter V (for Valentin). In 1860, Lorenz apparently still believed in the ether hypothesis, although that same year in the same issue of the Annalen der Physik he published a paper (Lorenz [1860d]) in which he demonstrated that the reflection experiments of Jamin [1850]) could be understood without involving longitudinal ether vibrations as done by Cauchy [1836]; see also the account given by Bouasse [1893a,b] and the description in § 4. The two papers are in fact both dated "Kopenhagen den 28 Juni 1860". When a light beam incident at the Brewster angle is reflected from an interface between two isotropic and transparent media, the reflected light is expected to be polarized perpendicularly to the plane of incidence according to Fresnel [1823], but polarized parallel to the plane of incidence according to Neumann [1835]; see also Bouasse [1893b]. The Fresnel theory in fact identifies the direction of "light vibration" with the direction of the £'-field, whereas the Neumann suggestion holds for the 5-field. Seen from our perspective the dualism between the electric and magnetic fields of a light field makes it illusory to determine a direction of light vibrations. When the frame of reference is changed from one inertial system to another, the electric and magnetic fields are changed, but this of course does not imply that the interaction of light with matter (e.g., a detector) in a given inertial system cannot be dominated by either the E- or ^-field. In an experiment where all parts of the setup are in relative rest, use of the inertial system in which the setup is at rest for the description implies that

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the matter response in general is dominated by the interaction with the electric field of the light (electric dipole approximation). By comparing his analysis of the diffraction of light from the aperture with his own experimental data Lorenz comes to the conclusion that the Fresnel point of view is the correct one. It is not evident that the proof established by Lorenz is correct, because the diffraction problem is complicated not least when it comes to a determination of the state of polarization of light, cf the discussion in § 3.3. Nevertheless, it is of interest to present the approach Lorenz used to attack the diffraction problem since his analysis involved important theoretical progress apparently overlooked today.

3.2. Bending of light at an opening in a screen: Lorenz enters optical research Lorenz' study of the diffraction of light from an opening in a plane screen (Lorenz [1860a,b] begins with a discussion of the relations to be fulfilled by the light vectors of the incident [V = (w, t;, w)], transmitted [V\ = (wi,t^i,wi)] and reflected [Vi = (w2, ^2, ^^2)] waves in the opening. If the screen is coincident with the yz-plane of the Cartesian coordinate system used, Lorenz assumes (correctly) that the prevailing field and its first-order derivative in the direction normal to the screen must be continuous everywhere in the opening, i.e., [ F + F 2 ] , = o = [^'i].v^o,

1^^^^^^^

dx

(3-5) (3.6) .v = 0

Stokes [1850] made a theoretical investigation of the diffraction problem earlier than Lorenz but neglected the reflected field in the opening, and (partly) because of this Lorenz found it necessary to reanalyze the problem. Thus, it might be correct to assume that Lorenz was the first to realize the importance of the reflected field in a diffraction problem. The integral theorem of Kirchhoflf [1882b, 1883], also known as the Helmholtz-Kirchhoff integral theorem because von Helmholtz derived the theorem for monochromatic acoustic waves (von Helmholtz [1859]), and based on Green's theorem, made it clear that the self-consistent field (not the incident field) basically appears in the opening. However, in the Kirchhoff diffraction theory, the Kirchhoff boundary conditions which set the field and its normal derivative equal to those of the incident field in the opening are used. Ever since its appearance the Kirchhoff diffraction theory

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has been important (and popular) in studies of optical diffraction problems, at least for apertures significantly larger in linear extension than the optical wavelength. For a modem account of the Kirchhofif theorem, the Kirchhoflf diffraction theory, and some of its applications, see Born and Wolf [1999]. Lorenz next assumed that the incident field is transverse (divergence-free), and showed thereafter that eqs. (3.5) and (3.6) can still be maintained if the reflected and transmitted fields are assumed to be transverse as well. Altogether, Lorenz then took V • K = V • K, = V • K2 = 0.

(3.7)

In the theory of Stokes [1850] the reflected field is neglected, and instead Stokes assumed that also in addition transmitted longitudinal waves are formed in the diffraction process. It is interesting that Lorenz in his first paper already abandons the generation of a longitudinal component in the light vibrations, a point of view he insisted on throughout his life, and a starting point which culminated in his establishment of a light theory which is equivalent to the relativistically invariant form of the Maxwell equations. Lorenz assumed that all components (generically called //) of the incident, reflected and transmitted fields had to satisfy a wave equation

^'nir,t)=h'^^

(3-8)

v^ atwhere v is the actual phase velocity of the wave. Lorenz later assumed this wave equation to hold also inside matter, provided the medium in question was isotropic, homogeneous and transparent. Starting from eq. (3.8), Lorenz established his general light theory a few years later (see § 6). In the famous integral theorem of von Helmholtz and Kirchhoff, which applies to diffraction of scalar waves, the wavefield in a given point inside a domain V bounded by a closed surface S is expressed in terms of the values of the prevailing field and its first-order derivative in the direction normal to S on S. To apply this integral theorem to the case of diflfraction from an opening in a plane screen (placed in the plane x = 0) it is necessary to extend (apply) the considerations to the case where V is the half space x ^ 0 (or x ^ 0). In 1891 Rayleigh published his important diffraction formulas of the first and second kind, expressing the scalar field everywhere in space in terms of the field and in terms of its first-order derivative normal to the plane x = 0, respectively (Rayleigh [1891]). In this connection it is interesting to note that Lorenz established a formula closely related to the Rayleigh diffraction formula

'3]

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of the second kind in his 1860 papers on the direction of the ether vibrations in polarized light (Lorenz [1860a,b]). Rayleigh's derivation considers strictly monochromatic fields (space-frequency domain study) whereas Lorenz deals with the problem for non-monochromatic fields (space-time domain study). Here, a brief account on the Lorenz approach is given. Let us assume that a well-behaved (see below) function q)(y,z,t), which depends on time, is given in each point of the plane of the screen. If we choose a point ro = (0,yo,zo) located in the opening of the screen, the function cp(yo,zo, t-\r- ro\/uyr will satisfy the wave equation (3.8), but it will also hold for the function (piyo,zo,t-\r-ro\/u)
1

= w(y, z, t) lim OX

r r X dyo dzp

(3.10)

2jr J Jop \r-ro\^

A—^0

A substitution (yo,zo) =» (y^ -\- y,ZQ -\- z) followed by integration in polar coordinates in the (yQ,ZQ)-plane lead to the result lim

1 2JT

f f xdyo dzp ^ r JJop roP /op k -\r-ro\

1 for for I

(3.11)

and therefore finally 1

d0(x,yo,zo,tdx

•ro\/u) = o±

dyo dzo. (3.12) I^'-^OI The scalar field 0(r, t) in the positive (jc ^ 0, upper sign) and negative (x ^ 0, lower sign) halfspaces thus can be obtained at a given time t if the normal derivative, d0{x,yo,zo,t - \r-ro\/uydx, of 0 is known in the opening at the relevant retarded times, t - \r-ro\/u. In the frequency (o)) domain eq. (3.12) takes the form exp(i^ \r-ro\) d(P(x,yo,zo',co) dyo dzo. 0(r; a)) = T 2JT 11,^ dx = o± r-ro (3.13) yielding essentially the Rayleigh diffraction formulae of the second kind (Born and Wolf [1999]). The integration in eq. (3.12) can be extended over the entire 0(r,O = =F

2JT J J/op ov

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X = 0-plane if needed, but Lorenz refrained from doing this. The only remark he gives on this point is that he assumes "the screen to be immobile". In accordance with the experience one already had at that time with elastic (sound) waves generated from vibrating media it is likely that he felt (in fact correctly) that no light waves could be generated by an immobile (non-vibrating?) screen, and in consequence took q)(y,z,t) = 0 for points outside the opening. In a later publication in Crelles Journal (Lorenz [1861a]) entitled "Memoires sur la theorie d'elasticite des corps homogenes a elasticite constante" Lorenz writes (on p. 335) about the above-mentioned solution: "Nous sommes done en etat de resoudre, par la methode indiquee, un probleme, que Ton n'a resolu jusqu'ici que d'une maniere inexacte et incomplete moyennant le principe de Huyghens." According to Pihl [1939], Lorenz probably was the first to give a mathematical proof of the principle of Huygens (in fact the Huygens-Fresnel principle). Lorenz' analysis of the diffraction problem goes further than described above, but we shall leave the first scientific publication of Lorenz at this point. 3.3. Understanding the diffraction of light: a continuous challenge It is characteristic of Lorenz that throughout his life he took up physical problems of central importance. This holds par excellence for his optical research as we shall see. Some of the problems central in his age still pose a challenge for physicists, and the description of the diffraction of light is a good example. Although the aim in Lorenz' first work from 1860 was to understand the ether vibrations in polarized light via the bending of light in its passage through an opening in a screen, he immediately realized that the diffraction problem had to be solved in a self-consistent manner with only a prescribed incident field. The self-consistency was expressed in a beautifiil manner much later in the Helmholtz-Kirchhoff scalar integral theorem (Kirchhoff [1882b, 1883]). The first satisfactory vectorial generalization seems to be due to Kottler [1923a,b]). Lorenz assumed in his paper (Lorenz [1860b]) that the screen was immobile (inactive). It is not clear what he meant by this, but for the transmitted light he added up only contributions from the hole in the screen. This in itself implies that the backside of the screen is passive in the sense that no (microscopic) currentdensity sources are active here. The front side of the screen cannot be inactive in the treatment of Lorenz since he takes into account the reflected field. The need for the reflected field arises in Lorenz' treatment from the necessity of fulfilling the conditions in eqs. (3.5) and (3.6) everywhere in the opening of

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the screen. In the early theories the diffracting obstacle (screen, disk, . . . ) was assumed to be perfectly "black", that is to say, all light falling on it is absorbed. This assumption leads to the KirchhofF boundary conditions: (i) the scalar field and its first-order derivative in the direction normal to the screen are both equal to the corresponding quantities for the incident field, and (ii) the same quantities are equal to zero on the backside of the screen. In the other limit, so to speak, the screen is assumed to be a perfect conductor, and therefore perfectly reflecting. The first rigorous solution to a diffraction problem where the obstacle has infinite conductivity was given by Sommerfeld [1896] who studied the two-dimensional case of plane-wave diffraction from an infinitely thin halfplane. Rigorous calculations of the diffraction of light from bodies (obstacles) that have a finite dielectric and/or conducting response in general can only be carried out in particularly simple situations, e.g., for a spherical body. It is interesting to notice that Lorenz in his last publication (see § 9) took up precisely a calculation of the diffraction of light fi-om a dielectric sphere, and indeed obtained a rigorous solution. To get an impression of the proportion of the diffraction problem challenge one may refer the reader to works of Bethe [1944] and Bouwkamp [1950, 1954]. These researchers investigated, on the basis of the Maxwell equations, the diffraction of light from a hole with an extension much smaller than the wavelength of light. In passing it is worth mentioning that these fifty-year-old studies, though carried out under the assumption that the screen is a perfect conductor, have recently become of substantial interest in near-field optics. From a microscopic point of view the diffraction of light from an obstacle is a scattering problem, where the scatterers are the charged atomic particles. In the spirit of the Huygens-Fresnel principle, the Maxwell equations can be cast into the form (Keller [1996]) E{r\w) = E^\r\w)-'\iiQO) [

Go{r-/;w)J{r';a))d\\

(3.14)

in the space-frequency domain, JLIQ being the vacuum permeability. The electric field, E (r; co), which prevails at a given point in space, thus deviates from the incident (inc) electric field, E^^^{r;a)), by a term which represents the radiation from the current-density distribution, J(r; co), induced in the obstacle (scattering medium), which is assumed to occupy the volume V. The dyadic electromagnetic propagator, Go(r - r'; w), which contains not only propagating parts but also important self-field parts, has a far-field dependence of the form ~exp(i^ \r - r'|)/ \r - r'\, i.e., precisely the form of the scalar propagator entering the Huygens-Fresnel and Kirchhoff formalisms. If the point of

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observation (r) is inside the scattering volume a spherical contraction (indicated by e ^ 0) around r is needed in eq. (3.14) to ensure the convergence of the integral with outgoing spherical waves. In the random-phase-approximation the induced current density is related to the local (prevailing) field by a constitutive equation of the form (see e.g., Keller [1996] and references herein) J{r,o))= I

G(r,r';a))E(r';co)d\\

(3.15)

provided the medium responds in a linear fashion to the electric field. The quantity a (r, r'; w) is the (linear) microscopic one-particle conductivity tensor of the obstacle. This tensor has to be calculated from the one-electron Schrodinger equation, or the Pauli equation if spin effects are of importance. By inserting eq. (3.15) into eq. (3.14) we are led to an integral equation for the localfield.At this stage it is important to emphasize that besides the (microscopic) Maxwell equations a dynamic equation (Schrodinger or Pauli equation) is needed to solve the optical diffraction problem. Evidently, the diffraction problem therefore is a major challenge for theoretical physics even today. When it comes to the diffraction ft-om small holes in a metallic screen, and the diffraction characteristics in the near-field zone of matter the often made assumptions that the screen (i) is a perfect conductor, and (ii) is infinitely thin, do not hold. In such cases, a good starting point for obtaining a qualitative understanding may be to assume that the screen is a non-simply connected metallic quantum well. When the quantum well has a thickness many times less than the optical wavelengths of interest it is possible, for geometrically simple aperture forms, to determine the relevant spectrum of current-density eigenmodes, and subsequently establish a microscopic expression (Keller [1997a] and references herein) ^(r;c(;) = ^-^(r;a;) + ^ F , ( r ; a ; ) ^ ,

(3.16)

h

for the local field inside and outside the metallic screen. In eq. (3.16) the summation runs over all relevant electronic transitions in the screen, and Fh(r; 0)) is a known vector field. The constants ^/, are determined by a set of inhomogeneous linear equations in which the incident field appears in the inhomogeneous terms. If the number of relevant quantum transitions is N, the coefficient matrix determining the /3/,'s has a dimension NxN. Once the local field inside the scattering medium (obstacle) has been determined the induced current density may be found from eq. (3.15). In the

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213

cases where the obstacle is an ultrathin object (screen, disk, . . . ) , and where ultrathin means that the thickness is many times less than the optical wavelength roughly speaking, the scattering medium acts as a current sheet, usually of the electric-dipole type. By treating the light-diffraction problem on the basis of eqs. (3.14) and (3.15) there is no need for supplementing the problem with (basically) unknown boundary conditions. The self-consistent boundary conditions are in fact inherent in the integral equation. If the Schrodinger (or Pauli) equation is solved for a model potential, model boundary conditions for the electrons are of course impressed indirectly. To select a good model potential it is often necessary to address a complicated many-body problem for the electron (particle) dynamics.

§ 4. Surface optics: the first theory 4.1. Light as transverse elastic vibrations: an indefensible theory When Lorenz as a young man began his optical research it was generally believed that the physical foundation for the wave picture of light should be based on the elastic ether theory. It was supposed in this theory that an elastic medium (the so-called ether) existed in all space, and that the vibrations of light were supported by local oscillations in the ether. Within such an intuitive framework it was natural to seek to understand the various properties of light along the same lines as those used to describe the propagation of mechanical waves in an elastic medium (solid). In an elastic medium, both longitudinally and transversely polarized mechanical waves can exist. Since it was well-established, however, that the vibrations of light waves were transversely polarized, it was necessary in a mechanical light theory to eliminate the longitudinal ether vibrations. Such an elimination turned out to be impossible without running into self-contradictions. For Lorenz, and others as well, one problem in particular took a central position, namely the problem of accounting in a satisfactory manner for the reflection and transmission of light at an interface between two transparent media. Therefore the mechanical theory of light should fulfil the boundary conditions known for elastic waves, i.e., the wave-field displacements and the normal projections of the stress tensor should match at the interface. This results in six equations, but for transverse waves only four unknown quantities exist, namely two amplitude components for, respectively, the reflected and transmitted light waves. An exact elimination of the longitudinal waves requires that the ether is incompressible, a feature that even in Lorenz' time was difficult for physicists to

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P? § 4

reconcile themselves to. It was therefore deemed important to examine whether the available experimental optical data did contain a fingerprint of a longitudinal light mode after all. The experimental results of Jamin in this respect draw attention. Jamin investigated the intensity of reflected and transmitted light beams at an interface between isotropic and transparent media and found that the data did not agree with the expectations of the Fresnel formulae for angles of incidence near the polarization (Brewster) angle. For linearly polarized incident light he observed that the reflected light in general was elliptically polarized (when the incident light was polarized parallel or perpendicular to the plane of incidence). In the light theory of Cauchy [1836] (see also Bouasse [1893a,b]), longitudinal waves were present, and in order to account for the data of Jamin, Cauchy introduced the longitudinal waves in the reflection/transmission problem, and assumed (using a complex refractive index) that these waves decayed very fast with the distance from the interface. Using this construction he was able to show that the reflected transverse wave did contain an elliptically polarized component, and since the calculations turned out to be in quite good quantitative agreement with the result of Jamin, the elastic light theory of Cauchy gained some support.

4.2. Reflection of light fi'om a transition layer: survival of the Fresnel formulae The theory of Cauchy implied that the Fresnel equations giving the amplitude reflection and transmission coefficients had to be modified. In a work with the Danish title "Om Lysets Tilbagekastning fi-a plane, gjennemsigtige, isotrope Legemer" (On the reflection of light fi'om plane, transparent, isotropic media) published in a Danish mathematical journal Lorenz [1860c] showed that it was possible to account for the results of Jamin without abandoning the Fresnel formulae. The calculations of the reflection and transmission of light at an interface had hitherto been carried out under the assumption that the interface is sharp, but to Lorenz such an assumption was an abstraction since he writes "sharp transitions do not exist in nature neither in space nor in time" [in Danish from Math. Tidsskr. 2, 116 (1860), p. 117: En pludselig Overgang, eller i det Hele alt Punktuelt i Tid og Rum, er en Abstraktion, der ikke existerer i Naturen, ... ]. The above-mentioned work of Lorenz, which was published essentially in the same form in the Annalen der Physik (Lorenz [1860d]) and in the Philosophical Magazine (Lorenz [1860e]) and therefore well-known to the international community, is important because it showed that there was no need

3, § 4]

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215

for introducing longitudinal light waves belonging to an incompressible ether. This does not imply that Lorenz himself did not believe in the ether hypothesis in 1860. For Lorenz, the work also opened the doorway for constructing a wave equation covering light propagation in heterogeneous media. Seen from where we stand today it might be correct to claim that Jamin and Lorenz are the fathers of surface (interface) optics, a field that has become of outmost importance in recent times. Let us assume that a light beam hits an interface between two isotropic and transparent media at an angle of incidence x. If we denote the angle of refraction by y, the ratios between the amplitudes of the incoming, transmitted, and reflected beams are given by

1 : ,^ : ,^ = 1 : i£2!liH^ : !!!^(iZZ), sm(x+7)

(4.1)

sin(jc+7)

if the electric fields are polarized perpendicular to the plane of incidence (s-polarization), and by 2 cos X sin y tan(x - y) 1 • ^P : ^P = 1 • _.„. . . . _ , / : : - . . . :, sin(x + y) cos(x - y) tan(x + y)'

(4.2)

when the beams are polarized parallel to the plane of incidence (p-polarization). As described in § 3.1, the dualism between the electric and magnetic fields of a light wave was unknown in Lorenz' time, and in fact Lorenz characterized the light as being polarized in the plane of incidence in eq. (4.1), and perpendicular to it in eq. (4.2). In a sense the "light vector" for Lorenz is what we call the magnetic field vector. In writing down the Fresnel formulae the sign convention of Lorenz has been used. Lorenz now assumes that the Fresnel formulae hold if the optical properties (refractive indices) of the two media deviate only infinitesimally fi-om each other. Thus, if 7 = x + dx, the ratios in eqs. (4.1) and (4.2) become djc

dx

1:1 + - ^ : — ^ , (4.3) sm Ix sm Ix dx dx 1:1 + — ^ : ^, (4.4) sm Ix tan Ix respectively. Lorenz next divided the transition layer between the two media into an infinite number of infinitesimally thin sheets, and at the boundary between consecutive sheets he used eqs. (4.3) and (4.4). We denote the angles at which the light beam enters the first sheet and leaves the last one by a and /?. The

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[3, § 4

quantities a and ji are therefore the angles of incidence and refraction in a sharp-boundary model. It seems that Lorenz was the first to take into account the transition layer in a theoretical calculation, though this refinement of the transmission-reflection problem is often ascribed to Drude. A model in which the optical properties of the transition layer are described in terms of a smooth (continuous) variation in the refractive index between the values of the indices in the two adjacent media has its limitations, cf the discussion in § 4.3, but within the framework of the experimental possibilities at Lorenz' time such a model was most likely sufficient (appeared correct). If we denote the (real) amplitude of the transmitted field at a depth where the local angle of incidence is x by A{x), it appears from eqs. (4.3) and (4.4) that M{X)

d^

Wl 1

\

,A r^

— ^ - —^ = d Q In tanx , (4.5) ax smzjc ^ ^ for both the p- and s-polarized case. In establishing eq. (4.5) it has been assumed that the transition layer (sheet) is so thin in comparison to the optical wavelength under consideration that the phase change across the sheet can be neglected. The solution of eq. (4.5) is given by A{x)=A{a)\r^, V tana

(4.6)

where A{a) is the amplitude of the light wave where it enters the first sheet. Lorenz now determined the resulting transmitted field by multiple reflection in the set of infinitesimally thin sheets. If one defines a function u{x) = | In tanx for the s-polarized case and u{x) = ^ In sin 2x for p-polarization, it follows from eqs. (4.3) and (4.4) that the local reflection coefficients in the two cases are given by dw(jc). A light wave transmitted to a depth in the transition layer given by the local angle of incidence x\ and then reflected thus has the amplitude A{x\)du\, where dwi = du{x\). If the reflected wave afterwards propagates back to the sheet located where the angle of incidence is X2, its amplitude has become ^(xi)dwi(tanx2/tanxi)^^^. Upon reflection in this sheet the amplitude now is ^(jci)dwi(tanx2/tanxi)^^^ (-dw2). The minus sign in front of dw2 originates in the fact that the second and first reflections occur in opposite directions. If the twice reflected light wave now proceeds across the sheet without any fiarther reflection its amplitude at entering the second medium is .4(jci)dwi(tanx2/tanxi)^^^ (-dw2)(tan/3/tanx2)^^^ = -^(a)(tan^/tan a)'^^ dwi dw2. To obtain the overall contribution to the transmitted field from waves twice reflected in the sheet one must make a summation

3, § 4]

Surface optics: the first theory

217

over all possible positions of the two sheets giving rise to reflections. By denoting this contribution by AiiP) we have ^2()S) = - ^ ( « ) \ / F ^ / " / 'dM2dM|.

(4.7)

The final expression for the transmitted ampHtude, A-x{ji), is the sum of the contributions reflected 0, 2, 4, 6 . . . , times, i.e.,

^T()S) = W | ^ X V tana piij]

-

rii\

/ J Ua

rUji

du2du\-\-

'J Ua

nil]

/ J 11(1 ^ ll((

piif]

rU}

/

/

^ H^

^ ltd

clw4dw3 dw2dwi + • (4.8)

At this point Lorenz defines afiinction/ ( « ) by the integral equation fiu)=l-

f" f'f{U2)dU2duu

(4.9)

and with the help of/(w) the infinite series in eq. (4.8) can be written in the compact form Aj{^) = A(a)yj^^f{ua\

(4.10)

By double differentiation it readily follows from eq. (4.9) that/(w) must satisfy the differential equation

and since/(w/j) = 1 and d/(w)/dw|^^^^^ = 0, as one realizes from eq. (4.9), the two arbitrary constants appearing when solving eq. (4.11) are easily determined. Altogether one obtains . . . ^ exp(t/ - ug) + exp(i/,, - u) exp(w/,> - Ua) + exp(w,, - w/0 * By combining eqs. (4.10) and (4.12), the amplitude transmission coefficient (0 according to Lorenz hence becomes ^_Aj{P) _ ^V tana A{a) Qxp(ufj - Ua) + exp(Wf^ - w/0

^^^^^

For s-polarized light u(x) = - ^ I n tanx, and with this inserted in Ua = u{a) and Uft = w(/?), eq. (4.13) gives t = ts, the s-polarized amplitude transmission

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Optical works of L V Lorenz

[3, § 4

coefficient. Likewise, the use of u{x) = ^ In sin2jc in eq. (4.13) leads to / = ^p after a number of elementary trigonometric manipulations. Starting from the differential forms given in eqs. (4.3) and (4.4), Lorenz thus showed that one comes back to the Fresnel formulae for transmission (eqs. 4.1 and 4.2) provided the thickness of the sheet is negligible in comparison to the wavelength of light. The Fresnel formulae for the transmission hence indirectly include the principle of multiple scattering. Lorenz also remarks that it is a peculiar property of the Fresnel formulae that they can be determined alone from the differential forms in eqs. (4.3) and (4.4) since these could be derived from many other forms than the Fresnel formulae themselves. Following the procedure above, Lorenz establishes the expression du\ -

A^{a) = A{a) J Ua

/ J Ua

/

du^du2du\

(4.14)

J 11(1 • ^ " 2

for the reflected amplitude, ^R(a), at the entrance to the sheet. With a function/(w) defined implicitly by f{u)=\-

f

f'f{u2)du2duu

(4.15)

one obtains A^{a) = A{a)) / ^owf(u)

f{u)du. f{u)

(4.16)

is given by

^(^) ^ exp(M-^/0 + exp(t//,>-t/) QXp(Ua - Uj]) + exp(W|,'

^^^^^

-UaY

and Lorenz hence finds that the amplitude reflection coefficient (r) is _ A^{a) _ exp(w/i - w«) - exp(w,, - w/0 A{a) exp(w« - Uf^) + exp(w/,> - w«)

(4.18)

By inserting the proper values for u^ and Uj^ Lorenz came back to the Fresnel formulae for the s- and p-polarized amplitude reflection coefficient (see eqs. 4.1 and 4.2). Although it is gratifying that the multiple-reflection procedure has led one back to the Fresnel formulae, the proper goal for Lorenz was to obtain the corrections to the Fresnel reflection formulae in the case where the thickness

Surface optics: the first theory

3, § 4 ]

219

of the transition layer cannot be neglected in comparison to the wavelength of the light beam, and afterwards to compare the results thus achieved with the experiments of Jamin. Phase differences must now be included in the calculation. We denote by 6/ the phase retardation of a monochromatic wavelet reflected from the layer characterized by the angle of incidence x/ in comparison to the wavelet reflected fi-om the (infinitesimally thin) sheet located at the boundary to the transition layer; then, at the sheet where the local angle of incidence is jci the (scalar) value of the reflected light field is A(a)cos(a)t - d\)dui, where t is time and a; is a constant, the angular ft-equency [the phase of the field reflected by the first layer was assumed to be wt at time t]. The phase retardation is 8\-62+ 63 for the beam reflected three times and the associated light field is -A(a) cos((ot -8\ -\-d2- <53)dwi dw2 dw3. The light field obtained after addition of the fields reflected 1, 3, 5 - - times (with reflection planes located in all possible (relevant) positions) thus is given by

r m AK(a) = A(a)

cos((jot- d\)du\ pUii

nil]

pUj^

cos,{(jL>t - d\ + 62- 82) dw3 dw2 dwi + • • JIIK

JIIa

J III

(4.19) This series can also be written in the form A^ia) = Re L ( a ) / ' Q'^'''-'^Y(u)du\ ,

(4.20)

where Re{- • •} stands for the real part of { • •}, i is the imaginary unit, and/(w) is defined (implicitly) by the integral equation pi

nun :\{c\-i\)

f(u)=l

j{U2)dU2dU\

(4.21)

J Ua ^ ll\

The result for ^R(a) in eq. (4.20) can be simplified with the help of the relation

^d - . ^ • , . , . , , „ , ,

(4.22)

which one can easily derive from the integral equation in (4.21). Thus, ^R(a) = Re<^-^(a)e'^

d/(w) dw

(4.23)

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Optical works of L. V. Lorenz

[3, § 4

since e"^"^' = 1 and d/(w)/dw|^^^^^ = 0 [this last relation readily follows upon taking d/(w)/dw from eq. (4.21)]. It is natural here to make the ansatz / ( . ) = ,iA e x p ( . - . . ) + e x p ( . , - . ) ^ exp(w/j - Ua) + exp(Wf, - w/0

^^^^^

remembering that with e'"^ = 1 one regains the already known result for/(w) in the limit where the light wavelength is infinitely small in comparison to the transition layer thickness, cf. eq. (4.12). By insertion of eq. (4.24) into (4.22) the following differential equation occurs for A: d^A ^^ d(2A-5) .dAd(A-(3) ^ , , ^^, --^+tanh(w-w^0^1 ^'-A— A— ^^(^-^^^ dw^ dw dw au This equation for A = A(w) is to be solved using the conditions A == 0 for w = w^ and dA/dw = 0 for w = u^ (this follows from d/(w)/dw = 0 for w = Uft). At this stage Lorenz assumed that the quantities dA/dw and dd/du are so small that the last term in eq. (4.25) can be neglected. The differential equation d^A ,, d(2A-5) ^ —r + tanh(w - un)-^ =0 dw^ au

,, , , , (4.26)

after integration from u to w/,' gives ^ = -o ITr ^ / " ' sinh[2((/' - u,,)]^ du'. au 2 cos ¥(u-uij) JII au'

(4.27)

By combining eqs. (4.23), (4.24) and (4.27) one gets ^ R ( « ) = -^(«) tanh(wa - w/0(cos ojt + tan A sin (X>0,

(4.28)

where tan

J^,^^ sinh[2(w« - w/0] dw

Remembering that u(x) = - ^ In tanx and u(x) = j\n sin 2x for s- and p-polarized light, respectively, we obtain the following amplitude reflection coefficients for the two polarizations: sm(a-B)^ . . . '^s = ^—. 7^7 (cos cot + tanzls sm cot), sm(a + p)

.. ^^x (4.30)

3, § 4]

Surface optics: the first theory

221

with sinacosa f'\ in -in xd(5 , tanZ\s = —7 J— / (cos ptanjc-sin pcotx)--dx, sin a - sin ji Ja OJC

,^^,^ (4.31)

and ^\y ""

tan(a-iS), . ; ;;T(COS (Dt H- tan Z\p sin cot), tan(a + p) ^

,, ^^, (4.32)

with sin 2a sin 2^ T^ / sin 2x sin 2/? \ d5 tan^p = —II I ; : p ^ _ -^^^ I — ^ ""^ sin^ 2a - sin^ 2^3 7^ V sin 2^ sin 2x / d^

1^4 33>^

The result given in eqs. (4.30)-(4.33) is the one obtained by Lorenz in 1860 (Lorenz [1860c-e]). With the assumption already made, that the phase retardation changes slowly with the change in the local angle of incidence Lorenz notes that A^ always is a small quantity. For p-polarized light the phaseshift zip is not small for angles of incidence around the Brewster angle. At the Brewster angle one now has a + ^ = n/2 and A^ = n/l. The relative phaseshift A^ - A^ thus is particularly large near the polarization angle, and this agrees with the experimental findings of Jamin [1850] who observed an elliptical component in the reflected light near the Brewster angle. Lorenz goes on and shows that all the results of Jamin can be understood starting from eqs. (4.30)-(4.33). The various relations found by Jamin between the relative phaseshift and the refractive index for different combinations of adjacent media could not be explained satisfactoraly by the elastic theory of Cauchy, but Lorenz showed that his formulae are able to account for these aspects. If we denote the cycleaveraged ratio between the s- and p-polarized energy reflection coefficients by k^ it appears fi-om eqs. (4.30) and (4.32) that _ cos(a + /3)cosZ\p c o s ( a - ^ ) cosZ\s'

(4.34)

a result which also appears in the theory of Cauchy. By using the experimental data of Jamin for k^, it was possible for Lorenz to estimate that the thicknesses of the various transition layers varied between approximately 1/10 and 1/100 of a wavelength, quite a good estimate in fact. Later on, Lorenz claims that his theory could provide an even better estimate of transition layer thicknesses if the uncertainty in the experimental data was smaller!

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[3, § 4

The brilliant work of Lorenz on the reflection and transmission properties of the transition layer has never received the merit it deserves. Immediately after the publication of Lorenz' work Christofifel published a critical account (see Pihl [1939]), in which he questioned the assumption that the nonlinear terms (dA/dw)^ and (dA/dM)(d(5/dw) appearing in eq. (4.25) are negligible; according to Pihl [1939] this, and the fact that Lorenz did not give an explicit formula for the phase difference A^ - A^ [from tan A^ and tan A^ it is easy to derive an expression for tan(Z\p - Z\s)] were contributory causes that Lorenz' work did not receive much attention in his own time. It is hard to believe that the last argument should have prevented the work from attracting attention. Whether or not the nonlinear terms in the differential equation can be neglected is a complicated question to answer in general, cf e.g., the discussion in Pihl [1939], and the references herein on this point. At the turn of the century (in 1900) Drude published his "Lehrbuch der Optik" (later translated into English under the title "The theory of optics" and published in several editions). Starting from the Maxwell equations Drude derived the elliptical polarization of the light reflected in the presence of a surface or transition layer; see, e.g., Drude [1959]. Drude was apparently not aware of the work of Lorenz from 1860, when he published his book in 1900. Thus, in the English Dover translation he writes on p. 287: "But strictly speaking there is no discontinuity in Nature. Between two media 1 and 2 there must always exist a transition layer within which the dielectric constant varies continuously from e\ to Ci. This transition layer is indeed very thin, but whether its thickness may be neglected, as has hitherto been done, when so short electromagnetic waves as the light-waves are under consideration, is very doubtful." In our time, where surface and interface optics have taken such a prominent position, it appears that most researchers refer to the book of Drude when it comes to the historical roots of the subject. Drude's result is essentially identical to that of Lorenz since the two can be brought in the same form. Bouasse [1893a] briefly describes the theories of Lorenz and Drude on pp. 225-235, and mentions that Drude in his original paper (Drude [1889]) used the Neumann reflection/transmission coefficients to study the role of the transition layer.

4.3. Transverse and longitudinal electromagnetic fields in the perspective of our time Some problems in physics have taken a prominent position for generations of scientists and appear ever challenging. Although the interest in understanding the

3, § 4]

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223

electromagnetic aspects of light-matter interaction at interfaces (surfaces) has waxed and wained over the years it has never disappeared. As we have seen, an understanding of the optical reflection and transmission properties at an interface between transparent media played a crucial role in the mid nineteenth century in finding out whether the theory of light propagation in matter could be formulated along the same lines as had been so successftil in accounting for the propagation, reflection, and transmission of elastic (mechanical) waves. After Fresnel together with Arago (in 1816) investigated the interference of polarized rays of light and observed that two rays polarized at right angles to each other never interfere, it became difficult to uphold the assumption of longitudinal light waves, and it appears that Young (in 1817) was the first to assume that the light vibrations were transverse. A comprehensive historical account of the elastic theory of light may be found in Whittaker [1952]. Since Lorenz, in the years where he developed his "light theory", had a firm believe that the theory of light should be based solely on a phenomenological description it was natural for him to try to establish a theory for the reflection and transmission of light at interfaces in which only transverse modes participated. Longitudinal light waves had never been observed directly, and the indirect fingerprint that Cauchy and others believed to be present in the experiments of Jamin turned out to be false. The refractive-index concept was used in Lorenz' time, and it was known that the refractive index (n) may depend on the (angular) frequency of light, i.e. n = n(co). The use of a refractive index in the description of the optical properties of matter for us means that we are within the framework of macroscopic electromagnetics. It follows from the macroscopic Maxwell equations that the electric field in the wave-vector-fi-equency (q-(o) domain, E(q, co), obeys the wave equation — j e(co) -q^lu^qqy

E{q, w) = 0,

(4.35)

when the medium under consideration is nonmagnetic, homogeneous, isotropic, and has a linear response function. In eq. (4.35), e(co) = n^(a)) is the relative dielectric constant, CQ is the vacuum velocity of light, and U is the unit tensor of dimension 3x3. In order for the homogeneous system of equations to have a nonvanishing solution the accompanying determinant must be zero. For homogeneous waves, i.e. waves where the real and imaginary (if any) parts of the wave vector are collinear, this gives |2

-

e(co)-q'

6(0)) = 0.

(4.36)

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Optical works of L. V. Lorenz

[3, § 4

Provided e{w) is differentfrom zero, only a doubly degenerate dispersion relation q = -n{(D)

(4.37)

follows, and it is easy to show that the two associated linearly independent eigenmodes are transversely polarized. The expectation of Young, Lorenz and others thus appears as a consequence of the macroscopic Maxwell equations. The above-mentioned analysis can be extended to anisotropic media, and again it is found that only two independent eigenmodes for light exist. Although these modes in general are only quasi-transverse they are divergence-free, and this is the genuine criterion for light waves. It is interesting here to note that for e{(D) = 0 a ^-independent longitudinal electric field fulfills the wave equation in eq. (4.35). In a metal, e.g., such a longitudinal solution in fact is important at (in the vicinity of) the plasma edge, as suspected already on the basis of the Drude free-electron model in which e{cD) = 1 - (JJ^/[(D{W + i/r)], r being the momentum relaxation time of the electrons. The longitudinal mode manifests itself in an important manner in those media where so-called spatial dispersion (spatial nonlocality) plays a role. Spatial dispersion means that the dielectric constant (tensor) depends not only on the frequency but also on the wave vector. The longitudinal mode (or strictly speaking rotational-free mode, if anisotropy is included) obeys the implicit dispersion relation 6L(^,a;) = 0,

(4.38)

where a subscript L has been added to the dielectric fiinction to emphasize that we are dealing with the longitudinal (irrotational) part of the underlying dielectric tensor. Spatial dispersion is important in many fields of field-matter interaction, e.g., in exciton systems, in superconductors and in metals. Optical activity thus would not exist without spatial dispersion, because to "see" the chirality of a structure (molecule, . . . ) the finite wavelength of light (and thus a finite wave vector) must be accounted for in the dielectric ftinction. In dyadic notation the dielectric tensor for homogeneous media can be written as ^{q, W) = eL(^, 0))qq + ej(q, (x)) {u - qq^ ,

(4.39)

where CL and ej are its longitudinal and transverse parts, and ^ = q/q is a unit vector in the ^-direction.

3, § 4]

Surface optics: the first theory

225

It is instructive now to relate the above-mentioned considerations concerning the boundary reflection problem to the studies of Cauchy and Lorenz. Neglecting single-particle excitations one obtains for a sharp interface (Keller [1988]),

'M\M=^^-^,

(4.40)

rp(,||, CO) = '^Y'^'''^ "^ - '^t " ^"^^'""-^''^"'^ ^^ [^^^^'" "^" '], q\ej{Kj, (D) + KI + {qw/K^yK^^e-xiKj,co) [ei\q\\,o))-

(4.41)

\\

using the sign convention of Keller [1988] for the fields. We have written the expressions for the s- and p-polarized reflection coefficients in terms of relevant wave vectors (and wave-vector components) because this gives the coefficients compact and physically appealing forms. It appears from eq. (4.41) that the reflection problem at a boundary between a spatially nondispersive medium (the medium in which the incident and reflected fields propagate) and a medium in which spatial nonlocality plays a role in the p-polarized case involves the presence of also a longitudinal (rotational-free) mode at the interface as in the elastic theory of Cauchy (!), but now on the basis of the Maxwell equations. The s-polarized dynamics does not involve longitudinal modes, as expected, and rs(^||,ft;) is the reflection coefficient obtained on the basis of the Fresnel formulae. If we put e]^{q\\,(j)) = 0 in eq. (4.41) we regain the Fresnel formulae for p-polarized light. In eqs. (4.40) and (4.41), q\\ and q\ = {{(o/cof- - qV\^^'^ are the components of the incident wave vector parallel and perpendicular to the interface, respectively, setting for (notational) simplicity the refractive index of the spatially nondispersive medium equal to unity. The magnitudes of the wave vectors which solve the (two) transverse and the longitudinal dispersion relations in the spatially dispersive medium for a given frequency have been denoted by K-X and ATL, and the components of these wave vectors perpendicular to the interface are K\ and K\. Longitudinally polarized fields hence may play an important role at interfaces, and in recent years these fields have turned out to be indispensable ingredients, e.g., in nonlinear surface optics and for electromagnetic surface waves. The pendulum therefore now seems to swing in favour of the presence of longitudinal optical modes at interfaces, but not in favour of the elastic ether theory, and yet, what is the situation like from the point of view of photons? Photons are introduced in the framework of quantum electrodynamics (QED), and are introduced via a canonical quantization procedure of the transverse field dynamics. In the Coulomb gauge the transverse field (EJ) and particle

226

Optical works of L. V. Lorenz

[3, § 5

current density {Jj) operators obey the inhomogeneous wave equation (CohenTannoudji, Dupont-Roc and Grynberg [1987], Keller [1998]),

Seen from the photon perspective, light waves therefore are always transverse (divergence-free). Thus, the pendulum has swung back once more! What about the longitudinal modes in QED. The longitudinal field operator, E]^{r, t), obeys the differential equation

^=4A(nO, at

(4.43)

6o

where JiirJ) is the longitudinal part of the (particle) current-density operator. The longitudinal field dynamics does not involve retardation effects (coming in via CQ) and the longitudinal electric field can be eliminated in QED in favour of the particle-position variables, and thereby transferred to the particle Hamiltonian. One may express this in a somewhat esoteric form by saying that a many-body matter (particle) system does not "see" its own longitudinal field. In the Lorenz (not Lorentz, see § 7.4) gauge four kinds of photons are constructed, namely two transverse photons, a longitudinal photon and a scalar photon but the combined effect of the scalar and longitudinal photons always vanishes. The so-called physical photons therefore are transverse even in the manifestly covariant Lorenz gauge. I have used the word physical photons, though photons are constructions of the free electromagnetic field and as such they only exist in space and time in an abstract matter-free world, but still these physical photons are important in the description of the birth process of photons (see Keller [2000]).

§ 5. Lorenz begins to doubt the elastic light theory The fact that the experimental results of Jamin could be explained solely by means of the Fresnel formulae once it was realized that "sharp transitions do not exist in nature neither in space nor in time", had convinced Lorenz of the correctness of these formulae, and subsequently he therefore tried to establish a theoretical basis for them. His first attempt was based on the theory of elasticity, and though it was doomed to fail, it definitely sharpened Lorenz' understanding of the problem with the ether theory, and within a year he had fiilly realized that

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the theory of light had to be built on formal grounds, abandoning mechanical hypotheses. Lorenz' attempt to state the reason for the Fresnel formulae in the theory of elastic waves (Lorenz [1861b]) was published under the title "Bestimmung der Schwingungsrichtung des Lichtathers durch die Reflexion und Brechung des Lichtes". It is remarkable that no one in 1861 had tried to derive the Fresnel formulae directly from the basic theory of elasticity. Lorenz himself writes on pp. 238-239 in the above-mentioned paper: "Die Brechung der Intensitat des reflectirten und gebrochenen Lichtes ist so oft gemacht worden, dass es unnotig scheinen mochte, dieselbe wieder aufzunehmen, wenn man nicht, je nach den gemachten Voraussetzungen, zu so entgegengesetzten Resultaten gekommen ware, dass fur den Augenblick der Frage in vollstandige Verwirrung geraten ist. Urn den endlichen Schluss riicksichtlich der Schwingungsrichtung des Lichtathers ziehen zu konnen, habe ich daher alle zweifelhaften oder bezweifelten Voraussetzungen, so wie die der Gleichheit der Druckkrafte und Verschiebungen an beiden Seiten der Granzflache vermieden, und die Rechnung auf die allgemeinen Gesetze der Bewegung elastischer Korper zuruckgefahrt." Lorenz starts his analysis from the wave equation for elastic waves in isotropic media, viz. (in modem vector notation), P^-^^

= aVV • u{rj)-pv

x (V x u(r,t)\

(5.1)

where u{r, t) is the displacement vector, p is the mass density, and a and ^ the two independent elastic constants. The elastic constants are related to the familiar Lame coefficients, ji and A, via a = A + Ij^i and jS = ^. In 1861 Lorenz still believed in the ether hypothesis and this means that pi = fi and A = a - 2^ are the two elasticity coefficients of the ether, and p the ether (mass) density. He again feels that the transition layer is of importance for understanding the nature of light, and therefore he assumes that the density and elasticity of the ether change across this layer. With the boundary between the two media placed in a Cartesian coordinate system the quantities p . A, and pi are assumed to be fiinctions of the x-coordinate within the transition layer and to attain constant values outside. For s-polarized light Lorenz essentially uses the same multiple scattering technique within the transition layer as he did in his successful explanation of the reflection experiments of Jamin, and he finds that only if the ether elasticity coefficient ji is the same in the two media the Fresnel formulae for s-polarized light will hold. For the p-polarized case he is only able to carry out his analysis under the assumption that the optical properties of the two adjacent media differ infinitesimally from each other, and for this special case he

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again finds that the ether elasticity coefficient /? must be the same all over space. From the result obtained for media which properties differ only infinitesimally Lorenz writes, with references to his earlier paper (Lorenz [1860d]), that he is able to determine the amplitudes of the reflected and transmitted light beams also for media with a finite difference in optical properties provided /? is constant, and the longitudinal mode is neglected. If Lorenz assumes that the ether density, p, is the same all over he obtains the results of Neumann, cf. the discussion in § 3.1. The conclusion reached by Lorenz on the basis of the rigorous theory of elasticity had already been presumed by Fresnel and Neumann who, in order to derive their formula for the reflection and transmission of light at an interface between transparent media, had to assume that either the ether elasticity (Fresnel) or the ether density (Neumann) was space independent. In contrast to Lorenz, Fresnel and Neumann had based their conclusions on heuristic principles of quasi-mechanical origin, cf Pihl [1939]. If the displacement field u{r,t), is divided into the uniquely determined transverse (T) and longitudinal (L) vector field parts, i.e. u{rj) = uj{r,t) + ui^{r,t),

(5.2)

where V • WT = 0 and V x WL = 0, eq. (5.1) may be split into wave equations for, respectively, uj and the scalar potential (j) related to I/L via I/L = - V 0 , viz..

= /5W(i',0, = aVV(r,0.

(5.3) (5.4)

Lorenz' paper on the possibilities of deriving the Fresnel formulae from the theory of elasticity is dated "Copenhagen den 28. Juni 1861", and there is no doubt that Lorenz had realized by the summer of 1861 that the presence of a longitudinal wave was a serious drawback for the elastic theory of light. Lorenz' desperate efforts to derive the Fresnel formulae from the theory of elasticity had all failed because of this longitudinal wave. In the Annalen der Physik paper he writes (Lorenz [1861b], pp. 248-249) after having carried out all the mathematical analyses: "Diese longitudinalen, oder wie wir sie lieber nennen mogen, Oberflache-Wellen, bringen wieder transversale Schwingungen von der ursprunglichen Art hervor; beriicksichtigt man diese, oder nimmt mann sogleich einen endlichen Unterschied der beiden Mittel an, so stimmen die Resultate nicht mehr mit der Erfahrung iiberein. Man ist also genothigt anzunehmen, dass die Oberflache wellen aus irgend

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einem Grunde nicht im Stande sind wieder transversale Schwingungen von der urspriinglichen Art hervorzubringen. Bekanntlich hat man immer eine Absorptionscoefficienten naher zu bestimmen gesucht. Allein jede Rechnung mit einem solchen Coefificienten schliesst implicite in sich eine Theorie der unvollkommen elastischen Korper und wir sind noch weit davon, eine solche begriinden zu konnen. Deshalb kann man auch mit diesem Coefficientenye^e^ beliebige Resultat [Lorenz' italics] erlangen, es kommt nur darauf an, wir man denselben einfiihren will und nach welchen Gesetzen man die Druckkrafte berechnen will. Auf einem solchen Boden lasst sich kein sicherer Schritt weiter machen." Towards the end of the paper Lorenz writes: "Das gewonnene Resultat schliesst die Moglickeit, die Dichtigkeit des Aethers sey constant, vollig aus, und wir machen also den Schluss, dass der Elasticitatscoefficient fi des Aethers (iiber den eigentlichen ZusammendruckbarkeitsCoefficienten A wissen wir dagegen gar nichts) in alien durchsichtigen, unkrystallinischen Korper und im leeren Raume derselbe ist. Daraus folgt nun weiter, dass die Schwingungen des Lichtdthers senkrecht zur Polarisationsebene sind [Lorenz' italics]." By the end of the year 1861 Lorenz in his mind is ready for a change of scenery basing his (phenomenological) theory of light on abstract conceptions freed from mechanical features. If the longitudinal part of the displacement field is neglected, so that u = uj in eq. (5.1), this equation is reduced to the form

-Vx(Vx«(r,0)-A^^#^=0'

(5-5)

where v = (p/pY^^ is the (transverse) phase velocity. As we shall see in §6, the phenomenological light theory of Lorenz leads to an equation of the form given in eq. (5.5) for the so-called light vector ualid also in inhomogeneous media where the phase velocity (of light) is a function of position, u = u(r). In inhomogeneous media the light-vector field is no longer a purely transverse vector field.

§ 6. The phenomenological light theory of Lorenz 6.1, Establishment of a coherent theory of light: wave equation for the light vector in inhomogeneous media In 1862 it had become clear to Lorenz that it was impossible to establish a mechanical theory of light on the basis of the theory of elasticity. Starting

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[3, § 6

from the differential equations describing the dynamics of elastic vibrations Lorenz had realized in his Annalen der Physik paper (Lorenz [1861b]) that the possibility existed that one might be able to account for the reflection, transmission, and diffraction properties of light if one was in possession of an adequate set of differential equations for the components of the light vector. Lorenz' confidence in the correctness of the Fresnel formulae, strengthened by the fact that he had been able to account for Jamin's experimental results without violating these formulae, made it natural for him to begin his search for the differential equations governing the propagation of light in arbitrarily inhomogeneous, but transparent media from the Fresnel formulae, which had shown their validity in describing the reflection of lightfi*oman inhomogeneous transition layer between two homogeneous and transparent media. The correct differential equations were obtained by Lorenz in 1862. In a popular article (Lorenz [1862]) written in Danish and published under the title "Om Lysets Theori" (On the theory of light) he describes in words these differential equations and the route he had followed to establish them. The following year the work was presented to the international scientific community under the German title "Ueber die Theorie des Lichts" (Lorenz [1863a]), and in that same year it was translated into English (Lorenz [1863b]). The Annalen der Physik paper is rather long (-35 pages) and describes not only the establishment of the differential equations themselves but also their application mainly in studies of birefringence and optical activity. In his popular Danish article Lorenz [1862] makes it clear from the outset that his endeavour was to establish a phenomenological theory of light, i.e. a theory based only on the available experimental data, and a theory which in a unified manner would enable one to describe all known optical phenomena. He also emphasizes that his goal is to set up a description relieved of any hypothesis of "the nature of light". By 1862, Lorenz had witnessed that in the course of time many mutually conflicting hypotheses had crept into the theory of light, and his objective certainly was to remove as many as possible of these weakly founded hypotheses. In another elementary article written in Danish and published under the same title as his popular 1862 paper, viz. "Om Lysets Theori" (On the theory of light), Lorenz [1863c] stresses the necessity of building a theory of light which does not invoke notions of molecular dynamics and the forces between the molecules constituting solid media. This does not mean that Lorenz in any way was in opposition to the atomistic theory, on the contrary (see §§8 and 9). His reticence alone was associated to the fact that so little reliable experimental information was available on the microscopic internal structure of condensedmatter systems. In a somewhat polemic fashion Lorenz writes that all these

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conceptions of the internal structure of matter are to be considered as will-o'the-wisp's for (natural) science. Let us follow the road which led Lorenz to the wave equation for the light vector in an inhomogeneous medium. Lorenz begins his Annalen der Physik paper (Lorenz [1863a]) with a discussion of the knowledge which forms the basis for his theory. Thus, the light vector E = (§, rj, C) which characterizes the oscillations (Schwingungsausschlag) in the optical field satisfies the partial differential equation

where f is a space- and time-independent quantity (the phase velocity) in isotropic homogeneous media. In passing, we again stress that the vector concept (here E) does not appear in the original article. Throughout, all relations were expressed in the (Cartesian) coordinate forms. The plane-wave solutions to eq. (6.1) Lorenz writes in the form ^ (a + ib) exp[i(^t;^ - q • r)], where q is the magnitude of q. The unit vector q = q/q gives the propagation direction of the wave. To make sure that the light field at all times is transversely polarized in every space point the condition V'E(r,t) = 0

(6.2)

must be ftxlfilled in isotropic homogeneous media. The two conditions in eqs. (6.1) and (6.2) valid for Lorenz at least in isotropic homogeneous media, together with the Fresnel formulae (eqs. 4.1 and 4.2) describing the reflection and transmission of plane light waves at an interface between two isotropic and transparent media, constitute the starting point for Lorenz. Since the intensity of light in the medium where the transmitted beam propagates was unknown (experimentally) at Lorenz' time, he argues that the Fresnel transmission formulae may contain an extra factor (sinx/sin7)^^ The basic analysis is carried out keeping an arbitrary (real) /7-value, and p = 0 corresponds to the Fresnel transmission formulae. Only if/? = 0 is energy conserved in the transmission/reflection process. Lorenz first assumes that the medium considered is inhomogeneous only in one direction (here the jc-direction) and that the light enters the medium at the plane x = 0. In an isotropic and homogeneous medium, where plane waves of the form exp[i(a;^-£x-my-nz)] ftilfil the wave equation, the dispersion relation can be written as a; = v{f- + w^ + «^)'^^, where q = (£, m, n) is the wave vector of light. If the plane of incidence coincides with the xz-plane, so that m = 0, the field

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[3, § 6

oscillations at the boundary plane {x = 0) can be expressed as P = Ae^^^'^^~^^\ Lorenz now proceeds essentially in the same manner as in his Annalen der Physik paper from 1860 (Lorenz [1860d]); see §4.2. Thus, without taking into account multiple reflections the refracted wave is modified to p^e~''\ where sm a \ ^ / tan a \ s m a / V tanfl at a plane where the angle of incidence is a, assuming the angle of incidence to be (3f at X = 0. If p = 0, one regains eq. (4.6). With multiple reflections included the refracted wave becomes Aj{a) = pPt-''^U{u\

(6.4)

where U{u) is determined via nil

U(u)=\-

pill,

/

/

JIIa

'J ll\

Q^'^'^^-'^'-^Uiu2)du2duu

(6.5)

remembering that u = u(a). At the exit plane the angle of incidence is named b, and for brevity the notations u{a) = Ua and d{a) = da have been used in eq. (6.5). The reflected wave is given by AK(a)=pP

f ' Q'^''-^'''-^U(u2)du2 = - p P e ' ^ ' ^ ^ . dw

(6.6)

With the sign convention used by Lorenz the resulting p-polarized "Schwingungsausschlag" in the x- and z-directions therefore are §(«) = pPQ~''' (u(u) - l ^ ^ J sin a, e(a) = -pPQ-''^ (u(u) + ^ ^

J cos a,

(6.7) (6.8)

respectively. For s-polarized fields, the resulting "Schwingungsausschlag" is given by r](a) = pPe-^^ ( U{u) - - ^

\.

(6.9)

The local phase of the wave is cot - nz - d{x), and to determine the local phase velocity, v{a), we use the fact that the phase must be robust against

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233

infinitesimally small space-time displacements. Thus, codt - {Ad{x)/dx) cbc = 0, and codt - ndz = 0. Since the phase velocity satisfies u(a) = (djc/dOcosa = (dz/dt) sin a one gets 0) cos a = u(a)^^, dx (JO sin a = v{a)n.

(6.10) (6.11)

By using eqs. (6.10) and (6.11) to eliminate sin a and cos a fi*om eqs. (6.3), (6.7), (6.8) and (6.9), Lorenz obtains for the scaled light-vector components, (1,7/, C) = v^\cL){^, V, t), the expressions m

=^ ^ ^ ^ ' - ^ - \

(6.12)

dd{x)

V

CLY

ri(a)=^^e^""-"--\

(6.13)

1(a) = -Aiu{a)s'J^^e'^""-"=\ V cue

(6.14)

where . =e - ( t / ( « ) - ^ ) , U(u)+-^\.

(6.15) (6.16)

Thefactor^i appearing in eqs. (6.12)-{6.14) is given explicitly by ^i = co^'^Ax sin''a/[«''" '''^(tana)'^^]. It follows from eq. (6.5) that the flinction U(u) satisfies the differential equation d'Uju)

,,d<5(M)dt/(M)

—-^

2i—

— = U(u),

(6.17)

and from eqs. (6.15)-(6.17) one may then easily derive the relations d(e"s)_ , ^ „ , -ieV, dd dKe^V) = -ie"i, dd

(6.18) (6.19)

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[3, §6

where d^ = {Ad/Au)du. From these one next obtains ' -2„d(e"^)+ e~"s = 0, Ad J.U A{s-"s') + e"^' = 0. Ad Ad

(6.20)

'Ad

(6.21)

By means of the relation u = -5ln(sin2a) [see the text immediately below eq. (4.6) and note that Lorenz uses opposite signs on the Fresnel reflection coefficients in the Annalen der Physik papers of 1860 (Lorenz [1860d]) and 1863 (Lorenz [1863a])] one gets, by the use of eqs. (6.10) and (6.11), 'ln(2co-V(a)/'^^''^ Ax and therefore (6.22)

Ax The constant C can be omitted, and upon doing this it appears that 1

(6.23)

'^i<^)W so that §(a) = Axn{v(a)fse"e^""'"-\ Utilizing that A/Ax = {Ad{x)/Ax)A/Ad a straightforward calculation gives, with the help of eqs. (6.20) and (6.23), d_ v\a)- d 'dx dx

\vHa)

Ax

(6.24)

A corresponding relation, viz.. d_ dx

-2,„, / dd{x)X

dC

-v-\a)C

(6.25)

is obtained for C by means of eqs. (6.21) and (6.23). By introduction of the function

_2, , /Ad(x)Y

dt

(6.26)

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235

eq. (6.25) can be written dcp _ dx

1 C, u^(a)

(6.27)

and hence ^ /^ 2/ ^ ^ ^ ^

2/^^ fdd(x)

From eqs. (6.10) and (6.11) it appears that

dx y

u^(a)

so that eq. (6.28) can be rewritten in the form d (u\a)^j dx

+ {w^ - n^u\a)) (p = 0.

(6.30)

At this point Lorenz makes the important observation that eq. (6.24) can also be brought into the form (6.30) (or 6.28) provided one now defines the function q) as (p=^—l

(6.31)

The homogeneity of eq. (6.24) implies that the factor i/n in eq. (6.31) can be replaced by another constant, but the choice i/n as factor in eq. (6.31) ensures that the p-polarized field is transverse, d^/dx + d^/dz = 0, in a homogeneous medium, where u is independent of a. With the correspondences d/dt <-^ ico and d/dz
The correspondence d/dz <-> -in transforms eq. (6.31) to dw

1

-

In a far-reaching jump, Lorenz at this stage assumed that eq. (6.32) is the basic partial differential equation for the p-polarized light vibrations. Once q) = q)(x, z, t) is determined, the light vector can be obtained from

(|,o,Q.„V)(S,o,-g

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[3, § 6

cf. eqs. (6.27) and (6.33). Differentiations of eq. (6.32) with respect to z and X, respectively, followed by the use of eqs. (6.27) and (6.33) enabled Lorenz to establish the following set of coupled equations among ^ and C"

n

^^3^^

n _ 1n

dz^

dxdz

v\x) df '

d^l dx^

d^ _ 1 d^l dxdz v^{x) dt^ '

(6.35)

where we have written u{x) instead of v{a) to emphasize the space-coordinate dependence of the phase velocity. The partial differential equation governing the light vibrations in the s-polarized case is established starting from the relation . r.\- n w = 1^wln(tan a) = C - 1^ 1In^ / ^ ^ W djc

[see the text below eq. (4.6) and combine eqs. (6.10) and (6.11)]. The constant C again can be omitted, and doing this one obtains

^ ^

(6.36)

djc

for the s-polarization. If eq. (6.36) is used to eliminate yjdd{x)/dx eq. (6.13), one may realize with the help of eq. (6.20) that

.x^n^)^=«'

^'-''^

and hence, by utilizing eq. (6.29) and the correspondences d/dt ^ d/dz ^ -in,

dx^ dz^ v\x) df'

from

io) and

^ ^ ^

In the special case where the phase velocity of the light is a function of the jc-coordinate only, Lorenz claims that eqs. (6.34), (6.35) and (6.38) give a

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237

complete description of the propagation of light. A slight alteration of the forms of eqs. (6.34) and (6.35) to

dz\dx^ dz]

vHx)df'

^''

allows Lorenz to claim that the basic equations in the case where the components of the light vector also depend on the >'-coordinate must have the form

ox

v^{x,y,z) ot^

„ 2 - d& 1 d^n ^ n - ^ = —, z ^ , dy v^{x,y,z) ot^

V^J_f = ^ ^ § ,

(6.42)

(6.43,

where

ox

oy

oz

The symmetric form of eqs. (6.41)-(6.43) finally enabled Lorenz to conclude that these equations govern the light propagation in inhomogeneous and transparent media in the general case where the phase velocity depends on all three space coordinates, i.e. u = u(x,y,z), as already indicated above. 6.2. The Lorenz Saltus conditions for the light vector Until Lorenz established the set of differential equations governing the light propagation in inhomogeneous media it seems that the boundary conditions for the light vector at a sharp interface had been added to the description as a necessary extra element (principle). Starting from the differential equations (6.41-6.43) Lorenz showed in his Annalen der Physik paper (Lorenz [1863a]) that the boundary (jump) conditions for the light vector can be obtained from the general theory without extra assumptions. As mentioned by Pihl [1939], it had generally been assumed (until 1939) that H. Hertz was the first to realize that

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the boundary conditions for the electromagnetic field follow from the differential equations. Following Lorenz, let us assume that the interface coincides with the jz-plane, and that the transition layer has an infinitesimal thickness extending from x = 0 to jc = 6 -^ 0^. Since the quantities d^Jj/dz^, d^l/dydz and uf(x,y,z)d^ri/dt^ are always finite everywhere in space, one has /•e-O^ n2 Jo

e^O^

dz^

dydz

Jo

' JQ JQ

d^rj 1 djc u^{x,y,z) dfi

= 0,

(6.45) and a term by term integration of eq. (6.42) from 0 to e —^ 0^ consequently gives the boundary condition x = e-^0^

dx

(6.46)

dy Jjc = 0

An analogous integration of eq. (6.43), and use of the results >o^

Jo

dy^ dy^

Jo

dydz

dx

f

>o^

d^^ djc = v\x,y,z) dfi 1

Jo

0,

(6.47)

yields the condition jc = e - ^ 0 *

dx

(6.48)

dz

An additional integration of eqs. (6.41) and (6.43) from x = 0 to x finally led Lorenz to the boundary conditions

ra^=r'^

x = e^O"

(6.49)

-0.

.T = 0

>o^

The boundary condition dWdy + dt/dz x=0

0 is derived from eq. (6.41)

by a single integration, but as Lorenz notes this condition is inherent already in eq. (6.49). A second integration of eq. (6.41) just gives the identity 0 = 0. Altogether, Lorenz thus found that his fundamental differential equations governing the time-space development of the three components of the light vector result in four independent boundary conditions for the field, and not six as required by the theory of elasticity.

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239

To regain the Fresnel formulae one must put /? = 0, cf. the discussion given below eq. (6.2), and this implies that (^,T/, t ) = (<^, ^, C). In modem notation the light vector is just the electric field, i.e. (§, r], t) = (Ey,Ey,E-). If the change in a given field component across the interface is denoted by A, and one introduces £"11 = (0,Ey,Ez), the boundary conditions in eq. (6.49) can be written in the form A^ii = 0.

(6.50)

The results in eqs. (6.46) and (6.48) show that (V x E), and (V x E)y, respectively, are continuous across the interface, and by means of the Maxwell equation V x E = -dB/dt, it is realized that one may write A^ll=0,

(6.51)

where B\\ = (0,^v,^z). The field boundary conditions of Lorenz, expressed in the forms given in eqs. (6.50) and (6.51), can hence be recognized as the textbook boundary conditions, valid in cases where the interface carries no ac currents either parallel (eq. 6.51) or perpendicular (eq. 6.50) to the boundary, cf, e.g., Keller [1995]. 6.3. On the road to the (microscopic) Maxwell equations To obey the Fresnel formulae we have seen that one must take (§, ^, t) = (S, ^, C). and the identification of the Lorenz light vector with the electric field, i.e. (§, r/, C) == (Ex,Ey,E::) = E hence shows that the quantity G = 0 given in eq. (6.44), and appearing in eqs. (6.41)-(6.43), is the divergence of the electric field vector. By multiplying eqs. (6.41)-(6.43) with unit vectors Ci (i = x,y,z) along the Cartesian axes, addition of the resulting equations gives

E

ai

/ = A', V, Z

-"'C-) E

^'^'

(6.52)

smce ^(=e)

= \/'E.

(6.53)

In vectorial notation eq. (6.52) reads V^£(r, 0 - VV • E(r, 0 = ^

^^^^,

(6.54)

and this is precisely the wave equation which follows from the macroscopic Maxwell equations when the light propagation takes place in an inhomogeneous

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[3, § 6

and transparent dielectric medium. The electromagnetic light theory established by Lorenz in 1867 is formally identical to the microscopic Maxwell-Lorentz theory in the relativistically invariant form, and since the wave equation in (6.54) played an important role for Lorenz in his search for a unified description of all electromagnetic phenomena it is worthwhile here to take a brief look at the Maxwell equations as they essentially appeared originally, but written now in Sl-units. In the macroscopic formulation the electric {E) and magnetic {H) field vectors thus were found to satisfy the equations (see, e.g., Whittaker [1952]) dH V xE = -ii—-, (6.55) dt BE \/xH = J^e—, (6.56) where the permittivity (e) and permeability (^0 were real material parameters characterizing the dielectric and magnetic properties of the medium considered. To describe the electrodynamic properties of inhomogeneous substances one had to assume that these parameters could vary from place to place in space, i.e. e = e(r) and jj. = i.i(r). The current density of the free charges ( / ) was related to the prevailing electric field through Ohm's law / = oE, o being the electric conductivity, a quantity which one would also in general allow to depend on the spatial coordinates, i.e. o = o(r). The so-called displacement current density, edE/dt, which Maxwell added (in eq. 6.56) to the already existing electrodynamic description is crucial even for field propagation in vacuum where e(r) = CQ, the vacuum permittivity. In inhomogeneous media it was found that the equations V • (eE) = 0,

(6.57)

V • (jnH) = 0,

(6.58)

were also obeyed. Thus, the fields themselves (E and H) are not divergence-free in general. The original Maxwell equations (6.55-6.58) are not the most general ones even within the framework of macroscopic electrodynamics, a point we shall return to in relation to the discussion of the electromagnetic light theory of Lorenz (see § 7). In the absence of free charges a combination of eqs. (6.55) and (6.56) leads, for nonmagnetic (f.i = /.IQ) media, to the following wave equation for the electric field: V X (V X E{r, 0) + c,'er(r)^^^^

= 0,

(6.59)

where CQ = (eoA^)"'^^ is the speed of light in vacuo, and e,(r) is the relative dielectric constant. Since the local phase velocity of light is u(r) = co/n(r), where

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The phenomenological light theory of Lorenz

241

the refractive index (a concept also used by Lorenz in several of his papers) n{r) is related to the dielectric constant via n^{r) = e(r), it appears that the wave equation published by Lorenz in 1863 (eq. 6.54) is identical to that obtained from the Maxwell equations, which were published in 1865.

6.4. Double refraction and optical activity as consequences Lorenz' light theory

of

It appears from the writings of Lorenz that his dissatisfaction with the existing framework (theory) for understanding the observed phenomena of light was an essential motivation in his search for a new and unified picture. For instance, in his own words (though in German) this is stated clearly in the introduction to the Annalen der Physik paper from 1863: "Wenn wir alle Voraussetzungen unserer jetzigen Theorie des Lichts zusammenhalten, namentlich alle diejenigen, die zur Erklarung der doppelten Brechung, der Farbenzerstreuung und der circularen Polarisation fiir notwendig angesehen worden sind, so konnen wir uns eines dergestalt zusammengesetzten Apparates gegenuber, dessen Haltbarkeit mit der wachsenden Anzahl der Voraussetzungen stark abnehmen muB, kaum gegen jedem Zweifel verwahren, selbst wenn wir von der Wahrscheinlichkeit jeder einzelnen Voraussetzung iiberzeugt sind (Lorenz [1863a])." Next Lorenz describes in words his goal: "Ich habe es daher versucht, unter den moglichst wenigen Voraussetzungen, sowohl in Bezug auf die Natur des Lichtes, als auf die des Lichtmediums und der Korper, die Theorie des Lichtes zu entwickeln, und es wird sich als Resultat der gegenwartigen Untersuchung ergeben, daB ein wesentlicher Theil der gewohnlichen physischen Hypothesen zur Erklarung der Phanomene des Lichtes unnothig ist, indem sich die Theorie auf einem andem Wege, als dem in diesen theoretischen Untersuchungen bisher befolgten, und namentlich durch eine weitere Entwickelung der formellen Seite der Theorie, durchfiihren lafst [laBt] (Lorenz [1863a])." In the wake of the establishment of the (correct) differential equations ( 6 . 4 1 6.43) for the description of light propagation in heterogeneous, transparent media, Lorenz first applied his theory to the double refraction phenomenon. When the material properties vary from place to place in space the light velocity becomes position-dependent, and the simple plane-wave ansatz ~ exp[i(a>^-^ ' ^)] is no longer a solution to the wave equation. Lorenz attacked the problem of

242

Optical works of L.V. Lorenz

[3, § 6

double refraction in a beautiful manner by expanding the square of the reciprocal local light velocity in a Fourier series, viz.,

where I have used partly Lorenz' own notation. Since v'^{r) = e{r)/cl, Lorenz therefore essentially makes a Fourier series expansion of the relative dielectric function (constant)! Seen from a mathematical point of view this kind of expansion is not so exciting, but from a physical perspective it turns out to be extremely gratifying for Lorenz (and for us) because it allowed him to classify the above-mentioned optical phenomena in orders of the ratio between all the relevant material periodicities and the (vacuum) wavelength of light. The approach used by Lorenz is essentially identical to that used much later, particularly by von Laue, Oseen and Ewald, to describe the refraction and diffraction of electromagnetic waves in condensed-matter media on the molecular level (see, e.g., Ewald [1962], Pinsker [1978], Whittaker [1952] and Born and Wolf [1999]). It appears that Lorenz' approach was unknown to (forgotten by?) the scientific community when these studies were carried out. The very fact that the spatial periods appearing in Lorenz' analyses of the above-mentioned optical phenomena all are assumed to be much smaller than the wavelength of light, unavoidably guides one to the lattice-periodic perception of crystalline (solid) structures, a picture not known when Lorenz wrote his 1863 paper in the Annalen der Physik. The quantity p^, which in modem notation is written as the scalar product of a structural wave vector Kp and the position vector r, i.e. pp = Kp - r, plus a phase constant dp/Gp, Lorenz writes in the form (yet without vector notation) Pp = — (Kp-r + dp), Gp \

(6.61)

/

where Kp = (ap,bp,Cp) is a unit vector in the A^;-direction, and a^ = K~^ is the magnitude of the reciprocal wave vector of Fourier component p, the amplitude of which is denoted by Cp. The mean value of u~^(r), which is nonvanishing, is named UQ^ in eq. (6.60). The criterion that the structural periodicities be small in comparison to the optical wavelength means that ap/X
I = loC + ^ ^(±Pp) C(±pp) + 5 ] 5 ] '^^^Pp ^ A/) C(±Pp ±p,) + • • •, p

p

^/

(6.62)

6]

The phenomenological light theory of Lorenz

243

where ^Q, ^{±pp), ^(±Pp ± p^), ... are constants, and C = cos(ft;/ - q r), C(±Pp) = cos((jot - q • r ±Kp • r ± dp/ap), etc. As previously, the Ught wave vector is denoted by ^ = (^, w, n) in terms of its Cartesian components, and Kp = Op^Kp, Following the notation of Lorenz a double sign (±) means that a summation over both of the occurring quantities is understood. Combination of eqs. (6.60) and (6.62) now gives _0Q_

v{r)

? = §o + i^e^5(±Pp) C

E P

^{±Pp) +\ephQ + \Y^

e,^(±pp ± pq) C(±pp) +

L

(6.63) writing out in explicit form only the terms of zero (~ C) and first (~ C{±Pp)) order. If expansions similar to the one given in eq. (6.62) are made for the y- and z-components of the light vector, viz. rj = IJQC -^ • • and ^ = So^ + ' *' ^^^ obtains from eq. (6.41) in lowest order

(6.64) and in the next order

^(p^)+\e,^,= (^y{{£l + ml + nf,)^ip,) -in

(6.65)

ip^iPp) + mptliPp) + np^iPp)\ } ,

where fH-^,

m+

(6.66) Gp

From eqs. (6.42) and (6.43) two equations analogous to eq. (6.65) can be established (easily done by appropriate letter interchanges in eq. (6.65). If eq. (6.65) and the analogous ones derived from eqs. (6.42) and (6.43) are multiplied by £p, nip and Up, respectively, and thereafter added one obtains ^/7|(Pp) + mpliipp) + npl{pp) = -\Cp (^^;;|o + mpliQ + «pCo) •

(6.67)

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Optical works of L. V. Lorenz

[3, §6

By combining eqs. (6.65) and (6.67) it is now possible to express |(Pp) in terms of lo, %, and ^Q- Hence, l^P

{<^-ii)^,-l„m„l\^-t„n„t.

l(Pp) =

(6.68)

ij + mj + nj-'jr

If this expression for ^(Pp) together with a corresponding one for ^(-Pp) [obtained via the replacements

-£+^,

rrin

-m +

-n+

•

cf. eq. (6.66)] are inserted into eq. (6.64) it follows that a linear relation of the form fliilo + fli2% + ai3?o= (—)

[{i^ + m^-^n^) l o - ^ (^lo+ 'w%+ «?o)] (6.69) is obtained. The explicit expression for a 12 is

--Ed) P

2

(ap± eap)(bp ± map) (up ± iGp)^ + (bp ± mapf + (cp ± napf - {f^ctp^

(6.70) and by appropriate interchange of letters one finds ajx = <212. In the limit ap/X -^ 0 one therefore gets a\2 =^21

=-\

Y^ ^l^pbp^

(6.71)

remembering that a summation over the upper and lower signs is implicit in eq. (6.70). By means of eqs. (6.42) and (6.43) two linear relations, namely ^21?0 + «22^0 + «23to

(6.72)

(S)

[(^^ + ^^ + «^)^o-^(^5o + ^^o + «Co)

« 3 1 § 0 + ^ 3 2 ^ 0 + ^33^0

= (^)'[(^'+^'+«')Co-«(^So+'^%+«eo)

(6.73)

can be obtained in the same manner as eq. (6.69), and the explicit longwavelength expressions for ^13 =^31 and ^23 ^ ^32 ^^Y easily be established by making the appropriate letter interchanges in eq. (6.71).

3, §6]

The phenomenological light theoiy of Lorenz

245

Starting from eqs. (6.69), (6.72) and (6.73), which Lorenz had shown followed from his basic differential equations for the light vector (eqs. 6.41-6.43), Lorenz was now prepared to take the last step in deriving the Fresnel formulae for the propagation of light in anisotropic media, which were only weakly founded theoretically in 1863. In compact form eqs. (6.69), (6.72) and (6.73) can be written

^•£'o=( —j [q^EQ-qq-E^],

(6.74)

where "a is the symmetric matrix ({«,>}) formed by the fl,y's, and £"0 = (^^ %' ^0) is the amplitude of the electric field. In the long-wavelength limit adopted here, the elements of ^ are independent of q, as we have realized above. In a new (NEW) coordinate system related to the old one by the rotation matrix S, eq. (6.74) takes the form ^NEW •^0,NEW ^ ( — j

[^NEW^O.NEW " ^NEW^NEW •^0,NEWj ,

(6.75)

where -0,NEW

- S • EQ,

^NEW

-

S

' q,

^NEW - S ' a ' S

This is so because the length of the wave vector {q) and the scalar product q • Eo are invariant under the rotation. Since ^ is symmetric one may choose the new coordinate system in such a manner that ^NEW is a diagonal matrix. Doing this, and omitting for brevity the subscript NEW on the various quantities, one obtains 0

0)

'EQ=

[U - qqj 'Eo,

(6.76)

0

with the definitions a = UQ a\\^, b = UQ «22 ' ^^^ ^ t;o^33^^. The homogeneous set of equations in (6.76) has a nonvanishing solution for the field only if the related determinant is zero. Thus, a-'s'

\+u' uu uw

b-^s^

-\+u^

uw = 0, uw c'^s^ - 1 + w^

(6.77)

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Optical works of L.V Lorenz

[3, § 6

where s = s(q) = (o/q is the directional dependent phase velocity, and q = (u,u,w). Since u^ -\- u^ -\- w^ = 1, it is easy to show that eq. (6.77) may lead to the following equation for the unknown phase velocity: 2

2

2

- T — ^ + T^—T + ^ ^ = 0 .

(6.78)

This equation is recognized as the Fresnel formula determining, for a given direction of the light wave vector, the phase velocity in terms of the three principal velocities of propagation a, b and c (see, e.g., Bom and Wolf [1999]). By the end of the analysis of the double-refraction phenomenon, Lorenz pointed out that the photoelasticity may also be explained by his theory. Hence, it appears from the theory of elasticity that a stress applied to a condensed medium gives rise to a strain. In turn, this strain in general will change the periods Pp and the amplitudes Cp in the Fourier series in eq. (6.60). Thus, even if the medium does not exhibit optical anisotropy before the stress is applied, double refraction will occur as a result of stress. In order to demonstrate that his light theory can account for optical activity (in Lorenz' time called circular polarization) Lorenz expands the Cartesian components of the light vector in a series containing not only cosine terms but also sine terms. Thus, for the jc-component of the light vector he writes (6.79) p

where §o? § (=tP/?)> • a r e new constants on the sine terms S = sm(a)t-qr),

S(±Pp) = cos I cut-q - r ±Kp - r ± — \

ctp

etc. Instead of eq. (6.74), Lorenz now obtains

where E^ = (^Q, %, JQ) and

EQ

= (§Q,7^Q, to) are the real (R) and imaginary (I) D

I

4r-^

parts of the now complex electric field amplitude, EQ = EQ -\- IEQ. The new b is antisymmetric, i.e.,

/ ^=

0

bn

-b3i\

-bn 0 623 , V ^31 -b23 0 J

(6.81)

and the individual matrix elements contain only uneven powers of o^ = K~\ In the long-wavelength limit (a^ -^ 0), eq. (6.80) therefore becomes identical

3, § 6]

The phenomenological light theory of Lorenz

lAl

to the long-wavelength limit of eq. (6.74) remembering that the elements of the "a matrix contain only even powers of a^. If §, r/ and t satisfy the differential equations in (6.41-6.43), the first-order time derivatives of these also constitute a solution to the fundamental wave equations. By a differentiation of eq. (6.79) with respect to cot, the terms with C go into -S and those with S into C. Another matrix equation between the real and imaginary parts of the complex field amplitude thus can be obtained from eq. (6.80) making the replacements D

I

I

D

EQ =^ -EQ and EQ^ EQ. Hence, one gets a . ^ 0 - : ^ .:EO = vl ( ^ ) ' iy-qq)

•%.

(6.82)

Starting from eqs. (6.80) and (6.82), Lorenz shows that the circular polarization (optical activity) phenomenon indeed can be described by his light theory. The D

D

D

I

I

J

six unknown field components E^^, ^^ ,., E^ ., E^^^, E^ ^., E^ - satisfy the set of six linear and homogeneous equations given in (6.80) and (6.82). To obtain a solution for these not equal to zero the associated 6x6 determinant must be zero, and the possible values of the phase velocity of light, co/q, follow from this condition for the determinant. Instead of following the rather cumbersome analysis of Lorenz let me briefly demonstrate using modern notation why the optical activity phenomenon is inherent in eqs. (6.80) and (6.82). Hence, if one multiplies eq. (6.82) by the imaginary unit, and thereafter adds the resulting equation to eq. (6.80), the D

I

following equation is obtained for the complex amplitude EQ = EQ -^ IEQ of the electric field: (a-lib) -E, = ul ( ^ ) ' {u-qq) -^o-

(6-83)

By the introduction of the gyration vector, with subscript L for Lorenz (Agranovich and Ginzburg [1984]), GL = (b23,b,ubuX

(6.84)

eq. (6.83) can be written in the form (^^

{a-Eo + iGLxEo)

= {q'U-qqyEo.

(6.85)

By means of the electric displacement field D(q, (O) = eo'eiq, co) • E(q, co)

(6.86)

248

Optical works of LV Lorenz

[3, § 6

the wave equation in eq. (4.35) may be generalized to take the form ^a;2/>(^, 0)) =(g^U-

qq) • E{q, co).

(6.87)

A comparison of eq. (6.85) and (6.87) [with the identification E^ip = 0) = £"0 = E{q, (JO) then shows that the electric displacement field in the present case is given by D(q,co)=(^\

eo('S'E{q,co)^iGtxEiq,w)).

(6.88)

In the long-wavelength limit where the elements of ^ are independent of ap(Xp/X -^ 0) and the elements of b are proportional to a^„ eq. (6.88) is reduced to D(q, CO) = Co i^eiw). E{q, co) + iG(q, a)) x E(q, co)),

(6.89)

where ?(a>) = (co/uof'aiX/X -^ 0) and G(q,w) = (co/uof Gt(X/}^ -^ 0). When written as in eq. (6.89) the constitutive relation takes precisely the form used in macroscopic optics to describe phenomenologically the optical activity phenomenon, see, e.g., Agranovich and Ginzburg [1984]. Throughout his life Lorenz searched for a unified theoretical description of the various optical phenomena. How far had he come towards such a goal in 1863 seen from where we stand today? The framework for Lorenz' light theory is the set of differential equations in (6.41-6.43). These contain the space-dependent function u^(r). In transparent media u{r) = co/n{r) is a local light velocity. It was known in 1863 that the refraction and reflection of light in the presence of absorption in the medium under study could be described by the Fresnel coefficients for the nonabsorbing case just by a replacement of the real refractive index n = CQ/U by a complex one. In the article written in the wake of the 1863 meeting of Scandinavian researchers in the natural sciences (Lorenz [1863c]), Lorenz emphasizes that his basic set of equations also holds for absorbing media. Lorenz was well aware of the fact that the particles of the medium in which the light propagation takes place are not necessarily at rest in the absence of light, as assumed when the ansatz in eq. (6.60) [with Pp given by eq. (6.61)] is used. Lorenz is not sure that the set of differential equations he has been using so far could cover this more general situation. In his Philosophical Magazine article (Lorenz [1863b]) he writes at the end: "A step further may still be made in the direction we have here entered upon, and I will briefly point out how. The velocity of light is regarded as a ftinction

3, § 6]

The phenomenological light theory of Lorenz

249

of X, y and z; it may, however, be taken still more generally as a function of the time t, for it is plainly a limitation to suppose the particles of the body originally at rest. This can be easily introduced into the calculation by giving to pp the value {kpt + QpX + bpy + CpZ + dp)/ap. Since, however, the differential equations (A) were not formed on this supposition, they cannot in this case be taken as a safe basis for the calculation, and the more generally valid differential equations would have to be deduced in another way." The differential equations numbered (A) in the quotation above are the basic ones in the light theory of Lorenz; i.e., eqs. (6.41-6.43). It seems that Lorenz was confident that the general theory of light could be based in a framew^ork w^here the concept of a local velocity of light alw^ays makes sense. Nowhere in his writings does he go beyond this concept nor does he indicate that a need for abandoning the concept might appear. It is interesting to mention in this context that in his theoretical description of the multiple scattering of light from a collection of point-like molecules situated in vacuum, Lorenz found it necessary to assume that the speed of light in vacuo in the vicinity of a molecule depends on the distance from the molecule in order to account for colour dispersion. We shall return to this very interesting aspect of his thinking in § 9. The ansatz Lorenz uses for t;~^(r) in eq. (6.60) does not depend on the light vector itself, which means that his theory is linear. Nowhere in his works are there suggestions to the possibility that the light-matter interaction might be nonlinear. Photoelasticity in fact is a nonlinear optical phenomenon. This is so because a stress applied to a medium gives rise to a quasistatic (static) electric field, and this field, in a nonlinear combination with the high-frequency optical field, is the basis of photoelasticity. To understand the generality of the light theory of Lorenz, let us take a brief look at modern linear dielectric response theory (see, e.g., Keller [1997b] and references herein). From the microscopic Maxwell equations one obtains in the space-frequency domain the following wave equation for the local electric field:

VV'E{r\o))-V^E{r\co)= ( - ]

/

? ( r , / ; co)-^(/; co) dV.

(6.90) The spatially nonlocal relative dielectric tensor, ^{r, /), which is a function of two position vectors (r,r'), is related to the microscopic conductivity tensor, 'd{r,r'\ co), via ?(r,/; w) = 8{r-r')U+^—^{r,r'\ o)).

(6.91)

250

Optical works of L.V. Lorenz

[3, § 6

In the wave vector-frequency representation eq. (6.90) reads

q^(u-qqyE{q,w)-{^\

J^ r(q,q',(o) • E(-q\co)d'q\

(6.92)

In general, the plane-wave component E(q, co) of the electric field is linked to all other field components E(-q', co) be means of the dielectric tensor r(q,q\co)=

I j

?(r,/;co)e-''' '^e-''-' "^'dV'dV,

(6.93)

but if the medium under study has a periodic structure, i.e. ?(r,r'\(D) = ? ( r + R,r' + R; w),

(6.94)

where R is an arbitrary lattice vector, one can easily show that only wave vectors q and q' connected by some reciprocal lattice vector K, that is q + q'=K,

(6.95)

appear in 'e{q,q'\ w). Thus, one may write "eiq, q\ co) = Y^ ^Kiq. co) d{q + q'- K)

(6.96)

K

By a combination of eqs. (6.92) and (6.96) one obtains

q^ (u-qq)

•E{q,Oj)=(~\

^

?^(^, co) . i ^ ( ^ - ^ , w),

(6.97)

and it appears that the assumed spatial periodicity of ^{r, r'\ co) has reduced the integral equation in eq. (6.92) to an algebraic relation between a given spatial Fourier component (q) of the electric field and those which are connected to this component by some reciprocal lattice vector, K. In the approach of Lorenz the dielectric fiinction is a local quantity, which in our terminology thus relates the dielectric displacement in a given space point only to the electric field which prevails in the same point. If the dielectric response tensor, ^(r,/; co), is nonvanishing only for point pairs ir,r') so close to each other that the

3, § 6]

The phenomenological light theory of Lorenz

251

(macroscopic) electric field is the same in r and r', SL local response formalism, based on the use of an inhomogeneous dielectric tensor oo

/

?(r,f';ft>)dV,

(6.98)

(X)

can be applied. Since oc

/

(6.99) OC

for a periodic medium, it appears that the macroscopic dielectric tensor is given by ?(r; CO) = (Iji)-^ Y^ ^K{K, a;)e'^ ^

(6.100)

K

In the macroscopic regime eq. (6.90) is reduced to the form VV'E{r\a))-V^E{r\CD)=(

— \ 7(r; (jo)E{r; w),

(6.101)

and eqs. (6.100) and (6.101) essentially constitute the framework of Lorenz' studies of the double refraction, optical activity, and colour dispersion to which we shall turn our attention now. 6.5. Chromatic dispersion and density dependence of the refractive index In 1863, the experimental data for the wavelength (AQ) dependence of the refractive index («) seemed to indicate that the chromatic dispersion had the form « = a + ^AQ^, a and b being wavelength independent constants. In his first treatise on the theory of light Lorenz [1863a] emphasized that it was possible to derive this experimentally established law of chromatic dispersion from his phenomenological light theory. Based on the assumption that the medium under study exhibits a structural periodicity that can be described by superposition of spatial Fourier modes which all have wave numbers K^ = a~^ much larger than the wave number of light, Lorenz had shown that the double refraction found in biaxial crystals could be obtained from his theory keeping only lowest (zero-)order terms in a^, and that the circular polarization (optical activity) phenomenon, which appears in crystals lacking inversion symmetry, followed if first-order terms in a^ were kept. The chromatic dispersion (n = a -\- bX^^) observed in media which exhibit inversion symmetry can also be obtained from

252

Optical works of L.V. Lorenz

[3, § 6

the Lorenz light theory keeping terms of second order in a^. In the nonrelativistic regime where an electromagnetic field cannot excite (virtual) electron-positron pairs fi-om the vacuum ground state, chromatic dispersion only appears as a property of material bodies. Existing light theories based on fundamental assumptions about the forces between molecules had led to the conclusion that unobserved scattering of light and chromatic dispersion should appear also in vacuum unless new hypotheses were added to these theories. For Lorenz it therefore was satisfactory to see that the chromatic dispersion, found in material media only, could be derived from his theory. In the English version of the light theory (Lorenz [1863b]) he concludes on p. 211: "According to this theory, chromatic dispersion appears as a property of material bodies, dependent on their periodic heterogeneity, whereas, on Cauchy's theory, the absence of chromatic dispersion in a vacuum can only be explained by new hypotheses." In his second treatise on the theory of light, published in 1864 in the Annalen der Physik (Lorenz [1864a]) and in the Philosophical Magazine (Lorenz [1864b]), Lorenz presents a quantitative calculation of the chromatic dispersion and in the wake of this calculation he examines the density dependence of the refractive index. These investigations are of some interest because they stand as forerunners for his later and important studies of the scattering of light from assemblies of molecules and from small spherical particles. Since the famous Lorenz-Lorentz relation also has it roots in Lorenz' 1864 papers, we now take a brief look at the part of his second treatise on the theory of light which deals particularly with the aforementioned subjects. In relation to the calculation of the chromatic dispersion Lorenz focuses his attention on the average refractive index n defined by n = yj\{n\ + nl + n]\

(6.102)

where n\ = a\[^co/uo, «2 = al'^^co/uo and «3 = a\^^^co/uo are the principal refractive indices. From inspection of eqs. (6.64), (6.66), (6.68) and (6.70) it appears that a\i to second order in 6p is given by

P

{a,, ± Ittpf + (bp ± mUpf + {cp ±napf-[J^^j

aj (6.103) remembering that summation over the plus and minus sign terms is implicit in the notation. Analogous expressions for 022 and 033 can be obtained replacing

3, § 6]

The phenomenological light theory of Lorenz

253

Up ± £ap in the nominator of eq. (6.103) (and only there) with bp ± map and Cp ± ndp, respectively. By addition one gets an +^22+^33 = 3

E(|) P

+2 P

(Up ± iGpY + (bp ± mapY + (cp ± riGpY - (^^^ aj (6.104) and hence to second order in a^, and with the summation over the plus and minus sign carried out, we obtain flii+«22+a33 = 3 - i ^ € ^ + ( ' ^ ' ) P

^

^"^

^ e X

(6.105)

P

By combining eqs. (6.102) and (6.104), and introducing the vacuum wavelength of light via AQ = IJICQ/W, Lorenz found the following result for the square of the average index of refraction: 2

{^'[-ii:4^'i(^)'ij:4< / ^0

(6.106)

The refractive index itself henceforth takes the form n = a -\- bk^^, where b = (2jr^/3)(co/t;o)^ Y^p ^pCCp is a positive constant in agreement with the experimental data available in 1864. If the chromatic dispersion is neglected (Gp -^ 0), the expression for n^ is reduced to

and this formula contains two material-dependent constants, viz. ul and Yip ^lIn the long-wavelength limit n = n{ap —> 0) is called the (average) reduced refractive index. From the general theory of Fourier series, Lorenz now showed by a beautiful mathematical analysis that these constants are related to integrals of/(i') = v'^{r) and/^(r) over the unit cell, which by stacking reproduces the

254

Optical works of L.V. Lorenz

[3, § 6

periodic structure of the medium. If the volume of the unit cell is denoted by F, he thus obtained /WdV,

(6.108)

V

2"^

(6.109) The experimental data available in 1864 indicated that the quantity M=

(6.110) P for a given medium was almost independent of the mass density p of the medium. On the basis of a simple model for the molecular structure of material bodies, Lorenz showed by means of eqs. (6.107)-(6.109) that the quantity M is almost independent of p. By examining the experimental data Lorenz saw that small variations of M with p were present and this came as a satisfaction to him because his theory also required that M exhibits a weak mass-density dependence. In the model suggested by Lorenz it is assumed that the material body consists of molecules inbetween which there is empty space. Furthermore, he assumed that the individual molecules possess a physical invariability so that the single molecule does not change its properties when the mass density of the medium is varied, i.e., only the magnitude of the vacuum region between the molecules is altered. The molecules are assumed to be transparent to light but the speed of light, and therefore also/(r) = u~^(r), is allowed to vary from point to point inside the molecule. In the vacuum regions between the molecules Lorenz assumed that the speed of light is identical to its value in empty space, i.e., CQ. This model of Lorenz in a sense points forward to a program for determining the range of validity of the macroscopic Maxwell equations. The goal of this program initiated by the Dutch physicist H.A. Lorentz was "to separate matter and aether". A fine description of the Lorentz program and the work of Lorentz in molecular optics has been given by van Kranendonk and Sipe [1977]. The electrodynamic theory of Lorenz which was published in 1867 and to which we shall return in § 7, essentially is identical to the theory of Maxwell, but it may be correct to say that in the understanding of Lorenz its physical interpretation seems to be more on a par with the microscopic interpretation of the Maxwell equations given much later by Lorentz. If one writes eq. (6.107) in the form

3, § 6]

The phenomenological light theory of Lorenz

255

it appears that the new integral only has contributions from the domains occupied by the molecules of the unit cell. Lorenz now rewrites eq. (6.111) as follows: i ^0

= ^(l+pP),

(6.112)

^0

where

^=^z

1

Uclf{r)-\)d\.

(6.113)

Since the mass per unit cell, p F , is independent of the mass density, and it is assumed in the Lorenz model that/(r) is independent of p, the quantity P will be independent of the mass density. By the same technique as used to rewrite u^^ = V~^ J^fd^r above, Lorenz next obtains 1

/ f\r)d'r=^(l^pS), .2..^.3 1

Jvy

(6.114)

^0

where 1 S-— jyj\r)-\)d'r

(6.115)

is a mass-density independent integral extending only over the parts of the unit cell which are occupied by the molecules. By combining eqs. (6.107)-(6.109), (6.112) and (6.114) it appears that the square of the reduced refractive index is given by

Since the last term in eq. (6.116) was presumed to be small Lorenz has justified that the quantity M = {n^ -\)/p = P is almost independent of p, in agreement with the experimental findings. For a mixture of molecules which have the mass densities pi, p 2 , . . . , and which occupy the volumes i;i, t;2,..., the Lorenz model leads in the first order approximation to

J_^ 1

^0 4

l + y(p\OlP\+p2U2P2+

")

(6.117)

where Pi, P2, • • •, are density independent quantities. With the constant P defined ^y P = (Y^iPi^iPiVipVX eq. (6.117) can be written in the form given in

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Optical works of L. V. Lorenz

[3, § 7

eq. (6.112). Thus, for a molecular mixture M = («^ - l)/p = P is the relevant constant in the Lorenz theory.

§ 7. The electrodynamic theory of Lorenz 7.1. Lorenz says: "the vibrations of light are themselves electrical currents" Throughout his life Lorenz acknowledged as his guiding principle the idea of the unity of natural forces. The culmination of his efforts towards establishing this unity came in 1867, when he published an electrodynamic theory in which the vibrations of light were identified with electrical currents. The theory was first presented in Danish in Oversigt over det Kongelige Danske Videnskabernes Selskabs Forhandlinger (Lorenz [1867b]), and it appears that Lorenz' paper was submitted on January 25, 1866. Shortly after its appearance in Danish it was translated into German and published in the Annalen der Physik (Lorenz [1867b]). In the same year a translation from Poggendorff's Annalen appeared in the Philosophical Magazine (Lorenz [1867c]) under the title "On the Identity of the Vibrations of Light with Electrical Currents". In a popular article written in Danish, and published in Tidsshrift for Physik og Chemi, Lorenz [1867d] describes in words his thoughts of the nature of light and his findings in regard to the unity of light and electrical vibrations. The article bears the brief title "Om Lyset" (about light). Before giving a detailed account of the electrodynamic theory of Lorenz it is worthwhile to present a brief overview of this important theory. Although it had become clear that light is a kind of wave motion the analogy between light and other types of waves had its limitations. The so-called ether, a medium physicists felt was needed in order to support the propagation of light, had in 1867 lost its reality for Lorenz. As we shall see in § 10, Lorenz had gradually come to this standpoint during the 1860s. It appears from his writings that for a long time he had felt that the ether had to be a rather peculiar substance due to the fact that it could only support transverse light vibrations. His successfiil establishment of a set of differential equations which in a unified manner had allowed him to account for all observations regarding the transmission and reflection of light at interfaces, the double refraction, optical activity, and colour dispersion without involving the ether hypotheses (or any other underlying hypotheses about the nature of light), had convinced him that it might be fruitful to try to establish the connection between the theories of light and electricity on a pure phenomenological basis. In his studies of the optical properties of condensed matter (and gases) Lorenz had realized the central importance of taking the finite

3, § 7]

The electrodynamic theoty of Lorenz

257

velocity of light propagation properly into account, and he had found that his phenomenological light theory became of great generality once he allowed this velocity to become nonuniform. To link light vibrations and electrical currents he fek it was necessary first to try to introduce the concept of retardation into the description of electrical current flows in matter. He achieved this goal in a manner characteristic for his thinking, and his line of reasoning is beautifully described in his popular Danish paper of 1867 (Lorenz [1867d]). In 1857 Kirchhoflfhad established a set of local equations from which, for instance, a quasistatic wave equation for current waves in a conducting wire could be derived; see Kirchhoff [1882a]. Lorenz was convinced that the Kirchhoff equations could account for all available experimental data concerning the propagation of electricity in matter but he felt that a theoretical basis was lacking. In the quasistatic theory of Kirchhoff the electrical interaction between charges placed in different points is instantaneous, but to Lorenz as well as to Faraday the concept of "instantaneous interaction-at-distance" was a doubtful one. However, if the velocity with which the electrical interaction propagates is very large the retardation would not show up in the experimental data. In consequence, Lorenz generalized the Kirchhoff formalism in such a manner that the Kirchhoff equations appear from his own equations as the first term in a series expansion. Lorenz next showed that his generalized set of equations can support transversely polarized current waves which can propagate over long distances, provided the medium is a poor conductor. But light waves can also propagate over long distances in (homogeneous) nonconducting media, and the light vector oscillates perpendicular to the direction of propagation in these media, Lorenz noted. In a far-reaching leap Lorenz now assumed that the velocity of light and the velocity of the electrical disturbances are the same. By this assumption he was then able to demonstrate that the set of differential equations for the components of the local electrical current density has the same form as the set determining the components of the light vector, provided the electrical conductivity is negligible. The electrodynamic theory of Lorenz also implied that good conductors should absorb light waves strongly, in good agreement with the experimental observations. Starting from the assumption that all electrical interactions are retarded Lorenz thus came to the conclusion that the vibrations of light are electrical currents. The electrodynamic theory of Lorenz is, as we shall see in § 7.6, equivalent to the theory Maxwell established a few years earlier. Lorenz apparently did not know Maxwell's theory in 1867.

7.2. The quasistatic theory of Kirchhoff: a good basis In his search for a connection between light and electricity Lorenz took as a

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Optical works of L V. Lorenz

[3, § 7

Starting point the laws which Kirchhoff in 1857 had established for the motion of electricity in bodies with constant conductivity. To facilitate the general view we shall write the set of Kirchhoff equations in vector form, and in a slightly modernized notation, moreover. The equations of Kirchhoff are written in local form, and as we shall see this offers Lorenz an extremely good basis for uniting light and current-density waves. Kirchhoff assumes that Ohm's law holds locally even if the distribution of electricity (electric charges) is non-uniform, and writes for the current density / ( r , t) the equation J{r,t) = o

Cl;

dt

(7.1)

Here o is the space- and time-independent conductivity of the assumed isotropic metal. In the notation of Lorenz (and Kirchhoff) the conductivity is denoted by k and related to our ohy o = Ak. The quantity cw is the constant introduced by Weber in 1846; see Weber [1893-94]. For Weber as well as for Kirchhoff the constant cw had the status of (merely) a conversion factor between two units of electric charge. The introduction of two distinct units of charge stemmed from the possibility of basing the measurement of charge on either a static or a dynamic interaction between charges (Weber [1893-94]). The works of Maxwell and Lorenz rose the Weber constant cw (divided by \/2) from the above-mentioned humble status to the dignity of a universal constant of nature, viz., the speed of light. The first vector, - V ( 0 / 2 ) , in eq. (7.1) represents the electromotive force (field) arising from the distribution of free charges. If the bulk and surface {S) densities of these are denoted by p and ps, Kirchhoff sets i r 2 ( M ) = / f ^ d 3 / . / ^ d 5 ' , J \r-r'\ J \r-r'\

(7.2)

where the two integrals run over the bulk and surface of the conductor, respectively. For the scalar potential, Q/2, Kirchhoff hence upholds the expression for a static charge distribution. The vector (-2/c]y)dE/dt in eq. (7.1) gives the electromotive force associated with the charge flow (current), and for E{r,t) Kirchhoff uses the expression

k-^'l

J Since (in Sl-units) AjT

yxE{r,t)=—B{r,t), Mo

(7.4)

3, § 7]

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259

where (i^ B(r,t)=^r'-'' 4JI J'

rj{r',t)xir-r')^,, . ^dV, '- -"^ \r-r%

(7.5)

is the standard expression for the magnetic field generated by a quasistatic current-density distribution, it is reaUzed that i.ioE(r,ty(4jt) has the status of a vector potential. The Kirchhoff expression for the self-consistent electric field, J/a, hence consists of a sum of weighted contributions from the gradient of the scalar potential and the time derivative of the vector potential, the weight factor being 2c^. Since the (normalized) potentials Q and E are related to the charge and current densities p(ps) and / in a time-local manner, the Kirchhoff formula describes the electrodynamics of a conductor in a quasistationary fashion, and therefore upholds the instantaneous-action-at-distance idea. Kirchhoff fiirther expressed the relation between the current density and the distribution of the free charges by means of the equation of continuity V./(M) + ^

= 0,

(7.6)

and fi"om this he also derives the relation n . / ( M ) - - ^

(7.7)

between the normal component of the current density at the surface and the surface charge density, the unit vector h in eq. (7.7) being directed into the conductor. It is from eqs. (7.1)-(7.3), (7.6) and (7.7) that Lorenz begins his attempt to find a link between the vibrations of light and electrical currents. From a microscopic point of view the use of eq. (7.7) is not essential since socalled surface charges are never completely confined in an infinitesimally thin shell at the conductor-vacuum (air) boundary. 7.3. The Lorenz (not Lorentz) retarded potentials It was obvious to Lorenz that eq. (7.1), which had been deduced in a purely empirical manner, did not necessarily give the exact expression for the actual law. Thus, he emphasizes that it is permissible to give this equation other forms provided the charges acquire no perceptible influence on the results which were established experimentally. Here again we meet Lorenz' characteristic way of thinking:fi-omconfidence in the correctness of the experimental data he seeks in

260

Optical works of L.V Lorenz

P? § 7

a phenomenological manner to develop a theory of general validity. By 1866-67 his trust in such an approach was pronounced because his successful light theory was established in precisely this manner, starting from the experimental results of Jamin and the Fresnel formulae. Lorenz began by considering the terms on the right-hand side of eq. (7.2) as the first members of a Taylor series expansion. By the equation

l^(r,0= / - ^

^-^dV'+ / — ^

US\

(7.8)

J \r-r'\ J \r-r'\ he thus defined a new ftinction 0(r, t). It appears that Q/2 is a retarded scalar potential and that a is the velocity with which the electrodynamic interaction propagates. It is of interest to note that Riemann already in 1858 had the idea to derive the law of electrodynamic action from the conception of a finite velocity of propagation of this action. Riemann thus suggested to replace the Laplace operator in the Poisson equation for the electrostatic scalar potential with the Dalembertian operator, and he gave the solution to the generalized Poisson equation in the form of a retarded potential. Riemann's idea was presented to the Gottingen Royal Society in February 1858 in a short paper which was withdrawn afterwards; see Riemann [1892]. The withdrawn paper was published posthumously in 1867 in the Annalen der Physik (Riemann [1867]), i.e., in the same year as Lorenz' work on the identity of light vibrations and electrical currents. Lorenz did not know of Riemann's paper since it appeared after Lorenz' work was written. Riemann's work is of a much more limited scope than that of Lorenz, and Riemann founders in the mistaken attempt to prove that his retarded scalar potential is equivalent to the electrodynamic (vector) potential derived from Weber's law. Apparently, the first published occurrence of a retarded potential is in the 1861 paper of Lorenz on the theory of elasticity (see §§3.2 and 5, and Lorenz [1861a]). By inserting a Taylor series expansion of the charge density, i.e.,

(7.9) and a similar one for the surface charge density, into eq. (7.8) Lorenz obtains upon differentiation with respect to x (and y and z)

VQir,t) = VQir,t) 1 d' ^ a2 dt^ J

Ir-r'l

J

\r-r'\ (7.10)

3, § 7]

The electwdynamic theory of Lorenz

261

In the next step Lorenz uses the equations of continuity given in eqs. (7.6) and (7.7) to eUminate dp{r', t)/dt and dpsir', t)/dt in favour of V • J(r\ t) and h' ' J{r',t). Upon a partial integration of the term containing V • J(r\t) the surface terms are seen to cancel each other with the result that a^ at

I

-dV

-s.{r,t)

(7.11)

So far Lorenz only let the retardation enter the relation between the scalar potential and the bulk and surface charge densities, but armed with the result in eq. (7.11) he is able to introduce the retardation in the relation between the vector potential and the current density. This is so because eq. (7.11) may be rewritten in the form J[r',tV

2

J

'

la^dtj

2

i

2fl2

\r-r'\ dt

(7.12) '

or with the definition J{r',t/

•

also as

It is easy to show that Vx(^^E(r,t))=B{r,tl

(7.15)

where B(r, t) is the magnetic field, provided the constant a is identified as the speed of light (in vacuo). In the theory of Lorenz the retarded vector potential therefore will be given by A{r,t)=^-E{r,t).

(7.16)

The right-hand side of eq. (7.14) is a series and if one puts a = c\^/2 the (shown) first two members of this series are precisely those which enter

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Optical works of L.V. Lorenz

[3, § 7

Kirchhoff's expression for the current density (see eq. 7.1). As Lorenz pointedly emphasized, the empirical basis of the Kirchhoff equations is in the experimental results obtained in the domain of quasistatic electrodynamics. Since it was known from the work of Kohlrausch that Weber's constant cw was of the order of the light velocity (Weber [1893-94]), Lorenz concluded that the possible presence of higher-order terms in Kirchhoff's expression for the current density would not show up in the available experimental data. In the firm belief that any electrodynamic action takes time to propagate, Lorenz therefore in accordance with eq. (7.14) suggested to replace, in the Kirchhoff eq. (7.1), the quasistatic potentials Q{r, t) and E{r, t) by the retarded ones. Hence, Lorenz postulates that

J{rj) = o

cl,

dt

(7.17)

with Q/2 and E given by eqs. (7.8) and (7.13). He also mentioned that the expression for E{rJ) is somewhat less complicated than the one for E{r,t). Lorenz upholds the equation of continuity given in eq. (7.6), and consequently also eq. (7.7). At this stage the electrodynamic theory of Lorenz is thus based on eqs. (7.6)-(7.8), (7.13) and (7.17), and it contains two yet unrelated velocities (constants) cw and a. Before explaining how Lorenz by postulating the decisive relation between cw and a brought his electrodynamic theory to its culmination we shall take a look at a consequence of his theory which for us has taken a central position.

7.4. The Lorenz (not Lorentz) gauge Since the equation of continuity links the charge and current densities, and since these in turn determine Q{r, t)/2 and E{r, t) a relation must necessarily exist between Q/2 and E, and Lorenz gives this relation in his paper on the relation between light vibrations and electrical currents (Lorenz [1867a-c]). To determine this relation, let us first take the divergence of eq. (7.13). This gives

v5('.')=/v(^)-.(/,,-t£l)<.V (7.18)

3, §7]

263

The electrodynamic theory of Lorenz

or equivalently

- ^ ^ - > - / - ' ( ^ ) - ( ^ ' - ^ ) ^ ^ ^ ' dj(r',tJ

\r-r'\

V' ( ^ ' I Z ^ ) d V .

dt

(7.19) A partial integration of the first integral in eq. (7.19) now leads to

J \r-r'\ i \r-r'\V

\

a

V-J{r',t\

) a

)

dj(r',tdt

•v

\r-r

dV,

(7.20) where «' is the inward directed unit normal vector at the surface. One may also write eq. (7.20) in the compact form

v..(.,)=/^.'..(.',,-trl)ay

^/^-{{.(.,-t.)}}aV,

(7.21)

where the notation {{•••}} around J means that when carrying out the operation V the implicit occurrence of r' in the retarded time [t] = t -\r- r'\/a is to be ignored. Since from eqs. (7.6) and (7.7) one has

-/ , / '

\r-r'\\

d

( ,

\r-r'

(7.22)

(7.23) a combination of eqs. (7.8) and (7.21)-(7.23) finally gives (7.24) The relation in eq. (7.24) can be found in Lorenz' 1867 papers and was derived originally essentially in the manner described above. The result in eq. (7.24) is

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Optical works of L.V Lorenz

[3, § 7

readily recognized if one translates E and Q/2 to modern notation. The vector E is related to our vector potential by means of eq. (7.16), and our scalar potential, 0(r, 0 is related to Q/2 via

*<-»=i^. and altogether eq. (7.24) reads in modem notation as follows:

V/r(r,0 +C o ' ^ ^ = 0 .

(7.26)

The relation in eq. (7.26) between the scalar and vector potential is an example of a so-called gauge condition. The relation in eq. (7.26) is commonly known as the "Lorentz gauge condition", named after the Dutch physicist H.A. Lorentz (18531928). Also, the retarded potentials discussed in the previous section go under the familiar name of "Lorentz retarded potentials". In both cases the paternity is wrong. When Lorenz wrote his 1867 paper Lorentz was only fourteen years old, and there is no doubt that the paternity must be assigned to L.V Lorenz. Readers interested in the history of electrodynamics in a broader framework are urged to study the monumental book "A History of the Theories of Aether and Electricity" by E. Whittaker, originally published in 1910 (Whittaker [1952]). On pp. 268-270 of volume 1, which deals with the classical theories, one finds a brief discussion of Lorenz' electrodynamic theory, including his introduction of the retarded vector and scalar potentials, and the gauge condition which follows from these, and the equation of continuity for the electrical charge. Scanning many modem textbooks, I found that only in Penrose and Rindler [1984], p. 321, was the relation in eq. (7.26) named the "Lorenz gauge condition". A small note published in "IEEE Antennas and Propagation Magazine and bearing the title "Lorenz or Lorentz?" emphasizes that the paternity suits for the retarded potentials and the gauge condition must be decided in Lorenz' favour (van Bladel [1991]). In our times gauge conditions have come to play a central role in theoretical physics, but it seems that for Lorenz the gauge condition in eq. (7.24), or equivalently eq. (7.26), (merely) was a mathematical consequence of his idea that all electrodynamic interactions must propagate with finite velocity, a principle he needed to take the final step in uniting light vibrations and electrical currents. Let us now, in the spirit of Lorenz, take this step.

3, § 7]

The electrodynamic theory of Lorenz

265

7.5. The culmination To obtain agreement with the quasistatic theory of Kirchhoff it appears from eq. (7.12) that Lorenz had to choose a = cw/2 (see also eq. 7.17). By means of eq. (7.11), eq. (7.17) can also be written in a series expansion as

Q{rj)\ 1 )

( 1 \c\y

\ \ dE(r,i) la-) at

1 dE(r,t) la^ ot

(7.27) which expression in the case oi a = cw/2 leads us to Weber's theory, whereas if a = cxo one obtains the electrodynamical theory of Neumann. This is so because E{r,t) = J \r-/\~^ J(r\t)d^/ for <7 -^ 00, a formula actually proposed by Neumann in 1845 already (see Whittaker [1952]). If one makes the choice a = cw/v^, the above equation represents a mean between Weber's and Neumann's theories, which both are in agreement with the available amount of experimental data Lorenz remarks. Thus, it is obvious to Lorenz that the constant a so far has to be regarded as an indeterminate magnitude, yet very great, i.e., of the same order as the vacuum velocity of light. At this stage Lorenz [1867c] writes on p. 292 in his Philosophical Magazine paper: "It now becomes necessary to obtain, in another manner, a determination of these undefined constants and if possible, seek a conformation or correction of the results found ... I have completely given up the idea of getting any good from physical (read: mechanical) hypotheses; and we can only develop the consequences from the results found, and inquire whether this does not furnish an indication towards answering the question." To find a possible link between the two constants cw and a, Lorenz now seeks to replace the integral equation for the current density given in eq. (7.17) with a differential equation in order to obtain a suitable framework for wavemotion analyses. He first cites a theorem he had proved previously in an article published in Crelles Journal (Lorenz [1861a]) under the French title "Memoire sur la theorie de I'elasticite des corps homogenes a elasticite constante". There he had shown that, for a given well-behaved function/(r,0,

v-il)//e'-^)F^=-/«->-

<-•

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Optical works of L.K Lorenz

[3, § 7

provided the point r is within the Hmits of the integral. If r is outside the domain of integration, the right-hand side of eq. (7.28) must be replaced by zero. By simple differentiations it is easy to show that

where [t] = t-^-

^

(7.30)

a is the retarded time. To employ eq. (7.29) to prove the theorem in eq. (7.28), it is necessary to move the V^-operator inside the integral sign. Since the integrand possesses a singularity atr' = r this cannot be done directly, and Lorenz therefore first considers the non- singular integral /[/(/•', [t]) -f(r', t)] \r - r'l" d V , for which one has

WAt])-f(r\t) J

\r-r'\

J

d'r\

(7.31)

r-r'

Because V^\r-r'\ = 0 and the 9V(9r^-operation can be moved under the integral sign directly, we thus get after rearrangement

(7.32)

The first integral on the right-hand side of eq. (7.32) is zero due to eq. (7.29), and for the second integral Lorenz cites the following result, well known in 1861: V

[{^^d'Z

= -4jr/(r,0

(7.33)

for r-points located inside the domain of integration (zero, else). In § 7.6 we shall see how important a role the theorem in eq. (7.28) plays in the physical interpretation of Lorenz' electrodynamic theory.

3, § 7]

The electwdynamic theory of Lorenz

267

By differentiation of eq. (7.17) with respect to time, and use of the Lorenz(!) gauge condition in eq. (7.24) one obtains dJ{r,t) ^ dt

2

d^S(r,t)

^w

*'

(7.34)

and with the help of eq. (7.28)

By elimination of the d^E/df- term between eqs. (7.34) and (7.35) one gets - V X (V X E{r, t))+(^-\\

V'S(r, 0

^^

a

at

(7.36)

c^

The curl of eq. (7.17) is 2o_^

V X /(r, 0 = -—V ctu

dE(rJ)

X~ ^ , at

(131)

and if one differentiates eq. (7.36) with respect to time and combines the resulting equation with eq. (7.37) one obtains

Cty

(7.38) 2fl2

SJTO

c^ c^ w w

dJ{r,t)

_ (2^

dt

' V r^ \ w

_ \ 2o ^ I r^ ^ / w

.d^ir.t) Bt

The phenomenological light theory of Lorenz led as we have seen to a wave equation for the light vector (JB'-field) which has the form given in eq. (6.59). Now, Lorenz recognizes in the left-hand side of eq. (7.38) the expression which, equated to zero, represents the propagation of the light vibrations, provided CW/A/2 is identified with the velocity of light. Lorenz also remarks that the quantity CW/A/2 is in remarkable agreement with the various experimental determinations of this velocity. In particular, he refers to Weber's electrodynamical experiments in which the electrical action passed from one conductor to another through air with the same velocity as the velocity of light

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Optical works of L. V. Lorenz

[3, § 7

in air. Lorenz is now prepared to draw his conclusion, viz. that the light and current waves are identical, and in consequence of this he takes ^=a(^co).

(7.39)

Since the speed of light in air (a) and in vacuum (co) are practically the same, the fundamental velocity of Lorenz, i.e. a, is essentially the vacuum velocity of light. The identification in eq. (7.39) reduces eq. (7.38) to the form

In the English translation of the 1867 paper, Lorenz [1867c, p. 295] comments as follows on the three scalar equations in eq. (7.40): "These equations for the components of the electrical current agree fully with those which I have already found for the components of light up to the last member, into which the electrical conductivity k [a/4 in eq. (7.40)] enters. This member indicates an absorption which will be greater the greater the electrical conductivity, ...." Lorenz further shows that eq. (7.40) has solutions in the form of transversely polarized and exponentially damped plane waves. For current-density waves propagating along the z-direction and polarized in the x-direction, the ansatz J(r, t) = Joe, e"^-" cos[^(6;^ -z)]

(7.41)

gives Lorenz the following relations between the damping coefficient, a, the wave number, q, and the phase velocity of the electrical action, u: a'cl = q\cl-u'),

(7.42)

acl = Ijiov.

(7.43)

From the two equations above one may determine the dispersion relation for the electromagnetic waves in a conductor of a given conductivity. By taking the divergence of eq. (7.40), Lorenz finds that d —V • J(r, t) + 4;raV • J(r, t) = 0.

(7.44)

In fact, one may only conclude from eq. (7.40) that the expression on the left-hand side of eq. (7.44) is equal to a time-independent constant, but being

3, § 7]

The electrodynamic theory of Lorenz

269

interested in wave motions only, one may put this constant equal to zero. For a possibly longitudinal (L) current-density wave, J^, one must require V • J = W ' JL ^ 0, but it is manifest from eq. (7.44) that V • / cannot be a periodic function of time. Hence, Lorenz concludes that longitudinal light vibrations cannot propagate, a result which to Lorenz clinches the argument for the identification of light and current vibrations. Remembering that V • / = -dp/dt, it appears form eq. (7.44) that dpirj) dt

_dp(rj) dt

Qxp(-4jTOt),

(7.45)

0

and the theory of Lorenz therefore shows him that: "... in the interior of a body with constant conductivity-power no development of free electricity is possible. This result is different from that which Kirchhoff has deduced from the original equations (1) [here eq. (7.1)] - that in the interior of a conductor there is in general free electricity; but from the whole of the present investigation it will be clear that at all events this conclusion cannot be drawn with any degree of certainty." (Lorenz [1867c], pp. 296-297) Kirchhoff's local equations for current waves are incorrect as we know today, and lead to wrong consequences notably on the above-mentioned point. Lorenz came to the differential equation for the current density (eq. 7.40) considering homogeneous bodies with a space independent conductivity. To extend his theory in such a manner that it would be valid for heterogeneous bodies, he suggested to regard the quantities a = c^ and o in eq. (7.40) as variable (space-dependent) magnitudes. With the notation a{r) = u{r), the electrodynamic theory of Lorenz hence predicts that the current-density wave (light vibrations) are governed by the differential equation

u^{r)

or

u^{r)

at

and in the case where the conductivity of the medium vanishes, he obtains exactly the wave equation describing the propagation of the light vector ((§, r/, t) <^ J) in homogeneous media. Lorenz is quite clear about the farreaching consequences of his theory, and by considering the local equation (7.40) as the basic one he abandons the whole concept of action-at-distance. In his own words from the Philosophical Magazine article (Lorenz [1867c]), p. 301:

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Optical works of L.V Lorenz

[3, § 7

"The theoretically important conclusion would thence follow which has been already indicated, that electrical forces require time to travel, and that these forces only apparently act at a distance (as would follow from equations (A) [eq. (7.17) here] if they were regarded as the fundamental equations), and that every action of electricity and of electrical currents does in fact only depend on the electrical condition of the immediately surrounding elements, in the manner indicated by the differential equations (B) [eq. (7.40) here]. This is well-known to be an idea indicated by Ampere, and which several physicists, more particularly Faraday, have defended."

7.6. Lorenz electrodynamics: the microscopic Maxwell theory in the covariant form To appreciate how far-reaching the electrodynamic theory Lorenz established in 1866-67 is, we just need to examine the consequences Lorenz obtained, by putting in retardation in both of the electromotive force terms entering the Kirchhoff expression for the local current density, from a perspective slightly different from that described in § 7.5. Remembering that the presence of a so-called surface charge distribution, ps{r, t), is an idealization from a microscopic point of view, one needs only to retain the first integral in eq. (7.8), and if the function f(r, t) in Lorenz' theorem (7.28) is identified with the charge density p(r, t) one realizes that Lorenz' scalar potential (given by eq. 7.25) must satisfy the differential equation n2 0(r,O = - ^ ^ ,

(7.47)

where D^ = V^ - C^d^/dt^ is the Dalembertian operator. With the identification in eq. (7.16), it appears from eq. (7.35) that the vector potential of Lorenz satisfies U^A(r,t) = -\i^J(r,t).

(7.48)

The two equations (7.47) and (7.48), taken together with the gauge condition in eq. (7.26) thus constitute the framework of Lorenz' electrodynamical theory in the form it took when the velocity of retardation was identified with the velocity of light in empty space. Equations (7.47) and (7.48), plus eq. (7.26), form a set of equations equivalent in all respects to the microscopic Maxwell equations, and they express the electrodynamics in the covariant form

3, § 7]

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SO important in relativistic studies. In the microscopic Maxwell equations, i.e., Vx£(r,0 = -

^

,

(7.49)

at V X B{rj) = ^ / ( r , 0 + ^ ^ ^ ^ ,

(7.50)

VE(r,t)

=^ ^ ,

(7.51)

VB(r,t)

= 0,

(7.52)

only microscopic charge and current densities appear together with the microscopic electric and magnetic fields, E(r,t) = -V0(r,t) - dA{r,t)/dt and B{r, 0 = V X A{r, t). Lorenz' attempt to generalize eq. (7.40) by replacing the speed of light in vacuo by v{r) = CQ/n{r) and letting the conductivity depend on position seems quite natural, but it does not give the correct result. Thus, fi*om the macroscopic Maxwell equations in (6.55) and (6.56), and the constitutive equation / ( r , t) = o(r)E(r, t), it follows that the current density must satisfy the differential equation - a(r)V X

a{r)

1 d'J{r,t) .. ^, ^dJjrj) = l.ioO(r)—-—-, u\r) df '^ ' ' dt

(7.53)

provided magnetic effects are absent, i.e. ILI = I^IQ. The wave equation in (7.53) is different fi-om that suggested by Lorenz, namely eq. (7.46) (with 4jr replaced by CQ*). Lorenz did not worry about magnetic effects in relation to electrodynamics, and therefore his "macroscopic" theory is of course of less generality than that of Maxwell. It should be remembered, however, that the Maxwell equations we nowadays consider to be the correct ones within the framework of macroscopic electrodynamics are more general than those suggested by Maxwell. Hence, nonlinear effects were not foreseen by Maxwell, nor did he realize that the constitutive relations could be nonlocal in space and/or in time. In the theory of Maxwell the electric field appeared to be split into the vector of the electric field strength (E) and the vector of the dielectric displacement (/)). In the simplest case E and D were linked together by the dielectric constant but in general they were regarded as independent entities. In the same manner the magnetic field intensity (H) and the magnetic induction (B) essentially were considered as independent magnetic fields, though sometimes linked by

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a magnetic permeability. It was the Dutch physicist H.A. Lorentz who first realized that there is only one electric and one magnetic field vector, and that the electromagnetic field is created by atomistic electric charges. According to Lorentz, the electromagnetic field acts back on the atomistic charges, and forces these to move in the prevailing field in a manner that can be described by Newton's second law. In the Newton-Lorentz equation the dynamics of each particle a, having mass m«, charge q^, and position ra{t), is described by dVa(0

E(ra{t)J)^^^xB{ra{t\t) at

(7.54)

and with the charge and current densities given by p{r,t) = Y,q„5{r-r„{t)),

(7.55)

/(r,0 = ^?a^6(/--r„(0).

(7.56)

and

a

Equations (7.49)-(7.52) and (7.54)-(7.56) constitute the framework for classical microscopic electrodynamics. H.A. Lorentz is considered as the physicist who separated ether and matter, see, e.g., van Kranendonk and Sipe [1977].

§ 8. The discovery of the Lorenz-Lorentz relation 8.1. Background and first appearance in 1869 It seems that Lorenz had confidence in the correctness of his electrodynamic theory of light, and after having accomplished the goal of "uniting the vibrations of light and electrical currents" in 1867 he apparently never returned to questions concerning the basic nature of light. Lorenz kept himself well-orientated in the progress of physics, and I believe he must have been aware of the importance of the electrodynamic theory of Maxwell yet not as early as in 1867, due to the fact that the scientific links between the European continent and Britain were not as tight as nowadays. The electrodynamic light theory of Lorenz and its phenomenological predecessor were well-known in the 1870s since the important works of Lorenz in this field were all published in some of the most acknowledged international journals (Annalen der Physik and the Philosophical

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Magazine). Maxwell himself knew Lorenz' electrodynamic theory of light and referred to it in 1868 (Maxwell [1868]), and in 1873 (Maxwell [1954]). In the paper from the Philosophical Transactions of the Royal Society of London (Maxwell [1868], p. 652), a note on the electromagnetic theory of light mentions the Riemann [1867] and Lorenz [1867b] papers from the Annalen der Physik. The comment as to Lorenz' paper is the following: "The second paper, by M. Lorenz, shows that, in Weber's theory, periodic electric disturbances would be propagated with a velocity equal to that of light. The propagation of attraction through space forms part of this hypothesis also, though the medium is not explicitly recognized." It might be surprising that Maxwell emphasizes only a consequence of Lorenz' electrodynamic theory [cf eqs. (7.41)-(7.43)]. The close relation between the basic theory of Lorenz and his own, he apparently did not discover. The equivalence of the electrodynamic theories of Lorenz and Maxwell was not realized by any of the two, nor by any of their contemporaries. The greatness of Lorenz' electrodynamic theory was not realized when it appeared, nor during his lifetime, and apparently it was forgotten soon after 1867. Although Lorenz had struggled to establish a phenomenological theory of light relieved of all mechanical (physical) hypotheses concerning the microscopic (molecular) structure of matter this did not mean that Lorenz in any way denied the existence of molecules (atoms); on the contrary, as we shall see below. The necessity of releasing the theory of light from physical hypotheses alone came from the fact that the knowledge about the microscopic structure of matter was so limited and uncertain in his day. The springboard for joining his light theory with molecular aspects, he pointed out, was the space dependence of the light velocity appearing in his basic differential equations. In 1869 Lorenz carried out a study of the wavelength dependence of the refractive index which culminated in the establishment of a dispersion formula which we know under the name "the Lorentz-Lorenz relation". The original paper of Lorenz [1869] was written in Danish and appeared under the title "Experimentale og theoretiske Underogelser over Legemernes Brydningsforhold" (Experimental and theoretical investigations of bodies' refractivity). In a second paper published under the same title in the year 1875 Lorenz [1875] extends the experimental studies reported in his 1869 paper. The theoretical (and experimental) results reported in 1869 in the Danish journal were published in the Annalen der Physik und Chemie in 1880 under the title "Ueber die Refractionsconstante" (Lorenz [1880]). Thus, more than ten years passed before Lorenz' theory on the refractive index was presented to the

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international scientific community! The derivation he presents of the "LorentzLorenz relation" in 1880 deviates somewhat fi-om his original 1869 version, though the final relation is the same. In his own words he writes in 1880: "In zweiten Theile der ersten dieser Abhandlungen (Lorenz [1869]) habe ich die Theorie der „Refractionsconstante" entwickelt. Indem ich hier mit dieser Theorie den Anfang machen werde, will ich die Rechnung in wesentlich geanderte Gestalt wiedergeben, indem ich theils einige Missgriffe berichtigt, theils durch Vereinfachungen die ziemlich weitlaufigen Rechnungen bedeutend erleichtert habe." The Lorentz-Lorenz relation was discovered independently by the Dutch physicist H.A. Lorentz and published in 1880 in Annalen der Physik und Chemie (Lorentz [1880]). It is interesting to note that H.A. Lorentz knew the work of L.V Lorenz because in his famous book "The Theory of Electrons" he writes (in the 1952 Dover edition on p. 145) the following about the Lorentz-Lorenz relation: "This result had been found by Lorenz, of Copenhagen, some time before I deduced it from the electromagnetic theory of light, which is certainly a curious coincidence." H.A. Lorentz obtained his results in 1878 (Lorentz [1878, 1936]) on the basis of the light theory proposed by von Helmholtz (essentially the Maxwell theory), and he knew that "Lorenz of Copenhagen" had reached his result from "his own light theory" (Lorentz [1952]). It is suggestive that H.A. Lorentz did not try to establish a link between the electrodynamic theories of Maxwell and Lorenz despite the fact that both theories produced the same Lorentz-Lorenz relation. If the chronology of independent scientific discoveries is important, it might be more correct to name the famous formula "the Lorenz-Lorentz relation" instead of "the Lorentz-Lorenz relation" as is common practice. A formula analogous to the Lorenz-Lorentz relation for static fields was derived earlier by R. Clausius, and by O.F. Mossotti; see Clausius [1879] and Mossotti [1850]. In the following section, I shall briefly outline Lorenz' [1880] derivation of the Lorenz-Lorentz relation. Although Lorenz had sought to free his light theory from assumptions concerning the internal structure of matter, this did not mean in any way that Lorenz did not believe that condensed matter consisted of atoms (molecules). In fact he writes in the introduction to his 1869 paper on the refraction of light in media that the goal he has in mind is to obtain a better understanding of the internal molecular state of matter through studies of the refraction and scattering

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275

of light (Lorenz [1869]). When Lorenz wrote his 1869 paper experiments suggested that a relation of the form {n-\)v

= const

(8.1)

existed between the refractive index, n, and the specific volume (inverse mass density), v, or may be («^ - 1) t; = const. For a refractive index close to unity, the two relations essentially become identical. The empirical relation in eq. (8.1), sometimes known as the Gladstone-Dale formula (after the British scientists John H. Gladstone and Thomas Dale) to Lorenz lacked a theoretical justification and in his 1869 paper he regrets this lack. According to Cauchy [1830, 1836] the relation between the refractive index and the vacuum wavelength of light, AQ, could be expressed in terms of a series «(Ao) = m + ^ + ^ + . . - ,

(8.2)

and in 1864 Lorenz had shown that the Cauchy dispersion formula could be derived from his light theory by expanding the square of the space-varying phase velocity of light in a Fourier series, cf. § 6.5 on the chromatic dispersion and density dependence of the refractive index. In the long-wavelength limit the refractive index is given by «(Ao ^ oo) = m, and m is called the reduced refractive index. The challenge for Lorenz in 1869 was to try to determine theoretically the reduced refractive index, and in particular to settle whether m was a fiinction of only the specific volume of the substance in question. Under the assumption that the medium is isotropic and the molecules are spherical and invariable in magnitude Lorenz finds that 'w^ - 1 —:.—-V = const. (8.3) m^ + 2 For a substance consisting of only one kind of molecule he fiirther deduced the relation m^ + 2

\

v^ J

where P and k denote two constants that depend on the molecular structure but not on the volume (or temperature) of the substance. For a gas, where m usually is close to unity, so that m^ - \ '^ 2{m - 1) and m^ + 2 « 3, eq. (8.4) reduces to the Gladstone-Dale formula (at long wavelengths) provided u is sufficiently large.

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[3, § 8

Lorenz' calculation of the reduced refractive index received experimental support in 1880 when the Danish physicist P.K. Prytz published extensive experimental data showing that Lorenz' formula in general was better than the Gladstone-Dale formula (Prytz [1880]). It was Lorenz who instigated Prytz to undertake the investigation, and Lorenz also supplied Prytz with the necessary optical apparatus and arranged financial support from the Royal Danish Academy; see Kragh [1991]. 8.2. The simplified derivation of 1880 It appears from the introductory remarks to the 1880 paper in the Annalen der Physik und Chemie (Lorenz [1880]) that Prytz until then had investigated seventeen different liquids and their gases and that the experiments represented a continuation of experimental studies of the refractive index begun by Lorenz himself. The experimental data confirmed the theory of Lorenz, and this was stated by Lorenz to be the reason for presenting to the international scientific community the results of his 1869 paper (Lorenz [1869]) and an extension pubUshed in 1875 (Lorenz [1875]). Lorenz assumes that the medium under consideration is isotropic and consists of spherical molecules. Inbetween the molecules Lorenz presumes that the light propagates with the same speed as in vacuum. The reader should note here that Lorenz in a sense starts from a point of view very similar to that put forward by the Dutch physicist H.A. Lorentz in his programme of separating matter and ether. Again Lorenz' approach to electrodynamics is more closely related to that based on the microscopic Maxwell-Lorentz equations than that appearing in the original (macroscopic) Maxwell equations. The framework of Lorenz' calculation is his basic differential equations (6.41)-(6.43) for the components of the light vector ($, r/, t)[= (5, /?, C)], together with the assumption that the velocity of light inside the medium is a rapidly varying function of the space coordinates. With the abbreviations C = cos(co^ - q - r - d) and S = sin(ft;^-q r-d), Lorenz begins with the ansatz ? = (§o + ^2)C + §i^, r} = (rio + m)C + mS, t - ( C o + t2)C + Ci^,

(8.5) (8.6) (8.7)

where (^o,^o,?o) is a constant (space-independent) vector and (^i,//i,Ci) and (?2,^2,?2) are vectors varying periodically with r = (x,y,z). In writing the

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The discoveiy of the Lorenz-Lorentz relation

277

coefficients of S in eqs. (8.5)-(8.7), constant quantities are omitted. This is allowed since these, if originally present, could be removed by a proper choice of the origin of the coordinate system. In an isotropic medium one can assume without limiting the generality of the calculation that rjo = Co = 0 (proper choice of the orientation of the coordinate axes). Doing this, one also has ^ = 0 in the ^-vector, q = (i,m,n), since q • ($o, ^o,to) = 0 in the isotropic case. By inserting eqs. (8.5)-(8.7) (with m = ^o = 0 md i = 0) into eqs. (6.41)-(6.43) Lorenz obtains from eq. (6.41) in the long-wauelength limit, where m and n can be considered as infinitesimally small quantities, an equation

d_{d^_drk\_d^fd^_dJA^^^ dy \dy

dx)

dz \dx

dz )

(8.8)

'

and two analogous equations are found from eqs. (6.42) and (6.43). From these three equations Lorenz can conclude that the three periodic functions 52, r]2, and ^2 are the partial differential coefficients of a certain periodic function F = F{x,y,x), i.e. fdF dF dF\ (§2,^2,C2)-^,^,^ . \ox ay oz J

(8.9)

By a differentiation of eqs. (6.41)-(6.43) with respect to x, y and z, respectively, followed by an addition of the resulting equations, Lorenz showed that F{x,y,z) must satisfy the equation 1 / dF{r) dx \_v^{r) \so + dx

.d_f^dF(r)\d dy \v\r) dy )

f^dFVy dz \u\r) dz

^ (8.10)

If the ansatz in eqs. (8.5)-(8.7) is inserted into eq. (6.41) one gets {m'^n^)^,^I=(^\

(5o + 52),

(8.11)

where 2' is a sum of terms which spatial mean values are zero. By an integration over the specific volume v, the terms in 1 disappear and one obtains «2(Ao^oo)$o= | ( l + V^(i'))(?o + 5 2 ) ^ ,

(8.12)

where ^ 0 / 2 , ^2x1/2 «(Ao ^ oo) =-{m^ + rrf^

(8.13)

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Optical works of L V. Lorenz

[3, § 8

is the refractive index in the long-wavelength limit, and « ( r ) = ^ = [l + V^(r)]'/^

(8.14)

is the local refractive index. Since /($o + C2)dt;/t; = So, eq. (8.12) can also be written in the form (n\Xo^^)-l)^o = J Uo^^]

V^(r)^.

(8.15)

In the Lorenz model it is assumed there is vacuum inbetween the molecules of the medium, as already mentioned. This implies that \p(r) is different from zero only inside the molecules. In the following, integrations extending only over the volume of the molecules are denoted by a superscript (i) on the integral signs, and integrations over the volume exterior to the molecules bear the superscript (e). The associated specific volumes are denoted by U[ and UQ, respectively, and U[ -\- UQ = V. Since ^2 ^ dF/dx is a rapidly varying periodic function one has J(dF/dx) du = 0 and therefore _

c=

&^dF(r)du -— ox

J

f'''dF(r)du

= -/ UQ J

-— OX

. Ue

,_ . ,.

(8.16)

Lorenz at this point introduces a new function dq){r)/dx related to ^2 = dF{r)/dx via ^ ^ = c + (c + § o ) ^ ^ ,

(8.17)

where c is the constant defined in eq. (8.16). By combining eqs. (8.16) and (8.17) it is easy to show that the mean value of dq){r)/dx over the exterior volume is zero, i.e.

I I

^^'^ dcp{r) dv dX

0,

(8.18)

UQ

and that the integral of dcp{r)/dx over the molecular domain satisfies ^^"^ d(p(r) du _ dx

V

c c + ^Q

(8.19)

By combining eqs. (8.15) and (8.17)-(8.19), Lorenz obtains the result

(«u-oo)-i)(^i.y ^ - j = y '/'w(i+^j(8.20) The functions i/^(r) and dcp{r)/dx are not independent but linked via eq. (8.10) as one may see by means of eqs. (8.14) and (8.17), and for Lorenz the goal at

3, § 8]

The discovery of the Lorenz-Lorentz relation

279

this stage is to find the relation between the two integrals appearing in eq. (8.20). To this end he now uses the assumption that the molecules are spherical, and after a long and beautiftil calculation he obtains the following result:

By combining eqs. (8.20) and (8.21) Lorenz finally reaches the important relation

z i = _ /'

n^(Xo —> oo) + 2

J

dx

Since the right-hand side of eq. (8.22) is a constant, we have contact with the result cited in eq. (8.3). To simplify eq. (8.22) Lorenz assumed that the refractive index inside a given molecule is a position-independent quantity, ni. This means that yj = n^ -I, and eq. (8.21) hence leads to the resuh

du = — j — - U i ,

j

dx ^^

nl + l

(8.23)

and hence

n^(Xo ^ oc) + 2

V,.

nj-\-2

(8.24)

If the substance consists of a mixture of different molecules (molecular label: /), eq. (8.24) is replaced by n^{XQ ^ cx)) - 1 _ v ^ «^ - 1 .2(Ao-oo) + 2 ' 4 - ; f T 2 ' ' -

.g ^^. ^^'^^^

Equation (8.25) is the relation later to be known as the Lorentz-Lorenz relation. If Ni is the number of molecules of the /th type per unit volume, one has 4jr

Oi = -jr^N^v,

(8.26)

where r/ is the radius of the molecule of type /. If the polarizability concept, a,, is introduced, we have for a sphere of constant refractive index placed in a

280

Optical works ofL V. Lorenz

[3, § 9

homogeneous electric field the well-known formula (Rayleigh [1871a,b, 1891], Bornand Wolf [1999]) ai = 4jteorf'\^.

(8.27)

A combination of eqs. (8.25)-(8.27) allows us to write the Lorenz-Lorentz relation in the familiar form (van Kranendonk and Sipe [1977], Born and Wolf [1999])

n^ -\-2

3eo ^-^ i

where the reference to long wavelengths has been dropped. After the LorenzLorentz formula became generally known in 1880, it soon turned out to be an important tool in the new physical chemistry (Partington [1953]), and as the reader undoubtedly knows, the relation has been used in numerous studies in physical optics since then. As Lorenz had hoped, via his relation a bridge was made between the phenomenological theory of light and the atomistic theory of matter. Lorenz further claimed that his result also holds without the assumption that the molecules are spherical. Unfortunately, he does not go into any specific considerations on this point. He ends the theoretical part of his 1880 paper with a very interesting remark on the assumption that there is empty space between the molecules: "Dagegen bekommt die Annahme, dass die Geschwindigkeit des Lichtes in den Zwischenraum der Molecule diejenige des leeren Raumes ist einen leicht zu iibersehenden wesentlichen Einfluss auf das Resultat, in der Weise, dass man berechtigt sein wird zu schliessen, dass sie auch der Wirklichkeit entspreche, wenn es sich in der That zeigen solhe, dass die „Refractionsconstante" wirklich eine Constante sei."

§ 9. Light scattering by molecules and a sphere 9.1. The size of a molecule as it is seen by light To gain insight in the molecular (atomic) structure of matter by optical methods Lorenz understood that it was necessary to study the wavelength dependence of the refractive index, and the scattering of light from (small) particles. A first

3, § 9]

Light scattering by molecules and a sphere

281

Step in this direction was taken in 1875, when he investigated the wavelength dependence of the refractive index for substances assumed to consist of spherical molecules of radius e and with a constant refractive index no (Lorenz [1875]). Without presenting a proof Lorenz claimed that his analysis gave the following result:

where t;o is the specific molecular volume, and AQ is the vacuum wavelength. Lorenz is fully aware that the simplified assumptions (constant refractive index of the molecules and spherical molecules) under which eq. (9.1) is derived are not fiilfilled for a real medium, but as he writes in his own words (translated) on p. 493 in the article (Lorenz [1875]): "This result [eq. (8.4)] is independent of the form of the molecules. To take the further step of calculating the refractive index at an arbitrary wavelength would be very difficult. I have therefore only accomplished the calculation in the simple case where the molecules are spherical in form, " It is interesting to note that Lorenz does not literally consider e as the radius of the molecule. Thus, he writes (in translation) on p. 493 in the above-mentioned paper: "More generally one may consider e as a lower limit for the radius of the molecule's zone of influence [virkningssfcere in Danish], i.e. a spherical surface around a molecule inside which the influence of the molecule on the propagation velocity of light is appreciable." Since {n\ - \)/{n^ + 2) < 1, available experimental results (Lorenz cited for sodium light a value of 0.22 for the second term in the parenthesis of eq. 9.1) allow Lorenz to conclude that 6 > 1.5xl0^^m, an estimate he mentions is in good agreement with the experimental determination of Quincke who found e?^5xlO-^m. It is amazing that Lorenz as early as 1875 distinguished between the particle (electronic) size of the molecule and its optical size, i.e. the size "seen" by light (the zone in which the molecule has an influence on the speed of light). The question of the (effective) speed of light in the molecular (atomic) zone of influence is a deep one which apparently only has been studied in a rigorous manner, on the basis of microscopic electrodynamics in the first- and secondquantized formulations, in recent years (Keller [1998, 2000, 2001]). Briefly summarized the conclusion is as follows: Seen with "the eyes of the photon" the atom is as big as the extension of the transverse part, Jj{r, t), of the induced

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Optical works of L V. Lorenz

[3, § 9

atomic current density, J{r, t), responsible for the light emission (or absorption). The /f-distribution extends over the near-field (r"^) zone of the atom in an electric-dipole active transition, and the "Lorenz zone of influence" therefore typically is of the order of AQ/IO ~ 6xl0~^m for visible (red) light, in fact essentially the Lorenz-Quincke estimate! In the birth process of the photon, which one has been able to follow theoretically in space-time recently, the photon embryo is never better confined (in an electric-dipole transition) than to the nearfield zone of the atom, and once the photon is born (decoupled from the atom) the speed of light is CQ (the speed in vacuo). Within a time ~ 10"^^ s the photon is completely out of the "Lorenz zone of influence". Before the photon is ftilly born, it turns out to be impossible to introduce a genuine velocity of propagation, even for the coupled atom-photon system. This is so because the photon generation (or destruction) process in the quantum statistical sense is spread over the entire near-field zone. Within certain limitations it is possible though to introduce an apparent photon (embryo) velocity. I call this velocity apparent because it is composed of a statistically random photon generation probability to which no velocity at all can be attached plus a genuine propagation with speed co provided the photon is emitted. Certainty about emission (absorption) is obtained only when the atom is no longer electrodynamically active. Without violating the Einstein causality the apparent velocity of field propagation may be superluminal in the near-field zone. Since there is no genuine photon embryo speed it is impossible rigorously to speak of a refractive index in the "Lorenz zone of influence". An apparent velocity, when adequate to introduce, corresponds to a real refractive index which depends on the distance from the atom (molecule) and which is less than unity in superluminal cases. The whole situation is complicated fiarther by the fact that quantum electrodynamics has taught us that this apparent speed also depends on the measuring process itself To introduce the speed of light in vacuo in a manifest manner in the formalism sketched above, it is necessary to describe the entire process by means of electromagnetic propagators (Green's functions). The transverse (T) part of the electromagnetic vector field, which describes the photon generation process, hence splits into a self-field part Ef{r,t) = - ^

f

Mrj')At\

(9.2)

describing the quantum statistics of the photon birth process, and a part £f (r, t) = ^ j ^

Dl(r -r',t-1')

• ^^^^At'd'r',

(9.3)

3, § 9]

Light scattering by molecules and a sphere

283

which would disappear if one let CQ —> oo. The tensor DQ{R, r) [R = r - r', T = t-t'\ which one may call the photon-embryo propagator, is given by DI{R, r) =

AnR 2

(P-M).g-.) .

X

(9.4)

where U is the unit tensor and R = R/R. Although the effect of the space-like term containing the two Heaviside unit step functions 6(T) and 6(R/co - r) is cancelled by the longitudinal field belonging to the atom Hamiltonian (Keller [2001]) it is needed for consistency. Once the photon is born only the part of the transverse field which is associated with the well-known photon propagator (first term in eq. 9.4) is present. No genuine (even local) speed of propagation, say u{r), can be attached to the "Lorenz zone of influence" because eq. (9.4) cannot be rewritten in such a manner that it appears as a function of R- UT, alone. Lorenz notes in his 1875 paper that observations show that the quantity u(n^ - iy(n^ + 2) is almost constant (it changes slightly with volume and temperature), and he names it the refraction constant. For Lorenz this experimental fact, considered together with his theoretical expressions in eqs. (8.4) and (9.1) indicated that the molecules possess a pronounced unchangeability {uforanderlighed in Danish), a conclusion he also notes can be inferred more directly from the observation that the positions of the spectral lines of molecular gases essentially are uninfluenced by changes in temperature and density. The substantial changes occurring in the refractive constant, the colour dispersion, and the position of the spectral lines when different molecular substances are mixed chemically to Lorenz also indicated that changes in the molecular structure (the molecules themselves as he writes) are much more important for the optical properties than changes induced by the light itself In a paper (Lorenz [1883a]) bearing the Danish title "Farvespredningens Theori" (The theory of colour dispersion) and published in the Kongelige Videnskabernes Selskabs Skrifter in 1883, and in the same year translated into German and printed in the Annalen der Physik und Chemie (Lorenz [1883b]), Lorenz attempts to solve the colour dispersion problem in a systematic manner. Again he starts from his phenomenological light theory. He assumes that atoms (molecules) are pointlike entities and that the light velocity is a function of the distance from the atom up to a certain distance (the range of the "Lorenz zone of influence"). For larger distances the velocity is assumed to be as in

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[3, § 9

empty space. In his previous studies he had assumed the velocity of hght to be independent of position in the zone of molecular influence. He further assumes that the atoms are randomly distributed in the medium, yet in such a manner that the medium exhibits isotropy on a macroscopic length scale. He fixes the positions of the atoms, and he mentions that in doing this he neglects the circumstance that thermal motions of the atoms and motions induced by the light itself probably exist. He doubts that it is necessary to include the thermal atomic motions which have speeds much less than CQ. AS far as the induced motions are concerned he claims that it would be most important to include the vibrations induced in the "zone of influence" and only later, as a finer detail, the center-ofmass motion. This indeed was a far-reaching conclusion. To establish the set of equations which describes the light vector inside the "Lorenz zone of influence" he divides the zone into concentric spherical shells. In each shell he assumes the light velocity to be constant, and he matches the light vector at the boundary between shells according to the boundary conditions implicitly inherent in the differential equations for the light vector, cf the method used in his 1863 papers (Lorenz [1863a,b]). He claimed that, if necessary, the calculation could also be carried out in the limit where the thicknesses of the shells approach zero. The refractive index is assumed to diminish with the distance from the atom, and to approach infinity when the distance goes to zero. The effective light velocity hence increases from zero towards the vacuum value when the distance from the atom increases. Although not correct, it must have seemed reasonable in Lorenz' time to assume that one could introduce a local speed of light and that this could be zero at the atomic positions! Lorenz knew that a plane polarized light wave injected into a medium would exhibit (periodic) amplitude modulations from atom to atom, and to study this modulation in greater detail he analyzes the situation where the propagation takes place between two atoms each possessing a near-field zone where the light velocity is a fiinction of the distance from the atom. There is no reason here to discuss Lorenz' mathematically difficult calculation since this would lead us far away from the central physics, and because no important physical conclusions seem to appear from this work. The theory ultimately leads to the relation

The result in eq. (9.5) cannot account for the anomalous dispersion discovered shortly before (see, e.g., Partington [1953] and Born and Wolf [1999] and references therein). To capture this, the light-induced motion of the "Lorenz zone of influence" must be included, e.g., in the framework of a harmonic oscillator

3, § 9]

Light scattering by molecules and a sphere

285

model (van Kranendonk and Sipe [1977]), a motion which Lorenz himself in fact had suggested might be of importance!

9.2. The Lorenz-Mie scattering theory In 1863, A. Clebsch published a beautiful paper in which he developed a theory required to describe the scattering of light from a completely reflecting spherical surface (Clebsch [1863]). The memoir by Clebsch was completed in October 1861, and his calculations were based on the elastic theory of light. In the elastic theory of light appear, as we have seen, both transverse and longitudinal waves and at an interface these are coupled. In Clebsch's theory the spherical surface is assumed to be perfectly rigid which means that the elastic displacement here must be zero. Despite the fact that Clebsch's paper was published in Crelles Journal, which was one of the most important mathematical journals in the middle of the nineteenth century, it was neglected by later writers, except for Lorenz, but as we shall see, Lorenz work in this field was destined to suffer the same fate as Clebsch's remarkable paper. Lorenz started his work on optical scattering from a spherical particle in 1885, and it was completed in 1890, the year before his death, with the publication of his great paper on the scattering of a plane wave from a homogeneous, isotropic and transparent sphere. Lorenz' paper was written in Danish and appeared under the title "Lysbevaegelsen i og uden for en af plane Lysbolger belyst Kugle" (Upon the reflection and refraction of plane light waves by a transparent sphere) (Lorenz [1890]). The paper was translated into French only in 1896 where it appeared under the title "Sur la lumiere reflechie et refractee par une sphere transparente" in his collected works "Oeuvres Scientifiques de L. Lorenz" (Valentiner [1896-1904]). According to Logan [1965] the English scientists Nicholson and Macdonald were greatly influenced by Lorenz' 1890 memoir, and make many references to it. Yet, in spite of the use of (parts of) Lorenz' paper in their own studies nothing in their writings indicate that they realized that Lorenz had given the exact solution for plane-wave scattering from a dielectric sphere. When Mie [1908] published his "classic" paper on the sphere-scattering problem in 1908, he apparently did not know of the 1890 paper by Lorenz, but he refers to Lorenz' 1880 paper "Ueber die Refractionsconstante" (Lorenz [1880]). Apart from the fact that Mie assigns to both the sphere and the surrounding medium a complex dielectric constant to account for absorption, his theory is identical to that of Lorenz published eighteen years earlier. In passing it is worth remarking that Walker in 1900 had already included in his analysis the case where the sphere

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[3, § 9

had a finite conductivity (Walker [1900]). Readers interested in the history of the scattering of plane waves by a sphere prior to World War II are urged to consult the interesting review by Logan [1965]. As we know, Mie was not the first to derive the solution to the sphere-scattering problem, but according to Logan [1965] Mie's paper is a good example of the fact that being first is not always what counts the most. Logan emphasized that Mie's paper caught the attention of his and later generations because he applied the results of his thorough paper to an interesting practical problem. Mie was concerned with the explanation of the colours displayed by colloidal metal suspensions. He started an ambitious computing program, in which he made new calculations involving the summation of several partial waves, thereby being able to obtain numerical results for spheres too large to be handled by the first partial wave, the Rayleigh scattering (Rayleigh [1881, 1899, 1899-1920]). A brief account of the history of the scattering of light from a sphere appeared in the book by Kerker [1969]. It has been suggested that the theory of plane-wave scattering of light from a sphere (should) be called the "Lorenz-Mie theory", instead of the "Mie theory". From a historical point of view this would be reasonable. Lorenz opens his 1890 paper with acknowledgement of Clebsch's memoir of the reflection of elastic waves from a rigid spherical surface. Lorenz starts from his own theory of light from 1863, expressed by means of the differential equations in eqs. (6.41)-(6.43), or in modern vectorial notation by the wave equation for the electric field given in eq. (6.54), and by doing so the longitudinal mode is abandoned from the outset. In Clebsch's paper extensive use of Cartesian coordinates is used and this in itself makes his paper difficult to read. Lorenz on the other hand immediately simplifies the problem by working in spherical coordinates. The polar axis is placed in the x-direction, so that x = r cos (p, y = rsinq)cos ip, z = rsm0sin \p, where (f and ip are the polar and azimuth angles, respectively. In the polar coordinate system Lorenz denotes the light vector, E, by {Er,E(i-,Ey,) = (5,^, t), and first he shows from his fiandamental differential equations that four quantities,

_

^

dirndl dr

d(rl) d(p'

dr

1 g| sin q)d(p'

must be continuous across the (sharp) spherical interface. In modern notation this is the same as the requirement that the tangential components of the electric (£"11) and magnetic {B\\ = (io;)~'(V x E)\\) fields must be continuous, as required by Maxwell equations in the case where the interface carries no currents. Remembering that the wave equation of Lorenz is identical to the one derived from the Maxwell equations, the boundary conditions must of course be identical. After

3, § 9]

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287

establishing in spherical coordinates the appropriate boundary conditions for the light vector at the interface, Lorenz returns for a while to the Cartesian coordinates. The incoming field, EQ = (^o, ^o, Co), he takes as a plane, monochromatic wave propagating in the positive x-direction, polarized along the ^-direction and normalized to unit amplitude, i.e., (£o,^o,Co) = (0,exp[i(a;^ - ix)],0). The total field outside the sphere he writes as £" = EQ + E^ or equivalently (5, rj, C) = (So, ^0, to) + (
(9.6)

which can be done since V • EQ = 0, and then he divides a into unknown components parallel and perpendicular to the radial direction, viz., a = rS-\-rxVQ.

(9.7)

With this choice Lorenz' fundamental wave equations for the scattered field components are satisfied if V ' e + ^ ' e = 0,

(9.8)

\/^S + fS = 0.

(9.9)

Similar equations are given for Q' and S\ Thereafter, Lorenz presents expressions for (le,7/e,£e) and (t.V^t) in terms of (K = £Q,S) and (K' = ^'Q\S'). With the definition a = ir this leads to

|,.«|P+«^, aocpoa

(9.10,

smq) oilJ

asmq) oaoip

ocp

and with a' = tr similar expressions for the internal field. Lorenz next expresses the incident field in the same manner in terms of the quantities

288

Optical works of L V. Lorenz

[3, § 9

(Ko,So) = (iQo,So). Next, he gives explicit series expansions expressions of (K,S), {K',S'), and {KQ.SO) in terms of the associated Legendre functions and the spherical Bessel and Neumann fiinctions. Finally, he reaches his goal by using the boundary conditions expressed in spherical coordinates. There is no need (or room) here to follow in detail the beautiful calculation of Lorenz. He obtains exactly the same result as given in a number of modem textbooks, e.g., that of Born and Wolf [1999]. Since his approach is so strikingly modern in every aspect it will be just as easy for the reader to follow the 1890 derivation of Lorenz as the derivation presented in the best new books. The Lorenz paper on light scattering from a sphere is sixty pages long, but the exact solution for the sphere is obtained within ten pages. The remainder of the article is devoted to analyses of the solution. Material of both physical and mathematical interest is found here, i.e., among other things the following: derivations of ways to sum series and evaluate integrals, new asymptotic estimates for Bessel functions, a calculation of the total scattered intensity of light, and not least, studies of the two cases in which the radius of the sphere is much larger and much smaller than the optical wavelength. For large spheres he calculated the intensities of the primary and secondary rainbows (he prepared a more detailed manuscript on the theory of the rainbow, but that was never published; see Kragh [1991]). Of particular interest is his analysis of the small sphere. He finds that the total scattered intensity relative to the incident intensity (denoted by L in his paper) is given by 128;r^ R^ f N^ - 1 where N is the refractive index of the sphere relative to the surroundings. The result in eq. (9.13) depends on the wavelength as AQ"^. Starting from the mechanical theory of light Rayleigh found two decades earlier the same wavelength dependence (Rayleigh [1871b]) and Lorenz cited Rayleigh's result in his paper. Using the electromagnetic theory of light Rayleigh [1881] later obtained precisely the result in eq. (9.13). Lorenz does not cite Rayleigh's 1881 paper in his own work, however. If, in a gas, one has A molecules per unit volume, the attenuation coefficient (related to scattering) of light would be h = AL, provided the spheres scatter independently of each other (incoherent scattering). Since the specific volume is Ui = ^R^Au, Lorenz used eq. (8.24) to give the following expression for h in terms of the refractive index (at long wavelength) and A:

24^/nU-^oo)-iy

3, § 10]

Lorenz and the aether

289

In the case where «(Ao —> co) is only sHghtly larger than unity, eq. (9.14) is reduced exactly to the result for h obtained by Rayleigh [1899] about ten years later. The priority for deriving this famous relation (eq. 9.14, possibly with « ^ 1) between the attenuation coefficient, the refractive index, the density of particles (molecules), and the optical wavelength thus must go to Lorenz and not to Rayleigh. From eq. (9.14) Lorenz is able to estimate the number of gas molecules per milliliter (the Loschmidt number ~ ^4^00 ^^ Avogadro's number). For atmospheric air at standard pressure he obtains A c^ 1.63x10'^ molecules/millimeter, i.e. a result too small, yet of the right order of magnitude. From the inequality {N^ - \)/{N^ + 2) < 1, Lorenz is able to estimate that the particle (molecular) radius is larger than ~ 1.4x 10"^ m. The theory of scattering of light from a spherical particle was Lorenz' last major paper, and as Logan [1965] says: "This was truly one of the most remarkable memoirs to be published in the 19th century. The history of scattering theory has been greatly enriched by the existence of this paper. This was Lorenz' last major memoir, but it alone should have been sufficient to have made him one of the famous figures in the last century."

§ 10. Lorenz and the aether When Lorenz on 28 June 1860 finished his Annalen der Physik paper on the bending of light at an opening in a screen (Lorenz [1860b]) he had apparently come to the conclusion that the vibrations of the aether took place in a direction perpendicular to the plane of polarization, and he finishes his paper by saying that this result confirms what was previously found. It is common practice in science to compare a result one has obtained with the findings of other researchers, but I have the feeling that the only thirty-one year old Lorenz in this case also had to strengthen the believe in his own conclusion. After all, the paper marked his debut on the international scene. There is no direct evidence for this point of view, but we know (Lorenz [1860d]) also from the Danish version of the paper (Lorenz [1860a]) that Lorenz had already realized that the experimental results of Jamin could be completely explained on the basis of the Fresnel formulae for reflection and transmission without assuming that longitudinal aether vibrations were formed at the interface. In his description of the bending of light he also proves that no longitudinal waves are generated in the process if the incident wave is transversely polarized. So, the dilemma must have been in his mind in the summer of 1860. He had no need for longitudinal

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[3, § 10

waves from a mathematical point of view, and yet from a physical point of view a medium seemed needed to support the vibrations of light. Lorenz' Annalen der Physik paper on the reflection of light at the boundary of two isotropic transparent media (Lorenz [1860d]) is also dated "Copenhagen, June 28, 1860" but neither in this paper nor in its Danish version (Lorenz [1860c]) does Lorenz mention the aether. A year after, when Lorenz' article entitled "Bestimmung der Schwingungsrichtung des Lichtathers durch die Reflexion und Brechung des Lichtes" appeared in Annalen der Physik (Lorenz [1861b]), he was not yet prepared to abandon the presence of the aether. In the introduction to this paper, dated "Copenhagen, 28 June, 1861", he emphasizes that the goal is to reach a conclusion concerning the direction of the aether vibrations: "Um den endlichen Schluss riicksichtlich der Schwingungsrichtung des Lichtathers Ziehen zu konnen, habe ich ...." Although the careflal analyses of the reflection and transmission of light at an interface on the basis of the elastic aether theory had ended up with his gnawing doubt in the elastic theory of light, he still concludes: "Daraus folgt nun weiter, dass die Schwingungen des Lichtathers senkrecht zur Polarisationsebene sindr With the successflil construction of a phenomenological theory of light in 1862-63, Lorenz achieved a goal which he himself had designated in the following manner (quotation from the introduction to Lorenz [1863b]): "I have therefore tried to develop the theory of light with the smallest possible number of hypothetical assumptions, whether in regard to the nature of light itself, to that of the luminiferous medium, or to that of material bodies, ...," and immediately after follows the conclusion: "and it will appear, as the result of the present investigation, that an essential part of the ordinary physical hypotheses are not needed for the explanation of the phenomena of light, inasmuch as the theory is capable of being carried through in a manner different from that which has been hitherto followed in the investigation of this subject, and consisting in the further development of the formal [Lorenz' italics] side of the theory." Although Lorenz does not state this explicitly in 1863, his phenomenological light theory must have convinced him that there was no need for the presence of a luminiferous medium (the aether). In his popular Danish article "Om Lyset" (about light) written in 1867 (Lorenz [1867d]) in several places he definitely

3, § 10]

Lorenz and the aether

291

abandons the aether hypothesis. Thus, he writes (in translation) the following conclusion about the aether on p. 8: "Altogether, it is a very unscientific course to invent a substance when its existence does not manifests itself in a much more definite manner." Just above this conclusion he emphasizes: "Besides [possible molecular movements] the assumption of the presence of an aether would be absurd, since this is a new substance, different fi-om the bodies, which only has been created, because one considered light in the same manner as sound, and therefore had to introduce a medium with extraordinary large elasticity and small mass density, just to be able to explain the large velocity of light." In his eminent 1867 article on the identity of the vibrations of light with electrical currents, he mentions his rejection of the aether hypothesis several times. Thus, in the English translation, Lorenz [1867c, p. 288] writes: "... for, apart fi-om the fact that this theory [he refers here to the elastic theory of Cauchy] necessitates the assumption of a special medium (the luminous aether, which moreover stands quite isolated and separate from any other observations or demonstrable connexion with other forces), even with this assumption, and the various hypotheses of Cauchy, it is scarcely possible to imagine a medium in which a wave-motion could travel without trace of longitudinal vibrations." On the last page Lorenz says: "The present general opinion regards light as consisting of backward and forward motions of particles of aether If this were the case, the electrical current would be a progressive motion of the aether in the direction of the (positive or negative) electrical currents. But it is impossible that the same equations which theory deduces for very small displacements from equilibrium should hold good for all kinds of displacement whatever; and it just follows from the whole of this investigation that the same equations hold for both cases. Light cannot, therefore, consist of vibrations of the kind hitherto assumed; and this last consequence of the theory of aether makes it untenable." This certainly is an unambiguous statement about the aether hypothesis. Within (less than) seven years, the consistent research program Lorenz had followed had led him from belief to disbelief in the existence of an aether. Towards the end of the paper Lorenz writes in a prophetic vision: "In this idea [here he refers to the idea that light in the interior of bodies may have the character of rotating vibrations] there is scarcely any reason for adhering to the hypothesis of an aether; for it may well be assumed that in the so-called vacuum there is sufficient matter to form an adequate substratum for the motion."

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[3

The fact that Lorenz, an autodidact by nature, developed his thinking essentially without any personal contact with the leading physicists in Europe may have helped to free his intellect from the aether idea already in 1867, if not before. Let me finish this description of Lorenz and the aether with the striking words with which Rosenfeld [1957] concluded his description of Lorenz' struggle to establish the electromagnetic theory of light: "Nothing could better illustrate how far the dialectical process which was going to transform the common view of the physical world by the end of the century had already accomplished itself in the mind of the lonely Danish thinker."

References Agranovich, VM., and V.L. Ginzburg, 1984, Crystal Optics with Spatial Dispersion, and Excitons (Springer, Berlin). Bethe, H.A., 1944, Phys. Rev. 66, 163. Bom, M., and E. Wolf, 1999, Principles of Optics (Cambridge University Press, London). Bouasse, H., 1893a, Ann. Chim. Phys. 28, 145. Bouasse, H., 1893b, Ann. Chim. Phys. 28, 433. Bouwkamp, C.J., 1950, Philips Res. Rep. 5, 321. Bouwkamp, C.J., 1954, Rep. Prog. Phys. 17, 35. Cauchy, A.L., 1830, Bull. Sci. Math. 14. Cauchy, A.L., 1836, Memoire sur la Dispersion de la Lumiere (Nouv. exerc. de Math., Societe Royale des Sciences de Prague, Prague). Clausius, R., 1879, Mechanische Varmetheorie, Vol. 2 (Braunschweig). Clebsch, A., 1863, J. Reine Angew. Math. 61, 195. Cohen-Tannoudji, C , J. Dupont-Roc and G. Grynberg, 1987, Photons and Atoms, Introduction to Quantum Electrodynamics (Wiley-Interscience, New York). Drude, P, 1889, Ann. Phys. (Wiedemann) 31, 327. Drude, P., 1959, The theory of optics (Dover, New York). Ewald, PP, 1962, Fifty Years of X-ray Diffraction (Oosthoek, Utrecht). Fresnel, A., 1823, Mem. de I'Acad. des Sciences 11, 393. Jamin, J., 1850, Ann. Chim. Phys. 29, 263. Keller, O., 1988, Phys. Rev. B 37, 10588. Keller, O., 1995, J. Opt. Soc. Am. 12, 987. Keller, O., 1996, Phys. Rep. 268, 85. Keller, O., 1997a, in: Progress in Optics, Vol. XXXVII, ed. E. Wolf (North-Holland, Amsterdam) p. 257. Keller, O., 1997b, in: Quantum Optics and the Spectroscopy of Solids, Concepts and Advances, eds. T. Hakioglu and A.S. Shumovsky (Kluwer, Dordrecht) p. 1. Keller, O., 1998, Phys. Rev. A 58, 3407. Keller, O., 2000, Phys. Rev. A 62, 022111. Keller, O., 2001, J. Opt. Soc. Am. B18, 206. Kerker, M., 1969, The Scattering of Light and Other Electromagnetic Radiation (Academic Press, New York).

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Kirchhoff, G., 1882a, Gesammelte Abhandlungen (Johan Ambrosius Barth, Leipzig): p. 131 [Ueber die Bewegung der Elektricitat in Drahten (1857)]; p. 154 [Ueber die Bewegung der Elektricitat inLeitem(1857)]. Kirchhoff, G., 1882b, Berl. Ber., p. 641. Kirchhoff, G., 1883, Ann. Phys. Chem. (Wiedemann) 18, 663. Kottler, F., 1923a, Ann. Phys. 71, 457. Kottler, R, 1923b, Ann. Phys. 72, 320. Kragh, H., 1991, Appl. Opt. 30, 4688. Logan, N.A., 1965, Proc. IEEE 53, 773. Lorentz, H.A., 1878, Verh. K. Ned. Akad. Wet. Amsterdam 18, 1. Lorentz, H.A., 1880, Ann. Phys. Chem. (Wiedemann) 9, 641. Lorentz, H.A., 1936, Collected Papers, Vol. II (Martinus Nijhofif, The Hague). Lorentz, H.A., 1952, The Theory of Electrons (Dover, London). Lorenz, L., 1860a, Skand. Naturf Forh. 8, 473. Lorenz, L., 1860b, Ann. Phys. (Pogg.) I l l , 315. Lorenz, L., 1860c, Math. Tidskr. 2, 116. Lorenz, L., 1860d, Ann. Phys. (Pogg.) I l l , 460. Lorenz, L., 1860e, Philos. Mag. 21,481. Lorenz, L., 1861a, J. Reine Angew. Math. 58, 328. Lorenz, 1861b, Ann. Phys. (Pogg.) 114, 238. Lorenz, 1862, Tidsskr. Phys. Chemi 1, 193. Lorenz, 1863a, Ann. Phys. (Pogg.) 118, 111. Lorenz, 1863b, Philos. Mag. 26, 81. Lorenz, L., 1863c, Skand. Nat.-Forskare Salsk. Mote, p. 225. Lorenz, L., 1864a, Ann. Phys. (Pogg.) 121, 579. Lorenz, L., 1864b, Philos. Mag. 28, 409. Lorenz, L., 1867a, K. Dan. Vidensk. Selsk. Skr. 1, 26. Lorenz, L., 1867b, Ann. Phys. (Pogg.) 131, 243. 1867c, Philos. Mag. 34, 287. Lorenz, 1867d, Tidsskr. Phys. Chem. 6, 1. Lorenz, 1869, K. Dan. Vidensk. Selsk. Skr. 8, 203. Lorenz, 1875, K. Dan. Vidensk. Selsk. Skr. 10, 483. Lorenz, 1877, Die Lehre vom Licht (Teubner, Leipzig). Lorenz, 1880, Ann. Phys. Chem. (Wiedemann) 11, 70. Lorenz, 1883a, K. Dan. Vidensk. Selsk. Skr. 2, 165. Lorenz, Lorenz, L., 1883b, Ann. Phys. Chem. (Wiedemann) 20, 1. Lorenz, L., 1890, K. Dan. Vidensk. Selsk. Skr. 6, 1. Maxwell, J.C, 1868, Philos. Trans. R. Soc. London 158, 643. Maxwell, J.C, 1954, A treatise on Electricity and Magnetism, Vol. 2 (Dover, New York) p. 450. Meyer, K., 1938, Lorenz, Ludvig Valentin, in: Dansk Biografisk Leksikon, Vol. 14 (Schultz, Copenhagen) p. 464. Mie, G., 1908, Ann. Phys. 25, 377. Mossotti, O.F, 1850, Mem. Soc. Sci. Modena 14, 49. Neumann, F., 1835, Abh. Berl. Akad. Math. Kl. 1. Partington, J.R., 1953, An Advanced Treatise on Physical Chemistry, Vol. 4, Physico-Chemical Optics (Longmans, Green, London). Penrose, R., and W Rindler, 1984, Spinors and Space-Time, Vol. I (Cambridge University Press, London) p. 321. Pihl, M., 1939, Der Physiker L.V Lorenz (Munksgaard, Copenhagen).

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Pihl, M., 1972, Centaurus 17, 83. Pihl, M., 1973, in: Dictionary of Scientific Biography, ed. C.C. Gillispie (Scribner's, New York) p. 501. Pinsker, Z.G., 1978, Dynamical Scattering of X-Rays in Crystals (Springer, Berlin). Prytz, PK., 1880, Ann. Phys. Chem. (Wiedemann) 11, 104. Rayleigh, Lord (J.W. Strutt), 1871a, Philos. Mag. 16, 274. Rayleigh, Lord (J.W. Strutt), 1871b, Philos. Mag. 41, 447. Rayleigh, Lord (J.W. Strutt), 1881, Philos. Mag. 12, 81. Rayleigh, Lord (J.W. Strutt), 1891, Philos. Mag. 43, 259. Rayleigh, Lord (J.W Strutt), 1899, Philos. Mag. 47, 375. Rayleigh, Lord (J.W. Strutt), 1899-1920, Scientific Papers (University Press, Cambridge). Riemann, B., 1867, Ann. Phys. (Pogg.) 131, 237. Riemann, B., 1892, Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, ed. H. Weber (Leipzig) p. 293. Reprinted: (Dover, New York). Rosenfeld, L., 1957, Nuovo Cimento, Suppl. 4, 1630. Sommerfeld, A., 1896, Math. Ann. 47, 317. Stokes, G.G., 1850, Fortschr. Phys. 1850-51, p. 349. Valentiner, H., ed., 1896-1904, Oeuvres Scientifiques de L. Lorenz, Vols. 1, 2 (Lehman and Stage, Copenhagen). van Bladel, J., 1991, IEEE Trans. Antennas Propag. 33, 69. van Kranendonk, J., and J.E. Sipe, 1977, in: Progress in Optics, Vol. XV, ed. E. Wolf (North-Holland, Amsterdam) p. 245. von Helmholtz, H., 1859, J. Math. 57, 7. Walker, G.W, 1900, Quart. J. Math. 31, 36. Weber, W, 1893-94, Werke (Berlin): p. 25, Vol. Ill [Elektrodynamische Massbestimmungen iiber ein allgemeines Grundgesetz der elektrischen Wirkung (1846)]; p. 609, Vol. Ill [(mit R. Kohlrausch) Elektrodynamische Massbestimmungen insbesondere Zuriickfuhrung der Stromintensitats-Messungen auf mechanisches Mass (1857)]; p. 105, Vol. IV [Elektrodynamische Massbestimmungen, insbesondere iiber elektrische Schwingungen (1864)]. Whittaker, E.T., 1952, A History of the Theories of Aether and Electricity, Vol. I, The Classical Theories, revised and enlarged edition (Nelson and Sons, London). First edition: 1910.

E. Wolf, Progress in Optics 43 © 2002 Elsevier Science B. V. All rights reserved

Chapter 4

Canonical quantum description of light propagation in dielectric media* by

A. Luks and V Pefinova Laboratory of Quantum Optics and Research Center for Optics, Faculty of Natural Sciences, Palacky University, Tfida Svobody 26, 771 46 Olomouc, Czech Republic

This article is dedicated to Professor Jan Pei^ina on the occasion of his 65th birthday. 295

Contents

Page § 1. Introduction

297

§ 2.

Origin of the macroscopic approach

304

§ 3. Macroscopic theories and their appHcations

323

§ 4.

385

Microscopic theories

§ 5. Microscopic models as related to macroscopic concepts . . . .

414

§ 6.

Conclusions

424

Acknowledgments

425

References

426

296

§ 1. Introduction The importance of quantum optics cannot be denied at present. Part of the investigations in this field can be interpreted as a quantum theory of nonhnear optics. Nonhnear optical effects are proportional to the nonlinear optical susceptibility (Boyd [1999, 2000]). Theories which have been used to describe the interaction of a quantized electromagnetic field with a nonlinear dielectric medium are either phenomenological or derived by quantizing the macroscopic Maxwell equations. The microscopic approach has become established as an alternative. Justification of the Hamiltonians used in the quantum theory of nonlinear optics is an important part of placing the theory on a firmer basis. The phenomenological (effective) Hamiltonians which were previously studied in quantum optics and still have their importance can in fact be derived in a "pre-quantal" form using the Maxwell equations. The question arises whether the quantization can be performed on a more fundamental level, say, on that of the quantized Maxwell equations in the Heisenberg picture or on that of an appropriate description in the Schrodinger picture, so that one can equally well or perhaps better arrive at an effective Hamiltonian. From the historical viewpoint, the problem of quantizing the electromagnetic field in vacuo was solved long ago by Dirac [1927], and the quantization of a nonlinear theory is due to Born and Infeld [1934, 1935]. With respect to propagation in linear dielectric media it is appropriate to refer first to Jauch and Watson [1948]. The revived interest in this problem in the last twelve years can be traced to some dissatisfaction with the situation following the advent of the laser in 1958. The new optical effects are analyzed both by nonlinear optical methods belonging to classical physics and by quantum optics (Shen [1969]). The normal-mode-expansion approach used in quantum optics is well-suited to systems in optical cavities, such as an optical parametric oscillator, but it is not appropriate for open systems such as a parametric amplifier. In nonlinear optics (Bloembergen [1965], Shen [1984]), the Maxwell equations completed by the constitutive relations are solved utilizing the slowlyvarying-envelope approximation, and the resultant equations are sometimes simplified using parametric approximation. In one particular case, this led to a quantum-optical theory of counterpropagation without introducing an effective 297

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Quantum description of light propagation in dielectric media

[4, § 1

Hamiltonian, which is usual in the case of copropagation (Milburn, Walls and Levenson [1984]). It has become standard practice in quantum optics to introduce phenomenological Hamiltonians without a quantitative connection to the classical equations describing nonlinear optical effects. This may be considered as unsatisfactory: while some processes occur in the classical regime, others in the quantum regime, the distinction is made only on the basis of the light intensity detected, and when the intensity changes continuously the boundary between the two domains cannot be determined. Philosophically, it would be desirable to replace any classical description with a quantum counterpart, but this goal has been considered either to be unreasonable or to have been achieved by the fathers of quantum theory in a form unsuitable for contemporary physicists. The quantization of the electromagnetic field in the presence of a dielectric is possible. This can be done in two ways: the macroscopic approach and the microscopic approach. In the macroscopic approach, the medium is completely described by its linear and nonlinear susceptibilities. No degrees of freedom of matter appear explicitly in this treatment. First a Lagrangian is sought which produces the macroscopic Maxwell equations for the field in a nonlinear medium; then the canonical momenta and the Hamiltonian are derived. Quantization is accomplished by imposing the standard equal-time commutation relations. In the microscopic approach, a model is constructed for the medium; degrees of freedom of both field and matter appear in the theory, and these are quantized. The result is a theory of mixed matter-field (polariton) modes, which are coupled by a nonlinear interaction. Hillery and Mlodinow [1984] used the electric displacement field as the canonical variable for nonlinear quantization, and they explored the macroscopic approach to the quantization of homogeneous nondispersive media. They pointed out that there is a difficulty in including the dispersion in the quantized macroscopic theory. The importance of a proper space-time description of squeezing has been recognized by Bialynicka-Birula and Bialynicki-Birula [1987]. The problem of a proper quantum-mechanical description of the operation of optical devices has been addressed by Knoll, Vogel and Welsch [1986, 1987]. In the past, many authors dealing with macroscopic quantum theories of light propagation wrote also on spatial displacements, shifts and translations of the electromagnetic field along with temporal displacements, shifts and translations, or simply the (time) evolution. Accordingly, they used the term "space evolution" in the former case. In the following, we will use the term space progression instead of space

4, § 1]

Introduction

299

evolution. Abram [1987] attempted to overcome the difficulties of conventional quantum optics by reformulating its assumptions. He based the formalism on the momentum operator for the radiation field and thus investigated not only the spatial progression of the electromagnetic wave, but also refraction and reflection. In addition to previous work devoted to the concept of quasinormal modes (Lang, Scully and Lamb Jr [1973], Barnett and Radmore [1988]), the modes of the universe have been used in the treatment of the spectrum of squeezing (Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990a,b]). A quantum description of linear optical devices is of interest provided it includes refraction and reflection. Similarly, a quantum description of nonlinear devices should include effects such as solitons and their dynamics. The dispersion has been treated on the assumption of a narrow-band field (Drummond [1990]). Besides this, an attempt at formulating a quantum theory of the propagation of an optical wave in a lossless dispersive dielectric material has been made by Blow, Loudon, Phoenix and Sheperd [1990]. These applications made use of the fact that nonlinear quantum-optical processes are described quantum-optically in the parametric approximation with linear mathematical tools, so that quantization procedures and solutions of the dynamics need not face such immense difficulties as with a really nonlinear formalism (Huttner, Serulnik and Ben-Aryeh [1990]). An original approach to the description of a degenerate parametric amplifier (Deutsch and Garrison [1991a]) has been related to the theory of paraxial quantum propagation (Deutsch and Garrison [1991b]). A formalism for the macroscopic approach to quantization was developed by Abram and Cohen [1991]. The space-time displacement operators have been related to the elements of the energy-momentum tensor (Serulnik and Ben-Aryeh [1991]). The macroscopic quantization of the electromagnetic field has been applied to inhomogeneous media (Glauber and Lewenstein [1991]). Huttner and Barnett [1992a,b] presented a fully canonical quantization scheme for the electromagnetic field in dispersive and lossy linear dielectrics. This scheme is based on the Hopfield model of such a dielectric, where the matter is represented by a harmonic polarization field (Hopfield [1958]). A microscopic theory of an optical field in a lossy linear optical medium has been developed (Knoll and Leonhardt [1992]). S.-T. Ho and Kumar [1993] have contributed to the derivation of the macroscopic field operators. They discussed the questions of light propagation across a dielectric boundary and of squeezing in a linear dielectric medium. Abram and Cohen [1994] developed a traveling-wave formulation of the theory of quantum optics and applied it to quantum propagation of light in a Kerr medium. Theoretical methods for investigating propagation

300

Quantum description of light propagation in dielectric media

[4, § 1

in quantum optics in which the momentum operator is used along with the Hamiltonian were developed by Toren and Ben-Aryeh [1994]. Drummond has presented a review of his theory and its applications (Drummond [1994]). Jeffers and Barnett [1994] modeled the propagation of squeezed light through an absorbing dispersive dielectric medium. A quantum theory of a field in 1+1 dimensions coupled to localized oscillators was developed and an analytic solution to the Heisenberg equation was given by Boivin, Kartner and Haus [1994]. Multimode consideration of nonclassical effects is needed when one wants to investigate the quantum fluctuations of light at different spatial points in the plane perpendicular to the propagation direction of the light beam (Kolobov [1999]). Matloob, Loudon, Barnett and Jeffers [1995] provided expressions for the electromagnetic field operators for three geometries: an infinite homogeneous dielectric, a semi-infinite dielectric and a dielectric slab. A microscopic derivation has shown that a canonical quantum theory of light at the dielectric-vacuum interface is possible (Barnett, Matloob and Loudon [1995]). Following Huttner and Barnett [1992a,b], Gruner and Welsch [1995] calculated the ground-state correlation of the quantum-mechanical intensity fluctuations. Dalton, Guerra and Knight [1996] dealt with the quantization of a field in dielectrics and applied it to the theory of atomic radiation in a one-dimensional Fabry-Perot resonator. Hradil [1996] considered "lossless" dispersive dielectrics, i.e., dielectrics with a thin absorption line. He formulated a canonical quantization of the electromagnetic field in a closed Fabry-Perot resonator with a dispersive slab. A simple quantum theory of the beamsplitter, which can be applied to a Fabry-Perot resonator, was introduced by Barnett, Gilson, Huttner and Imoto [1996] and developed by Barnett, Jeffers, Gatti and Loudon [1998]. Extensions of the previous work on the propagation in absorbing dielectrics took linear amplification into account (Jeffers, Barnett, Loudon, Matloob and Artoni [1996], Matloob, Loudon, Artoni, Barnett and Jeffers [1997], Artoni and Loudon [1998]). Artoni and Loudon [1997] applied the Huttner-Barnett scheme for quantization of the electromagnetic field in dispersive and absorbing dielectrics to calculations of the effects of perpendicular propagation in a dielectric slab and to the properties of the incident light pulse. Their approach has provided a deeper understanding of antibunching (Artoni and Loudon [1999]). Brun and Barnett [1998] considered an experimental set-up using a two-photon interferometer, where insertion of a dielectric into one or both arms of the interferometer is essential. Gruner and Welsch [1996a] performed an expansion of the field operators which is based on the Green fianction of the classical Maxwell equations and preserves the equal-time canonical commutation relation of the field. They found that the spatial progression

4, § 1]

Introduction

301

can be derived on the assumption of weak absorption. Dutra and Furuya [1997] considered a single-mode cavity filled with a medium consisting of twolevel atoms that are approximated by harmonic oscillators. They showed that macroscopic averaging of the dynamical variables can lead to a macroscopic description. Dutra and Furuya [1998a,b] observed that the (full) Huttner-Barnett model of a dielectric medium does not comprise all the dielectric permittivities of the medium expected from classical electrodynamics, although the field theory in linear dielectrics should have such a property. Schmidt, Jeffers, Barnett, Knoll and Welsch [1998] extended the microscopic approach to the quantum theory of light propagation to nonlinear media; a generalized nonlinear quantum Schrodinger equation well-known from the description of quantum solitons was derived for a dielectric with a Kerr nonlinearity Dung, Knoll and Welsch [1998] developed a quantization scheme for the electromagnetic field in a spatially varying three-dimensional linear dielectric which causes both dispersion and absorption. The well-known Green function was used for the case of a homogeneous dielectric, and it was shown that the indicated quantization scheme exactly preserves the fundamental equal-time commutation relations of quantum electrodynamics. The Green function has also been used in the more complicated case of two dielectric bodies with a common planar interface. The quantization of the full electromagnetic field in linear isotropic inhomogeneous Kramers-Kronig dielectrics based on the integral representation of the field with the Green tensor yields exactly the equal-time commutation relation of quantum electrodynamics (Scheel, Knoll and Welsch [1998]). Spontaneous decay of an excited atom in the presence of dispersing and absorbing bodies has been investigated using an extension of this formalism (Dung, Knoll and Welsch [2000]). Electromagnetic field quantization in an absorbing medium has been readdressed, and the Casimir effect both for two lossy dispersive dielectric slabs and between two conducting plates was analyzed by Matloob [1999a,b] and by Matloob, Keshavarz and Sedighi [1999]. A quantum scattering-theory approach to quantum-optical measurements has been expounded by Dalton, Barnett and Knight [1999a]. In addition to Lang, Scully and Lamb Jr [1973], and along with an independent work (K.C. Ho, Leung, Maassen van den Brink and Young [1998]) devoted to the concept of quasinormal modes, a quasimode theory of macroscopic canonical quantization was invented and applied by Dalton, Barnett and Knight [1999b,c,d]. A macroscopic canonical quantization of an electromagnetic field and a system of a radiating atoms, involving classical, linear optical devices, based on expanding the vector potential in terms of quasimode fimctions, was carried out by Dalton, Barnett and Knight

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Quantum description of light propagation in dielectric media

[4, § 1

[1999b]. The relationship between the pure-mode and quasimode annihilation and creation operators was determined by Dalton, Barnett and Knight [1999c]. A quantum theory of the lossless beamsplitter is given in terms of the quasimode theory of macroscopic canonical quantization. The input and output operators that are related via the scattering operator are directly linked to multi-time quantum correlation functions (Dalton, Barnett and Knight [1999d]). Brown and Dalton [2001a] have generalized the quasimode theory of macroscopic quantization in quantum optics and cavity quantum electrodynamics developed by Dalton, Barnett and Knight [1999a,b]. This generalization admits the case where two or more quasipermittivities are introduced. The generalized form of quasimode theory has been applied to provide a fully quantum-theoretical derivation of the laws of reflection and refraction at a boundary (Brown and Dalton [2001b]). Using the microscopic approach, Hillery and Mlodinow [1997] devoted themselves to the standard optical interactions, and derived an effective Hamiltonian describing counterpropagating modes in a nonlinear medium. On considering multipolar coupled atoms interacting with an electromagnetic field, a quantum theory of dispersion has been obtained whose dispersion relations are equivalent to the standard Sellmeir equations for the description of a dispersive transparent medium (Drummond and Hillery [1999]). The Green-fijnction approach to the quantization of the phenomenological Maxwell theory was used by Gruner and Welsch [ 1996b] in the derivation of the quantum-optical input-output relations in the case of propagation in dispersive absorbing multilayer slabs. The behavior of short light pulses propagating in a dispersive absorbing linear dielectric, with a special attention to squeezed pulses, was studied by Schmidt, Knoll and Welsch [1996]. Scheel, Knoll, Welsch and Barnett [1999] found quantum local-field corrections appropriate for the spontaneous emission by an excited atom. Knoll, Scheel, Schmidt, Welsch and Chizhov [1999] investigated quantum-state transformation by dispersive and absorbing four-port devices. Under the usual assumptions on the dielectric permittivity, quantization of the Hamiltonian formalism of the electromagnetic field using a method close to the microscopic approach was performed by Tip [1998]. A proper definition of band gaps in the periodic case and a new continuity equation for energy flow was obtained, and an ^-matrix formalism for scattering from absorbing objects was worked out. In this way the generation of Cerenkov and transition radiation have been investigated. A path-integral formulation of quantum electrodynamics in a dispersive and absorbing dielectric medium has been presented by Bechler [1999], and has been applied on the microscopic level to the quantum theory of electromagnetic fields in dielectric

4, § 1]

Introduction

303

media. Results concerning quantum electrodynamics in dispersing and absorbing dielectric media have been reviewed by Knoll, Scheel and Welsch [2001]. Tip, Knoll, Scheel and Welsch [2001] have proven the equivalence of two methods for quantization of the electromagnetic field in general dispersing and absorbing linear dielectrics: the Langevin noise current method and the auxiliary field method. In linear optical couplers both co- and counterpropagation have been considered, and an attempt at quantization was made by Pefinova, Luks, Kfepelka, Sibilia and Bertolotti [1991]. A theory of the electromagnetic field with the time axis replaced by one of the spatial axes was outlined. This replacement corresponds to replacing the Hamiltonian by a momentum operator (BenAryeh, Luks and Pefinova [1992]). The flexibility of the classical canonical description and the obstacles to quantization in the case of counterpropagation in a nonlinear medium were analyzed by Luis and Pefina [1996]. Di Stefano, Savasta and Girlanda [1999] extended the field quantization to those material systems whose interaction with light is described, near a medium boundary, by a nonlocal susceptibility. Suggestive is a comparative study of fermion and boson beamsplitters (Loudon [1998]). Fermions can be studied in analogy with bosons (Cahill and Glauber [1999]). Independently, the theory of light propagation in a Bose-Einstein condensate and a zero-temperature noninteracting FermiDirac gas has been developed (Javanainen, Ruostekoski, Vestergaard and Francis [1999]). It is appropriate to mention here work concerning the photon wave function (Bialynicki-Birula [1996a,b, 1998], Inagaki [1998], Hawton [1999], Kobe [1999]), although it is relevant mainly to the electromagnetic field in vacuo. In this chapter we will review spatio-temporal descriptions of the electromagnetic field in linear and nonlinear dielectric media, applying macroscopic and microscopic theories. We will treat both macroscopic theories appropriately generalized from the free-space quantum electrodynamics and microscopic theories as related to the macroscopic descriptions. We will deal with the quasimode theory of macroscopic canonical quantization and the scattering operator theory in this chapter. We refer to an excellent review of linear and nonlinear couplers by Pefina Jr and Pefina [2000], where the restriction to mere spatial behavior of interesting optical fields has been accepted. We will use units following the original papers, and although the system of international (SI) units prevails, there are exceptions: some of the relations (2.14)-(2.47) and (3.192)-(3.216) are in Gaussian units, relations (3.217)-(3.273) are in rationalized cgs units, and relations (2.1)-(2.13), (3.13)-(3.93) and (3.274)(3.337) are in Heaviside-Lorentz units.

304

Quantum description of light propagation in dielectric media

[4, § 2

§ 2. Origin of the macroscopic approach 2.1. Nondispersiue lossless homogeneous nonlinear dielectric The approach to a quantum theory of Hght propagation considered standard at least until the critique by Hillery and Mlodinow [1984], has been assessed by Drummond [1990, 1994] in a somewhat critical way. Concerning this approach, let us consider papers by Shen [1967, 1969]. The theory is restricted to steadystate propagation taking place in one dimension along the z-axis. On quantizing in a volume L^ and assuming that the field does not vary appreciably over a distance d large compared with the wavelength, and associating the discrete values of the wavevector k with d (instead of I), the localized annihilation and creation operators bk{z) and b\{z) have been proposed. An appropriate component of the vector potential operator has the expansion of the form

Mz,t) = cJ2J

2^^^13

{h(z)Qxp[-i{co^t - kz)] + H.c.Y

(2.1)

where z is the propagation distance, c is the speed of light, (O/, is the frequency, h is the Planck constant divided by 2jr, e^ = ^{(O/,) is the value of the dielectric function e at co^, and H.c. denotes the term Hermitian conjugate to the previous one. The annihilation and creation operators S^(z) and bl(z), respectively, satisfy the equal-space commutation relation

h(z)/bUz)

= 4A-'1.

(2.2)

The localized Hamiltonian and momentum operators H{z) and V(z) have been defined in terms of a Hamiltonian density H{z). It has not been specified whether these operators can always be written simply in terms of the localized operators bf,(z) and bjiz). Shen [1969] suggested that the momentum operator plays the role of the translation operator. The translation is interpreted as space "evolution" and appropriately described (cf eq. 2.29). We call this transformation a space progression. In analogy with the time-ordered product, the space-ordered product is introduced. An analogue of the Heisenberg and Schrodinger pictures is expounded. A localized density matrix (statistical operator) p(z) is defined. According to the pioneering paper of Hillery and Mlodinow [1984], the standard macroscopic quantum theory of electrodynamics in a nonlinear medium is due to Shen [1967] and has been elaborated upon by Tucker and Walls [1969]. Hillery and Mlodinow [1984] have pointed out some problems with the standard

4, § 2]

Origin of the macroscopic approach

305

theory, above all the fact that it is not consistent with the macroscopic Maxwell equations. One approach to the derivation of a macroscopic quantum theory would be to begin from a quantum-microscopic theory, as will be reviewed below. The other approach is to take the expression for the energy of the radiation in a nonlinear medium, which differs from the free-field Hamiltonian in part, and to keep interpreting the electric field (up to sign) as the variable canonically conjugated to the vector potential. (Note that this differs from Shen [1969].) In order to understand the problems of a previous theory, Hillery and Mlodinow examined its Hamiltonian formulation. This is the noncanonical Hamiltonian ^noncan(0 "^ ^ E M ( 0 + ^lnoncan(0?

(2-^)

where HEuit) is a free-field Hamiltonian and Anoncan(0 is an interaction Hamiltonian, ^EM(0 = \ j{E^+B^)d'x, ^lnoncan(0

=-

h p d ' x ,

(2.4) (2.5)

where E^ = E • E, with E = E{x, t) the magnitude of the electric field strength operator E{x, t), B^ = B B, with B = ^(jc, t) the magnitude of the magnetic induction field operator B{x,t), P = P{x,t) is the polarization operator of the medium, and we have used Heaviside-Lorentz units. The polarization of the medium is a fiinction of the electric field strength operator which may be written as a power series. It can be seen easily that, as an undesirable "quantum effect", we obtain an improper expression for the time derivative of the magnetic induction field operator B. Hillery and Mlodinow [1984] assumed that the medium is lossless, nondispersive and homogeneous. A Lagrangian is considered which gives proper equations of motion. The electric field E = E(x, t) and the magnetic induction field B = B{x, t) are expresssed in terms of the vector potential A ^ A{x, t) and the scalar potential AQ = Ao{x, t), dA E = ——-VAo, at

B^VxA.

(2.6)

The appropriate Lagrangian density depends on first partial derivatives of the four-vector A = A{xj) = {AQ,A). The momentum canonical to A is

Quantum description of light propagation in dielectric media

306

[4, §2

n = 77(jc, t) = (TTo, n), where TIQ = /7o(JC, t) = 0. The vanishing of i7o indicates that the system is constrained. It has been shown how to utiHze the quantization procedure developed by Dirac [1964] for constrained Hamiltonian systems. It can be derived that the canonical momentum is 77 = n(x,t) = -D, where D = D(x, t) is the electric displacement field. The canonical Hamiltonian has the form (2.7)

H(t) = HEM(t)^HM, where HEMit) = \

(2.8)

f(EE-^BB)d^x,

H,(t) - j E- \p-

j P{XE)(M

d^jc,

(2.9)

with P = P(jc, 0, P{E) = P[E{x, t),x, t] being the polarization of the medium. In order to simplify the quantization of the macroscopic Maxwell theory, the dual potential A = A(jc, t) has been introduced along with A = A(x, t) and Ao = Ao(jc, t)\ these we call the dual vector and scalar potentials, respectively. The relations in eq. (2.6) are replaced by Z) = V X A,

^^f.VAo.

(2.10)

It can be shown that the canonical momentum is 77^ = n^(x,t) = B. Finally, upon expressing the canonical Hamiltonian functional in terms of the electric displacement and magnetic induction fields the results are the same, H = H^(x, t) = H^. We can compare

AiixJXtlAx'A

=id^{x-x')\.

(2.11)

with A,(x,0,iT/(V,0

=id^{x-x')\.

(2.12)

where 6^{x) = d{x)6ij +

d" 1 dxidxj An\x\

(2.13)

are components of the transverse tensor-valued b function (Bjorken and Drell [1965]). Hillery and Mlodinow [1984] do not mention propagation, except in

4, § 2]

Origin of the macroscopic approach

307

a paragraph on interpretation problems, where they recommend to confine the medium to part of the quantization volume and to place the field source and the detector outside the medium, being aware that these require propagation to be taken into account. It is added that different diagonalizations indicated by the quadratic part of the total Hamiltonian generate different kinds of normal ordering. Doubt is expressed regarding the existence of a suitable ordering, and the microscopic approach is propounded. Dispersion is also considered a reason for contemplating a microscopic theory.

2.2. Nondispersive lossless linear dielectric 2.2.1. Momentum operator as translation operator In the late 1980s, the problem of propagation did not seem to be typical for quantum optics. Abram [1987] addressed the problem of light propagation through a linear nondispersive lossless medium. Although this model can be an appropriate limit of the Huttner-Bamett model, we expound the main ideas of (Abram [1987]). Abram criticized the modal Hamiltonian formalism, especially the inclusion of a linear polarization term in the Hamiltonian:

^(0-w-

8jr 7/ /(E'+H'-^4jTxE')dV,

(2.14)

where H^ = H - H, with H = H(x, t) the magnitude of the magnetic field strength operator H{x, t), x is the (linear) susceptibility of the material, and V is the quantization volume. This would lead to an incorrect result, mainly to a frequency change of modes which does not occur. Abram decided to extend the traditional theory of quantum optics in such a way that it could describe propagation phenomena without invoking the modal Hamiltonian; he observed that one of the propagation phenomena, refraction, suggests the momentum to be the appropriate concept for describing these phenomena. Quantum-mechanically, space and momentum are canonically conjugate variables. Huttner and Bamett [1992a,b] have demonstrated in a microscopic theory that a Hamiltonian including the light-matter interaction can be chosen. It is a good remedy against the idea that space and momentum are canonically conjugate variables like time and energy.

308

Quantum description of light propagation in dielectric media

[4, § 2

The propagation of the electromagnetic field is described by the Maxwell equations VxH

= - ^ , c at ^ ^ ^OB VxE =—^, c at V B =0,

(2.15) ,, ,,, (2.16) (2.17)

V-Z) = 0 ,

(2.18)

where the electric displacement field in the chosen system of units isD = E + 4jtP. It is assumed in the following that there are no free charges or currents and that we are dealing with nonmagnetic materials, so that B = H. For simplicity we consider only the case of plane waves propagating along the z-axis, with the electric field polarized along the x-axis, and the magnetic field along the 7-axis. This reduces the Maxwell equations to scalar differential equations, the directions of all vectors being implicit. Further it is assumed that light is propagating in a linear dielectric, where the induced polarization is at all times proportional to the incident electric field, P(z,t) = xE(z,t), where the susceptibility of the material is assumed for simplicity to be a scalar (neglecting its tensorial properties), independent of frequency (no dispersion). It is convenient to define also the dielectric function e of the material, e = 1 +4jrx, and the refi-active index, n = y^. The change in the total energy which is given by the integrated energy flux (the Poynting vector) over the surface of a body or volume is proposed in (Abram [1987]) as the proper quantum-mechanical Hamiltonian. The change in the total momentum is given as the integrated flux of the Maxwell stress tensor. The momentum is treated on the same footing as the Hamiltonian. However, the enigma of the Hamiltonian (2.14) is solved. It is possible to consider a square pulse which enters a dielectric. The total energy is conserved, but the energy density is increased by a factor of «, because the volume V reduces to F' = V/n. In volume V^ the wavelengths of the modes become A' = A/«, but the oscillator frequencies remain unchanged. It is interesting that in the absence of reflection, the electric and magnetic fields of the transmitted (T) waves in the dielectric are related to the corresponding incident (I) fields in free space by Ej(z, t) = -^Ei(z, t\

Hj(z, t) = V^mz,

t).

(2.19)

This change in the energy density implies a similar increase for the total momentum of the pulse, the components of which are always proportional to the

4, § 2]

Origin of the macroscopic approach

309

wavevectors of the excited modes. In propagation along the z-axis the Maxwell stress tensor is replaced by the energy density. When both directions of the propagation along the z-axis in free space are considered with the electric field polarized along the x-axis and the magnetic field along the>^-axis (x = 0,e= 1), the electromagnetic vector potential operator A = A(z, t) is usually written as i(z,0 = c Y , \ \ ^

(a/e'^'^'^-'^>" +a;e-*"'^^''-) ,

(2.20)

where Oj and hj are the creation and annihilation operators for a photon in the yth mode of wavevector kj (with k-j = -kj) and frequency (DJ = c\kj\ ftilfilling the boson commutation relations. In order to simplify notation, unit vectors are omitted. It is convenient to rearrange eq. (2.20) in a manner that is familiar to solid-state physicists, 2jr

i(z,0 = ^ E y ^

(a)e'^'^'^ + a,,e-'^'^'^) e-'^>-.

(2.21)

The electric and magnetic field operators may be obtained as

£(z, 0 = 4 | i ( . , 0 = E ^/ = - E V ^ ( ^ J - ^.')

(2.22)

H{z, t) = |i(z, 0 = E ^/ = -' E ^' V F ^*'^ ^ ^-'^'

^^-^^^

where Sj = sgny and bj = aj^'^'^'^-^'^K

(2.24)

When products of these operators are envisaged, it is supposed that they are symmetrized. The Hermiticity of the operators E = E{z, t) and H = H(z, t) can be verified using the relations e/ = e.j,

h] = Lj.

(2.25)

The energy density operator u = u(z, t) can be written as

'

'

= ^ E ^v {y]bi+^!,L,+i) = - ^ 0,, {b]bj+i i ) .

(2.26)

310

Quantum description of light propagation in dielectric media

[4, § 2

The energy densities u± = u±{z, t) due to the forward and backward waves alone can be expressed uniquely:

"^ = E f (b]k + iO '

«- = E f (b]k + ii) .

i(>0)

(2.27)

iX<0)

The total momentum operator G is then

J

Abram [1987] proposed the following equation of motion {h= 1): ^=-i[G,Ql

(2.29)

where Q is any operator. We would prefer to define the operator Q. Let us consider Q = Q(z, t) = Q[Eiz, t% Hiz, t)l

(2.30)

where |2[*, •] is a formal series in E and H. Since the differential operator ^ has the same formal algebraic properties as the superoperator -i[G, • ] , it suffices to verify the relation (2.29) for Q = E,H. This is true at least in the situations considered by Abram [1987]. Although the operators bj = bj(z,t) are studied using eq. (2.29), the Heisenberg equation of motion and the initial condition S,(0,0) = a„

(2.31)

as appropriate for any operator Q(z,t), we perceive that the operators do not obey our definition of the operator Q. We may calculate the Poynting vector operator as J

j

The Poynting vector operators due to the forward and backward waves alone can be expressed uniquely:

^ ^ = E f (^)^.-ii)' y(>0)

^ - - E f (^;^.-ii)-

(2.33)

i(
The total energy operator of the free field inside the volume of quantization is thus

n = u = ~(^s^-L^ = Y^ ^j {^h + f^)'

(2.34)

J Investigation of the case ^ ^ 0^ ^ ^ 1 does not lead to any new expansions of the field operators E and H. The individual components of the rearranged

4, § 2]

Origin of the macroscopic approach

311

electric and magnetic field operators according to eqs. (2.22) and (2.23) satisfy a modified operator algebra relative to that of the harmonic oscillator: [ej,ei] = [hjM

= 0,

[CjM = s^j ( ^ )

S.jjl

(2.35)

where djj is the Kronecker 6 function. The knowledge of these commutators and of the generalized total momentum operator G = Grefr obtained via appropriate modification of the relations (2.26) through (2.28), where, e.g., the relation (2.26) becomes

"=i(^^'^^^) = 8^E(-^-/-M-/),

(2.36)

enables one to derive the Maxwell equations via both the temporal derivatives and the spatial derivatives. The energy density operator (2.36) has been generalized to the quantization volume V that is entirely included in a homogeneous medium. In the expansion (2.26) we can set Uj = ii/refr, Uj refr

2V

[hytj + L/blj - 2JTX (b^ - b.^ (b\j - hi) ] .

(2.37)

The energy density operator iirefr may be diagonalized through a Bogoliubov transformation. To this end we introduce an anti-Hermitian operator R of the form

R = R{z, t) = Y, {bjb-j - b]P_^

(2.38)

J

and the operators Bj = Q-^%Q^^ = (cosh 7) bj - (sinh y) blj,

(2.39)

where y=\\ne={\nn.

(2.40)

Upon the substitution bj = Q^^BjQ-''^ = (cosh y)Bj + (sinh y)^!-,

(2.41)

312

Quantum description of light propagation in dielectric media

[4, § 2

the operator R takes the form R = Y,{BJB^J-BJB[,),

{IAD

j

and the energy density operator has the diagonal form «,refr = ^ ( 5 ; 4 + M - / ) -

(2-43)

The momentum operator is then given by G,efr = ^ (M+ - M-) = 5 1 ^/ (^/^/ + I ' ) '

^'^'^^^

J

with Kj = nkj, and the Hamikonian can be calculated as Wrefr = ^ 0 ; ; ( ^ / ^ , + i i ) .

(2.45)

By inserting eq. (2.41) into (2.22) and (2.23), respectively, we can obtain the electric and magnetic field operators inside the dielectric:

^(-'0 =-'E\/^(^/-^.')'

(2.46)

Hiz,t) = - i ^ . , Y ^ ( « t + ^ _ ^ ) .

(2.47)

Similarly as above, these relations can be interpreted as a result of the replacement bj \-^ Bj and a consequence of the quantized classical equations (2.19). For normal incidence on a sharp vacuum-dielectric interface, both reflection and diffraction occur. This more general case has also been treated by Abram [1987]. 2.2.2. Wave-functional description of Gaussian states Bialynicka-Birula and Bialynicki-Birula [1987] made the first attempt to define squeezing as a generalization of the standard definition for one mode of radiation. This definition can be reformulated referring to Bialynicki-Birula [2000]. The Riemann-Silberstein-Kramers complex vector has been introduced as

""•" = 7!

D{r,t)

.B{r,t)

(2.48)

where the division by yJT^, yjjx^ is appropriate for SI units. It has been shown how the Green-function method can be used for solving linear equations for the

4, § 2]

Origin of the macroscopic approach

313

field operator F(r, t). This approach allows the medium under investigation to be inhomogeneous and time dependent. It has been suggested that the periodicity of the electric permittivity tensor e(r, /) or the magnetic permeability ^^(r, t) can be important for the generation of squeezed states. Only the dispersion of the medium has not been considered. It has been derived that photon-pair production is a necessary condition for squeezing. It is tempting to generalize the concept of a Gaussian state of the finitedimensional harmonic oscillator to the case of an infinite oscillator. BialynickaBirula and Bialynicki-Birula [1987] treat the time development of the Gaussian states in the free-field case. The Schrodinger picture is adopted and an analogue of the Schrodinger representation in quantum mechanics is introduced. Let us recall the quadrature representation in quantum optics. This representation is a wave functional W[A,t]. Let us observe that, contrary to the operator A{r, t), the argument A{r) of the wave functional does not depend on t, but the wave fianctional does depend on t. The Hamiltonian in this representation has the form ^ 7 1^ €o dA(ry ^io J Bialynicka-Birula and Bialynicki-Birula [1987] presented the wave functional of the vacuum state, i.e., the simplest Gaussian state of the electromagnetic field, as well as that of the "most general" Gaussian state. Thus, the exposition is confined to pure Gaussian states while it is possible to generalize it also to mixed Gaussian states of the electromagnetic field. The pure Gaussian state is determined by a complex matrix kernel, i.e., by two real matrix kernels. It is shown that the expectation values {B) = B and {D) = V (equivalently, {E) = £) evolve according to the free-field Maxwell equations; in addition, the equations for the matrix kernel W{r,r'\ t) can be found there. The entire electromagnetic field is treated as a huge infinite-dimensional harmonic oscillator. The wave function and the corresponding Wigner function then become fiinctionals of the field variables. Mrowczyhski and Miiller [1994] have considered only the scalar field. Bialynicki-Birula [2000] starts from the wave ftinctional (Misner, Thorne and Wheeler [1970]) <^oM = Cexp

^

^//fiW—L^.5(/)d^.dVl Ih J J \r-r'Y J

(2.50)

and from the wave functional (we change A -^ -D) ^ o [ - ^ ] = Cexp

1

[^

f

/•^(^)_L^.^(/)d3;.dV (2.51)

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Quantum description of light propagation in dielectric media

[4, § 2

The normalization constant C is an issue not completely solved by BialynickiBirula [2000]. The analogy with the one-dimensional harmonic oscillator leads to other notions. The Wigner functional of the electromagnetic field in the ground state is Wo[A,-D] = Qxp{-2N[A,-D]},

(2.52)

where N[A,-D] = (2.53) d^rd^r Expression (2.53) also plays the role of a norm for the photon wave function (Bialynicki-Birula [1996a,b]). The Wigner functional for the thermal state of the electromagnetic field has been presented. This state is mixed and it even has infinitely many photons in the whole field. In each of the subsequent cases, the wave ftinctional and the Wigner fianctional have been introduced. The exception, the mixed state, has no wave functional. Let us remark that for (the statistical operator of) such a state a matrix element can be considered which is a functional of two arguments, A and A\ In particular, the Wigner functional for the coherent state of the electromagnetic field \A,-V) has been presented, where A(r) and V(r) are the vector potential and the electric displacement vector, respectively, which characterize the state. The exposition is related to the hot topic of the superposition of coherent states of the electromagnetic field. The exposition continues with the Wigner functional for the states of the electromagnetic field that describe a definite number of photons. An example of the functional for the one-photon state with the photon mode function/(r) is included. The norm (2.53) has not been related to any inner product of the photon wave functions, but these notions are connected. In contrast to (Bialynicka-Birula and Bialynicki-Birula [1987]), we introduce 1,[P] = jb(r,0)f(r)d'r,

(2.54)

X2[B] =

(2.55)

B(r,0)g(r)d'r,

- /*<•••'

(2.56) ^ W = JZJT I xr^T-—;^B(r')d^r\

(2.57)

4, § 2]

Origin of the macroscopic approach

315

The commutator of the X\ and X2 operators is [X, [VIMB]]

= ih jfir).

[V X gir)] d'r 1.

(2.58)

Let us note that the right-hand sides of eqs. (2.56) and (2.57) comprise the operator |V|"^ up to a certain factor (cf. Milbum, Walls and Levenson [1984]). Without resorting to this notation, we obtain that [X,[VIX2[B]] = ^7 f V(n)

AirOd'nl

(2.59)

We see easily that the usual commutator - ^ i l results from the field (T>,B) (or (A, -T>)) with the property

/•[-V(n)]'A(r0d'n=2k

(2.60)

We have not deepened the contrast by introducing the notation Xi [-V] andX2[^] on the left-hand sides of relations (2.54) and (2.55). Bialynicki-Birula [2000] presents the Wigner functional for the squeezed vacuum state, ^sq[^,-/>] = exp

—B • KsB ' B

^—DKODD'^B

KBD

' D' I dVd^f

Co

(2.61) where K^B, ^DD and A'BD are real matrix kernels. The kernel A'BD is not independent of A^BB and A^DD, but must obey a condition that is reminiscent of the Schrodinger-Robertson uncertainty relation (Bialynicki-Birula [1998]). The problem of time evolution is also discussed. Using a straightforward procedure, Mendonga, Martins and Guerreiro [2000] have quantized the linearized equations for an electromagnetic field in a plasma. They have determined an effective mass for the transverse photons. An extension of the quantization procedure leads to the definition of a photon charge operator. Zalesny [2001] has found that the influence of a medium on a photon can be described by some scalar and vector potentials. He extended the concept of the vector potential to relativistic velocities of the medium. He derived formulas for the photon mass in a resting and moving dielectric, and the velocity of the photon as a particle.

316

Quantum description of light propagation in dielectric media

[4, § 2

2.2.3. Source-field operator Knoll, Vogel and Welsch [1987] have compared the problem of quantummechanical treatment of action of optical devices with the input-output formalism (Collett and Gardiner [1984], Gardiner and CoUett [1985], Yamamoto and Imoto [1986], Nilsson, Yamamoto and Machida [1986], cf also Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990a,b]), concluding that apart from the fact that only a very particular setup is considered in the input-output formalism, the theory does not take into account the full space-time structure of the field. Knoll, Vogel and Welsch [1987] have further elaborated an approach based on quantum field theory and applied to the problem of spectral filtering of light (Knoll, Vogel and Welsch [1986]). The only assumptions are that the interaction between sources and light is linear in the vector potential, that the optical system is lossless, and that the condition of sufficiently small dispersion is fiilfilled. First, the classical Maxwell equations with sources and optical devices are formulated and solved by the procedure of mode expansion, and a quantized version is derived. The classical Maxwell equations comprise the relative permittivity e{r) = rr{r), where n{r) is the space-dependent refractive index. The charge density and the current density are written in terms of point charges, Qa being the charge of the aih particle. Canonical structure is imposed on the field and the matter, starting from a Lagrangian including the masses of the particles, rua being the mass of the aih particle. The mode functions Ai{r) are introduced as the solutions of the following equation: V X (V X Ax{r)) - e{r)^Ax{r)

- 0,

(2.62)

with (ji)\ the separation constant for each A, from which the following gauge condition can be derived: V.(6(r)^,(r)) = a

(2.63)

It is assumed that these solutions are normalized and orthogonal as follows: / •

e{r)A,{r)-Ax'{r)dh

= d,A-

(2.64)

The vector potential can be decomposed in terms of these fianctions. The destruction and creation operators ai and a\ are defined in a standard way, with the properties [axraU = Su'l

[ax,ax'] = 6 = [alall

(2.65)

4, §2]

Origin of the macroscopic approach

317

Upon inserting the operators a^ and a\ into the decomposition of the vector potential, the operator of the vector potential A(r, t) is defined:

A{r,t) = Y,A,{r)\ax{t)

+ a\{t)

(2.66)

The source quantities, i.e. the position vectors r^ and the generalized momenta Pa, are related to point charges and taken into account as operators r^ and Pa obeying the standard commutation relations. [ha.Pk'a']

= ihdaa'^kk' 1,

[rka. ^kuA = 0 =

[pka.Pk'a'].

(2.67)

and the commutation relations [ha,

^A]

= 0 = [Vka, :^t 5 ! ],

[pka,

«A]

ti = 0 = [pka, Aa[]

(2.68)

The operator ^(r, t) can be used for the derivation of the electric field strength operator, which is associated with the radiation field by the relation

E{r,t) =

--A{rj\

(2.69)

and to the derivation of the magnetic induction field operator, B{r, 0 = V X A{r, t).

(2.70)

The mode fiinctions may approach the notion of photon waveftinctions if redefined so that they obey the normalization condition

/

t{r)Ax{r)Ai,(r)d'r 3„

-

le^ojx

5AA'.

(2.71)

The quantum-theoretical Hamiltonian is written as the sum of a field Hamiltonian, a source Hamiltonian and an interaction Hamiltonian. The latter is formulated in terms of appropriate operators, with the Hermitian operator

J{r, t) = T ^

Wr\ - ra)Pa + A.5(H - h) - QcAih) (5(H - h)

being the current density operator. The form of the normalization conditions (2.64) and (2.71) is tailored to realmode functions, and ways to modify some fundamental relations are commented

318

Quantum description of light propagation in dielectric media

[4, § 2

on by Knoll, Vogel and Welsch [1987]. All of these field operators may be written in the form F(r, t) = J2 [^AW ax(t) + Fl(r) al(t)

(2.73)

Dependent on the choice of the operator F(r,t), the fianctions Fx(r) can be derived fi-om the mode functions of the vector potential Ax(r). It is often convenient to decompose a given field operator F(r, t) into two parts by the relation F(r, 0 = F^^\r, t) + F^-\r, t\

{2.1 A)

where F^^\r, t) = ^F,(r)a,(t),

F^-\r, t) = F'^\r,t)

(2.75)

In the following, the vector components are introduced by mere labeling by an index k, and repeated indices indicate summation. Further, the Heisenberg equations of motion for the field operators are derived, so that the field operators can be expressed in terms of the free-field and source-field operators. It is typical of the approach of Knoll, Vogel and Welsch [1987] that any field operator F{^^ is decomposed into a free-field operator and a source-field operator as follows Fi%, t) = Fj;ljr,

t) + h s(/-, 0,

(2.76)

where ^IfreeC^O =Y.^kx{r)aur..{t\

(2.77)

A

Fks{r,t) = j j e{t-t')K,,>{r,t',r\t')Jk'{r'j')d'r'dt'.

(2.78)

The operator a A free (0 is defined relative to the condition ^Afree(OL = /o ^ ^A(^) for t = t{), and the dynamics for ^ ^ ^ can be found in (Knoll, Vogel and Welsch [1987]). In eq. (2.78), the kernel K^k' is defined by Kkk'{r,t'yj')

= - ^ ^ F a ( r ) ^ ^ ^ ( / ) exp[-ia;A(^-0].

(2.79)

4, § 2]

Origin of the macroscopic approach

319

Inserting eq. (2.78) into (2.76) yields the following representation of F^^ : ^ r V ^ 0 = J J e(t - t')K,k'{r. t; r\ t')Jk'{r\ t') &'r' dt' + F{;^,(r, 0(2.80) In particular, if F^^^ is identified with the vector potential Aj, , it holds that Fkx = Akx\ the kernel Kkw takes the form ^ , , . ( r , r ; / , O = -T^5]^^^W^^A(''0exp[-ia;A(/-O]-

(2-81)

A

Analogously, if one is interested in the electric field strength operator of the radiation E^j^\ the appropriate form of the kernel Kkk' is Kkk'{r.t-r\t')

-~\Y.

^xAkx{r)Al,,{r')

exp[-ia;A(^-^')].

(2.82)

A

Thus the symmetry relations ^;,,(r, n r\ t') = TK,'k(r\ t'; r, 0

(2.83)

are valid for Aj^ and E[^\ respectively. The information on the action of the optical instruments on the source field is contained in the space-time structure of the kernel K/^k', which may be regarded as the apparatus function, similar to what is used in classical optics. Further, the commutation relations for various combinations of field operators at different times are studied and relationships between field commutators and source-quantity commutators are derived. These commutation relations are used to express field correlation functions of free-field operators and source-field operators and to describe the effect of the optical system on the quantum properties of light fields. 2.2.4. Con tin uum frequency-space description Blow, Loudon, Phoenix and Sheperd [ 1990] have formulated a quantum theory of optical wave propagation without recourse to cavity quantization. This approach goes beyond the introduction of a box-related mode spacing and enables one to use a continuum frequency-space description. In two papers. Blow, Loudon, Phoenix and Sheperd [1990] and Blow, Loudon and Phoenix [1991] developed a continuous-mode quantum theory of the electromagnetic field. As usual in

320

Quantum description of light propagation in dielectric media

[4, § 2

quantum field theory, they considered box-related modes whose creation and destruction operators satisfy the usual independent boson commutation relations [ai,aj] = Sijl. Different modes of the cavity, labeled by z andy, have frequencies given by different integer multiples of the mode spacing A a;. The mode spectrum becomes continuous as Aoj —> 0, and in this limit the transformation to continuous-mode operators is convenient, a, -^ y/Acoa(a)). The authors considered a complete orthonormal set of functions which may describe states of finite energy. The set is numerable infinite, and a destruction operator is assigned to each function in it. Such operators have all properties usual for the operators of the monochromatic mode. Blow and colleagues treated additional specific states of the field, such as coherent states, number states, noise and squeezed states; with the use of noncontinuous operators, they proved a generalization of the single-mode normal-ordering theorem. They treated field quantization in a dielectric including the material dispersion, and the theory was applied to pulse propagation in an optical fiber. A comparison with results by Drummond [1990, 1994] would be in order. Let us consider the fields in a lossless dielectric material with real relative permittivity e((jo) and the refractive index n((ji)) related by e(co) = [n(a))]^. Let us recall the definition of the phase velocity, VFi(o) = ^ = - ^ , k

and that of the group velocity * VQ{W)

^^ dco

(2.84)

n{(jj) UQ{W),

1 ^ [^„(^)j c do)

(2.85)

The vector potential operator has been modified for the dispersive lossless medium, and compared with (Loudon [1963], Drummond [1990]). The positivefrequency part is

(2.86) where e(k. A) are the orthogonal polarization unit vectors and w=-^\kl Expression (2.86) can easily be converted to one-dimensional form.

(2.87)

4, § 2]

Origin of the macroscopic approach

321

It is shown how the formulation of the quantum field theory is modified for a one-dimensional optical system. The fields are defined in an infinite waveguide parallel to the z-axis, but of finite cross-sectional area A of rectangular form with sides parallel to the x- and j-axes. The x and y wavevector components are thus restricted to discrete values, and any three-dimensional integral over this spatial region is converted according to

On the assumption that modes with ky^ ^ 0 ox ky ^ 0 are vacuum modes, one can exploit a reduced Hilbert (namely Fock) space. The summation in eq. (2.88) can, therefore, be removed and, putting k- = k, the other conversions are ^(3)(^ _ k') ^ ^ S i k - k'),

a{k. A) - . ^a(k.

A).

(2.89)

The field operators are obtained in accordance with the relation E^^\z, t) = -l/"\z, t) B^-\z, t) = ^A'"\Z, t) at oz and with the expansion of the vector potential operator

(2.90)

^W(z,0= r / ^^°^^\ V e(k,X)a{k,X)tx^[-\{wt-kz)-\dk, J~oo V 4;reoccon(cL>M ^ f f ^ (2.91) which is taken to be oriented in the x-direction, A^^\z, t) = A (z, t) e^. Operators a(w) are introduced, whose normalization is fixed by requiring the normally ordered total energy density operator (/(z, t) to have a diagonal form: ^free(0 = ^ /

^(^, t)dz=

I hcoa\(o) a{co) dco.

(2.92)

On noting that dco dk = — - - ,

a(k, A) = .yuGiaj)aico)

(2.93)

UG(CO)

and taking the polarization to be parallel to the x-axis, it follows from eq. (2.90) that the field operators are E^^\z,t) = i / W

^-—-a(co) expi -iw

n(co)z

do)

(2.94)

322

Quantum description of light propagation in dielectric media

[4, § 2

and 5'

*'<-•=•//lI^«»>-{-'"h=v^]}-

•-^'

Alternatively, the propagation constant can be expanded to the second order in frequency and a partial differential equation can be obtained (cf Drummond [1990]). Assuming a narrow bandwidth, the slowly varying field envelope can be represented by the operator a(z, t), which obeys the equation d k" d^ i - 5 ( z , 0 + y g^a(z, t) = 0,

(2.96)

where k^' is the second derivative with respect to the frequency of the propagation constant evaluated at the central frequency. The equation has been simplified by transforming the envelope into a frame moving with the group velocity, which is necessary for the envelope to be slowly varying. In classical nonlinear optics the stationary fields have envelopes too, but those are defined in a different way. The treatment of this problem in a noncontinuous basis proceeds from the replacement «(z,O = ^ 0 , ( z , O 9 ,

(2.97)

where 0j(z, t) form a complete orthonormal set of functions of z, and Cj are destruction operators obeying the usual commutation relations. The advantage of this treatment is that the functional dependence on z and t is contained in the c-number functions rather than the operators 5(z,/) as in the propagation equation (2.96) for instance. It is not emphasized by Blow, Loudon, Phoenix and Sheperd [1990] that the solution of eq. (2.96) preserves equal-space, not equaltime commutators. Similarly, the set of fijnctions 0/(z, t) enjoys orthonormality and completeness only as equal-space, not as equal-time properties. The propagation equation (2.96) now yields the following equations for the noncontinuous basis fiinctions: i - 0 , ( z , 0 + — ^ 0 ; ( z , t) = 0.

(2.98)

Finally, the process of photodetection in free space is considered and the results are applied to homodyne detection with both local oscillator and signal fields pulsed.

4, § 3]

Macroscopic theories and their applications

323

McDonald [2001] has considered a variation of the physical situation of "slow light" to show that the group velocity can be negative at central frequency. A Gaussian pulse can emerge from the far side of a slab earlier than it hits the near side and the pulse emission at the far side is accompanied by an antipulse emission, the antipulse propagating within the slab so as to annihilate the incident pulse at the near side.

§ 3. Macroscopic theories and their applications 3.1. Momentum-operator approach 3.1.1. Temporal modes and their application Huttner, Serulnik and Ben-Aryeh [1990] have developed a formalism that describes in a full quantum-mechanical way the propagation of light in a linear and in a nonlinear lossless dispersive medium. At first, they assume a similar situation as Abram [1987], i.e., they consider only the one-dimensional case restricting themselves even to fields propagating in the +z-direction only. They take for granted that quantum field theory has a generator for spatial progression, i.e., that relation (2.29) holds for any operator. They remark that the change in the quantization volume pointed out by Abram [1987] is not defined when the medium is dispersive, i.e., when the refractive index depends on the frequency, but they develop Abram's idea of the use of the energy flux not being dependent on the medium (cf. Caves and Crouch [1987]). In their opinion, the classical analysis of nonlinear optical processes shows that in order to obtain simple propagation equations it is usefial to introduce a photon-flux amplitude, i.e., a quantity whose square is proportional to the photon flux. At present we hesitate to accept the consequences of their approach (cf., however, Ben-Aryeh, Luks and Pefinova [1992]). Specifying the state at a given point (e.g., z = 0) and within a time period T cannot substitute for specifying the state at an initial time (t = 0) and within a quantization length L. Temporal modes of discrete frequencies a)„j, where a>,„ = 2mJt/T, cannot substitute for spatial modes. The equal-space commutation relations [a(z,Wi),a\z,ojj)]

= Syl

(3.1)

cannot substitute for the usual equal-time commutation relations. In comparison with (Abram [1987]), we see the following changes. In (Huttner, Serulnik and Ben-Aryeh [1990]), the MKSA (SI) system of units is

324

Quantum description of light propagation in dielectric media

[4, § 3

used. Instead of immediately reducing the Maxwell stress tensor to a single component, the tensor is first reduced to the Minkowski vector. The normal ordering is used where necessary. The notation ceases to express the dependence on both z and t, and states the dependence on z only. "The generalization" of the relation for the free-field momentum flux operator g(z, t) = c[b^-\z, t)B^^\z, t) + H.C.]

(3.2)

to the form g(z,t) = 0~\z,t)E^^\z,t) + Kc.]

(3.3)

is rather a modification, which remains correct in a dielectric medium. Its integration over T gives the momentum operator G{z)= r^

g{zj)dt

(3.4)

In the case of a linear dielectric medium, in contrast with (Abram [1987]), the electric field operator E{z, t) is dependent on the refractive index n(oJnj):

^

Y 2eQcTn{(On,) ^

^

while in (Abram [1987]) the operator is independent of the medium. The same paper does not present a pure Heisenberg picture, so that the equivalence of the two theories (for the refractive index n independent of o)) is not excluded. On substituting the relation (3.3) with appropriate b^~\z,t), D^^\z,t) into the relation (3.4), the momentum operator is obtained as Giin(z) = ^(hkn,) a\z, a),n) a(z, w„,),

(3.6)

m

where k^j = n((jo„j)(jOnj/c is the wavevector in the (linear) medium. The equalspace commutation relations are conserved. For such a medium, the equal-time commutation relations can be derived as A(z,t),-b(z\t)\ = ihd{z-z') i.

(3.7)

In an attempt at quantization in a nonlinear medium, Huttner, Serulnik and Ben-Aryeh [1990] concentrated on the propagation of light in a mukimode

4, § 3]

Macroscopic theories and their applications

325

degenerate parametric amplifier. The postulated relation (3.3) then leads to the nonlinear part of the momentum flux operator gnonlin(^, 0 = X^^^ U^"\z,

0 [E^'K^,

O ] ' + H.C. | ,

(3.8)

where £^^\z,t) = \£\Q-^i('¥-^P=) is the positive-frequency part of the pump field, with pump frequency Wp. From relations (3.4) and (3.8), the momentum operator is obtained as GnonvUz) = Y . ^ ^

[a\z,

COo + €,„) a\z,

COo - €,,) c'^-" + H.C.] ,

(3.9)

where e^ = o^m - COQ, a)o= ^-^, and X ( e „ . ) . ' - ^ , r ^ " - ^ - ' " '

(3.10)

is the coupling constant between different modes. It is assumed that the phase-matching condition at (JOQ, n((jOp) = n(a)o), is satisfied. It is found that the phase mismatch Akie^i) is proportional to e^,. As far as |AA:(e„;)| < A(€„,), the Bogoliubov transformation for squeezing emerges, and amplifying behavior can be recognized. When |AA:(6,„)| > A(e,„), the evolution is not essentially different from that in a linear medium: the squeezing effect is band-limited. For equality |AA:(e,„)| = ?i(e,„), amplification is present, but the increase is only linear, not exponential. For the nonlinear medium, the equal-time commutation relations are

i^"V,0,-I)^%',0] =

'~d{z-z')l

(3.11)

and relation (3.7) can be recovered only approximately. In relation to the experiment, a standard two-port homodyne detection scheme is assumed, where the light is mixed at a beamsplitter with a strong local oscillator e{z,t) of frequency COQ. For the correlation function gs{T) of the photocurrent difference and its Fourier transform y(r]) = j gs(r)c-'''dT, /

(3.12)

we refer to Huttner, Serulnik and Ben-Aryeh [1990]. It has been shown that the values of y(rj) can be minimized sufficiently uniformly by an adequate choice

326

Quantum description of light propagation in dielectric media

[4, § 3

of the local oscillator phase. The result is comparable with that of (Crouch [1988]), where the usual interpretation of homodyne detection in terms of the field quadratures is used. 3.1.2. Slowly-varying-amplitude momentum operator In spite of the above, there is a class of problems for which the modal approach is very convenient. This is cavity quantum electrodynamics. We mention its use in the development of the input-output formalism for nonlinear interactions in a cavity (Yurke [1984, 1985], Collett and Gardiner [1984], Gardiner and Collett [1985], Carmichael [1987]). The modal approach can describe many of the features of traveling-wave phenomena, but, in principle, it mixes effects related to spatial progression of the beam with the spectral manifestations of the nonlinearity. For example, for traveling-wave parametric generation (Tucker and Walls [1969]), wavevector mismatch appears as energy (frequency) nonconservation. Several authors have tried to reformulate the quantum-mechanical propagation in direct space. One technique (Drummond and Carter [1987]) involves partitioning the quantization box into finite cells. Another technique considers the spatial progression of the temporal Fourier components of the local electric field (Yurke, Grangier, Slusher and Potasek [1987], Caves and Crouch [1987]). The propagation of light in a magnetic (dielectric) medium is not usually considered in quantum optics. We proceed with the field inside an effective (linear or nonlinear) medium and the direct-space formulation of the theory of quantum optics as presented by Abram and Cohen [1991]. Their approach is an alternative to the conventional reciprocal-space approach to quantum optics; it relies on the electromagnetic momentum operator as well as on the Hamiltonian, and is restricted to a dispersionless lossless nonmagnetic dielectric medium. They derived an operatorial wave equation that relates the temporal evolution of an electromagnetic pulse to its spatial progression. As an illustration, they applied the theory to the generation of squeezed light by parametric down-conversion of a short laser pulse. This approach does not use the conventional modal field description. The appeal of classical optical theory may be in the fact that it considers a material as a continuous dielectric characterized by a set of phenomenological constants. Classical nonlinear optics has given rise to the slowly-varyingamplitude approximation for the electromagnetic wave equation. An important simplification of quantum optics results when the microscopic description of the material is replaced by a macroscopic description, in terms of an effective linear or nonlinear polarization. In spite of the phenomenological treatment of the

4, § 3]

Macroscopic theories and their applications

327

medium, such an effective theory still permits a quantum-mechanical description of the field (Jauch and Watson [1948], Shen [1967], Glauber and Lewenstein [1989, 1991], Hillery and Mlodinow [1984], Drummond and Carter [1987]). In propagation problems, one examines the interactions undergone by a short pulse of light. Abram and Cohen [ 1991 ] use Heaviside-Lorentz units and take h = c = 1. They simplify the geometry for the electromagnetic field so that the electric field E is polarized along the jc-axis, the magnetic induction field B along the 7-axis, while propagation occurs along the z-axis. In this simple geometry, the Maxwell equations reduce to two scalar differential equations, d^^dB dz dt'

dB^dD dz dt '

where the electric displacement field D is defined by D = E +R

(3.14)

P is the polarization of the medium, which can be expressed as a converging power series in the electric field £", P = /^^E^x^^^E^ + • • -+/"^E" + . . . ,

(3.15)

where x^"^ is the «th-order susceptibility of the medium. The dispersion cannot be taken into account rigorously within a quantum-mechanical theory based on the effective (macroscopic) Hamiltonian formulation (Hillery and Mlodinow [1984]), but it can be introduced phenomenologically (Drummond and Carter [1987]). To impose canonical structure on the field, Abram and Cohen [1991] introduce the vector potential A and adopt the Coulomb gauge in which the scalar potential vanishes and^ is transverse. In the assumed geometry, the vector potential is polarized along the Jc-axis, A = Ae^, with Cv the unit vector in the +jc-direction, and is related to the electric field and the magnetic induction field by

The effective Lagrangian density has been chosen in the form (Hillery and Mlodinow [1984], Drummond and Carter [1987]) C = \{E^-B^) + {x^'^E^ + \x^^^E' + \x^'^E' + • • •,

(3.17)

which is known to provide the most general density dependent only on the electric field and having the gauge invariance.

328

Quantum description of light propagation in dielectric media

[4, § 3

Let us note that the theory with the effective Lagrangian density (3.17) is not renormaHzable (Power and Zienau [1959], Woolley [1971], Babiker and Loudon [1983], Cohen-Tannoudji, Dupont-Roc and Grynberg [1989]). The canonically conjugated momentum of ^ with respect to the Lagrangian density (3.17) is the electric displacement n=--=-D. (3.18) oA The Lagrangian density is then transformed to some components of the energymomentum tensor of the electromagnetic field inside a nonlinear medium, O/^iy, namely, the energy density dA

e, =n—-c = \{B^ +A^) + ^/'^E'

(3.19) + l/'^E^ + ^^/'^E' + .. •,

(3.20)

and the momentum density dA er- = -n—=DB. (3.21) oz In setting up the Hamiltonian functional, the electric field E is to be expressed in terms of the electric displacement, which is the canonically conjugated momentum of A according to relation (3.21). It is assumed that E = P^^^D + P^^^D^^P^^^D^ + .. •,

(3.22)

where the coefficients 13^^^ may be expressed in terms of the susceptibilities x^"^ through definition (3.14) and relation (3.15) (Hillery and Mlodinow [1984]). The Hamiltonian functional is then written as

H= f Ond^r Jv

(3.23)

/'V. and the momentum has the form

G= I Ot,dV = I BDd^r,

(3.24)

where the integration is over the cavity obeying periodic boundary conditions, and the lower and upper limits extend to -oo and oo, respectively.

4, § 3]

Macroscopic theories and their applications

329

The field can now be quantized by replacing each field variable by the corresponding operator, and by replacing the Poisson bracket between the displacement D and the vector potential A by the (-i) multiple of the equal-time commutator [Z)(r, t\A{r\ t)] = id^ir - /) 1,

(3.25)

where the transverse delta fiinction d^Xr - r') reduces to the ordinary d ftmction, and where the three-dimensional position vector r can be replaced by the coordinate z. The vector potential A replaced by the operator A does not appear explicitly in the Hamiltonian (3.23) and momentum (3.24) operators, but rather in terms of its spatial derivative B. Taking the curl according to r' of both the sides of the canonical commutation relation (3.25) and using relation (3.16) in the simple geometry we obtain that [b{z,tXB{z',t)] = -id\z-z')

i,

(3.26)

where

b\z-z')=-b{z-z') oz

Chll)

is the derivative of the b fiinction. Ignoring divergencies, Abram and Cohen [1991] consider any product of noncommuting operators appearing in an expression to be fiilly symmetrized, i.e., to include all possible permutations of the individual field operators, such as, BD^ ^

.

(3.28)

3 In contrast, Abram and Cohen [1994] performed renormalization, i.e., normal ordering and elimination of divergencies. The description of propagative optical phenomena has been discussed within the framework of a direct-space formulation of quantum optics and the operatorial (or better commutator) equivalent of the Maxwell equations and the electromagnetic wave equation (Abram and Cohen [1991]). It is emphasized that in the Hamiltonian formulation of mechanics the time variable plays a prominent role.

330

Quantum description of light propagation in dielectric media

[4, § 3

The integrals in eqs. (3.23) and (3.24) and the equal-time commutator (3.25) correspond to the requirement that the field be specified over all space at one instant of time (e.g., at ^ = 0). Abram and Cohen [1991] use the Kubo [1962] notation for the commutator, or more exactly for a corresponding superoperator. The superoperator assigns operators to operators. Respecting this, the Heisenberg equation can be written as ^=i[H,Q]

= iH''Q,

(3.29)

where Q is any field operator and the superscript x denotes a superoperator, namely the commutator of the operator it indexes with another operator which follows. Equation (3.29) has the solution Q(t) = Qxp(itH'')Q(0) = e'^'^(0)e~'^^

(3.30)

The following Heisenberg-like equation involving the momentum can be considered: ^=-iG-Q.

(3.31)

This equation has the solution Q(z) = exp[-i(z-zo)G^]e(^o)-

(3.32)

Apart from the obvious similarity of eqs. (3.29) and (3.31), there is also a difference. The Hamiltonian of the electromagnetic field relates the desired spatial distribution of the field at an instant / + dMo its spatial distribution at t, but the momentum operator G relates the translated and non-translated fields only (at the same instant of time). In analogy with (3.13), two commutator equations can be derived: G''E = H''B,

G''B = H''b.

(3.33)

On the assumption that the medium is homogeneous, so that the Hamiltonian and momentum operators commute with each other, i.e. G^H = 0, relations (3.33) may be combined into the commutator equivalent of the electromagnetic wave equation: G''G''E = H''H''b.

(3.34)

Abram and Cohen [1991] illustrate the direct-space description of propagation (i.e., without passing over to a modal decomposition of a propagating pulse)

4, § 3]

Macroscopic theories and their applications

331

by examining the propagation of light (of a short light pulse) through a linear medium and through a vacuum-dielectric interface. For a linear medium, the commutator wave equation (3.34) reduces to 'H'')E = b,

CC'-eH

(3.35)

where 6 = I-^ X^^^ is the dielectric function of the medium. It is also convenient to define u = 1/ y/e, the velocity of an electromagnetic wave in a refractive medium. In the following exposition, the notation c will be used, because the convention c = 1 is not observed. The wave equation (3.35) enables one to rewrite Abram and Cohen's [1991] equations (4.2a) and (4.2b) in the form E(z, t) = cos\i{-\utG'')E(z, 0) - w sinh(-ii;^G^)^(z, 0),

(3.36)

B{z,t) = --sinh(-ii;^G^)£'(z,0) + cosh(-id;^G^)^(z,0).

(3.37)

V

Equations (3.36) and (3.37) indicate that the linear combination ^ ; ( z , 0 = ^(z, 0 + vB{z, t)

(3.38)

evolves in time as ^ ; ( z , 0 = exp(it;^G^) W^{z, 0) = ^ ; ( z - vt, 0).

(3.39)

Similarly, the linear combination W;{z,t) = E{zj)-vB{z,t)

(3.40)

evolves as W;{z, t) = Qxpi-iutG'') W-{z,0) = W;{z^vt,0).

(3.41)

To examine the problem of the interface, we could now consider two halfspaces such that the negative-z half-space is empty, while the positive-z halfspace consists of a transparent linear dielectric. We could then consider three waves: incident, W^{z, t), reflected, W~{z, t), and transmitted, W^^(^, t\ and the relations they obey. Abram and Cohen [1991] derive the commutator equivalent of the slowlyvarying-amplitude wave equation, on which the classical theory of nonlinear optics is based.

332

Quantum description of light propagation in dielectric media

[4, § 3

Not even classical optics provides a general solution to the problem of propagation of a short pulse in a nonlinear medium. In classical nonlinear optics, the assumption of a weak nonlinearity permits the slowly-varyingamplitude (SVA) approximation of the electromagnetic wave equation (Shen [1984]). Abram and Cohen [1991] examine a perturbative treatment of the time evolution of the field in a nonlinear medium that corresponds to propagation within the slowly-varying-amplitude approximation. For simplicity, a single nonlinear susceptibility x^"^ is considered. In the perturbative treatment of nonlinear propagation it is assumed that the optical nonlinearity of the medium is absent at ^ = -co, and is turned on adiabatically. In the absence of the nonlinearity, the electric strength and magnetic induction fields in the medium, EQ and ^o, as well as the displacement field Do, Do = CEQ, propagate under the Hamiltonian and momentum operators

Ho = \j

[BUIDI] 6

dV,

(3.42)

^3 = j Bobo dV,

(3.43)

respectively, of zeroth order in the nonlinear susceptibility ;^^^'^ Following standard perturbation theory (Itzykson and Zuber [1980]), the exact field operators in the nonlinear medium, D and B, can be derived from the zerothorder fields by the unitary transformation b{z,t)

= U-\t)bo(z,t)U(t\

(3.44)

B(z,t)

= U-\t)Bo(z,t)U(t).

(3.45)

Here tj(t) is the unitary operator which is the solution to the differential equation -U{t)

= -{Xh,(t)U{t),

(3.46)

with H\ the nonlinear interaction part of the Hamiltonian H,

A, = - ^

/yS'"'l);;+' dh =

^

/z""£'r' dV,

(3.47)

or f{dt) = expiitH,^)Hu

(3.48)

and obeys the initial condition U(t)\

= i. 1 / —>•

(3.49)

- o c

The Hamiltonian ^ i is of first order in the nonlinear susceptibility x^"'' which is expressed also by A, a dimensionless parameter introduced for the bookkeeping

4, § 3]

Macroscopic theories and their applications

333

of this and higher powers of x^'^^- The exact Hamihonian (3.23) can then be expressed perturbatively up to the first order in A as H = Ho + kHis^O(?i^X

(3.50)

where ^15 is the "diagonal part" ofH\,S with the Hnear Hamihonian HQ,

stands for stationary, which commutes

H,s=H\"^ = -^^js„,,Ah

(3.51)

with [f]

,

"^^ = 2""' E ^ / M O ,,^~"'Bl"'Er"", ^

(3.52)

{n - 2m)\{2m)\

where [JC] is the integer part of the number x. The neglect of O(A^) leads to decoupling of opposite-going fields. In the context of eqs. (3.51) and (3.52), a connection with the standard modal approach has been mentioned by Abram and Cohen [1991]. A method for deriving standard "effective Hamikonians" has been proposed by Sczaniecki [1983]. According to eq. (3.30), the time evolution of the displacement operator D can be written in the form D(z,0 - exp(iM^i^^)I)o(z,0 + AZ),(z,0,

(3.53)

where A = 1 and Dx{z,t)

i /

//i(r)dr

Do(z,0

(3.54)

is the first-order correction to the displacement field. The action of the superoperator on DQ in relation (3.53) can be compared with the mukiplication of the fast-varying ("carrier") wave by a slowly-varying-envelope fianction. On introducing the nonlinear polarization NL

-e^^^^Dl=x^'^El

(3.55)

it can be shown that the exact commutator wave equation (3.34) can be written up to order A^ as {G^G^-eH^H^)b, = b,

(3.56)

334

Quantum description of light propagation in dielectric media

[4, § 3

and that to order A^ leH^H^.bo = -eH,^H,^b, + G^ G^D, - G^ G^P^t-

(3.57)

The nonlinear polarization PNL consists of two parts, Pw and its complement, and Pw obeys the zeroth-order wave equation (G^G^-eH^H^)p^-b,

(3.58)

namely, Pw=P%^=/'%.

(3.59)

This partition again eliminates all terms that couple opposite-going waves in /*NL- Relying on the relation 6 = -eH^H^b, + G^G^b, -G^G^{P^L-PWI

(3.60)

we can derive the commutator equation leH^H^s^o = -G^G^Pw.

(3.61)

which has been compared to the classical slowly-varying-amplitude (SVA) wave equation, which is written as (Abram and Cohen [1991]) 2ik-E=-^P>r,

(3.62)

or, more often, in terms of the temporal Fourier components of E and Pw as ^E(w)=^P^(co),

(3.63)

where E is the envelope ftinction of the electric field. The connection between Piv and the modal approach has been shown by Abram and Cohen [1991]. The commutator equivalent of the slowly-varying-amplitude wave equation will be applied to the quantum-mechanical treatment of propagation in a nonlinear medium. Let us consider eq. (3.61) whose right-hand side, unlike the

4, § 3]

Macroscopic theories and their applications

335

left-hand side, does not contain DQ. This problem can be remedied by defining an effective "SVA" momentum operator obeying G^^^bo = {G^Pw

(3.64)

It follows that GsvA is the stationary part of the effective "interaction" momentum operator Gi = {jBopNLd'r,

(3.65)

namely, GsvA = G^svA = ^

IRn.x d^,

(3.66)

with [?]-!

Rn = 2 - ^ ' Y. ^

,

^-^ ntn e-'^^B'f^'Et'"^-\ {n - 2m - \)\{2m + \)\

(3.67)

Again, in the context of eqs. (3.66) and (3.67) for GSVA, the connection to the modal approach can be shown. With the definition (3.64), the commutator wave equation (3.61) can be written as (^S'VA^O'' + ^H^sH^ )bo = b.

(3.68)

In this form, the commutator SVA equation directly relates the slow component of the temporal evolution of a short pulse of the displacement field DQ to the long-scale modulation of its spatial progression. In order to clarify the role of eq. (3.68), the forward (+) and backward (-) polarization waves are defined in analogy with eqs. (3.38) and (3.40), V^=b±VeB,

(3.69)

which in the absence of the nonlinearity have the form Ko± = e r ± ,

(3.70)

where in accordance with perturbation theory the forward and backward electromagnetic waves are defined as W^=Eo±vh-

(3.71)

336

Quantum description of light propagation in dielectric media

[4, § 3

Relation (3.53) now becomes V^(z, t) ^ QxpiitH^s) ^^u(^^ 0 + VH^^ 0, V'(z, t) ^ exp(i/^,^) e^;(z, 0 + ^fC^, 0,

(3.72) (3.73)

where V^ are the first-order corrections to V given by equations analogous to (3.54). Equation (3.68) simplifies to VeH^sfV: =-G^^^w:,

(3.74)

VeH^sK =G^VAK,

(3.75)

for the forward-going and backward-going waves, respectively. These equations provide a simple rule for converting the temporal evolution of the modulation envelope to the spatial progression. That is, relations (3.72) and (3.73) can be written as F^(z, 0 = e expi-iutG^^^) W^(z - ut, 0) + F,^(z, /),

(3.76)

V-(z, t) = e cxpiiutG^y^) Wj(z + ut, 0) + V;{z, t).

(3.71)

In most practical situations, the first-order terms V(^ may be neglected and Wj^ can be introduced also on the left-hand sides of eqs. (3.76) and (3.11), using eq. (3.70). Nevertheless, V^ play an important role in that they incorporate the coupling to the wave going in the opposite direction and do give rise to the nonlinear reflection. As an illustration of the above quantum treatment, the traveling-wave generation of squeezed light by parametric down-conversion of a short pulse is examined. For the case of a classical pump, this problem was treated through a modal analysis by Tucker and Walls [1969]. More recently, Yurke, Grangier, Slusher and Potasek [1987] and Caves and Crouch [1987] treated this problem by using spatial differential equations for appropriately defined creation and annihilation operators. 3.1.3. Space-time displacement operators Serulnik and Ben-Aryeh [1991] have discussed the general problem of electromagnetic wave propagation through nonlinear nondispersive media. They have used a four-dimensional formalism of the field theory in order to develop an extension of the formalism introduced by Hillery and Mlodinow [1984]. The complications following from the common definitions for the vector and scalar

4, § 3]

Macroscopic theories and their applications

337

potentials are indicated. It is shown that the scalar potential can be neglected only by using alternative definitions. First, Serulnik and Ben-Aryeh [1991] show that the conventional approach that uses the standard potentials A and V is not appropriate for treating the general case of nonlinear polarization when V • P ^ 0, since in this case V does not vanish. As a solution to this problem they propose to use the vector potential ip, D = -Wxtp,

(3.78)

which fulfils the relation V • D = 0. This choice enables them to work in the new Coulomb gauge, where V • i/^ = 0, so that from the condition V • ^ = 0 it follows that the dual scalar potential $ obeys the equation V^§ = 0.

(3.79)

It is then consistent to assume ^ = 0 everywhere in a nonlinear medium, and the dual scalar potential need not be taken into account. Serulnik and Ben-Aryeh [1991] derive the Lagrangian and Hamikonian densities from the Maxwell equations by using nonconventional definitions for the scalar and vector potentials. The general form of the energy-momentum tensor is derived and explicit expression for its elements is given. The relation between this tensor and the space-time description of propagation is analyzed. Further the quantization is performed and the properties of space-time displacement operators are presented. Space-time is described by a Lie transform (Steinberg [1985]). Serulnik and Ben-Aryeh obtain the displacement operators from their energy-momentum tensor with an alternative definition for the vector potential. They are able to obtain explicit expressions for all the elements of the energy-momentum tensor and to discuss their physical meaning. In the following we will show that the actual relationship between the energymomentum tensor and the space-time description of propagation is different from that derived by Serulnik and Ben-Aryeh [1991]. Let us restrict ourselves to the usually treated one-dimensional case, where only the fields E\, D\, B2 and A2 are significant, and we use A2 = -^2 according to Drummond [1990, 1994]. The arguments of these fields are JC3 and ct. The corresponding quantum fields obey the commutation relations [A2(x3,cO,^2(y3,cO] =ihcd(x,-y,)l

(3.80)

[b,(x,,ct),B2(y3,ct)]

(3.81)

= -ihcd'ix, -y,) i.

Considering for this case the Hamikonian density

H=U-^b]+B\\,

(3.82)

Quantum description of light propagation in dielectric media

338

[4, §3

where the right-hand side is symmetrically ordered (cf. Abram and Cohen [1991]), we obtain the equations of motion in the Heisenberg picture:

'-•I"djci

dt dB2 dt

B

•I"

(3.83)

= CB2,

=

(3.84)

-C-

dX3

Relation (3.83) explains the role of the dual vector potential, and relation (3.84) is essentially the second of the Maxwell evolution equations. We could obtain the first of them as the equation of motion for the quantum field D\. It is a question whether the tensor element 1

(3.85)

c

is a correct quantum density for generation of the displacement as indicated by Serulnik and Ben-Aryeh [1991]. The presumable equations of the spatial progression are i

dA2 dx2

he i

dB2 dx2

he

(3.86)

Al, • //<(£>,fi2)5 dx3 ^ •

«

.

/

(P,B2)s^^

_ dB2

(3.87)

Relation (3.86) expresses the role of the dual vector potential, and relation (3.87) is a mere tautology. The same is obtained for the quantum field D\. This failure of the application of the ordinary presentations of quantum field theory has been published by Ben-Aryeh and Serulnik [1991]. Considering, in contrast, the equal-space commutators [A2(x3,cO,I>i(x3,cO] = 6 ,

(3.88)

[A2(jC3,cO,^2(x3,cO] =ihed{et-et')\,

(3.89)

[B2{x3,et\B2{x3,et')\

(3.90)

= ihed\et - ct') 1,

[D,(x3,cO,^2(x3,cO] = 6 ,

(3.91)

[Z)i(x3,cO,Z>i(x3,cO] ='\heed'{et-et')\,

(3.92)

we obtain peculiar equations for the spatial progression: dA2 dx3

8X2

A2, f{b,B2hdt ^b,,j{b,B2)sdt

= -D^,

\dB2 ' c dt '

(3.93)

(3.94)

4, § 3]

Macroscopic theories and their applications

339

Relation (3.93) expresses the role of the dual vector potential, and relation (3.94) is essentially the second of the Maxwell evolution equations. We could obtain the first of them as the equation of the spatial progression for the quantum field B2. Since we often have to make a guess about previously unknown commutators, the above example is a warning against excessive trust in the spatial progression technique. For a medium with nonlinear polarization, the global nature of creation and annihilation operators is lost. By consistently following this idea, Serulnik and Ben-Aryeh [1991] have introduced the shift operators which, by their definition, are based on the energy-momentum tensor. They have followed in their treatment Peierls' solution of the problem of momentum conservation in matter (Peierls [1976, 1985]) by which the atoms or the bulk matter are considered to be at rest while the electromagnetic field is propagating. They show that it is always possible to relate the external field in front of the medium to that behind it by the use of the shift operators, that is by a Lie transformation. As we can see from relations (3.83), (3.84) and (3.93), (3.94), the transition from the so-called time-displacement operator to the displacement operator in the X3 direction must be accompanied by a change of integration variable from JC3 to ^ Leonhardt [2000] has determined an energy-momentum tensor of the electromagnetic fields in quantum dielectrics. The tensor is Abraham's [1909] plus the energy-momentum of the medium characterized by a dielectric pressure and enthalpy density. While the consistency of this picture with the theory of dielectrics has been demonstrated, a direct derivation from the first principles has been announced only.

3.1.4. Generator of spatial progression Theoretical methods for treating propagation in quantum optics have been developed in which the momentum operator is used in addition to the Hamiltonian. A successful quantum-mechanical analysis has been given for various physical systems which include amplification and coupling between electromagnetic modes by Toren and Ben-Aryeh [1994]. Distributed-feedback lasers have been described, but an overarching generalization of both successftil analyses has not been developed. The authors have paid attention to distributedfeedback lasers (Yariv and Yeh [1984], Yariv [1989]) in which contradirectional beams are amplified by an active medium and are coupled by a small periodic perturbation of a refractive index.

340

Quantum description of light propagation in dielectric media

[4, § 3

The energy and momentum properties of the electromagnetic field can be described, in four-dimensional form, by the energy-momentum tensor F^, where 7,A: = 0,1,2,3 (Roman [1969]), W g, gy g= JT/'^ =

Sx O^, 0,y O,.Sy Gy-, Oyy Oy-_

(3.95)

S- 0-v O-y 0--

The tensor element T^^ represents the energy density. The vector {gx,gy,gz) represents the density of the vectorial momentum (proportional to D x B). Let us take further, for example, the fourth row. The tensor element T^^ is the component of the Poynting vector representing the flux of energy in the z-direction. The vector (a-v, O-y, a--) refers to a flux of momentum in the propagation direction of z. In the conventional approach (Roman [1969]), the four-vector/?^^ is defined as /^=

f I f T^''(x,y,z,t)dxdydz.

(3.96)

The energy p^^ is used as the Hamiltonian for the description of time evolution; the momentum component/?^^ does not describe a real propagation equation (but cf eq.3.87). Ben-Aryeh and Serulnik [1991] have shown that for the description of the spatial progression, the four-vector p^^ can be used given as 3^ _ f f f T^k P = r\x,y,z,t)cdtdxdy.

(3.97)

We must assume that the commutators (3.88)-(3.92) have been simplified in analogy with the relations (3.80) and (3.81). The momentum component in the z-direction, p^^, can be used as the generator of the spatial progression, and the energy p^^ is expected to translate the field in time. Toren and Ben-Aryeh [1994] treat propagation problems by expanding the field operators in terms of mode operators associated with definite fi*equencies. Starting with (Caves and Crouch [1987]), the approach has been associated with the conservation of commutation relations for creation and annihilation operators, which are space dependent (cf. Huttner, Serulnik and Ben-Aryeh [1990]). Imoto [1989] has developed the basic equation of motion by using a modified procedure of canonical quantization in which time and space coordinates are interchanged in comparison with the conventional procedure.

4, § 3]

Macroscopic theories and their applications

341

Ben-Aryeh, Luks and Pefinova [1992] did the same by using a slightly different notation. Toren and Ben-Aryeh [1994] dissociate themselves from this approach, but they are not very explicit about whether the use of the integrals (3.97) is compatible with the canonical quantization in which the time coordinate plays the usual role. Linear amplification is treated by the use of momentum for space-dependent amplification. Traveling-wave attenuators and amplifiers can be treated as continuous limits of an array of beamsplitters (Jefifers, Imoto and Loudon [1993], Ban [1994]). According to Toren and Ben-Aryeh [1994], the propagating modes are coupled to a momentum reservoir. The Hamiltonian of this system is given by H = h (coa^a - ^

W/resi-/^)'

(3-98)

and the total momentum operator is G = h 0 a - Y, I^Ms'bjbj + Yji^i^^bj + >^;abj)

(3.99)

where the subscript res stands for the reservoir, K/ are appropriate coupUng constants, and a and bj represent (in the zeroth order) modes which are propagating in the positive direction of the z-axis. The equations of motion obtained from the momentum operator (3.99) are ^ = i[a, G] = i/5a + i 5 ^ Kjb], dz

(3.100)

db] ^ = -^\b], G] = -xK*a + ip,;J].

(3.101)

By using the spatial Wigner-Weisskopf approximation, the Heisenberg-Langevin equations can be obtained:

^ = [ i O S - 4 / 3 ) + i y ] « + Zt,

(3.102)

dz

where .«

y

_

vn

f

l'^(/^es)IV(fes)

= {2jr|/C(/^es)|V(/?res)}|/^^.^.,,,

.^

(3.103) (3.104)

342

Quantum description of light propagation in dielectric media

[4, § 3

with Vp. the principal value of the integral, p(ftes) the density function of the wave propagation constants ^yres in the reservoir, and

V = Y^iKjb]S^-\

(3.105)

The codirectional coupling is also analyzed. It is assumed that two modes are propagating in the same direction and they are coupled by a periodic change in the refractive index. For a classical description, we refer to Yariv and Yeh [1984] and Yariv [1989]. The Hamiltonian is given by Ho=H = ha) a\a\ +a\a2

(3.106)

where the classical relation a)\ = (O2 = (O has been used. The total momentum operator is G = h P\a\a\ + ^a\a2 + ka\a\ + k*a\a2

(3.107)

where ^\ and ft are components of the wavevectors of the two modes in the propagation direction of z and

;r = /rexp

/ im2jr \ —-z ,

^^ ,^^^ (3.108)

with K 3. coupling constant, m an integer, and A the "wavelength" of the spatial periodic change in the index of refraction (a perturbation in the dielectric constant). In this connection papers by Pefinova, Luks, Kfepelka, Sibilia and Bertolotti [1991] and Ben-Aryeh, Luks and Perinova [1992] are criticized by Toren and Ben-Aryeh [1994] for not taking account of the spatial dependence (3.108). The equations of motion obtained from the momentum operator (3.107) are ^ dz ^ dz

- k « i , 6 ] = iA«i+i^*«2, n = ^[a2,G] = i/cai+i/32a2. n

(3.109) (3.110)

We define slowly varying operators of the form ii(z) = al(z)e-^^^•^

A2(z) = a2(z)e-*^^\

(3.111)

4, § 3]

343

Macroscopic theories and their applications

Substituting the operators (3.111) into equations (3.109) and (3.110) we get d4i ~d7

(3.112) (3.113)

~d7 where

AI3 =

l3,-k-m^

(3.114)

is the mismatch. A "field" mismatch may be cancelled by a medium component. For the input-output relations we refer to Yariv and Yeh [1984], Yariv [1989] and Pefinova, Luks, Kfepelka, Sibiha and Bertolotti [1991]. Pefinova, Luks, Kfepelka, Sibilia and Bertolotti [1991] introduced m = 0 and 26 = -AI3 in an application to the codirectional coupler. In general, the solution to eqs. (3.112) and (3.113) coincides with the classical solution, Aj «-> AJ, J = 1,2, where A\ and A2 are the amplitudes of the waves propagating in the +z-direction. The counterdirectional coupling is analyzed in (Toren and Ben-Aryeh [1994]). The total Hamiltonian is given by ^ Pico \a\a\-a^a2

Hn=H

(3.115)

We consider the momentum operator in the form = h P\a\a\ - (hci2^2 + ^^1^2 "^ ^*^i^2 « 2 ^^

(3.116)

^T

It is reasonable that the Hamiltonian and the zeroth-order operator are related, respectively, to the flux of energy and that of the component of momentum in the z-direction. Compared to Toren and Ben-Aryeh [1994], we have interchanged the operators ^2 and al. Toren and Ben-Aryeh criticize our assumption [S2, ^2] - - 1 , which we obtained by this interchange (Perinova, Luks, Kfepelka, Sibilia and Bertolotti [1991]) fi*om the usual equal-space commutator [^2,^2] ^ ^- ^^ ^^ tempting to have the same alternation between the opposite-going modes as can be seen in comparison of eq. (3.303) with eq. (3.305) (cf Abram and Cohen

344

Quantum description of light propagation in dielectric media

[4,

[1994]). The equations of motion obtained from the operator (3.116) are given by dfli

i

"dF ~ li da2

i

"d7 ~ h

a\, G

= iP\a\ + iK*a2,

(3.117)

«2, G

= -ika\ + '\f^a2-

(3.118)

Using the slowly varying operators (3.111), in contrast to Toren and Ben-Aryeh [1994], we obtain that

"d7

^ i^*i2e-'^^^-".

- = -iKA,e^'''

(3.119)

with A^ defined by eq. (3.114). In the work of Perinova, Luks, Kfepelka, Sibilia and Bertolotti [1991], 26 = -AP still holds in an appHcation to the counterdirectional coupler. The solution to eqs. (3.119) coincides with the classical solution, Aj ^ AJ, j = 1,2, where A\ and A2 are the amplitudes of the waves propagating in the +z- and -z-directions. Yariv and Yeh [1984] obtained the solution to the corresponding classical equations for the boundary conditions A\(Z)\_^Q = A[(0), ^2(^)\- = i "= M{L)' First, however, one obtains the solution for the usual condition at z = 0. Perinova, Luks, Kfepelka, Sibilia and Bertolotti [1991] obtained the output operators in terms of the input ones. While we simply determine the operators A\{L), ^2(0) from two equations for the operators ^i(O), ^2(0), A\{L), AiiL), we observe that in this procedure the equal-space commutator [Ai.Aj] - - 1 must depend on both z and Z in a complicated manner, and simplifies to [^2,^2] ^ 1 for z = 0,1. Since the commutators correspond to the Poisson brackets, much is illustrated by the appropriate classical theory (Luis and Pefina [1996]). One must be aware of the fact that in formulating the theory, Luis and Pefina [1996] avoided the above considerations on the z-coordinate and the generator of spatial progression, and they used in the main part of their paper the usual time dependence and the Hamiltonian fijnction. Although still obscure in the case of commutators, the situation is clear in the classical case, when the input-output transformation is characterized by the usual Poisson brackets and the solution for the usual boundary conditions at z = 0 requires the noncanonical transformation ai ^ cC{, with the complex amplitude ai. The richness of their theory is due to nonlinearities, whereas

4, § 3]

Macroscopic theories and their applications

345

it is shown that in the quantum case only a poor linear theory is possible. The difficulty lies in the formulation of an appropriate dynamical operator. Tarasov [2001] has defined a map of a dynamical nonlinear operator into a dynamical superoperator. He had in mind quantum dynamics of non-Hamiltonian and dissipative systems. A quantum-mechanical treatment of the distributed-feedback laser using the momentum operator in addition to the Hamiltonian has been developed by Toren and Ben-Aryeh [1994]. They start from the classical description based on two coupled equations M

= IvA,\yAi-iKA2&' -\i^J-.P^'^f' = — 1=

(3.120) ^=i/c*^,e-^/^'---i7^2,

where A \ and A2 are the amplitudes of the waves propagating in the +z- and -z-directions, respectively, K is the coupling constant, 7 is the amplification constant, and AjS is given by eq. (3.114), with /?i = ^, ft == -/S. The solution of the classical equations is well-known (Yariv and Yeh [1984], Yariv [1989]) and it shows that under special conditions the amplification becomes extremely large. The classical theory does not include the quantum noise which follows from the amplification process. Unfortunately, Toren and Ben-Aryeh [1994] did not develop an overarching generalization of the analysis of amplification and that of the counterdirectional coupling. To the best of our knowledge, such a quantum-mechanical theory is not in hand. The treatment of parametric down-conversion and parametric up-conversion by Dechoum, Marshall and Santos [2000] is interesting with its use of the Wigner representation of optical fields, but it starts just from the Maxwell equations for the field operators and the lossless neutral nonlinear dielectric medium. Using the common approximation of treating the laser pump as classical, they obtain classical equations of nonlinear optics. 3.2, Dispersive nonlinear dielectric 3.2.1. Lagrangian of narrow-band fields Drummond [1990] has presented a technique of canonical quantization in a general dispersive nonlinear dielectric medium. Contrary to Abram and Cohen [1991], Drummond creates an arbitrary number of slightly varying copies of the free electromagnetic field for the nonlinear dielectric medium; essentially, the

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Quantum description of light propagation in dielectric media

[4, § 3

number required by the classical slowly-varying-amplitude approximation. But Abram and Cohen work with a single field. The paradox of both approaches being valid can be resolved only by a detailed microscopic theory. Drummond [1990] generalizes the treatment of a linear homogeneous dispersive medium (Schubert and Wilhelmi [1986]). Until 1990, the reference papers for the theory of inhomogeneous nondispersive linear dielectrics were Knoll, Vogel and Welsch [1987], Bialynicka-Birula and Bialynicki-Birula [1987] and Glauber and Lewenstein [1989]. Hillery and Mlodinow [1984] were attractive with their use of the idea due to Born and Infeld [1934] for the quantization of a homogeneous nonlinear nondispersive medium. Macroscopic quantization is a route to the simplest quantum theory compatible with known dielectric properties, unlike the microscopic derivation of the nonlinear quantum theory of electromagnetic propagation in a real dielectric. Drummond [1994] compares the quantum theory obtained via macroscopic quantization with the traditional quantum-field theory. He concedes that most model quantum field theories prove to be either tractable but unphysical, or physical but intractable. The tractable model quantum-field theory ceases to be unphysical when it is tested experimentally in quantum optics. An excellent example of this is provided by fiber-optical solitons whose quantization is given in detail by Drummond [1994]. In agreement with theoretical predictions (Carter, Drummond, Reid and Shelby [1987], Drummond and Carter [1987], Drummond, Carter and Shelby [1989], Shelby, Drummond and Carter [1990], Lai and Haus [1989], Haus and Lai [1990]), experiments (Rosenbluh and Shelby [1991]) led to evidence of quantum solitons. More recent experiments (Friberg, Machida and Yamamoto [1992]) demonstrate that solitons can be considered to be nonlinear bound states of a quantum field. In addition to the quadrature squeezing in Rosenbluh and Shelby [1991], quantum properties of soliton collisions were measured (Watanabe, Nakano, Honold and Yamamoto [1989], Haus, Watanabe and Yamamoto [1989]). Similar nonlinearities are encountered in photonic-bandgap theory (Yablonovitch and Gmitter [1987]), microcavity quantum electrodynamics (Hinds [1990]), pulsed squeezing (Slusher, Grangier, LaPorta, Yurke and Potasek [1987]) and quantum chaos (Toda, Adachi and Ikeda [1989]). It is advantageous to begin with the treatment of a classical dielectric by introducing the nonlinear response function in terms of the electric displacement field D. Contrary to the usual description (Bloembergen [1965]), which uses the dielectric permittivity tensors in a decomposition of this field, the inverse expansion is necessary here. For simplicity, the dielectric of interest is regarded as having uniform linear magnetic susceptibility. The charges are assumed

4, § 3]

Macroscopic theories and their applications

347

to occur only in the induced dipoles of polarization. The field equations are therefore V x : £ ( x .0 V x H{x

=

dB(x,t)

dD(x,t) dt ' = 0, =

,0 =

V D{x ,0

(3.121)

V B(x ,0 == 0, where D{x, t) = eoE(x, t) + P{x, t\ (3.122) B{x, t) = iiH{x, t). Here /»CX)

^"(^,0= /

X{x.T)'E{xJ-T)dT

Jo /•CXD

+ /

/»CXD

/

X^^\x,Tur2)

: E(x,t - T^)E(x,t - T2)dTidT2

Jo Jo /»(X)

/»(X)

/"CX)

+ / / / Jo Jo Jo

X^^^(x,ri,r2,r3):£'(Ac,/-ri)£'(A:,/-r2)£'(x,^-r3)dridr2dr3

+ •••, (3.123) where the tensor of rank 2 and, in general, the (n + l)th-rank susceptibility tensor read, respectively,

Xix,r)=^lx{x,(o)c"'"daj, X^"\x, Tu...,r„)=

QAJ-

• jt"\x,

w\...,

«")e-'<'"' ^' ^ - ^"'"^"' dw' • • • dw". (3.124)

348

Quantum description of light propagation in dielectric media

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After adding the vacuum electric displacement eoE(x,t) to both sides of eq. (3.123), we express the electric vector in the form

E(x,t)=

/ Jo

+ / / Jo Jo /»CXD

/»CXD

i(x,T)-D(x,t-T)dT

i^\x,ruT2)

: D{x,t - TOD(x,t - T2)dT, dT2

/>CXD

+

^^^\x,ruT2,T3y:D{x,t-Ti)D(x,t-T2)D(x,t-T3)dr^dT2dT3 Jo Jo Jo

+ •••, (3.125) where i(x,r)=^

JUx,CO)c-'^'^'do),

i^'\x, ri, . . . , T,)= (iy\j..p^\x,

co\...,

co'Oe-'^^'^'^' " -^^'^"^"Ma;^ • • -do;'',

(3.126) and the tensors in the right-hand sides of eqs. (3.126) are given by the recurrence relations e(x,w)'l(x,a))

= l,

e(x, 0)^ + 0)^) • l^^\x, a)\w^) + x^"\x, w*, w^) : Ux, co^) Ux, co^) = 0^^\ e(x, (0* + w^ + 0)^) • l^^\x, a)\ (XT, (X?) + 2x^'^\x,(o\o?

+ a?) :

l{x,o)^)l^'\x,a;^o?)

+ x^^\x, (D\ a;^ 0?): l{x, w^) l{x, a?) l{x, o?) = 0^^\

with e(x,(jo) = eol + Xix,(^) permittivity. In particular, l(x,co) = [e(x,co)r\

(3.127) the usual frequency-dependent tensor of

(3.128)

4, § 3]

Macroscopic theories and their applications

349

Here 1 and 0^"^ are the second-rank unit tensor and the {n + l)th-rank zero tensor, respectively. Introducing the Fourier transforms of the electric strength field and electric displacement field, respectively,

^(jc, CD) = I E{x, t)e'''' d^,

b{x, CO) = f D(x, t)e'''' d/,

(3.129)

and performing the Fourier transform of both sides of eq. (3.125), we obtain that E{x,

(D) = l(x,

-^

w) • ^(jc, (JO) ll^^\x,(o\w-w^):b(x,w^)b(x,co-(o^)dw^

+ / l^^\x,co\(o^,co-co^-w^)':b(x,(D^)b(x,co^)b{x,(jo-o)^-0)^)d(o^

&o?

+ • • •. (3.130) The inverse relation of b{x,w) involves the terms /(jc, co) = X^^\X,-(D\(JO), X^^\x, a)\(o-(o^) = X^^\x, -co; co\co-co^),.... A similar extension of notation is conceivable also in tensors l(x,co), l^^\x, co\co- co^),.... Let us note that Pefina's [1991] similar relation for P{x, co) comprises sums instead of the integrals. Relation (3.130) may be matched to relation (2.4) of Drummond [1990] on the condition that the integrals be replaced by the sums. Such a change does not only affect the meaning of the tensors /^"^ and l^"\ but also (and above all) the physical unit of their measurement. We will treat the time-averaged linear dispersive energy for a classical monochromatic field at nonzero frequency co. For a permittivity e(x, co), this can be written in terms of a complex amplitude £(x) (Bloembergen [1965], Landau and Lifshitz [1960], Bleany and Bleany [1985]),

{//) = J

U^x)

• ^[coe(x,co)]-£(x)^-^ ^ [coe(x, CO)]. £(x) + ^^ {B(x, t) • B(x, t)) ) d'x, (3.131)

where E(x, t) = 2 Re[f (jc)^'^'^']. It is important to distinguish the monochromatic case from the case of quasimonochromatic fields. In the more general case, the displacement D is expanded in terms of a series of complex (envelope) ftinctions, each of which

350

Quantum description of light propagation in dielectric media

[4, § 3

has a restricted bandwidth. The relevant non-zero central frequencies are then o;-^,...,a>^, thus yv

Z>(jc,0= 5^/)^(Jc,0,

(3.132)

v = -N

where D~^ = (Z)^)* and in the monochromatic case D\x,t)

= V\x)Q-'''^^.

(3.133)

The notation we use here differs slightly from that of Drummond [1990]. Again, the electric field vector can be expanded as N

E(x,t)=

^

E\x,t),

(3.134)

where E~^ = (E^y and in the monochromatic case E''(x,t) = £\x)Q-'"'^''.

(3.135)

In the case of quasimonochromatic fields, relations (3.133) and (3.135) should be replaced by 1

pC) +0

D\x, t)=:r-

b(x, 0)) e'"'" dco, *^"'"r'

E\x,

t)=—

(3.136) E(x, w) Q-''" d(o.

Bloembergen [1965] presented the relation (3.131) as being sufficiently accurate for this case. While relation (3.131) is exact for monochromatic fields, it must be modified for a quasimonochromatic field as follows:

mt')){t)=/ (i E ^"'(^'') • a^t'^'^^^' ^"^] • ^'(^' '^ (3.137) +

^^{B{x,t')B{x,t')){t)\d'x.

4, § 3]

Macroscopic theories and their applications

351

By modifying the summation, we obtain the energy integral in terms of the electric displacement fields

^"•(^'') • [^(^')-w''^i'{x,oj'')]-D\x,t) ^''

{H{t')){t) = IUY. •^ V \' = -N +

~{B(x,t')B(x,t'))(t)\A'x.

(3.138) To complete the description, we supplement relations (3.132) and (3.134) with the expansion of the magnetic induction field: N

B{x,t)= Y.

^"(^'')'

(3.139)

where B^ B\x,t)

= (By,

(3.140)

= — /

B(x,(D)Q-'"'d(D.

(3.141)

Next, ^(jc, a;) can be approximated near w = co^' by a quadratic Taylor polynomial, l(x, CO) ^ Ux) + wlUx) + {w'l[!(xl

(3.142)

so that ^d^(x,Oj)

^ix,co)-co ^y^

_ ^

^ ^

,,<://,

' ^ Ux)-\co^iy{x).

(3.143)

Using the notation D = (d/dt)D, we rewrite relation (3.138) in the form

(//(/'))(o=^E / v =

D\x,

t) • ly(x) • D'(x, t)

-N'

\b-\x,

t). ?:;(jc). b\x,

1 t) + ^,B\x, ^^

t) • B\x, t) d'x.

(3.144) Here we deviate slightly from Drummond [1990]. Drummond speaks of time averages, and he indicates time averaging on the left-hand side and partially on

352

Quantum description of light propagation in dielectric media

[4, § 3

the right-hand side in (3.138), but he does not remove the time dependence from the right-hand side. A canonical theory of linear dielectrics will be obtained using the causal local Lagrangian. Drummond [1990] considers a Lagrangian L[A~^,... ,A^], which is a functional of (components of) the dual vector potential. This is defined as A, for which Z)(jc, 0 = V X A(x, t),

Bix, t) = iiA{x, t).

(3.145)

We introduce also A{x,o))

= f A(x,t)e'''dt,

A"{x,t)

= ;^ / lit

1

(3.146)

MO +0

A{x,(D)e-""'do).

(3.147)

Each quasimonochromatic field obeys the Maxwell equations

V X E\x, t) = -B'(x, t), V X H\x, t) = b'(x, t), VD\x,t) = 0, V-B\x,t) = 0,

(3.148)

where E\x, t) = Ux) • D\x, t) + il',(x) • b'ix, t) - {l';(x) • b\x, 1

t), (3.149)

H\x,t)=-B\x,t). The components of the dual vector potential fulfil linear wave equations. On the basis of (3.144) we can infer a Hamiltonian fianction of the form W = Wo = ^ / ^

I [V X A-\x,

t)]. Ux) • [V X A\x, 0]

- i[V X A-\x,

+

t)] • ^(^(jc) • [V X A\x, t)]

(3.150)

IAA-\x,t)'A\x,t)[d^x.

In order to quantize the theory, a canonical Lagrangian must be found that corresponds to eq. (3.150) and generates the Maxwell equations (3.148) as Hamilton's equations. The linear Maxwell equations can be recast as a wave equation. Wave equations for wave functions are considered. It is next necessary

4, § 3]

Macroscopic theories and their applications

353

to derive a Lagrangian whose Lagrange's variational equations correspond to these wave equations and whose Hamiltonian corresponds to eq. (3.150). Since A^ can be specified to be transverse fields, the variations can also be restricted to be transverse. These restricted variations can be realized using transverse ftinctional derivatives (Power and Zienau [1959], Healey [1982]). Drummond [1994] derived the Lagrangian using the method of indeterminate coefficients in the form

N

-\

Y.

{^^'\x,t)A\xJ)-[V - i[V X

A-'(A:,

- i[V X A-\x,

X A-\xj)]

• Irix) • [V X A\x,t)]

0] • ?;(x). [V X A > , 0] 0] • ly{x)' [V X A\x, t)]} d^x. (3.151)

The canonical momenta are n(x,

t) = B\x,

0 - IV X i$;(jc) • D\x,

t) + i[[x)' D \x, t)

(3.152)

where for brevity we re-introduced the fields (3.145) again. Analogously, we can rewrite the Lagrangian of Drummond in the form

=\

C ^ 1 Y.{ -^'(^^ 0 • B\x, t) - D\x, - \D\x,

t) • l[ix). b\x,t)

t). Ux) • D(x, t)

t). l'^,{x). b{x,O} d^x. (3.153) The Legendre transformation, i.e., a substitution of JV' in the Hamiltonian (3.150) by an expression in W and A^\ was not performed by Drummond [1990]. The reason is that each JV' is to be found from (3.152) considered as a partial differential equation. Also, this theory simplifies a great deal if planewave one-dimensional propagation is considered. The local Lagrangian method is used as the foundation for a nonlinear canonical Lagrangian and Hamiltonian. The objective is the total Lagrangian and Hamiltonian of the form L = LQ- f U^(x,t)d^x,

- \b-\x,

n = HQ+ f U^(x,t)d^x,

(3.154)

354

Quantum description of light propagation in dielectric media

[4, § 3

where U^(x,t) is a nonlinear energy density, N

N

Vi =-N

N

V2= - / V V'3 = - A '

N

N

N

N

Vi = -N

V2 = -N

V3 = -N

V'4 =

-N

:Z)^'^(A:,0/>'-(A:,0^''(^,05-aV.,r,/'2W3+./'4 + • • ••

(3.155) In order to give an example, a one-dimensional case is treated and the nonlinear refractive index, being the lowest nonlinearity, is of most universal interest. For N = \, one has U''{x,t)=\l^'^\D\xJ)\\

(3.156)

Drummond [1990] has presented the quantization of the nonlinear medium using a treatment of modes defined relative to the new Lagrangian. The canonical momenta have the form (3.152) in the nonlinear case too. In the corresponding quantum theory, the field operators A^' and W are introduced, which obey transverse commutation relations of the form [Aj{xj\ni\x\t)] - ihd^{x-x')d,X

(3.157)

Since these operators are not Hermitian, it is also interesting to note that A-^ = (Ay,

ii'

= (my.

(3.158)

This entails that AJ and (fl-)^ commute. Then, a set of Fourier-transformed fields is defined and the annihilation operators a^ and b^ are introduced. The operators a^ correspond to the normal modes while b^ generate additional necessarily vacuum modes. This feature of the theory is due to the dependence of the Hamiltonian (3.150) on both the real and the imaginary parts of the components A^(x,t) (A^'(xJ)). 3.2.2. Propagation in one dimension and applications Drummond [1994] discusses in detail a simplified model of a one-dimensional dielectric, where l(x, (D) = l(co) 1, l^"\x, co\...,w") = l^"\co\..., co") 1^"\

4, § 3]

Macroscopic theories and their applications

355

with 1^"^ the (n + l)th-rank unit tensor for n odd and l^"\x,a)\...,co'') = 0^^^ for n even, « ^ 3. Instead of the time average of the energy (3.131), eq. (3.137), Drummond [1994] presents the total energy in the length L, I \\liH^{t)+ [ E(T)b(T)dT dx. Jo I J to

^(t)=

(3.159)

The Hillery-Mlodinow theory, which does not take account of dispersion (S.-T. Ho and Kumar [1993]), has the commutation relation of the electric field with the magnetic field modified from its free-field value. Drummond [1994] points out that this commutator problem is solved when one does take account of dispersion as an important physical property of a real dielectric. The ingenious analysis of the linear dielectric permittivity has not been generalized to the nonlinear dielectric permittivity. Traditionally, the description of the nonlinear medium assumes that the dispersion terms are negligible. Neglecting unphysical modes, the dual vector potential has the expansion

A\x, t) = y J

^5

ake''-\

where a^ and al, have the standard commutators [aA,5f|.,] = the solutions of a,, = J ^ ^ .

(3.160) 5A,A'1?

and Wi^ are

(3.161)

This enables one to write no = J2^^kalak

(3.162)

k

and (reintroducing D^) n = ^h(Dka\cik^k

[ U^(b^)dx.

(3.163)

J

When there is a nonlinear refractive index or a C^^^ term, the free particles interact via the Hamiltonian nonlinearity. It is this coupling that leads to soliton formation. It is also possible to involve other types of nonlinearity, such as Q^^ terms, that lead to second harmonic and parametric interactions.

356

Quantum description of light propagation in dielectric media

[4, § 3

With respect to practical applications, it is necessary to define photon-density and photon-flux amplitude fields. The photon-density amplitude field reminds us of the so-called detection operator (Mandel [1964], Mandel and Wolf [1995]). A polariton-density amplitude field is simply defined as

^(jc,o=4- y^e^^^"'^'^'"''^^/^-'

p-164)

where k^ = k{a)^) is the center wave number for the first envelope field. This field has an equal-time commutator of the form [W(xut), ^U^2,0] = 5(jci -JC2) i,

(3.165)

where S is an L-periodic version of the usual Dirac delta function

S(xi -X2) = jJ2^'^''^"'~"'\ ^

(3166)

Ak

where the range of Ak is equal to that of A: - A: ^ The total polariton number operator is « = /*.„,„*(.,-)d,.

(3.167,

The polariton-flux amplitude can also be approximately expressed as 0(x,O= A/|^e't<^'-^'^-^-^^'^'%,

(3.168)

where u is the central group velocity at the carrier frequency co^ Thisfluxhas an equal-time commutator of the form [0(xut), 0\x2,t)]

= u8{x, -X2) i.

(3.169)

Operationally, {0^(x,t)0(x,t)) is the photon-flux expectation value in units of photons/second. A common choice is to define the dimensionless field t/^(x, t) by the scaling ^(x,t)= W(xj)J—, (3.170) Vn where n is the photon number scale and to is a time scale, defined in such a way that {\l)\x,t)\l){xj)) is appropriate for the system. This scaling transformation

4, § 3]

Macroscopic theories and their applications

357

is accompanied by a change to a comoving coordinate frame. The first choice of an altered space variable gives

?.-V^,

T=-.

(3.171)

Here XQ is a spatial length scale introduced to scale the interaction times. An alternative moving frame transformation is

?--,

^. = ^ -

(3.172)

The quantization technique developed by Drummond [1990] was later applied to the case of a single-mode optical fiber (Drummond [1994]). On simplified assumptions, the nonlinear Hamiltonian is (cf. eq. 3.156)

n=

[hw(k)a\k)a(k)dk-i-\l^^^ hw(k)a\k)a(k)dk+\t^^^ / D\x)d^x. f D'

(3.173)

Here (o(k) are the angular frequencies of modes with wavevectors k describing the linear photon or polariton excitations in the fiber including dispersion. a(k) are corresponding annihilation operators defined so that, at equal times, [a(k'),a\k)] = d(k -k')l. In terms of modes of the wave guide and neglecting the modal dispersion, the electric displacement b{x) is expressed as

Dix) W = i iI /\/ /

fte(A:)fo(A;).^,_^__^,_ __^ ^,4^,

^'

' ' aik)uik,r)e"''dk

+ H.c.,

(3.174)

where

/ '

u{k,r)\'d'r=\.

(3.175)

Here u(k) is the group velocity and e{k) is the dielectric permittivity. The mode fianction u{k,r) is included here to show how the simplified one-dimensional quantum theory relates to the vector mode theory. When the interaction Hamiltonian describing the evolution of the polariton field W(x, t) in the slowly-

358

Quantum description of light propagation in dielectric media

[4, §3

varying-envelope and rotating-wave approximations is considered, the coupling constant Xe is introduced as

1V2 4e(A:')c

JHrtd'r.

(3.176)

After taking the free evolution into account, the following Heisenberg equation of motion for the field operator propagating in the +;c-direction can be found: d_ d_ 1'{x,t) = ^dx^ dt

2 dx^

+

\Xc'i'\x,t)W{x,t) V(x,t),

(3.177)

where u = v(k^) = dco/dk\^^j^,, CD" = 5^co/9A:^|^^^|. In a comoving reference frame, this reduces to the usual quantum-nonlinear Schrodinger equation:

— i:7-;^e^/(x„r)^,(x„0 ^ife,0, 2 ac2

(3.178)

where W\{xu,t) = ^(x^; + ut,t). In the case of anomalous dispersion which occurs at wavelength longer than 1.5 ^am, allowing solitons to form, the second derivative o)'' can be expressed as w'^ = h/m, where m is the effective mass of the particle. Similarly, the nonlinear term Xe describes an interaction potential

V(x„x'j = -X,S(x,-xly

(3.179)

This interaction potential is attractive when Xe is positive as it is in most Kerr media. It is known that this potential has bound states and is one of the simplest exactly soluble known quantum field theories (Ben-Aryeh [1999]). The repulsive and attractive cases were investigated by Yang [1967, 1968]. This theory is onedimensional and tractable and does not need renormalization, while two and three-dimensional versions do need renormalization. In calculations, it is preferable to substitute the flux amplitude operator

Wix,t)=^0(xj)

(3.180)

into equation (3.177). Drummond associates the idea of spatial progression with the flux amplitude operator. Upon modifying the time variable, he obtains an "unusual form" of the quantum-nonlinear Schrodinger equation, which he reduces to a more usual form again. Since the operators there have their standard meaning, they must have equal-time commutators. In contrast, the resulting

4, § 3]

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359

equation (Drummond [1994]) appears as the quantum-nonlinear Schrodinger equation with time and space interchanged. This means that the operators have equal-space commutators. The problem is whether these commutators are well defined. An important physical effect in propagation is the Raman scattering fi-om molecular excitations. For this reason, the nonlinear Schrodinger equation requires corrections due to refi-active-index fluctuations for pulses longer than about 1 ps, especially when high enough intensities are present, and fails for pulse durations much shorter than this. Korolkova, Loudon, Gardavsky, Hamilton and Leuchs [2001] have studied a quantum soliton in a Kerr medium. They have simplified, implicitly, the classical propagation equation for the slowly varying electric field envelope by introducing a new time measurement in dependence on a position. In changing to dimensionless variables they make the new time a "position" and the position a "time" variable and then get a classical nonlinear Schrodinger equation.

3.3. Modes of the universe and paraxial quantum propagation 3.3.1. Quasimode description of the spectrum of squeezing Toward the end of the 1980s it had become clear that the use of squeezed states (Walls [1983], Loudon and Knight [1987]) in interferometry can lead to enhanced signal-to-noise ratios. Milbum and Walls [1981] have shown that the cavity of a degenerate parametric oscillator admits only 50% squeezing (in the steady state). Yurke [1984] was the first to realize that this pessimistic conclusion does not hold, as the noise reduction in the transmitted field can be quite different from that in the intracavity field; the first step is to relate the field operators inside and outside the cavity. Whereas it was obvious at the time that the field operators inside the cavity remain the usual quantum-mechanical annihilation operators of one or a small number of harmonic oscillators, the connection of the field operators outside the cavity with the "Langevin-noise operators" was not established until the 1980s (Collett and Gardiner [1984], Gardiner and Collett [1985], Carmichael [1987]). These authors have clarified the relation between this subtle property of squeezed light and its generation and the concept of light propagation. Not only the interpretation, but also the derivation of the Langevin-like "noise" terms was presented by Lang and Scully [1973], after they had introduced and studied the "modes of the universe" (Lang, Scully and Lamb Jr [1973], Ujihara [1975, 1976, 1977]). Ley and Loudon [1987] studied the "mode-strength function", which exhibits peaks near the wavevectors

360

Quantum description of light propagation in dielectric media

[4, § 3

of the cavity quasimodes. Barnett and Radmore [1988] have defined a similar mode-strength function for the phenomenological coupling between the cavity quasimodes and the external environment and found that their function has peaks at the frequencies of the quasimodes. It is appropriate here to mention a book by Scully and Zubairy [1997], where the results of Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990a] are expounded or formulated as exercises. The latter discuss the modes of the universe, which include the interior of the imperfect cavity of interest, and use them to define the intracavity quasimode, the incident external field mode and the output field mode. The mutual coupling of these modes emerges naturally in this formalism. Following Lang, Scully and Lamb Jr [1973], the one-sided empty cavity is described also by the relation t:J,(t) = r£W (t _ '^\ +lEl:\t),

(3.181)

where / is the cavity length, r is the real amplitude reflection coefficient and 1 is an appropriate transmission coefficient, 1 = \ / l - r ^ . Here £'•„ (0 is the positive-frequency part of the input field; it obeys the commutation relation El^\t),El;\s)\ =Kd(t-s)l,

(3.182)

where £|;>(0 = [El^\t)V and

with Q the quasimode frequency. For the full Fox-Li quasimode (Fox and Li [1961], Barnett and Radmore [1988]), a single-mode annihilation operator a(t) is defined: «(0 = y ^ ^ * : ^ ( O e ' " ' .

(3.184)

It is convenient to use the slowly varying amplitudes £l^\t), fout(0 ^^^ fcav(0 for the input, output and cavity fields related to the cavity frequency Q, respectively: ^''\t)

= £J^\t)Q''^^\

7 = in, out, cav.

(3.185)

4, § 3]

Macroscopic theories and their applications

361

In the situation where it holds that

(,_|)»(,_|;.),(„_|1„„

(3.186)

in the short cavity round-trip time limit, we get a quantum Langevin equation

^a(0 = - r a ( 0 + y ^ C ( 0 ,

(3.187)

where

^ = 'i ^

(3.188)

Further, Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990a] define, for arbitrary measurement times, the spectrum of squeezing of the output field via the quadrature variances. They present a microscopic effective Hamiltonian model of balanced homodyne detection. They refer to the fundamental papers of Collett and Gardiner [1984], Gardiner and Collett [1985], Caves and Schumaker [1985] and Yurke [1985], where this concept of spectral squeezing was originally treated. As an approximation, the following operator is introduced: ^out(6a;) =

1 /KT

I ^0

^-'out (Oe'^^^'^^d/,

(3.189)

where bo) is the (level) separation between two nondegenerate ground states of the atom, and T is the measurement time or the inverse of the detection bandwidth. As shown also by Yurke [1985] and Carmichael [1987], with a balanced homodyne detector one measures the combinations with appropriate phase shift 6: ^OUtfAO

2

e'^^C(0 + e-'^^C(0

(3.190)

where £oJ(t) = [oouliOY^ ^^^ fr^i^i this the natural generalization of the singlemode quadrature concept is ^out,(8w) •

e'AU^aj)^Q'''^Aa-^co)

(3.191)

One might wonder why a non-Hermitian operator is taken for such a generalization of the Hermitian operator. Finally, the connection between single quasimode

362

Quantum description of light propagation in dielectric media

[4, § 3

squeezing and spectral squeezing is explored and the difference in the noise reduction inside and outside the cavity is clarified in a way that lends itself to simple visualization. Gea-Banacloche, Lu, Pedrotti, Prasad, Scully and Wodkiewicz [1990b] have first analyzed measurements of small phase or frequency changes for an ordinary laser, and calculated the extracavity phase noise for a phase-locked laser. These analyses are based on the mean values and normally ordered variances of quantum operators for which classical Langevin equations may be written down. The classical Langevin formalism is fiarther replaced by the alternative Fokker-Planck formalism for the calculation of the spectrum of squeezing. This general Fokker-Planck formalism has been applied to the twophoton correlated-spontaneous-emission laser. It was seen that without onephoton resonance and initial atomic coherences involving the middle level, the maximum squeezing of the intracavity mode is 50%, while the detected field can be almost perfectly squeezed. Almost the exact reverse holds, however, if one-photon resonance and initial atomic coherences involving the middle level are present. In particular, the intracavity field may be perfectly squeezed while the outside field shows almost no squeezing in the same quadrature, but still has, in fact, increased noise in the conjugate quadrature. Finally, the effect of finite measurement time on the quadrature variances has been briefly analyzed. Dutra and Nienhuis [2000] have unified the concept of normal modes used in quantum optics and that of Fox-Li modes from semiclassical laser physics. Their one-dimensional theory solves the problem of how to describe the quantized radiation field in a leaky cavity using Fox-Li modes. In this theory, unlike conventional models, system and reservoir operators no longer commute with each other, as a consequence of the use of natural cavity modes. Aiello [2000] has derived simple relations for an electromagnetic field inside and outside an optical cavity, limiting himself to one- and two-photon states of the field. He has expressed input-output relations using a nonunitary transformation between intracavity and output operators. 3.3,2. Steady-state propagation Deutsch and Garrison [1991a]) assumed that in the case of an ampHfier, one is usually interested in the spatial dependence of temporally steady-state fields. They do not attempt to reformulate one-dimensional propagation, cf Abram and Cohen [1991], where the temporal evolution by the Hamiltonian is supplemented by the spatial progression with the momentum operator. The

4, § 3]

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363

alternative proposal is made that the quantum-mechanical equivalent of the classical steady-state condition is the description of the system by a stationary state of a suitable Hamiltonian. There is a formal resemblance to nonrelativistic many-body theory for a complex scalar field (Deutsch and Garrison [1991b]), which helps determine the Hamiltonian. In this theory a non-Hermitian envelopefield operator W{z, t) with the property W{z,t\ W\z',t)\ = 6{z-z') i

(3.192)

is introduced. In the application to the optical field, the vector potential operator (or electric-field-strength operator in the lowest order) corresponding to a carrier plane wave of a given polarization unit vector e is expressed as

t:\z,t) = eJj^^^W(z,t)

exp[i(^z-a;0],

(3.193)

where A is the beam area and n(a)) is the dispersive index of refi-action. In contrast to Deutsch and Garrison [1991a], we make a simplification, i.e., we will not consider a carrier-wave Hamiltonian. For a single wave interacting nonlinearly with matter, the total Hamiltonian can be written as

^ ( 0 =4nv(0 + ^mt(0,

(3.194)

where Hem(t) is the Hamiltonian governing the free progression of the envelope and Hmi(t) is a general interaction Hamiltonian. In fact, the generality will not be exercised and we will treat only the vacuum input and the case of a degenerate parametric amplifier. In the standard Heisenberg picture, the equation of motion for the envelopefield operator reads dW{z,t) _ i 'nz,t),H(t) dt h

(3.196)

or for a linear medium dW(z,t) _ c• d4^(z,t) dt n(w) dz '

(3 197) y' J

364

Quantum description of light propagation in dielectric media

[4, §3

The solution is ct

(3.198)

•^(^'^^=^f^-;K^'«

In the standard Schrodinger picture, the state |
dt

(3.199)

H,n,(t) + Hm(t) | 0 ) .

Introducing the carrier-wave Hamiltonian has invoked the consideration of the Schrodinger picture (Deutsch and Garrison [1991b]), along with the envelope picture which we have confined ourselves to. Relation (3.193) is the positivefrequency component of the electric field in the envelope picture similarly as relation (4.5b) of Caves and Schumaker [1985] is this component in the interaction picture. The envelope picture is essentially the modulation picture of Caves and Schumaker [1985]. For the application under consideration, the steady state (ss) solutions are identified with the stationary solutions to eq. (3.199),

J^env ' ^ i n

(3.200)

l^)ss = ^l<^)s

For the stationary solutions, the label (ss) will be omitted. In the case of the degenerate parametric amplifier, the interaction Hamiltonian can be written as (Hillery and Mlodinow [1984])

//int(0

- ^ / / / ( ^

/'\z)£;(z)cxp[-i(k,z

a)^t)][El:\z,t)f

+ }lx.}

dxdydz,

(3.201) where a^p is the pump frequency, (J0p = 2co, X^^\^) is the second-order susceptibility coupling the pump to the degenerate signal and idler fields, £p(z) is the pump •(+)/ M+) 7(+)/ amplitude, and [Elo\z,t)Y = £';,7(z,0 • E\o (z,t\ with E\o\z,t) given by the 7( + )/ formula E^^\z,t) = e- E\,\z, t). Substituting for E'^^z.0 from relation (3.193) gives the interaction Hamiltonian in the envelope picture: ih c H\r^x — 2 n{w)

I

K''(Z)W\Z)-K(Z)W^-(Z)

dz.

(3.202)

4, §3]

Macroscopic theories and their applications

365

with (3.203)

J^(^) = ^go(^)exp[i0(z)],

(3.204)

n((x))c (l>(z)

Jt

(3.205)

+ Mz^l3(zy

Here go(z) is the standard power gain coupHng constant (Yariv [1985]), Ak = 2k- kp is the phase mismatch at the degenerate frequency, and I3(z) is the remaining phase originating from the product /^^^(z)f*(z). To solve the time-independent Schrodinger equation, Deutsch and Garrison [1991a] assume that the eigenstate is a squeezed vacuum state corresponding to a two-photon wave function. They define a functional squeezing operator S[^] = exp

0/

^iz)W^\z)-r{z)W\z)

dz

(3.206)

with z-dependent squeezing parameter ^(z) = -r{z) exp[i6{z)]. The squeezed vacuum is defined as |0){s}=5[|]|0).

(3.207)

For the squeezed vacuum to be a stationary solution, ^(z) and A should obey ^^[?](4nv -f^int) m

|0) = A |0).

(3.208)

Applying the operator W(z) to both sides of eq. (3.208), and taking into account that A«l^(z)|0)=0 (3.209) =

S^[^](^H,,,+H,r,,)s[^W(z)\0),

we rewrite the eigenvalue problem in the A-independent form [S^[^]{H,,,+H,n,^

S[^l W(z)\ |0)^.j = 0 = ^ |0),

(3.210)

366

Quantum description of light propagation in dielectric media

[4, § 3

where the commutator C is n(co) { - exp[i0(z)] ( exp[-i0(z)] cosh[r(z)] ^ (exp[i0(z)] sinh[r(z)]) d - -— (cosh[r(z)]) sinh[r(z)] dz + /f*(z)exp[i0(z)]sinh^[r(z)] - K(z) exp[-i0(z)] cosh^[r(z)]) W\z) (3.211) ( d + cosh[r(z)]-—(cosh[r(z)]) Mz V dz - exp[i0(z)] sinh[r(z)]-^ (exp[-i0(z)] sinh[r(z)]) dz + /c*(z) exp[i0(z)] sinh[r(z)] cosh[r(z)] - K(z) exp[-i0(z)] sinh[r(z)] cosh[r(z)]^ W{z)

+ dz^^(^)}. The eigenvalue condition requires that the real and imaginary parts of the coefficient of ^ ^ vanish, yielding the desired propagation equations £

= igocos(0-0),

%- =-goCOth(2r)sin(0-0). dz

(3.212) (3.213)

Upon introducing the complex amplitude C(z) = - exp[ie(z)] tanh[r(z)],

(3.214)

we can write a propagation equation for it in the compact form ^±&-

= K{z)-K*{z)l;\z),

(3.215)

which may be useful for guessing the boundary condition r(z)Uo = 0,

0(z)Uo = 0(O).

(3.216)

When /3(z) = 0 and the phase difference Q(z) - 0(z) is small, the squeezing parameter r(z) is a weighted integral of the experimental parameter ^go(^0? ^' ^ [^,2:];

4, § 3]

Macroscopic theories and their applications

367

when moreover go{oo) > \Ak\, the squeezing parameter 6{z) converges to a function of the experimental parameters {0(z)-arcsin[A^go(oo)]}. Direct solution of eq. (3.208) requires that the real and imaginary parts of the coefficient of W\z) vanish, yielding the propagation equations (3.212) and (3.213) again. The presence of the singular operator W{z)^f\z) indicates that A generally has no finite value. 3.3.3. Slowly-uarying-enuelope approximation A macroscopic approach to the quantum propagation aims at a quantum version of the slowly-varying-envelope approximation. Such an envelope implies that the wave is paraxial and monochromatic. The problem of quantum propagation of paraxial fields was considered first by Graham and Haken [1968]. Revived interest is indicated by Kennedy and Wright [1988]. Deutsch and Garrison [1991b] begin with generalizing the results of Lax, Louisell and McKnight [1974], which develop the classical theory of a strictly monochromatic wave in an inhomogeneous nonlinear (perhaps amplifying) medium. The generalization is made only to a quasimonochromatic wave, and the quantum theory is presented in the simplest system of codirectional propagation considering only the freefield dynamics. In the Coulomb gauge, the positive-frequency component of the vector potential satisfies the free-field wave equation W'A^-\xJ)-\^A^^\x,t) = 0.

(3.217)

The approximation of the slowly varying envelope is introduced by expressing A^^\x,t) as an envelope modulating a carrier plane wave propagating in the z-direction with wave number ko and frequency COQ ^ ck^, A^'-\x, t) = Ao W(x, t) exp[i(^oz - oj^t)].

(3.218)

Here, ^{x, t) is a vector-valued function, henceforth referred to as an envelope field, and ^o is a normalization constant, which we will specify above relation (3.226). The initial positive-frequency component can be expressed as "'*'*^^'' = ^ ^ = r 2 i ^ / r a

5]^A(*)^AWe'*-M^ft.

(3.219)

Here ex{k) are the orthogonal polarization unit vectors, and the reduced Planck constant is introduced in view of the possible later quantization. Tx{k) are thus momentum-space wave functions.

368

Quantum description of light propagation in dielectric media

[4, § 3

The intuitive notion of a paraxial field is that it is composed of rays making small angles with the main propagation axis. In other words, a paraxial wave function {Tx(k)} is concentrated in a small neighborhood of the wavevector ko = koei, of the carrier wave. We define/A(^) by the relation fx{q) = ^x{q^h\

(3.220)

where q is the relative wavevector. Let us observe that q = {qj,qz), where qj is the transverse part of ^, ^ = ^j + ^r^3. In contrast to Deutsch and Garrison [1991b], we stress that we express the concentration in a small neighborhood of qo = 0, by letting the wave fiinction {fx(q)} depend on a small positive parameter 0\f}{q) ^fx(q, 0). Let us assume that /.(,„,..)= V / V 7 , ( ^ , ^ , e ) ,

(3.221)

where

and we have introduced the notational convention that an overbar indicates a dimensionless fianction of the scaled variables (and perhaps 6). This relates to defining a dimensionless "momentum" vector i; == (i^, r/-), where

The fimctions of interest are those that have a convergent power-series expansion in 6, oc

7,(i/,0) = ^ 0 " / r ( i j ) .

(3.224)

n=0

In contrast to Deutsch and Garrison [1991b], we note that

7,(t;,0) =7f(i;).

(3.225)

Relations (3.223) and (3.225) lead to the wave ftinction being ^-dependent, different from Deutsch and Garrison [1991b].

4, § 3]

Macroscopic theories and their applications

369

Substituting the integral (3.219) for the envelope field defined by eq. (3.218) at ^ = 0 by the momentum-space wave fianction given by eq. (3.220), and choosing Ao = \/hc/2ko, we find that

•^(-.^ - 0,0) = ^

/ 7 ^ »

E ..(. +W.(.,a)e-'d3,. A — 1.2

(3.226) Here, the parameter 6 has been introduced, which is not present in the integral (3.219), where Tx(k) = ^A(^, ^), and A^^\x, t) = ^^^^(jc, t, 6). Deutsch and Garrison [1991b] investigated the integro-differential form of the wave equation for A^^\x, t, 6): i-^'^»(jc, t, 6) = c(-V-)''- A^*\x, t, 0),

(3.227)

where (-V^)''^^ is an integral operator defined by

{-VY'F(x) = ^^_

J \k\ F(k) e'*- d^*,

(3.228)

with F(k) the Fourier transform. Let us note that V = (Vj, §:). Substituting from eq. (3.218) into (3.227) gives i— W(x, t, d) = {cQ - (i)Q)W{x, t, e\ at where

(3.229)

with the transverse Laplacian

The scaled configuration-space variables | = (|T, 0 are §T = Okoxj,

t = d^h)Z,

(3.232)

and the dimensionless time variable is T = e^oMit.

(3.233)

370

Quantum description of light propagation in dielectric media

[4, § 3

After expressing the envelope field in the form W(x, t, 6) = W(xj,z, t, 6) = -^WiOkoXj, vV

e^koz, d^coot, 0),

(3.234)

we can rewrite relation (3.229) as

i—W(iT,e) = n(e)V(iT,e),

(3.235)

where Hid) = ^^{d)-l],

(3.236)

Q(d) = l^Qfek,VT,0'ko^Y

(3.237)

with Uko

OL, 6~ko OZ

This provides the possibility of expanding the differential operator H(d): CX)

Hid) = J2 d"n^"\

(3.239)

n=0 7(")

where the differential operators H are just defined by the formal expression. The dimensionless amplitude ^ ( | , r, 0) has the expansion V(l T,e) = Y, 0"V^"\l T).

(3.240)

n=0

It is evident that the terms satisfy the following equations: i^W'\l dr

r)=Y^ n''-"'W''\l r),

« - 0,1,2,....

(3.241)

In the article by Deutsch and Garrison [1991b], the discussion of the classical equation of motion is completed by considering the initial-value problem. We rewrite eq. (3.226) as "Pix, 0) = — ^ f ^

K,(q)fM 0) e'»- d'q,

(3.242)

4, § 3]

Macroscopic theories and their applications

371

where the function Kx{q) is defined by

with ex{q) = ex{q + h\

(3.244)

Re-expressing eq. (3.242) in terms of the scaled variables gives l P ( | , r = O;0)= — ^ 3 ^ / Y. ^^^^

•'

jf,(i;, 0)7,(1/) e'-'-^d^I/.

(3.245)

A =1.2

The scaled kernel function is ifA(i;,e)=^#%,

(3.246)

where w(iy, 0) = yJ\ + e\2r]-_ + r]l) + d^r]l

(3.247)

Considering the expansion oo

Kx{n,e)-Y.^''K';:\t,),

(3.248)

«=0

we obtain the initial expansion of the envelope fields:

^'"*^^^= ( 2 ^ /

^

KT{n)f,{n)e'''d?t,.

(3.249)

A=l,2

Deutsch and Garrison [1991b] claim that the preceding arguments can be used to identify the subspace of the photon Fock space consisting of the paraxial states of the field. They resort to the space S, the infinitely differentiable functions that decrease, when |i/| -^ oo, faster than any power of |i/| ^

372

Quantum description of light propagation in dielectric media

[4, § 3

Let us recall the standard plane-wave creation and annihilation operators al(k), ax(k), and the wave-packet creation and annihilation operators

^h^=

f Yl ^x(k)al(k)d'k ^

(3.250)

X=\,2

and its conjugate. We now define

"^

A=l,2

where cl{q) = aliq + ko)

(3.252)

are the creation operators corresponding to the envelope field. For the subsequent analysis, a unitary operator T(d) is of interest, f(0) I/'; m) = \f(dy, m),

(3.253)

such that f(a)|0) = |0),

f(0)ct[/]7't(0) = c t [ y v / ( ^ , | ) ]

(3.254)

where (cf. eq. 3.221) /A, • A,„(?T 1, ^z I, • • • , ?Tm, ^zm, 6)

(3.255) /A,-A4 0 '

g2'--'

0 ' 02 ; •

In the description of the dynamics using the Schrodinger picture, the paraxial approximation means mainly the evolution of the initial state \(p{t,0)), ^^\
(3.256)

where m,e))\,^_, = m\0'{t = O)).

(3.257)

Defining the state \cp'{t,e)) = Pie)\0(t,d)),

(3.258)

4, § 3]

Macroscopic theories and their applications

373

for all times, we can rewrite eq. (3.256) in the form

I \0\t, 0)) = -'-H'{d) \^\t,6)),

(3.259)

where H'{d) = f\d)Hf{e\

(3.260)

Using the expansion

H'{d) = Y^ 0"'H'^"'\

(3.261)

we may expand eq. (3.259) into a set of coupled equations for the coefficients of the series oc

|0'(^, 6)) = Y^ e'" \0'^"'\t)).

(3.262)

w=0

Describing the dynamics in the Heisenberg picture, we should generalize the relations (3.260) and (3.261) to an arbitrary operator M(t), M\t,d) = f\e)M{t)f{e\

(3.263)

m=0

We can then rewrite the equation of motion _ M ( 0 = -^[M(/),^(/)]

(3.265)

in the form -M'{t,6)

= -'-[M\t,e\H'{t,

0)1

(3.266)

We may expand this equation into coupled equations similarly as eq. (3.259). Since f(l)=i,

M'{t,\) = M{t\

(3.267)

relation (3.264) simplifies for 0 = 1, or any operator M{t) can be expressed as CXD

M{t) = Y^M^"'\t).

(3.268)

In both pictures, definition (3.233) can be used whenever it is advantageous.

374

Quantum description

of light propagation

in dielectric media

[4, § 3

In accordance with Deutsch and Garrison [1991b], we introduce the operator ^f{x) by relation (3.226), with »F(jc, / = 0,0) »-^ ^{x) on the left-hand side and fxin^ S) "-^ Q ( ^ ) on the right-hand side. This operator can be expanded as oo

Wix) = Y^ W^"\xl

(3.269)

n=0

where •^^^^W = 7

^

/ E

^

^

-^ A=l,2

^1"^(^) ^A(^) e'^ ^ d^iy.

(3.270)

Using this expansion, we can compute the commutators between fields of different orders:

'^'^ '

•'

A =1,2

(3.271) As expected, the nth-order commutator can be expressed as n

Wi(x), W^(x')

E

[^^""\^), ^y "^^(JcO] .

(3.272)

m=0

The equal-time commutation relations are preserved by the dynamics in each order of the approximation scheme.

dt

Wi{xj\WJ{x\t)t^

= b.

(3.273)

In zeroth order, the theory yields a quantized analogue of the classical paraxial wave equation, and formally resembles a nonrelativistic many-particle theory. This formalism is applied to show that Mandel's local-photon-number operator and Glauber's photon-counting operator reduce, in zeroth order, to the same truenumber operator. In addition, it is shown that the 0{d^) difference between them vanishes for experiments described by stationary coherent states. 3.4. Optical nonlinearity and renormalization Abram and Cohen [1994] mainly applied a traveling-wave formulation of the theory of quantum optics to the description of the self-phase modulation

4, § 3]

Macroscopic theories and their applications

375

of a short coherent pulse of Hght. They seem to have been the first to use renormaUzation (Itzykson and Zuber [1980], Zinn-Justin [1989]). The renormalized theory successfully describes the nonlinear chirp of the propagating pulse, and permits the calculation of the squeezing characteristics of self-phase modulation. The description of the propagation of a short coherent light pulse inside a medium with an intensity-dependent refractive index (Kerr effect) has become relevant to optical-fiber communications, all-optical switching, and optical logic gates (Agrawal [1989]). The neglect of dispersion and of Raman and Brillouin scattering leads to the description of self-phase modulation. In classical theory it is derived that, in the course of propagation, the pulse becomes chirped (i.e., different parts of the pulse acquire different central frequencies), which also influences its spectrum. Abram and Cohen [1994] have pointed out many difficulties in the investigation of the quantum noise properties of a light pulse undergoing self-phase modulation. The traditional cavity-based formalism truncates the mutual interaction among the spatial modes to the selfcoupling of a single mode (or only a few modes) and cannot give a reasonable approximation to the frequency spectrum produced by self-phase modulation. In spite of the difficulties, papers based on a single-mode field description have indicated that the self-phase modulation can produce squeezed light (Kitagawa and Yamamoto [1986], Shirasaki, Haus and Liu Wong [1989], Shirasaki and Haus [1990], Wright [1990], Blow, Loudon and Phoenix [1991]), and others have treated squeezing in solitons (Drummond and Carter [1987], Shelby, Drummond and Carter [1990], Lai and Haus [1989]), an effect that has been verified experimentally (Rosenbluh and Shelby [1991]). Blow, Loudon and Phoenix [1991] have shown the divergence of nonlinear phase shift which Abram and Cohen [1994] treat through the process of renormalization. Let us review the basic features of the quantization of the electromagnetic field in a Kerr medium and discuss the relevance of the renormalization procedure to the treatment of divergences of effective medium theories. We consider a transparent, homogeneous isotropic and dispersionless dielectric medium that exhibits a nonlinear refi-active index. We examine a situation similar to that in Abram and Cohen [1991]. The Hamiltonian for the electromagnetic field in a Kerr medium is ^ ( 0 = J {^2 [^'(^'0 + ^E\z,O]

+ lxE\z,O}

dz,

(3.274)

where the integration along the direction of propagation, z, is written explicitly, while the integration over the transverse directions x and y will be implicit. We use the Heaviside-Lorentz units for the electromagnetic field without passing

Quantum description of light propagation in dielectric media

376

[4,

over to atomic units, i.e., h = c = I, and x is the nonlinear (third-order) optical susceptibility. From the perspective of substitution of eq. (3.22), the Hamiltonian (3.274) can be written as (Hillery and Mlodinow [1984]) H(t) =

/{

B^{z,t)+-b\z,t)

-^£)\z,/)|dz,

(3.275)

where the displacement field has been defined as b(z,t) = eE(z,t) + xE\zJ).

(3.276)

The canonical equal-time commutators are (cf. eq. 3.25) [A(zutXb(z2,t)]

= -ihcd{z, -Z2) i,

(3.277)

[B{zut)Mz2.t)]

= -ihcd'iz, -Z2) I

(3.278)

It is convenient to adopt a slowly-varying-operator picture in which the zerothorder dynamics of the field governed by the linear-medium Hamiltonian are already taken into account exactly, while the optical nonlinearity can be treated within the framework of perturbation theory. In such a picture, a field operator Q(z, t) evolving inside a nonlinear medium is related to the corresponding linearmedium operator go(^, 0 by Q{zj)=U-\t)Q^{z,t)U{t).

(3.279)

The unitary transformation Lf{t) is given by

^(0 = Texp

4/>'

(r)dr

(3.280)

where T = T denotes the time ordering and

m) = -^,jbtiz,t)
(3.281)

is the interaction Hamiltonian. The full nonlinear Hamiltonian (3.275) in which the exact displacement and magnetic induction field operators D(z,t) and B(z, t) are replaced by the corresponding zeroth-order (linear-medium) operators Z)o(z, t) and Bo{z, t) can be written as H{b^X)

= m ) + H,{t\

(3.282)

where

Ho{t)= \ j \Bl{z,t)+-^^^^^

(3.283)

is the linear-medium Hamiltonian. Following the traditional modal approach to relation (3.279), Kitagawa and Yamamoto [1986] developed a singlemode treatment of the self-phase modulation. Clearly, such an investigation

4, § 3]

Macroscopic theories and their applications

317

is valid only inside an optical cavity with a sparse mode structure. In this situation, the time evolution cannot be interpreted as space progression. In developing a traveling-wave theory for self-phase modulation, Blow, Loudon and Phoenix [1991] obtained a solution similar to the single-mode solution. They encountered a nonintegrable singularity upon normal ordering, a phenomenon known in quantum-field theory as an "ultraviolef divergence. To avoid an infinite nonlinear phase shift due to this simple description of the Kerr interaction. Blow, Loudon and Phoenix [1991, 1992] introduced a finite response time for the nonlinear medium as regularization in the Heisenberg picture. Alternatively, Haus and Kartner [1992] considered group-velocity dispersion for pulse propagation in the medium as a regularization. At any rate, regularization is a subsequent sophistication of the simple model known from classical theory. The response time as well as the group-velocity dispersion are necessary ingredients of a complete description of the propagation of an electromagnetic excitation in fibers. But they have no influence on the effects associated with the vacuum fluctuations under study. A systematic way of dealing with the vacuum fluctuations in quantum field theory is the procedure of renormalization (Itzykson and Zuber [1980], Zinn-Justin [1989]). In order to obtain finite results, the procedure of renormalization redefines all the quantities that enter the Hamiltonian. The renormalization point of view is that the new Hamiltonian is the only one we have access to. It contains the observable consequences of the theory, and the parameters are the ones we obtain from experiments. The bare quantities are only auxiliary parameters that should be eliminated exactly from the description (Stenholm [2000]). The re-defined (renormalized) quantities are able to incorporate the (infinite) effects of the vacuum fluctuations. We will provide the definitions of broad-band electromagnetic field operators and treat the propagation of light in a linear medium. The normal ordering is considered as the simplest renormalization, e.g., in the case of the effective linear Hamiltonian (3.283). To this end, the Hamiltonian Ho{t) is to be written in terms of the creation and annihilation operators. The normal ordering allows us to subtract the vacuumfield energy up to the first order from the effective Kerr Hamiltonian (3.275). However, when this Hamiltonian is used to describe propagation disregarding the richness of the quantum-field theory, the normal ordering gives rise to additional divergences that can be attributed to the participation of the vacuum fields. Upon renormalization, involving also the refractive index, the divergences are removed. As the Kerr nonlinearity involves the fourth power of the derivative §-fA(z, t), it cannot in general be renormalized to all orders with a finite number

378

Quantum description of light propagation in dielectric media

[4, §3

of corrections. Inspired by nonlinear optics, the slowly-varying-amplitude approximation decouples counterpropagating waves and renormalization to all orders becomes possible. All types of optical nonlinearity x^"^ give rise to divergences which require renormalization. In the treatment of parametric downconversion (Abram and Cohen [1991]), the problem of divergences and the need of renormalization were not formulated. Abram and Cohen [1994] defined the broad-band electromagnetic field operators, and treated the propagation of light in a nonlinear medium. In the absence of optical nonlinearities, / == 0 and the linear-medium displacement field has the usual proportionality relationship with the electric field: (3.284)

Do(z,t) = eEo(z,ty

The magnetic field and the displacement field in the linear medium obey the equal-time commutation relation Bo(zut),Do(z2,t)

(3.285)

= -\hcd\z\ -zi) i.

The operators V^ (cf eqs. 3.70, 3.71) from Abram and Cohen [1991] reappear as the operators

Mz,t)

1

V^(z,tl

1

^-(z,t)

Vo(z,t).

(3.286)

For V^±(z, t), the equations of motion in the Heisenberg picture may be calculated by use of the commutator (3.285),

l^-(^''^4

//o,t/;±(z,0

d^

(3.287)

where u = c/^/e is the speed of light inside a dielectric with refractive index e. Their solutions are t/;+(z,0

xjj+iz - ut, 0),

(3.288)

t/;_(z,0

V^_(z + t;^0).

(3.289)

The equal-time commutators of the copropagating field operators can be obtained from the definition and the commutator (3.285) as t/;+(zi,0,V^+(z2,0

= -ifiud\z\ -zi) i.

(3.290)

V^-(zi,0,V^-(z2,0

= ihud'{z\ -Z2) 1.

(3.291)

4, § 3]

379

Macroscopic theories and their applications

For the fields, the corresponding operators commute with each other, (3.292)

0.

\l}+{Zxj\\l)-{Z2j)

The operators (3.286) permit us to express the hnear-medium Hamiltonian (3.283) as

m) = \j\l)l{z,t)-^xi)l{zj)

(3.293)

dz.

thus separating it into a sum of two mutually commuting partial operators, one for each direction of propagation. In the homogeneous medium it is possible to separate the electromagnetic field operators V^±(z,0 into positive- and negative-frequency parts. (3.294)

i/;±(z,o = 0±(z,O + 0±(^,O, defined as h{zj)=\

\p±{z,t)±

i

r v^±(z^o,^,

(3.295)

z -z' The operators 0^(z, t) and 0±(z, t) can be considered as creation and annihilation operators, respectively, for a right- (or left-) moving electromagnetic excitation which at time t is at point z. The equal-time commutators of 0±(z, 0 are somewhat complicated, 0±(Zi,O,0±(Z2,O

hv d

1 Z\

-Zi

=F '\7lb(z\ -zi) 1, (3.296)

0+(Zi,O,0!(Z2,O = 0, where V refers to the familiar generalized fianction V-_. Nevertheless, an important simplification results when only unidirectional propagation is considered. Upon introducing the operators

b',\z,t)0+(z,O-0-(z,O ,

B'-\z,t)-

it

D':'{z,t) B';\z,t)

,

(3.297) (3.298)

[4, §3

Quantum description of light propagation in dielectric media

380

and considering the relations Do(z,t)

xl)^{z,t) + xl).{zj)

Bo(z,t) =

V2

xp^iz, t) - ip^z, t) (3.299)

and relation (3.294), we verify that Z)o(z, 0 = D^^\z. t) + b^^\z, t\

Bo(z, t) = B^^\z, t) + B^,\z, t).

(3.300)

Using the new operators, the equal-time commutation relation (3.285) can suitably be modified as B^^\zut\b',\z2.t) = --hcd\zi-Z2)l.

(3.301)

For a right-moving electromagnetic excitation, we observe that 0^(^)(z,,O = V2B[\l^(zutl

0.(^)(22,O = )J^b^oil)(^2,tl

(3.302)

where the subscript (+) refers to A: > 0. Using eq. (3.302), we obtain that

0+(zi,o,0!(z2,o

= -ihuS\z\ -zi) 1(+ (+)•

(3.303)

(+)

Similarly, for a left-moving electromagnetic excitation, we note that 0_(_)(zi,O = -y2^|^!^(zi,0,

0-(-)fc,O = ]J-^&oU^2,t%

(3.304)

where the subscript (-) refers to A: < 0. From this, (3.305) = i^t;^'(zi -Z2) i (-)• The electromagnetic creation and annihilation operators allow us to speak of the normal order, for instance, when we write the Hamiltonian (3.293) in the form 0_(zi,O,0!(z2,O

J(-)

Ho(t) = [ te(z, 0 Mz, t) + 0!(z, 0 0_(z, t)] dz.

(3.306)

We can define annihilation and creation wave-packet photon operators F^{z,t) = fF(z-z)Mz,t)dz, F|(Z,0

= /F*(z-z)0|(z,Odz,

respectively, with F(z) a complex function: 1 F(z) = exp(L^) F(z),

(3.307) (3.308)

(3.309)

where uK is the central (carrier) fi-equency and F(z) is the wave-packet envelope fianction peaked at z = 0 and A: = 0.

4, § 3]

Macroscopic theories and their applications

381

Under the usual assumption of narrow bandwidth and ihv I F\z)F\z)dz

= \,

(3.310)

where F' denotes the spatial derivative, we obtain that the operators F+ and F} follow the boson commutation relation [F4zJXFl(z,t)\=l

(3.311)

Let us remark that the commutation relation (3.311) is relation (A5) of Milbum, Walls and Levenson [1984], where the formalism of counterdirectional coupling was derived or rather this pitfall was underestimated. Now we can consider a coherent pulse whose shape is described by pF(z) with a scaling factor p. A coherent state appropriate to pF is defined as |pF)=exp[p(F,^-F,)]|0).

(3.312)

It satisfies the "single-mode" eigenvalue equation F^(z,t)\pF)=p\pF)

(3.313)

and, at the same time, it obeys the approximate quantum-field eigenvalue equation 0+(z,O \pF) = h\^pF\z-z)Qxp[-iK{z-z)]

\pF).

(3.314)

The approximation made in the derivation of eq. (3.314) has kindled the interest in the Glauber factorization conditions and the theory of coherence (see also Ledinegg [1966]). When we examine right-moving pulses, we can introduce a moving-frame coordinate, r] = z-ut,

(3.315)

and simplify relation (3.288), Mz,t)

= 0(ri,O) = m ,

(3.316)

dropping the subscript + whenever we use the coordinate t] explicitly. Similarly, the commutation relation (3.303) can be modified. The right-moving narrowbandwidth wavepacket operators (3.307) and (3.308) can now be written as F(fj) = jF(r]-f])mdr], where f] = z-ut,

(3.317)

and

F\r]) = j F\r]-m\r])dr]. In the moving-frame representation F{r], t) ^ F(rj, 0).

(3.318)

382

Quantum description of light propagation in dielectric media

[4, § 3

It is feasible to find a connection with the approaches leading to narrow-band field operators (a) appearing in papers by Shirasaki and Haus [1990], Drummond [1990] and Blow, Loudon, Phoenix and Sheperd [1990], and used in papers (Blow, Loudon and Phoenix [1991], Shirasaki and Haus [1990]). An important feature of these operators is that their commutator is a 6 fiinction ^>to(^l),«l,fe) = hukod(zi-Z2)l

(3.319)

Under the same narrow-bandwidth condition, the commutator of the AbramCohen operators, which is a delta-function derivative, can be approximated by d\z,-Z2)^-ikod(z,-Z2).

(3.320)

Abram and Cohen [1994] have analyzed the approximations that enter the quantum treatment of propagation in a Kerr medium and outlined the corresponding renormalization procedure. The slowly-varying-amplitude approximation according to Abram and Cohen [1991] is used by Abram and Cohen [1994]. According to eq. (3.299), the interaction Hamiltonian (3.281) can be written as

HM =16e2 -lL j\l)+{z - vt) + V^_(z + vt) dz.

(3.321)

The exact Hamiltonian (3.275) may be written up to the first order in x as H(t) = Ho(t) + H^s^(t) + H,s-(t) + d(x'l

(3.322)

where ^1S±(0 - - ^ ^ 2 / V^i(^ ^ ^^)d^

(^-^2^)

are the parts of the Hamiltonian (3.321) that commute with HQ, and the operator O(x^) involves all terms with x", « = 2,3,4,.... The authors do not content themselves with a partial formulation of substitution (3.315) and, appropriately, they complete the formulation with a conservation of the time t or of the spatial coordinate z. So we see that the application of the substitution (3.315) resulting in the solution (3.316) is incomplete. In this case, the transformation leaves the time coordinate

4, § 3]

Macroscopic theories and their applications

383

unchanged. In view of the approximation (3.322), the equation of motion of a right-moving field operator can be written in the interaction picture as

d^

i

(3.324)

His-„^iV,t)

This first-order approximation to the equation of motion can be solved formally using the corresponding time-evolution operator (cf eq. 3.280)

I7s40 = ^exp

-ll>

(r)dr

(3.325)

The classical slowly-varying-amplitude approximation has its quantum counterpart on a double assumption: (1) the initial state of the field is a narrowbandwidth state, and (2) the nonlinearity is weak enough so that the fiill nonlinear Hamiltonian (3.321) may be approximated by its first-order stationary component ^ i s , H,s=H,s^+His-

(3.326)

Therefore, ^is+ will be referred to as the slowly-varying-amplitude Hamiltonian. Now we turn to the renormalization. In the framework of the rotating-wave approximation, we obtain that

where (3.328) Upon normal ordering, the perturbative Hamiltonian (3.322) for the electromagnetic field in a Kerr medium can be written as ^(0= /0^(z,O^(z,Odz-y f0\z,t)^\zj)kz,t)0(zj)dz ^ -^

-hKZ I

(3.329)

0\z,t)0{zj)dz,

where K = 3;^/(4e^). We give the ftinction Z only asymptotically (cf. Abram and Cohen [1994]): Z

^

-^A^

(3.330)

A ^ oo ZJt

where A is a high-frequency cut-off. Whereas the first two terms in equation (3.329) are familiar, the third term, which is divergent, arises in the normal ordering procedure. For A fixed, this last term vanishes if ^ ^ 0.

384

Quantum description of light propagation in dielectric media

[4, § 3

In the renormalization procedure, a formal series (in h) of "counterterms" is added to the Hamiltonian in order to remove the divergences that arise upon normally ordering the results of calculation (Itzykson and Zuber [1980]). The Hamiltonian itself exemplifies that it is not sufficient for removing divergences, but at the same time renormalized parameters and renormalized field operators are introduced. In particular, a renormalized Kerr Hamiltonian //R may be defined by introducing a counterterm of order h as H^{t) = H{t) + 2hKZ / 0^(z,O0(z,Ociz.

(3.331)

• / * ' ( - - ,

The third term in eq. (3.329) changes sign, and for A ^ oo it is an infinite change in the inverse of the refractive index. The renormalized field operators 0 R ( Z , O = \ / 1 + /^Z0(Z,O,

0^(Z,O =

Vl+/cZ^t(z,0

(3.332)

are further quantities which, or at least whose Hermitian parts, etc., would relate to an experiment. Such a relationship is no more required from the bare quantities. At the same time, a renormalized refractive index is defined

The renormalized Kerr Hamiltonian can be written in terms of the renormalized field operators as ^ R ( 0 ^ ^OR ( 0 + H\ S+. R (0,

(3.334)

with

^OR(0 = J 4>i(z,t)M^,t)dz

(3.335)

OC

^IS.,R(0

=Y,h^'K-^'^'H\il,,(tl ./ = o

(3.336)

where

(3.337) where KH[llj^(t) is the "usual" Kerr term and h^K^'^^HlUj^it) is the yth quantum correction.

4, § 4]

Microscopic theories

385

Abram and Cohen [1994] have calculated the quantum noise properties of a coherent pulse undergoing self-phase modulation in the course of its propagation by eliminating the vacuum divergences through the renormalization procedure. The one-point averages were first determined. Two-point correlation fimctions were examined too. The result is similar to that obtained by linearizing the self-phase-modulation exponential operator exp(iy^]^Qfl'Ao) around the mean field (Shirasaki and Haus [1990]).

§ 4. Microscopic theories 4.1. Method of continua of harmonic oscillators 4.1.1. Dispersive lossy homogeneous linear dielectric Huttner and Barnett [1992a] started from the observation that the macroscopic approach to the theory of the electromagnetic field in a medium is a quantization scheme that does accept dispersion, but does not accept losses. Thus it does not deal with a fundamental property of the susceptibility, the Kramers-Kronig relations. Losses in quantum mechanics are treated by coupling to a reservoir, and thus a quantization scheme to describe the losses must introduce the medium explicitly. Huttner and Barnett [1992a] use the model of Hopfield [1958] and Fano [1956], having first treated the quantization of light in a purely dispersive dielectric (Huttner, Baumberg and Barnett [1991]) using a simple version of this model (Kittel [1987]). Their analysis is restricted to a one-dimensional model and to transverse electromagnetic fields. After introducing the Lagrangian densities, they discuss the effect of choosing the type of coupling between light and matter on the definition of the conjugate variables for the components of the vector potential. The matter is not quite identifiable with the reservoir, but there is a chain of couplings: the radiation is coupled to the matter (this is a field again) and the matter is coupled to the reservoir (this is a field of the dimension of the matter field increased by unity). Diagonalization is performed via the Fano technique (Fano [1961], Barnett and Radmore [1988], cf Rosenau da Costa, Caldeira, Dutra and Westfahl Jr [2000]). Huttner and Barnett [1992a] work, as usual, with fields in reciprocal space. Only in the very beginning do they consider radiation and matter in direct space and the reservoir in the Cartesian product of direct space and frequency space are considered. The description of the matter and reservoir is first diagonalized. This diagonalization gives rise to the (dressed) matter field B{k, w, t), whose operator exhibits the same dependence on the wavevector and the frequency as the

386

Quantum description of light propagation in dielectric media

[4, § 4

operator of the reservoir field. It is proven also that a coupling constant dependent on at least the fi*equency or the reservoir's "elementary" mode fijlfils the conditions for further diagonalization. This diagonalization gives rise to the field Cik, w, t) for polaritons. The operator of this field shares the dependence on the wavevector and the frequency with the operator of the reservoir field. In contrast with the free-field theory (the theory of the electromagnetic field in a vacuum), a macroscopic field emerges here whose operator depends also on the frequency. The vector potential depends on the spatial coordinate and time as usual, and it has the form of an integral of the vector potential for a unit density of polaritons with the wavevector k and the frequency o) multiplied by the polariton operator C{k, €0, t). The appropriate relation contains the complex relative permittivity of the medium e{o)) as a linear transform of the coupling constant g{o)) between the light and the dressed matter field B{k, co, t) (cf eq. 4.55). The complex relative permittivity e{o)) ftilfils the Kramers-Kronig relations. Taking into account the frequency decompositions of the fields E{x, t) and B{x,t) (see Huttner and Barnett [1992b]), one can introduce, in an "almost" conventional manner, the positive and negative propagating components. These differ almost negligibly due to the imaginary part of the refractive index (see eq. 3.286 or 3.69), and are proportional to the fields c+{x,cx),t) and c_(x, 0), t), respectively. Respecting the frequency decomposition of the field D(x, t), the fields c±(x, (o, t) are used and the spatial Langevin force/(x, O), t) is introduced. Using these definitions, two Maxwell equations are transformed into two spatial Langevin equations. The equal-space commutation relations between the operators at the frequencies co and co' can also be derived. Huttner and Barnett [1992a] list the papers devoted to the phenomenological approach to quantization (Levenson, Shelby, Aspect, Reid and Walls [1985], Potasek and Yurke [1987], Caves and Crouch [1987], Lai and Haus [1989], Huttner, Serulnik and Ben-Aryeh [1990]). Let us note that Huttner and Barnett [1992b] in their introduction mention also the popular approach (Huttner, Serulnik and Ben-Aryeh [1990]) in which spatial progression equations are derived and quantization of the field is performed by imposing the equal-space commutation relations. In contrast with the macroscopic theories, this technique is not derived from a Lagrangian and has not been justified in terms of a canonical scheme. Huttner and Barnett [1992b] provide the derivation of such equal-space commutation relations for the case of a linear dielectric. The canonical scheme and losses cannot be easily unified, but this has been solved by Huttner and Barnett [1992b]. The one-dimensional model (Huttner and Barnett [1992a]) has been expanded to three dimensions.

4, § 4]

Microscopic theories

387

The Hamiltonian is first derived, then diagonalized, and the expansions of the field operators are transformed. The propagation of fight in the dielectric is analyzed, the field is expressed in terms of space-dependent amplitudes, and their spatial equations of evolution are obtained. Huttner and Bamett [1992b] started the canonical quantization from a Lagrangian density

where -E

B

£e™ = | ( ^ ' + V f / ) 2 - — ( V x ^ ) 2

(4.2)

is the electromagnetic part expressed in terms of the vector and scalar potentials A and U; C^,t = ^(X'-wlx')

(4.3)

is the polarization part, modeled by a harmonic oscillator field X of frequency (OQ (the polarization field);

-H

(Yf,-w'Y;;,)dw

(4.4)

0

is the reservoir part, comprising a field F^,, of the continua of harmonic oscillators of frequencies co, used to model the losses (reservoirs); and C,ni = -a(A'X+UV'X)-X-

[

u(a))Kdw

(4.5)

is the interaction part with coupling constants a and u(a)). The interaction between the light and the polarization field has the coupling constant a, and the interaction between the polarization field and other oscillator fields used to model the losses has the coupling constant U((JO). In general, a could be generalized to a tensor. The displacement field is defined by Z)(r, 0 = eoE(r, t) - aX{r, t).

(4.6)

As (j does not appear in the Lagrangian, U is not a proper dynamical variable, but it can be written in terms of the proper dynamical variable X. The former has

388

Quantum description of light propagation in dielectric media

[4, § 4

an integral expression and that is why we go to reciprocal space. For example, the electric field is written

J_

E(r,t) = j : ^

jE{k,t)e"'d'k.

(4.7)

We shall underline the newly introduced quantities in order to differentiate between quantities in real and in reciprocal space. Let us recall that E*(k,t) = E_{-k,t). It comprises both the annihilation and creation operators, see below. The total Lagrangian can be written in the form ^

/

V^em "•" ^mat + £res + £int j ^ ^ '

(4.8)

where the prime means that the integration is restricted to half reciprocal space, and the Lagrangian densities become

/•OC

io

(4.9)

£int = - «

-L As usual in quantum optics, we choose the Coulomb gauge, k • A(k, t) = 0, so that the vector potential ^ is a purely transverse field. The scalar potential in reciprocal space can be obtained as

where K is a unit vector in the direction of k. The polarization field X and other oscillator fields Y(,j (the matter fields) are decomposed into transverse and longitudinal parts. For example, X can be written as X(k,t)=X^(k,t)+xHk,t)K, (4.11) and Y^^^ can be expressed similarly. The total Lagrangian can then be written as the sum of two independent parts. The transverse part, containing only transverse fields, is L^ = / ' {^L + £ i . + £ris + ^,n,) d^A,

(4.12)

4, § 4]

Microscopic theories

389

where

4-i, = eo(Up-cV|4|'), 4 t s = p / (lZ;lp-o>^|ZiP)d(y,

(4.13)

-'0

£^ = - aA'X

+/

ir^

i;(a;)X^*-y,,da;

+ C.C.

JO

The longitudinal part, containing only longitudinal fields, is also given by Huttner and Barnett [1992b]. It can be derived that /) is a purely transverse field. For convenience, one can restrict oneself to transverse components of other fields and omit the superscript ^. Unit polarization vectors ex{k), X= 1,2, are introduced, which are orthogonal to k and to one another, and the transverse fields are decomposed along them to get

A{Kt)= Y,A\Kt)ex{k)

(4.14)

A=l,2

and similar expressions for the other fields. C can now be used to obtain the components of the conjugate variables for the fields: -eo£^ = ^ dA dC

=eo/,

(4.15)

P^ = - ^

= pf- - aA\

(4.16)

at^^

=pit-^(«)r.

(4.17)

A famous ambiguity is worth mentioning: The conjugate of 4 can be -CoE_ (with the coupling ^A • P), as well as -D (with the coupling £ • X_). Thus, the choice of gauge determines the type of coupling. The Hamiltonian for the transverse fields is // = !

(2iem+2i„a. + 2i,n.)d'A,

(4-18)

where H,^ = eo{\Ef + c^P\A\^)

(4.19)

390

Quantum description of light propagation in dielectric media

[4, § 4

is the electromagnetic energy density; k is defined by ^ = y/k^ + k^ with kc = cOc/c = y/a^/pc^Co;

do)

Ziimaf

Jo

P

r

r

(4.20) is the energy density of the matter fields, including the interaction between the polarization and the reservoirs; 0)1 = 0)1 + j ^ ^^^ do; is the renormalized frequency of the polarization field; and (4.21)

Wi„, = - ( 4 * - ^ + c.c.)

is the interaction energy between the electromagnetic field and the polarization. Part of the interaction energy with the matter, namely ^ | 4 p , has already been classified into eq. (4.19). Fields are quantized in a standard fashion (Cohen-Tannoudji, Dupont-Roc and Grynberg [1989]) by postulating equal-time commutation relations between the variables and their conjugates.

A(k,tlE

(k\t)

(4.22)

Jlsu'6{k-k')l ^0

X\k,t),p''\k',t)

=

tlikJXQ'Ak'j)

ihd,^.d{k-k')\. ihdxx>d{k-k')6{o)-o)')\.

(4.23) (4.24)

with all quantized operators identified by a caret. As usual, one introduces the annihilation operators

a{X,k,t)

=

b{X,k,t)

=

\o(X,k,t)

=

_eo_ ~kct{k, t) - iE\k, t) 2h~kc (boX\k,t)+-p\k,t) P ~P~

-i(oYlik,t)

+

(4.25) (4.26)

^Qlik,t)

(4.27)

391

Microscopic theories

4, § 4 ]

From the equal-time commutation relations for the fields (4.22)-(4.24), one obtains the equal-time commutation relations for the creation and annihilation operators:

(4.28) ko{KK t\ bla',k\

t)] = 6,,.8(0) - co')6(k

-k')l

The normally ordered Hamiltonian for the transverse fields is (4.29)

^ ( 0 = 4 m ( 0 + ^mat(0 + 4 n t ( 0 ,

where

4m(0== / Yl "^

(4.30)

^^ca\k,kJ)a{X,kj)d^k,

A=l,2

ffmai(t)= / ^

lh(bob\X,k,t)b(X,k,t)+

/

h(x)bl{^,k,t)b,o(X,k,t)dw

A=l,2

[b(X,k,t)bla,k,t) +b\x,-k,t)bia,kj)^R.c.'\

(4.31) dco } (Tk,

HUt) = il f Yl Mk)[a{KKt)b\KKt) *^

(4.32)

A=l,2

+ a\X, -k, t) b\X, k, t) + H.c] d^k, where

Via,)

u{o)) jlo (Wo

AW^J^.

and the k integration has been restored to the fiall reciprocal space. It is worth mentioning that the Maxwell-Lorentz equations can be derived from the Hamiltonian. It is important that the matter can be formally decoupled from the reservoir by the Fano technique, obtaining a dressed matter field. Following Fano [1961],

392

Quantum description of light propagation in dielectric media

[4, §4

the polarization-reservoir Hamiltonian can be diagonalized. The dressed matter creation and annihilation operators B\X, k, co, t) and B{X, k, o), t) are introduced, respectively, which satisfy the usual equal-time commutation relations. B{X, k, 0), t\ B\X\

k',0)', o] = ^kk' ^{k - k')6{(o

(4.33)

-(D')\,

B(X, k, (JO, t) = a^io)) b(X, k, t) + ji^iw) b\X, -k, t) , (4.34)

+ / \ai((o,w')b,,AX,kj)-^P\{w,a)')bl,(X,-k,t)\ do;' Jo ^ ^ The coefficients ao(co), fio(co), a\(co,aj^) and li\{w,co') are defined in such a way that the polarization-reservoir Hamiltonian ffmaxiO is diagonal in the operators B^X, k, co, t) and ^(A, /r, co, t). It is interesting that the diagonalization is performed once for the polarization-reservoir Hamiltonian and once for the total Hamiltonian. From relation (4.34) it can be seen that the diagonalization is performed independently for every pair of counterpropagating modes of the polarization field (these "modes" are only formally similar to those of the electromagnetic field) and that it is performed using a Bogoliubov transformation. The dressed matter-field annihilation operator is an "eigenoperator" (Barnett and Radmore [1988]), = ha)B(X,k,co,t).

B(X,k,co,t\Hm,r(t)

(4.35)

The coefficients of the Bogoliubov transformation are calculated by a Fanotype technique, cto(a)) = l3o{w) = aiico.w')

V{w) co^ - ajQzicoY V(co) co^ - cb^zicoY

(O + O^o

2 CO-

Cbo

= 6(0)-w')^

(4.37) V(cjo)

( — CO-

l3,(co,w') =

(4.36)

CO'

iOj co^ - cb^zico)

V{co^) V(co) co + co') CD- - cbh{coy

(4.38) (4.39)

where z(co) is defined by z(co) = 1

1 lim 2cbo

/

V(co^) CO'

dco'

(4.40)

C0-\-l8

From the study, V(co) = \V(co)\^ independent of the frequency is excluded by the assumption that the analytical continuation of V(co) to negative frequencies is an odd fiinction.

4, §4]

393

Microscopic theories

As usual with substitutions, we are also interested in the inverse transformation. It is given by the relations

(4.41)

ao (CO) B(?i, k, CO, t) - Poio)) B\X, -k, co,d(X>, t) Jo

a\{w',(D)B{X,k,w\t)-P\{(jo',(jo)B\X,-k,(D\t)\

dw'. (4.42)

Jo The conditions /•DC

/ = /

[|ao(a;)|' - |A)(^)|'] do; = 1,

(4.43)

/•OC

I{a),(D')= /

[a\{v,(x))ax{v,o)')-l]x{v,(D)P;{v,(ji)')\

dv = d{co-w')

(4.44)

for the coefficients of the Bogoliubov transformation look familiar. Relation (4.31) becomes

H^^x{t)=y^ 11 . , J Jo X=\2

hcoB\Kk,co,t)B{X,k,co,t)do)d?k.

(4.45)

It has been shown that the diagonalization cannot be performed on the common assumption of white noise (Markov-type coupling). We note that free charges and a conducting medium are beyond the scope of Huttner and Barnett [1992b]. The diagonalization of the total Hamiltonian is formally very similar to the diagonalization of its matter part. The expression for the total Hamiltonian H{t) has the same form as the expression of the matter-reservoir Hamiltonian HmaM in terms of the initial creation and annihilation operators, when the parameters a>o and

V((JO)

in relation (4.31) are replaced by kc and yS-^(a))^

with

^(a>) = i^/a)Q[ao(cJO) + ft(w)], and the operators b and ba, are replaced by the operators a and B(a)). The creation and annihilation operators C\?i,k, co, t) and C(A, k, CO, t) are considered: C(A, k, CO, t) = ao(k, CO) a(X, k, t) +ft^{k,co) a\X, ~k, t)

f Jo

a\ (k, CO, co') B{X, k, co' ,t) +ft\(k, co, co') B\X, -k,

dcx)', CJJ' , t) (4.46)

[4, §4

Quantum description of light propagation in dielectric media

394

where the coefficients ao(k, w), ^{k, w), a\(k, a), o)') and ^i(A:, o), o)') appear in slightly modified form, \ 0)}

OoCA:, a;) = an

(ji)-\-kc

t{^)

2

I e*(ft;)co2-Fc2'

kc \

^

/ ^c l 0) - kc

kc \ a,ik,w,co')

2

= 6(0)-0)')+

P\ (k, W, 0)')

(4.47)

(4.48)

e'^(co)co^-k^c^' ^^' ^ ^*^'^'^ 2 \oj-a)'-iOj

0)1 ( K{0)') \ 2 \o) + o)')

U(o) e''(a))a)^-k^c^

t{0)) e''{o))o)^-k^c^'

,

(4.49) (4.50)

as the normalization condition j ^ [\t{o))\^/o)) do; = 1 has been used and the complex relative permittivity e{o)) is introduced: '^2 ^^ \t{0)'f 0)^ e(o)) = 1 + —^ lim , ^ , . ^ , da;^ 20) e^+O y _ ^ (O)^ - 0)-

(4.51)

18)0)'

Relation (4.29) (-4.32) becomes ^ ( 0 = y2

I I

^^C\X, k, 0), t) C(A, k, 0), t) do; d^k.

(4.52)

A=l,2

The operators C(A, k, o), t) and C^(A, k, o), t) also satisfy the usual commutation relations, C(A, k, 0), t\ C\X', k\ o)\ 01 = ^U' ^{k - k')d{oj

-o)')\,

(4.53)

and they have a harmonic time dependence (4.54)

C(A, A, CO, 0 = C(A, ^, a;, 0) e-*^'^^ The vector potential is now given by A{r,t)

1 (2;r)3^2

A=l,2

ho)l

/ / 26^

r(co) -C(A,)t,a;,0)e O)^e(o))- k^c-

-i((0/-A-r)

+ H.C. do)d^k.

(4.55) Huttner and Barnett [1992b] restrict themselves to a one-dimensional case when justifying the temporal mode approach to the propagation in the dielectric.

4, § 4]

Microscopic theories

395

The complex refractive index n(a)) is introduced as the square root of the relative permittivity e(co), with a positive real part r]((i)). The vector potential is considered in the simpler form 1

A(x, t) = —== /

V4jr Jo

f^

A{o)) [c+(jc, a;, 0) e'"'' + C_(JC, O;, 0) e'^"' + H.c.l dco,

(4.56) with a normalization factor

^ ^

Y 6o5cft;|«(a;)P'

<S being a cross-sectional area, and the temporal mode operators for forward and backward propagating fields c,(.,a,,0= ^ / ^ ^ i ^ S e - ^ " - r ^ ( ^ - ^ - O f d . , V n J_^ K(w)Tk

(4.58)

with Kico) = " ^ ^ ,

(4.59)

Mco) ^ t*(c^) | K ^ ) | |C(a;)| n(co) ' C(k,a),t)

^

(4.60)

^C{X,k,co,tl

(4.61)

the complex wave number, a phase factor and a polariton operator, respectively. Since the magnetic field can be expressed similarly as the vector potential, the spatial quantum Langevin equations of progression can be obtained: d

_ c±(x,CO,t) = ±iK(w)c±(x,CO,0±

y/2lm{K{w)}f(x,co,t),

(4.62)

(JX

where fix, 0), t) = --=e'^(^'^) / C(k, CO, t)e'^^ dk v2jr J-oo

(4.63)

is the Langevin noise operator, which enters a rather similar expression for the electric displacement operator. Let us note that Im{A^(co)} > 0. Equations (4.62) have been obtained from the Maxwell equations for monochromatic fields.

396

Quantum description of light propagation in dielectric media

[4,

Huttner and Barnett [1992b] remind of the simple commutation relations f{x,o),t\f\x\a)\t)

d{x-x')d{(i)-o)')\,

(4.64)

of the equal-space commutation relations \c±{x,(j),t),c^j^{x,o)' ,t)\ = d{(ji)-(o')\,

\c±{x,(jo,t),c\^{x,(j)\t)\

=0, (4.65) and, finally, that the temporal mode operator c+(x, o), t) commutes with all the Langevin operators/(x^ o)', t) and/^(x^ o)', t) for all x' > x, while the operator c_(x, 0), t) commutes with all the Langevin operators/(x^ co', t) and/^(x^ w', t) for all x' < X. Wubs and Suttorp [2001] have solved the initial-value problem for the damped-polariton model formulated by Huttner and Barnett [1992a,b] and have found that for long times all field operators can be expressed in terms of the initial reservoir operators. They have investigated the transient dynamics of the spontaneous-emission rate of a guest atom in an absorbing medium. Hillery and Drummond [2001] have studied the scattering of the quantized electromagnetic field from a linear dispersive dielectric in the limit of "thin" absorption lines. The field is represented by means of the dual vector potential. Input-output relations are unitary and no additional quantum-noise terms are required. Equations specialized to the case of a dielectric layer with a uniform density of oscillators are usual expressions. 4.1.2. Correlation of ground-state fluctuations The quantization of the radiation imbedded in a dielectric with a space-dependent refractive index has been expounded in the book by Vogel and Welsch [1994]. A canonical quantization scheme for radiation fields in linear dielectrics with a space-dependent refractive index has been developed by Knoll, Vogel and Welsch [1987] and later by Glauber and Lewenstein [1991]. For application see, for example. Knoll, Vogel and Welsch [1986, 1990, 1991], Knoll and Welsch [1992] and a related work Knoll and Leonhardt [1992]. Gruner and Welsch [1995] have contributed to the stream of papers aiming at a description of quantum properties of dispersive and lossy dielectrics including the vacuum fluctuations, i.e.,fluctuationsof the radiation field in the ground state of the coupled light-matter system. They study it in terms of a symmetrized correlation fianction. They try to expound and supplement the paper by Huttner and Barnett [1992b] from the point of view of the quantization of the phenomenological Maxwell theory.

4, § 4]

Microscopic theories

397

First, the quantization of radiation in a dispersive and lossy dielectric is performed. This begins with the classical Maxwell equations (3.121), with the constitutive relations (3.122) and a constitutive relation comprising an integral term. ^ ( ^ 0 = ^0 E{r,t)^

/I

xir)E(r,t-T)dT

(4.66)

0

Jo Everything is transformed into Fourier space, where relation (4.66) becomes b(r, 0)) = 6o6(w) E{r, w), (4.67) and the Helmholtz equation is presented. The Huttner-Bamett quantization scheme is introduced with a diagonalized Hamiltonian (4.52). The effect of the medium is entirely determined by the complex permittivity e{co). It still has no tensorial character. Let us refer to original relations (4.3), (4.5), (4.6) and (4.7) after Huttner and Barnett [1992b]. In these relations, one should make use of the identity ^ r ( w ) = (o^/lm{e(a))}. Frequency-dependent field operators are introduced in the three-dimensional case. Not only the equal-time commutation relations, but even the most general ones are presented. The vector-field operators a(r,co,t) and f(r,a),t) are introduced, the vector a(r, co, t) being a generalization of the component c(x, co, t) from (Huttner and Barnett [1992b]). Using the operators a(r,co,t) a n d / ( r , co,/), an analogue of relation (5.21) of Huttner and Barnett [1992b] (cf eq. 4.56) is written. Then, analogues of their frequency decompositions of the vector-potential operators, electric-fieldstrength operators, etc. are presented. The operator constitutive equation (in Fourier space) b(r, w, t) = eoe(a>) k{r, w, t) - eoT(aj)f{r, (o, t\

(4.68)

where eo^(co) = y ^eo Im{e(a;)},

(4.69)

differs from the classical equation (4.67) by an additional term. On substituting into the phenomenological Maxwell equations, the partial differential equation

398

Quantum description of light propagation in dielectric media

[4, § 4

for the operatorfl(r,co, t) is obtained, which is a Helmholtz equation with a righthand side. In the three-dimensional case, there exists no decomposition into firstorder equations. The canonical commutation relations are Ai{rJ\Ej{r'A = --d^(Ar) 1, J

(4.70)

Co

with the abbreviation Ar = r-r'.

(4.71)

A test of the consistency of the theory in the limit e((jo) —> 1 has been accomplished. Let us recall the usual annihilation and creation operators entering the expansion for A(r,t), i.e. a(X,k,t) and a^(X,k,t), respectively; they satisfy the commutation relations [aa,k,tXa^(X\k\t)]=d^;,^d(k-k')l

(4.72)

The operators a(r, w, t) and / ( r , w, t) derived from C(A, Ar, (o, t) are not independent operators. Compare Huttner and Bamett [1992b] who in the onedimensional case introduce forward- and backward-propagating fields and show that such a definition ensures the causal (one-sided) independence of the respective temporal mode operators of the operator / ( r , co, t). In the threedimensional case, there exists no generalization of relation (4.58) and no equation for such quantities. The theory is applied to the determination of the correlation of the groundstate fluctuations of the electric field strength. A symmetric correlation ftinction of the electric field strength is considered: Kmn{Ar, r) = \ (0| [^^(r, t + T)E,{r + Ar, t) + E,{r + Ar, t)E,,{r, t + r)] |0). (4.73) The influence of absorption, phase and group velocities and group-velocity dispersion on the dynamics of the field fluctuations within a frequency interval have been studied. The absorption causes a spatial decay of the correlation of the field fluctuations. The light cone of strong correlation, which in empty space is determined by the speed of light in vacuum, is now given by the group velocity in the medium, provided that the spatial distance is not too large. With increasing distance, the dispersion of the group velocity should be taken into account. 4.13. Green-function approach If one allows a frequency-dependent complex permittivity that is consistent with the Kramers-Kronig relations and introduces a random operator noise source

4, § 4]

Microscopic theories

399

associated with the absorption of radiation, the classical Maxwell equations can be considered as quantum operator equations. Their solution based on a Green-function expansion of the vector-potential operator seems to be a natural generalization of the mode expansion applicable to source-free radiation in nearly lossless dielectrics. (i) Dispersive lossy linear inhomogeneous dielectric. Gruner and Welsch [1996a] have expounded a quantization scheme which starts from the phenomenological Maxwell equations instead of the Lagrangian densities, and is consistent with the Kramers-Kronig relations and the familiar (equaltime) canonical commutation relations for the vector-potential and electricfield-strength operators. This is realized for homogeneous and inhomogeneous (especially multilayered) dielectrics. In the phenomenological classical Maxwell theory, the equations comprise e(a;), the frequency-dependent complex relative permittivity introduced phenomenologically. This function has an analytical continuation in the upper complex half-plane, e(0), which satisfies the relation 6(-0*) = e*(^). The real and imaginary parts of the relative permittivity satisfy the well-known KramersKronig relations: Re{e(a,)} - 1 = I v . p . H

'-^^^^^

lm{e(co)} = -iv.p. r

da,',

M^(^!^:)hl do.;

(4.74) (4.75)

The quantization scheme is based on the Helmholtz equation with the source term AA{r, 0), t) + K^{a))A{r, w, t) =j^{r, w, t),

(4.76)

where A(r, w, t) = A(r, o), 0) e'^'^^ with A(r, co, 0) the "Fourier transform" of the (known) operator-valued vector potential A{r, t), andy^(r, w, t) =jjj', (o, 0) e'^'^^ with 7j^(r, w, 0) the "Fourier transform" of the operator-valued noise current. In fact, from the exposition it can be seen that the vector-potential operator is introduced by the relation A(r,0)=

/

A(r, w,0)da)^H.c.,

(4.77)

where quantum-mechanically also the frequency-dependent operators can be time dependent.

Quantum description of light propagation in dielectric media

400

[4, §4

When \m{e{Ci))} > 0, a (hypothetical) addition of a nontrivial solution of the homogeneous wave equation would violate the boundary condition at infinity. Hence, the operator A{r, co, t) is uniquely determined by a linear transformation of the source operator/^^(r, (0,0• This operator can be chosen in the form (cf. Gruner and Welsch [1995]) CO

(4.78)

j^(r, 0), t) = Hco) —f(r, CO, t\

with T{cji)) given in eq. (4.69). The Hamiltonian H{t) is diagonal in the operators / ( r , co, t). H{t) =

hcof\r, CO, t) • / ( r , CD, t) dco dV,

(4.79)

and these operators have the usual properties \fi{r,coj)J^{r\co',t)\^

=

\fi{r,co,t)Jj{r\co'j)]

=

d^{r-r')d{cj)-co')\, \f^{r,co,t)/f^{r',co',t) = 0.

(4.80) (4.81)

From the foregoing considerations it follows that (when all appropriate conditions are fulfilled) the operator of the vector potential can be defined by the relation A{rj)=

/

j

G(r,r\co)}\{r\co,t)d^/dco^H.c,

(4.82)

where the Green function G(r,r\co) satisfies AG(r, / , CO) + K\CO) G(r, r\co) = d(r - /)

(4.83)

and the boundary condition that it vanishes at infinity. Another required property is E(r,0) = -A(r,0)

(4.84)

and the canonical field commutation relations

i,(^0),£;(/,0)l

=--S;^(r-r')l

(4.85)

Relations (4.85) must be verified by straightforward calculation. For the sake of clarity, Gruner and Welsch [1996a] illustrate this procedure in linearly polarized

4, §4]

Microscopic theories

401

radiation propagating in the x-direction. Relations (4.85) are replaced by the relation (4.86)

A(x,0),E(x,0)

where A is the normalization area perpendicular to the x-direction. It is shown that when losses in the dielectric may be disregarded, lm{e((j))} -^ 0, the concept of quantization through the mode expansion can be recognized. The operators f(x, 0), t) are replaced by the operators a±(x, co, t), which satisfy the commutation relations a±{x, CO, t), al_(x, (JO\ t) exp

\m{n{(x))} — \x - x' 6{co-co')\, c

(4.87)

CO

a±{x,co,t),ciL{x ,CD'J)\ = 2lm{n(co)}— expH=i^(<^) ^(x-\-x^) ^ J c I ^ lm{n(co)} — \x-x^ X exp c sm[RQ{n(co)}'-^\x-x'\] "" Rc{n(co)}^-f X

(4.88)

6[±{x-x)]dico-co')l,

where d(x) is the Heaviside function. These operators become independent of x in the limit lm{n(co)}f\x - x ' | -^ 0. As the commutation relation (4.86) is in obvious contradiction with the macroscopic approach, it is important that Gruner and Welsch [1996a] have derived the relation AAa)(x,0%EAa)(x\0)

ih 8(x-x)l, AeR(co,)eo

(4.89)

where cOc is the center frequency for suitably defined operators, A/^a)(x,0), The theory further reveals that the weak absorption gives rise to spacedependent mode operators that spatially progress according to quantum Langevin equations in direct space. As could be expected, the operators a±(x,co,t), representing the forward and backward propagating fields, are governed by quantum Langevin equations. In other words, the operators a±(x, co,t) progress in space. As an example of inhomogeneous structure, two bulk dielectrics with a common interface are considered. The problem of determining a classical Green function reappears. The verification of the commutation relation (4.86) is performed by straightforward calculation, more complicated this time. A general

EA(O(X, 0).

402

Quantum description of light propagation in dielectric media

[4, § 4

proof of this relation is not presented, causality reasons are only pointed out. There exists a straightforward generalization of the quantization method based on a mode expansion (Khosravi and Loudon [1991, 1992], Agarwal [1975]). (ii) Dispersive lossy nonlinear inhomogeneous dielectric. Emphasizing the important differences from the linear model, the Lagrangian and Hamiltonian for the nonlinear dielectric are introduced by Schmidt, Jeffers, Barnett, Knoll and Welsch [1998]. The Lagrangian density (4.1) is denoted by C\{r), and this relation with C replaced by C\{r) has been utilized in a more general Lagrangian density: C{r)-^C^{r) + C,^{r\

(4.90)

where moreover C^x{r)=f[X{r)l

(4.91)

While in the linear case it is sufficient to quantize only the transverse fields, in the nonlinear case such a procedure would result in a loss of generality. The result of substitution from relations (4.30)-(4.32) into (4.29), which we have denoted by H, is denoted here by H^^. The normally ordered Hamiltonian for the longitudinal fields is

Hl\t) = J(ficoob\lkj)klkj)^ h j j V{o)) \^b\l-Kt) 2 ./ ./o

J + klkj)]

hajblXlkj)bM^k,t)d€o)d'k \bl{lKt)-^h,{l-Kt)\

da;d^/r,

(4.92) where the components b{\\,k,t), b^o{\\,-k,t) must be defined appropriately (see Schmidt, Jeffers, Barnett, Knoll and Welsch [1998]) for b\^{k\ b^\{k,o))). The total Hamiltonian can be written as H(t) = HM + HM,

(4.93)

where the nonlinear interaction term ^ni(0 is given by Hni(t) = -jf[Xir,t)]d'r,

(4.94)

and the Hamiltonian ff\(t) that governs the linear dynamics can be written as ^ , ( 0 = A" (0 + ^1^(0-

(4.95)

In general, Hn\{t) couples the transverse and longitudinal fields, cf, the relation (4.11).

4, § 4]

Microscopic theories

403

Schmidt, Jeffers, Barnett, Knoll and Welsch [1998] have derived evolution equations for the field operators, showing that additional noise sources appear in the nonlinear terms. Linear relationships between quantum (operator-valued) fields are introduced following Huttner and Barnett [1992b] as well as Gruner and Welsch [1995]. The relations hold for all times and for both linear and nonlinear cases. We now add the following representations of the matter fields. The longitudinal matter field X^r, t) can be expressed in terms of the field/"(r, co, t) as

'•''-{ik[^"'

(4.96)

and the transverse matter field X^{r,t) / ( r , 0), t) as

can be expressed in terms of the field

*"<•••'>" l / v i T r /

X^{r,t)=

/ Jo

K<<>')-fi;(»')]/"(MO,<)'lo' + H,c.,

X (r,ft>,Oda; + H.c.,

(4.97)

where X

{r,o),t)=-

c -\a)[e{w)- \]A{r,o),t)+

h W — Im{e(co)}/(r, (0,0

(4.98) with^(r, 0), t) connected with the field/(r, co, t) as the solution of eq. (4.76) and by the explicit relation (4.78). The vector-potential field has the representation A{r,t)^

/

^(r,co,Oda; + H.c..

(4.99)

Relating the validity of expressions (4.99), (4.96) and (4.97) to a time evolution like eq. (4.54), we might suspect that it will not survive the change to the nonlinear case. This change is reflected in the equations of motion for the basic fields and the vector-potential field in the Heisenberg picture, i ^ | / l l ( r , CD, t) = [/ll(r, CO, 0 , ^ ( 0 ] - hojfhr, co, t) + [fHr, w, 0 , 4 i ( 0 ] , (4.100) i ^ - / ( i - , (o, t) = [/(/•, w, t),H{t)] = hwf(r, w , 0 + [/(^ 0), t),HM],

(4.101)

ih—^(r,(jo,t)=

(4.102)

[2(r,a),t),ffit)]

= ha)A{r,(i),t)-^ [k{r,a),t),Hn\(t)]-

404

Quantum description of light propagation in dielectric media

[4, § 4

Among the relations which do hold in both linear and nonlinear cases is the nonhomogeneous Helmholtz equation:

A^(r,a),t) + K\co)Air,(o,t)=

%J—lm{e(a))}f{r,a),t). c^ y Jteo

(4.103)

Considering the notation K^{w) = c~^co^e((jj) (cf. eq. 4.59), we can see that K\a))A(r,

(O, t) = K\co\)A{r,

co, t\

(4.104)

where 1 is the identity superoperator and relation (4.102) implies that w\ = \^^+^-H,,{tr,

(4.105)

where for the sake of clarity we have written ^ to the right of 1, and the action of ^ni(0^ 01^ ^^ operator 0{t) is defined by

H,x{trO{t)

Hn\{t\0{t)

(4.106)

Relation (4.105) can be written in the form

(4.107) where the elimination of the field X{r, t) using relations (4.96) and (4.98) indicates new noise sources. All of the fields/"(r, a;, t),f{r, w, t) and A{r, a), t) obey the nonlinear dynamics. By integration of eq. (4.107) over w, an equation adequate for both linear and nonlinear cases is obtained. The wealth of operatorvalued fields facilitates the expression of dispersion and absorption in the nonlinear medium. The basic equations are applied to the one-dimensional case and propagation equations for the slowly varying field amplitudes of pulse-like radiation are derived. The scheme is related to the familiar model of classical susceptibilities, and is applied to the problem of propagation of quantized radiation in a dispersive and lossy Kerr medium. In the linear theory it is possible to separate the two transverse polarization directions from each other and from the longitudinal direction. As has already been stated, this is not possible for nonlinear media. In practice, in a single-mode optical fiber, only one transverse

4, § 4]

Microscopic theories

405

polarization direction will be excited. Then the total Hamiltonian (4.93) can be reduced to a one-dimensional single-polarization form. Let us consider the propagation in the x-direction of plane waves polarized in the j;-direction. The one-dimensionality of the problem permits one to decompose the field A{x, w, t) into components A+(x, w, t) and A^{x, co, t), respectively, propagating in the positive and negative x-directions, A(x, 0), t) = A+{x, CO, t)-\-A.(x, 0), t), where A±(x,w,t)

(4.108)

are the solutions of spatial equations of progression:

—A±(x, (o, t) = ±iK(co)k:t(x, CO, t)^\MJ (jx y

^"^^^^^^^ /(;c, CO, t\ e{co)

(4.109)

with a normalization factor M = y/h/AjreoAc^. Similarly as from relations (4.103) and (4.107), one can arrive from relation (4.109) at the relation d ^

/^9

1.

H„,it)xy^(x,co,t)Ti^fJ^^^^^^fix,OJ,t). (4.110)

In analogy with eq. (4.99), the operators A±. {x,t) can be introduced as -M)

A^(x,t)=

'"" -

I

A±(x,co,t)dco.

(4.111)

Integrating eq. (4.111) over co yields an equation appropriate for both linear and nonlinear cases. Adequately to the derived equations which we consider to be mere approximations in the nonlinear case, Schmidt, Jefifers, Barnett, Knoll and Welsch [1998] study the narrow-bandwidth field components and narrow-bandwidth pulses. The theory has been applied to narrow-bandwidth pulses propagating in a dielectric with a Kerr-like nonlinearity. (Hi) Elaboration of the linear theory. Dung, Knoll and Welsch [1998] have developed a three-dimensional quantization presented in part by Gruner and Welsch [1996a] concerning a dispersive and absorbing inhomogeneous dielectric medium. The approach starts directly from the Maxwell equations in the frequency domain for the macroscopic electromagnetic field. It is shown that the classical Maxwell equations together with the constitutive relations, except relation (4.66), can be transferred to quantum theory. In considering the charge

406

Quantum description of light propagation in dielectric media

[4, § 4

and current densities, one concentrates on the noise-charge and noise-current densities. An operator-valued noise-charge density p(r, w, t) and an operatorvalued noise-current density j{r, co, t) are introduced, related to the operatorvalued noise polarization P{r, (o, t): p{r, CO, 0 = - V • P{r, 0), t\

(4.112)

y(r, CO, t) = -ia)P(r, co, t).

(4.113)

It follows from relations (4.112) and (4.113) that p(r,a),t) md](r,co,t) the continuity equation V j(r, (O, t) = ico^(r, co, 0-

fulfil

(4.114)

The source term j{r, co, t) is related to a bosonic vector field / ( r , co, t) by a relation like (4.78). The commutation relation (4.81) remains valid, and relation (4.80) must be modified to the form fi{r,CO,t),fj^{r',co',t)\ = dij8(r-/)d(co-co')

I

(4.115)

It is pointed out that the current density j{r, co, t) is not transverse, because the whole electromagnetic field is considered. Hence, the vector field f{r, co, t) assumed here is not transverse either, and the spatial 6 fiinction in the relation (4.115) is an ordinary d function instead of a transverse d fiinction. Relation (4.77) is an integral representation of the vector-potential operator. Dung, Knoll and Welsch [1998] start from the partial differential equation C

C02

C

C

V X V XE(r,co,t)-—^e(r,co)E(r,co,t)

= icofi^jir,co,t),

(4.116)

whose solution can be represented as (different notations will be used here and below)

) I/ G{r,s,co)j{s,co,t)<\ G{r,s, E{r,co,t) ='\coiM)

s,

(4.117)

where G{r, s, co) is the tensor-valued Green fiinction of the classical problem. It satisfies co^

V , V , - l M , + -ye(r,co)

G(r,s,co)^8(r-s)l

(4.118)

4, § 4]

Microscopic theories

407

together with appropriate boundary conditions. Dung, Knoll and Welsch [1998] have derived commutation relations 'Urj)Mr\t)] = ^ Y , e , „ ~ m,7

'"

H

^Gij(r,r\ w)dw,

(4.119)

-^-^

where ekmj is the Levi-Civita tensor, Gij{r, r^ (o) = a • G(r, r\ co) • Cj,

(4.120)

and [£K^O,^K^',0] - 6 = [Ur,t\B,{r'j)\

.

(4.121)

In the sense of the Helmholtz theorem there exists a unique decomposition of the electric field E(r, co, t) into a transverse part E (r, co, 0 and a loncll gitudinal part £" {r,aj,t), i.e., the Coulomb gauge can be introduced, where E (r, CO, t) = i(joA(r, co, t) and E (r, a>, t) = -Vqp{r, co, t). In the Coulomb gauge, the vector and scalar potentials A{r, co, t) and (p{r, co, t), respectively, are related to the electric field as Ai{r,CD,t) = — / dlr{r-s)Ei{s,Ci),t)dh,

(4.122)

ILL/ J

—-^(r,co,t)

= - I dl(r-s)l:j{s,a),t)dh,

(4.123)

where 6^- and djj are the components of the transverse and longitudinal tensorvalued d fiinctions d^(r)

= d(r)l + VS/(4jT\r\y\

S\\(r) = -VV(4jr|r|)-*.

(4.124) (4.125)

It is recalled that A(r, t) and eoA{r, t) are canonically conjugated field variables. In contrast, the complexity of the commutation relation (4.119) suggests that the "canonical" commutators are not as simple as one would expect from the definition. The commutation relation between the vector potential and the scalar potential is as complicated, when one and only one of these quantities is

Quantum description of light propagation in dielectric media

408

[4, §4

differentiated with respect to the time or comprises such a derivative. The simple commutation relations are Ai(r,t)Jj(r\t)

(4.126)

= 0 = Mr,t)Ji{r\t)

[q)(r, t\ qp(r\ t)] =0= \qp(r, t)Ji(r\

t)

(4.127)

Then, the theory is applied to the bulk dielectric such that the dielectric function can be assumed to be independent of space: e(r, (o) = e(oj) for all r. In this case, the solution of eq. (4.118) that satisfies the boundary condition at infinity is (cf Tomas [1995]) G(r,r\a))=

[V,Vr+K\a))l]K-\aj)g{\r~/\,aj),

(4.128)

where g(r, CO) =

Qxp[iK(a)) r] 4jtr

(4.129)

Relation (4.119) can be simplified: \Ei(r,tlMr\t)

— ei/„„-—d(r-/)U Co dx„,

(4.130)

and the "canonical" commutator corresponds to the definition,

J

ih eo

(4.131)

= 0.

(4.132)

Moreover, \qp{r,t)Jj(r\t)

The commutation relations presented are in the equal-time Heisenberg picture, and therefore it is emphasized that they are conserved. To make contact with the earlier work. Dung, Knoll and Welsch [1998] define the vectors (4.133) fhr,coj)

= f

dHr-s)f{s,a),t)dh.

(4.134)

4, § 4]

Microscopic theories

409

The commutation relations (4.115) and (4.81) imply that

't^%,co,t),{f^^^%',aj',t))'

(5,^""(/--/-')<3(w-w')i,

f,^^%,co,t),f.^^%\oj',t))

(4.135) = 0.

(4.136)

The representation of the transverse vector potential simplifies to k';o},t) = ^k>fg(\r-r'\,w)]^{r',(o,t)d'r'.

(4.137)

It can be derived that the scalar potential operator ^^r,co,t)=---^^ f^^^dh, 4jteoe(a))

(4.138)

where p(r, co, t) = (ico)~^ V j (r, w, t). Another application is the quantization of the electromagnetic field in an inhomogeneous medium that consists of two bulk dielectrics with a common interface. The determination of the tensor-valued Green function for threedimensional configuration of dielectric bodies is in general a very involved problem. Dung, Knoll and Welsch [1998] return to the simple configuration mentioned by Gruner and Welsch [1996a]. The reader is referred to (Tomas [1995]) for the classical treatment of multilayer structures. It is shown that for the configuration under study, the commutation relations (4.130)-(4.132) hold. The necessity of a new calculation of the quantum electrodynamical commutation relations for a new three-dimensional configuration (cf Dung, Knoll and Welsch [1998]) is not absolute. Scheel, Knoll and Welsch [1998] have proven that the fiindamental equal-time commutation relations of quantum electrodynamics are preserved for an arbitrarily space-dependent Kramers-Kronig dielectric ftinction. Let us recall that the complex-valued dielectric function 6(r, of) depends on frequency and space: e(r,a;)^l

if

co-^ oo.

(4.139)

It is assumed that the real part (responsible for dispersion) and the imaginary part (responsible for absorption) are related to each other according to the

410

Quantum description of light propagation in dielectric media

[4, § 4

Kramers-Kronig relations, because of causality. This also implies that e(i*, oj) is a holomorphic function in the upper complex half-plane of frequency ~—6(r, (o) = Q,

Im ft; > 0.

(4.140)

aft;*

Scheel, Knoll and Welsch [1998] study relation (4.119). By comparison of the right-hand sides of this relation and (4.130), they arrive at the following identity to be proved: -^G(r,r\co)dco x Vr' = -ijnd(r-r)

x W-

(4.141)

oo ^

Here the left arrow means that the operators d/dx',^, in the expansion of the nabla operator with this upper limit will first be written on the right-hand side as

d/dxl. Based on the partial differential equation (4.118) for the tensor-valued Green fianction, an integral equation will be presented in what follows. The partial differential equation and the boundary condition at infinity determine the Green ftmction uniquely. By comparison of relation (4.117) with a constitutive relation, we could derive that ijHoCoGijir, s, (o) are holomorphic functions of co in the upper complex half-plane, i.e., - - ^ [coGkjir, 5, ft;)] = 0 ,

Im ft; > 0,

(4.142)

with (oGkj(r,s, CO) ^ 0

if

|ft;| -^ oo.

(4.143)

Second derivation of the Cauchy-Riemann equations (4.142) consists in the application of 9/9ft;* to relation (4.118). The left-hand side of relation (4.142) is then the unique solution of the homogeneous problem. Knoll and Leonhardt [1992] calculate the time-dependent Green ftanction. Scheel, Knoll and Welsch [1998] have derived the relation i!J^a)Gij(r,s,co) = bijir,s,co)=

f

e'^'^^Ay(r,5, r)dr,

(4.144)

where Dij(r,s,T) are components of the tensor-valued response function that causally relates the electric field E(r, t) to an external current yext('S', t - r), so that Dij{r,s, ^)=^J

^~'"''D,ir,s, w)dco.

(4.145)

From the theory of partial differential equations it is known (see, e.g., Garabedian [1964]) that there exists an equivalent formulation of the problem in

4, § 4]

Microscopic theories

411

terms of an integral equation. For eo(co) = J e(r, (o) d V / / d^r, an appropriately space-averaged reference relative permittivity, the integral equation for the tensor-valued Green function can be written as G(r, s,a))= I K{r, v, o)) • G{v, s, aj)d^v-^ G^^\r, s, co),

(4.146)

where G^yr,s, (o)=[l-

VrVsK-\s,

co)] gi\r - s], (o),

(4.147)

K(r,V, w) = [V,g(|r-1;|, co)] [V, In^^^^^ ^^j +

[K\v,co)-K^{co)]g{\r-v\,co)]l.

Here g(r, co) = go(r, co) is given by eq. (4.129), where K(co) = Ko(co), K\r,co)

= ^e(r,a>),

(4.149)

J-

K^(co) = —6o(a>).

(4.150)

It can be seen that the components of the kernel Kik(r,v,co) are holomorphic functions of co in the upper complex half-plane, with Kik(r,v,co)-^0

if

\co\ ^oo.

(4.151)

To prove the fundamental commutation relation (4.130), we first decompose the tensor-valued Green function into two parts: G(r,s,co) = Gi(r,s,co) + G2(r,s,co),

(4.152)

where G\ (r, s, co) satisfies the integral equation Gi = j K-Gxd^v + Gf\

(4.153)

G | V ^ , W ) =g{\r-slco)h

(4.154)

C72(r,5,co) =r(r,5,a>)Vv

(4.155)

with

In relation (4.155), F is the solution of the integral equation

r= j K-rd^v-^r^^\

(4.156)

with r^yr.s,

CO) = -V,[K-\s,

aj)g{\r-sl

co)].

(4.157)

Scheel, Knoll and Welsch [1998] derive that i^^o^^i and /.IQCO^F are the Fourier transforms of the response functions to the noise-current density and the noisecharge density, respectively.

412

Quantum description of light propagation in dielectric media

[4, § 4

Combining relations (4.152) and (4.155), and recalling that Vr' x Vr' = 0, we see that the left-hand side of relation (4.141) can be rewritten as f^

-

0)

-^G(r,r\w)da)x\/,^ J-oc

^

^

f^

=-

CO

-

— G,(r,/,a;)dw x Vr'J-oc

(4.158)

^

Thus, only the noise current response function 'm^(x)G\ contributes to the commutator (4.119). Muhiplying the integral equation (4.153) by the function ^ and integrating over co, we obtain, similarly as in the derivation of relation (4.86), from the holomorphic properties of the tensors K and (j)G\ that

f

%Gx{r,r\o))di(D ^

'\JT\d{r-r').

(4.159)

J -o

The outer product of this equation and the operator (-VrO can be taken, and together with relation (4.158) implies relation (4.141). In addition, it is shown that the scheme also applies to media with both absorption and amplification (in a bounded region of space). An extension of the quantization scheme to linear media with bounded regions of amplification is given, and the problem of anisotropic media is briefly addressed: there the permittivity is a symmetric complex tensor-valued ftinction of w, eij{r,CD) = eji{r,w).

(4.160)

Di Stefano, Savasta and Girlanda [2000] have developed a quantization scheme for the electromagnetic field in dispersive and lossy dielectrics with planar interface, including propagation in all the spatial directions, and considering both the transverse electric and transverse magnetic polarized fields. Di Stefano, Savasta and Girlanda [2001a] have presented a one-dimensional scheme for the electromagnetic field in arbitrary planar dispersing and absorbing dielectrics, taking into account their finite extent. They have derived that the complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the classical light modes, they have obtained a generalization of the method of modal expansion (e.g.. Knoll, Vogel and Welsch [1987]) to absorbing media. Di Stefano, Savasta and Girlanda [2001b] have based an electromagnetic field quantization scheme on a microscopic linear two-band model. They have derived for the first time a noise current operator for general anisotropic and/or

4, § 4]

Microscopic theories

413

spatially nonlocal media, which can be described only in terms of an appropriate frequency-dependent susceptibility. (iu) Modification of spontaneous emission by dielectric media. Dung, Knoll and Welsch [2000] have developed a formalism for studying spontaneous decay of an excited two-level atom in the presence of arbitrary dispersing and absorbing dielectric bodies. They have shown how the minimal-coupling Hamiltonian simplifies to a Hamiltonian in the dipole approximation. The formalism is based on a source-quantity representation of the electromagnetic field in terms of the tensor-valued Green function of the classical problem and appropriately chosen bosonic quantum fields. All relevant information about the bodies such as form and dispersion and absorption properties is contained in the tensor-valued Green function. This function is available for various configurations such as planarly, spherically, and cylindrically multilayered media (Chew [1995]). The theory has been applied to the spontaneous decay of a two-level atom placed at the center of a three-layer spherical microcavity, the wall being modeled by a Lorentz dielectric. The tensor-valued Green function of the configuration is known (Li, Kooi, Leong and Yeo [1994]). The calculations have been performed on the assumption of a dielectric with a single resonance. For simplicity, it has been assumed that the atom is positioned at the center of the cavity. Weak and strong couplings are studied, and in the study of the strong couplings both the normal-dispersion range and the anomalous-dispersion range associated with the band gap are considered. Whereas in the range of normal dispersion the cavity input-output coupling dominates the strength of the atom-field interaction, the most significant effect within the band gap is photon absorption by the wall material. Dung, Knoll and Welsch [2001 ] have studied nonclassical decay of an excited atom near a dispersing and absorbing microsphere of given complex permittivity that satisfies the Kramers-Kronig relations laying emphasis on a Drude-Lorentz permittivity. Among others, they have found a condition on which the decay becomes purely nonradiative. For a transparent dielectric, theoretical studies can take a traditional approach. Inoue and Hori [2001] have developed a formalism of quantization of electromagnetic fields including evanescent waves based on the detector-mode functional defined in terms of those for the widely used triplet modes. They have evaluated atomic and molecular radiation near a dielectric boundary surface. Matloob and Pooseh [2000] have discussed a fully quantum mechanical theory of the scattering of coherent light by a dissipative dispersive slab. Matloob and Falinejad [2001] have calculated the Casimir force between two dielectric slabs by using the notion of the radiation pressure associated with the quantum

414

Quantum description of light propagation in dielectric media

[4, § 5

electromagnetic vacuum. Specifically, they have used the fact that only the field correlation fiinctions are needed for the evaluation of vacuum radiation pressure on an interface. Matloob [2001] has postulated an electromagnetic field Lagrangian density at each point of space-time to be of an unfamiliar form comprising the noise current density. He has expressed the displacement D(r, t) merely in terms of the electric field E{r, t'), t' ^ t, without adding a noise polarization term. Leonhardt and Piwnicki [2001] have analysed the propagation of slow light in moving media in the case where the light is monochromatic in the laboratory frame. The extremely low group velocity is caused by the electromagnetically induced transparency of an atomic transition. § 5. Microscopic models as related to macroscopic concepts 5.7. Quantum optics in oscillator media A quantum-optical experimental setup may consist of active and passive devices: active devices to generate light with certain properties (e.g., nonclassical light), and passive ones to modify and distribute it. It is an interesting nontrivial problem to study how quantum-statistical properties of light are influenced by passive optical devices like mirrors, resonators, beamsplitters or filters. Knoll and Leonhardt [1992] have elaborated on the paper by Knoll, Vogel and Welsch [1987] where the medium is nondispersive and lossless, but they now intend to consider dispersion and losses. On introducing the Hamiltonian for the complete system, the Heisenberg equations of motion for field operators and medium (not source) quantities are derived. The complete system under consideration consists of the following subsystems: optical field, medium atoms and sources. The field is described by the electric-field-strength operator E(x,t) and the electromagnetic vectorpotential operator A(x,t) in the Coulomb gauge. The medium is modeled by damped harmonic oscillators {qa(t),pa(t)}, namely, oscillators coupled to reservoirs composed of bath oscillators {quB(t), J&//B(0}? whose energy quanta may be, for example, phonons. The medium oscillators are localized at x^i, and they all have the same mass m and elasticity (force) constant k. The bath oscillators are characterized by masses m^ and elasticity constants ks, and the coupling is expressed by the coupling constants o^. The atomic sources are described by a current operator j(x, /), of which the dynamics need not be specified, and e is the electron charge. For simplicity, a one-dimensional model is considered only.

4, § 5]

Microscopic models as related to macroscopic concepts

415

The Hamiltonian of the complete system is (5.1)

^ ( 0 = ^ R ( 0 + ^ M ( 0 + ^RS(0 + ^S(0,

where ^ R ( 0 is the Hamiltonian of the optical field, Hu{t) is that of the medium atoms, and^Rs(0 describes the interaction between the optical field and sources: ,. ,, , 2 / dA{xJ) E\x,t) + c' dx

^R(0 = J

dx.

(5.2)

n2

Hu{t) - 2 ^ i

—

+ -q,{t) (5.3)

B

L

(5.4)

^Rs(0 = - / Kx, t)A(x, t) Ax,

and ^ s ( 0 is the Hamiltonian of the atomic sources which is left unspecified. The usual commutation rules are i(x, 0, -eQE{x, t) = ifid(x

-x)\. (5.5)

The Heisenberg equations of motion for field operators, medium operators, and bath operators have been obtained. As a result of a Wigner-Weisskopf approximation for the interaction of medium oscillators with the bath operators, quantum Langevin equations have been obtained. By eliminating the medium quantities from equations for field operators and otherwise by a common procedure, a generalized wave equation for the vector potential is obtained. Using a Green function, this wave equation is solved. The decomposition of the timeordered quantum correlation functions into time-ordered correlation functions of the source operators and the free-field operators has been derived. The time-dependent Green function for a dielectric layer as the simplest optical device is calculated. The field behind the layer is discussed and represented by the negative-frequency part of the field, its expectation value and the normallyordered quadrature variance determined for the sake of squeezing analysis.

416

Quantum description of light propagation in dielectric media

[4, § ^

5.2. Problem of macroscopic averages 5.2.1. Conservative oscillator medium Dutra and Furuya [1997] have investigated a simple microscopic model for the interaction between an atom and radiation in a linear lossless medium. It is a guest two-level atom inside a single-mode cavity with a host medium composed of other two-level atoms that are approximated by harmonic oscillators. The intention is to show, in general, that ordinary quantum electrodynamics suffices, at least in principle, and that there is no need to quantize the phenomenological classical Maxwell equations. If a macroscopic description is possible, it should appear as an approximation to the fundamental microscopic theory under certain conditions. Such a "macroscopic" approximation is obtained and conditions for its validity are derived. All of the medium harmonic oscillators are represented by a collective harmonic oscillator, then two modes of a polariton field are defined. The microscopic average is regarded as filtering out higher spatial frequencies. The field that influences the guest atom is modified and the characteristics of the effect of such a microscopic field are calculated. A condition is pointed out under which the contribution of the atoms to the quantum noise appears only through a frequency-dependent dielectric constant. An effective description is obtained by leaving out the polariton mode which is approximately equal to the collective mode of the medium. Dutra and Furuya [1997] introduced a microscopic model for a material medium that they have adopted: A^ two-level atoms having the same resonance frequency WQ in a single-mode cavity with resonance frequency co. They consider a guest two-level atom with resonance frequency o)^ and strongly coupled to the field so that it will not be approximated by a harmonic oscillator. The operator of electric displacement field in the cavity is given by

b(x, t) = y

^

[a(t) + at(0] sin (^.v) ,

(5.6)

where L is the length of the cavity. It is assumed that co and L for the singlemode cavity are related as (D/C = JT/L. The operator of the polarization of the medium is given by A'

P(x, t) = Y^ dj \f)j{t) + ^/(O] d{x-xj\

(5.7)

4, §5]

Microscopic models as related to macroscopic concepts

417

where (5.8) 2womo^J are electric dipole moments. In eq. (5.8), mo is an effective mass, q, are effective charges, and products dj[bj{t) + ^J(0] are the electric dipole moment operators of the atoms of the medium that are located at Xj. The operators a{t), a\t), bj(t) and bj(t) satisfy the bosonic commutation relations, in particular dj =

\bj(tlb]^(t)\ = d-.l.

(5.9)

= 0,

bj(t),br{t)

and a(t),a\t) commute with bj(t),b^{t). The Hamiltonian is given by the relation H(t) = ha)a\t)a(t) + Picooy^bUt)bj(t)

/

b(x,t)P(x,t)dx (5.10)

+ ^o,(t)

+ hQ [a{t) + a\t)] [a(t) + a\t)] ,

where a-(t) and a(t) are the pseudo-spin operators, and Q

-d^l—rrSin CoLfi

(5.11)

\ c

is the Rabi frequency of the guest atom located at x^ whose electric dipole moment is d and the effective mass Wg. We notice that 1 /• ~ j b{x,t)P{x,t)dx

^ = hJ28j[^iO

+ ciHt)][bj(t) + b]{t)],

(5.12)

where como

Sj

Hh)-

(5.13)

In simplifying the Hamiltonian, Dutra and Furuya [1997] denote N

G=

H^h

V

(5.14)

the coupling constant between the field and the collective harmonic oscillator described by the annihilation operator

m = -^Y.^Mt).

(5.15)

7=1

In the Hamiltonian (5.10), the self-energy terms (Cohen-Tannoudji, DupontRoc and Grynberg [1989]) have been neglected. This resuhs in the condition

418

Quantum description of light propagation in dielectric media

[4, § 5

4G^ ^ a^cao. Further, the original problem is reduced to the case of a single atom coupled to two polariton modes. The frequencies of these modes are denoted by ^1 and Qj so that Q\ :^ coJl - 4 ^ , co and Q2 -^ <^o when co ^ 0, oc, respectively, with G' = J^G.

(5.16)

In other words, Q\ < Q2 for (x) < (JOO, Q\ > Q2 fox o) > co^. The dressed operators are denoted by Ck{t) and c\{t), k = 1,2, and the operators a(t) and B(t) are expressed in their terms (Chizhov, Nazmitdinov and Shumovsky [1991]). The problem of the extra quantum noise introduced by the atoms of the medium is discussed. When the atoms of the medium are coupled only weakly to the field, i.e., G\ co ^ ^ , where A[a(t) + a\t)] permittivity: e r ^ 1+4—^.

= a(t) + a\t) - {a(t) + ci\t))l,

(5.17) and er is the relative

(5.18)

From relation (5.17) the variance of the electric displacement field D(x,t) (cf. Glauber and Lewenstein [1991]) can be calculated. Adopting a continuous distribution of atoms in the medium instead of the realistic discrete one implies a greater variance (Rosewarne [1991]). Let us address the problem of macroscopic averages. The macroscopic theories of quantum electrodynamics in nonlinear media have often, by "definition", avoided discussing the macroscopic averaging procedure. The quantummechanical averaging advocated by Schram [1960] removes the quantum fluctuations from the macroscopic theory. The problem of what macroscopic averaging procedure to use had evaded solution for many years. Lorentz, in the early twentieth century, was the first to attempt such a derivation (cf. the chapters by de Groot [1969] and van Kranendonk and Sipe [1977] in previous volumes of Progress in Optics). Robinson [1971, 1973] proposed a different kind of macroscopic average. He regards a macroscopic description as a description where spatial Fourier components of the field variables are irrelevant above some limiting spatial frequency ko. Dutra and Furuya [1997] consider Fourier components with spatial frequencies above oj/c irrelevant in a macroscopic

Microscopic models as related to macroscopic concepts

4, §5]

419

description. They arrive at the following expression for the operator of the macroscopic polarization: P(x,t) = -2G^

IJieo_

\B(t)-^B\t)\

(5.19)

sin(^-x^

(joLmo

The macroscopic "average" does not change the electric-displacement-field operator b(x, t), and the macroscopic electric-field-strength operator is given by 1 E{x, t) = (5.20) D(x,t)-P(x,t) eo The calculation of the variance of a quantity typical of the operator E(x, t) yields a larger value than e~^^^, which agrees with Rosewame's [1991] result. Thus, it has been shown that the contribution from the atoms to the quantum noise of the field does not restrict itself to inclusion of the dielectric constant. We will now report a suitable macroscopic theory of electrodynamics in a material medium which does not suffer from the problems which are discussed here. It is shown that under certain conditions a macroscopic description incorporating the frequency dependence of the relative permittivity provides a good approximation. In this domain, Milonni's [1995] result has been recovered. The guest atom is not affected by the polariton mode if the frequency of the atom is far from O2. Analysis of the probability of this mode inducing transitions shows that this probability is negligible when ^2^0

\Q^-o)'\~-

ft>2)2

7«l^2-

+A0)QWG-

(5.21)

0)^

In the regime described by relation (5.21), if one leaves out the polariton mode described by C2{t) and ^2(0 and preforms macroscopic averaging, the resulting macroscopic Hamiltonian is / r ^ a c ( 0 = ^ ^ l ^ | ( 0 ^ l ( 0 + ^ ^ r ( 0 - - ^ m a c ( X a , 0 [d{t) + d\t)\

,

(5.22)

with the macroscopic displacement field ^macv-^? 0

\hQ\eQe,JTr Ly

c,(0 + c (0 sm

er—X

(5.23)

where

y-do;(^'^)

(5.24)

By relation (5.24), y is the ratio between the speed of light in the vacuum and the group velocity in the medium. The macroscopic polarization Pmac(^, 0 is given

420

Quantum description of light propagation in dielectric media

[4, § 5

by relation (5.19), simplified by leaving out its C2-cl polariton component. Then, the macroscopic electric field is obtained ft-om the relation 1 ^macv-^? 0

^macl-^? * j

^macv-^5 * j

(5.25)

^0

It is stated that the results of Dutra and Furuya [1997] for the macroscopic fields coincide with those derived by Milonni [1995] for the case of one and more modes. 5.2.2. Kramers-Kronig dielectric Dutra and Furuya [1998a] have pointed out that the Huttner-Bamett model at the stage after the diagonalization of the Hamiltonian for the polarization field and reservoirs can operate with a larger class of dielectric fianctions than that admitted by the original microscopic model. At this stage, the relative permittivity is expressed dependent on the dimensionless coupling function ^(o;) as introduced in eq. (4.51). For example, the permittivity obtained in the Lorentz oscillator model (Klingshirn [1995]) can be recovered by adopting C(w) = —.—z-r—z

1=.

w^ - <J)Q- \2K(D

(5.26)

yjn

where K is the frequency-independent absorption rate. The relative permittivity for the original Huttner-Barnett microscopic model has the form e(o;) = 1

—^.

,

(5.27)

where F ( a ; ) - lim

r

l^^^'^l'

dco^

(5.28)

f -^ +0 7_oc (^ - CO - If

It is indicated that, in the Lorentz oscillator model, eq. (5.27) yields the solution F(cy) = i ^ ,

(5.29)

but the integral equation (5.28) with this left-hand side (= replaced by =) cannot be solved to yield a coupling ftinction V{o)). This is the main difficulty, because from the relation t(a» = i^^;^

" ^

(5.30)

4, § 5]

Microscopic models as related to macroscopic concepts

421

we obtain that |K(a»|^ = ^

(5.31)

or we could determine v{a)) = p\ — V(a)) = ± p ^ .

(5.32)

y/jT

V (O

This presents a restriction to the Huttner-Barnett microscopic model. 5.2.3. Dissipative oscillator medium Let us recall that in the microscopic model, the electromagnetic-field operators are given by integrals both over k and co. Huttner and Barnett [1992a] themselves say that they lose the relationship between the frequency and the wavevector k. This observation relates to the macroscopic theories as well: the Dirac delta fianction suitable for the expression of such a relationship is never replaced by another (generalized) function. Quantities such as (4.47)-(4.50) are formulated dependent on the relative permittivity e{(x)). Dutra and Furuya [1998b] suggest a simplified expression: e(a;)=l + ^ l i m / " / ^ ^ ' ^ . da;^ 2a; f -^ +0 J_^ oj' - (D-\£

(5.33)

where (Dutra and Furuya [1998b]) S(co) = cboco

l^^^jf

(5.34)

with F(a)) defined by relation (5.28). They try to calculate the relative permittivity for the Huttner-Barnett microscopic model by means of classical electrodynamics. The Huttner-Barnett approach is applied to the particular case where the coupling strength is a slowly varying function of frequency. In continuation of Dutra and Furuya [1997], a modified version of the simple model takes account of absorption. The inclusion of losses necessarily introduces a continuum of modes in the model. The consequences are minimized by adopting the standard elimination of the reservoir. The interaction between the radiation field and the medium is described by a dipole-coupling Hamikonian, where the canonically conjugated field is

422

Quantum description of light propagation in dielectric media

[4, § 5

the displacement field, instead of a minimal-coupling Hamiltonian, where the canonically conjugated field is the electric field. For simplicity, a Lorentzian shape is assumed for |F(a>)p, given by the relation V((0)=

^^—-ri/^,

(5.35)

where o^o > ^ > ^. The Hamiltonian incorporating absorption is assumed to be N

H(t) = h(D^a\t) a(t) + hcoo ^

b^.(t) bj(t)

7=1 1

--

/•

poo

I b{x,t)P{x,t)dx-^h

^J

N >

/

y^QWJ{Qj)Wj{Qj)dLQ

Jo

j=,

(5.36) where Wj{Q,t), Wj{Qj) are reservoir creation and annihilation operators that commute with every other operator, but the commutation relation holds: \Wj{Qj\

Wj\Q',t)\

= djj>d{Q-Q') i.

(5.37)

Upon substitution of relations (5.6) and (5.7) into the Hamiltonian (5.36) and introducing appropriate collective operators, the total Hamiltonian becomes a sum of two uncoupled Hamiltonians H(t) = Hi(t)^H2(t).

(5.38)

The second Hamiltonian H2(0 describes A^^ - 1 damped collective excitations of the medium to which the field does not couple. The field and the single damped collective excitations of the medium, to which the field couples, are described by the Hamiltonian H\(t) alone. This Hamiltonian is given by Hi (t) = HUt) + ^mat(0 + ^int(0, (5-39) where the Hamiltonian of the field, that of medium and their interaction Hamiltonian are expressed as follows HUt) = hw,a\t)a(t), H^!,tit) = fi(t}oBHt)B{t) + f' / +h I

\V{Q)B\t)Y{Q,t)+

HUt) = hG[a\t) + a{t)][B{t) + B\t)\,

(5.40) QY\Q,t)Y(Q,t)dQ V*(Q)Y\Q,t)B{t)\

dQ, (5.42)

4, § 5]

Microscopic models as related to macroscopic concepts

423

respectively. The collective annihilation operators B{t) and Y{Q, t) are given by N

B(t) =

Y,0jbj(tl 7=1

'"'

(5.43) N

where (frj = gj/G and gj and G are given by eqs. (5.13) and (5.14), respectively. Dutra and Furuya [1998b] have a (conventional) strictly microscopic model, where the medium is not continuous, but discrete. They admit the practicality of the macroscopic average of the physical quantities. Following Robinson [1971, 1973], they understand the macroscopic averaging as filtering out of higher spatial frequencies. Considering a classical Hamiltonian which is a modification of relation (5.39), they derive a relative permittivity. A further topic in the article by Dutra and Furuya [1998b] is essentially relation (4.68) due to Gruner and Welsch [1995]. In particular, it is shown that also in the case of the simple microscopic model of the medium used by Dutra and Furuya [1998b], it is possible to first diagonalize HmatiO a^d then the total Hamiltonian (5.39). The diagonal form of the Hamiltonian ffmaM is achieved in terms of continuous operators B(v,t), /•OC

B(v,t) = a{v)B(t)+

/

l3(y,Q)Y(Q,t)dQ,

(5.44)

where a(v) and ^(v, Q) are coefficients like (4.36) and (4.38). The diagonal form of the total Hamiltonian (5.39) is achieved in terms of continuous operators h3i((0, v)^(v, 0 + ^ ( 0 ; , v)^^(v,0 dv. '(5.45) where a\((o), a2((x)), l5[((o,v), ft(a;, v) are coefficients like (4.47)-(4.50). Suitable operators, namely those of the electric displacement field and of the macroscopic electric field strength, are defined such that

A(a),t) = ai(a))a(t) + a2((o)a\t)-\- / Jo

eoe((i))

/ (^^QL

2G'a*(a;)yi(co,0dwsinf—jcj+H.-

JO

(5.46) The difference between this and relation (4.68) arises because Dutra and Furuya have only a single mode of the field, use a dipole-coupling Hamiltonian instead

424

Quantum description of light propagation in dielectric media

[4, § 6

of a minimal-coupling Hamiltonian, and have defined their field operators in terms of different quadratures of the annihilation and creation operators. Ruostekoski [2000] has theoretically studied the optical properties of a FermiDirac gas in the presence of a superfluid state. He also considered diffraction of atoms by means of light-stimulated transitions of photons between two intersecting laser beams. Optical properties could possibly signal the presence of the superfluid state and determine the value of the Bardeen-Cooper-Schrieffer parameter in dilute atomic Fermi-Dirac gases. § 6. Conclusions In this chapter we have mainly reviewed canonical quantum descriptions of light propagation in a nonlinear dispersionless dielectric medium and in linear and nonlinear dispersive dielectric media. These descriptions have regularly been simplified by a transition to one-dimensional propagation, which has been illustrated also by some original simple descriptions. Besides this we have reported criticisms of the description of light propagation in a nonlinear medium using a spatial variable instead of and similarly as the quantum-mechanical time parameter. We have adopted the standpoint that macroscopic quantum electrodynamics arises both through reduction of a microscopic description and through immediate application of a quantization scheme to macroscopic fields. The origins of immediate macroscopic theories are connected with the possibility of determining the electrical displacement field as the canonical momentum (up to the sign) to the vector potential. One may also consider as a member of this class the possibility of starting from the assumption that the canonical momentum is the magnetic induction field with the dual vector potential whose curl is the electric displacement field. Due to the relatively large number of various fields and their components, and the relative complexity of commutators, discrete and continuous expansions in terms of annihilation and creation operators must be presented. In the papers reviewed here, the ground (vacuum) state of the electromagnetic field has mostly been assumed. The Heisenberg picture of time evolution has been preferred. In one of these papers the opinion has been formulated that the quantum description of sources is to be treated separately, but a detailed treatment cannot be found there. As a criterion for the correctness of the theories, equivalence between the Heisenberg equations of motion for the electric displacement and magnetic induction fields and the Maxwell equations is required.

4]

Acknowledgments

425

The role of the momentum operator has been analyzed: its limited domain of application, one-dimensional propagation, is still appealing for its simplicity. The integral expression for the momentum operator has been given. It has been stressed that the use of this operator as the generator of spatial progression depends on the knowledge of appropriate equal-space commutators between field operators. Results on the connection with the slowlyvarying-envelope approximation have been presented, including some important applications. The possibility has been examined of determining a phenomenological Hamiltonian, and the quantization scheme with dual potential has been expounded and appropriately modified, so that the real relative permittivity of the lossless medium is expressed at least locally (in terms of a quadratic Taylor polynomial). In simplifying to the one-dimensional case, an application to a nonlinear dispersive (Kerr-like) medium and quantum solitons has been presented. Various notions of the slowly-varying-envelope approximation and the quantum-paraxial approximation have been presented, along with applications of these concepts in both linear and nonlinear cases. A new application to a nonlinear dispersionless (Kerr-like) medium has been concerned with the propagation of arbitrary pulses. The need for renormalization has been declared even in this, one-dimensional case. Derivation of a macroscopic description from a microscopic model has been performed; it is not possible - in the prevailing Heisenberg picture - to eliminate the matter fields completely. The argument has been provided not only with the description of the dispersion up to any order of accuracy, but also with losses, as follows from the Kramers-Kronig relations. The exposition of this microscopic model has been delivered from the viewpoint of the macroscopic variables and the operator-valued noise current, which is a form of matter field. A nonlinear modification of this description of the linear dispersive absorbing medium has been performed for the Kerr medium. The application to a two-level atom positioned in the center of a spherical cavity has been mentioned. A different notion of the macroscopic fields has been mentioned as well: these are not comprised in a microscopic model in advance, but approximately equal operators of measurable modes of the dressed matter fields. Acknowledgments This work under project number LN00A015 was supported by the Ministry of Education of the Czech Republic.

426

Quantum description of light propagation in dielectric media

[4

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E. Wolf, Progress in Optics 43 © 2002 Elsevier Science B. V. All rights reserved

Chapter 5

Phase space correspondence between classical optics and quantum mechanics by

D. Dragoman Univ. Bucharest, Physics Dept., P.O. Box MG-U, 76900 Bucharest, Romania

433

Contents

Page § 1.

Introduction

435

§ 2.

The phase space in classical optics and quantum mechanics

§ 3.

Definitions and properties of phase space distribution

§ 4.

Nonclassical states in phase space

§ 5.

Measurement procedures of phase space distribution functions in quantum mechanics and classical optics

.

fiinctions

439 447 455

461

§ 6.

Propagation of classical fields and quantum states in phase space 469

§ 7.

Interactions of classical fields and quantum states as phase space overlap

474

§ 8.

Classical and quantum interference in phase space

477

§ 9.

Universality of the phase space treatment

488

§ 10.

Conclusions

489

References

491

434

§ 1. Introduction Quantum mechanics was born from the need to quantify the energy of the emitted or absorbed electromagnetic radiation in order to explain the black body spectrum and the photoelectric effect. Light was thus considered, from the beginning of the quantum revolution, either as an extended wave or as a bunch of particles with definite energy. The subsequent evolution of quantum mechanics has not succeeded to settle the century old controversy regarding the corpuscular or wave-like nature of light, but rather deepened the mystery, extending the wave-corpuscle controversy to material particles. Quantum mechanics, and more recently quantum optics, is able to correctly calculate the essential characteristics of a quantum system, such as its energy levels, quantum statistics and so on, but still longs for a proper interpretation of its calculations. The widely accepted probabilistic interpretation of quantum mechanics is still an open question, the meaning and even existence of a wavefiinction for photons, for example, being still subject to debate. One of the main reasons for this paradoxical situation of the most successfiil recent theory in physics is that although it has initially borrowed a lot of concepts from classical physics, in particular from classical optics, the subsequent evolution of quantum mechanics has focused towards developing its own language, manifestly different from classical mechanics. The aim of the present paper is to show how much is classical and how much is quantum in quantum mechanics, and why and where quantum and classical mechanics agree in their predictions. The success of this approach is maximized in the phase space realm, where both classical and quantum theories are expressed in the same mathematical language.

1.1. Existing analogies between classical optics and quantum mechanics Classical and quantum physics are fiandamentally different from a conceptual point of view. For example, the wavefunction is only a probability amplitude in quantum mechanics, whereas in classical optics its analog - the electric field is a measurable quantity. From the beginning, the founders of the quantum 435

436

Phase space correspondence between classical optics and quantum mechanics

[5, < 1

theory tried to find at least a formal connection to classical mechanics. In Schrodinger's view, for example, classical dynamics of point particles should be the "geometrical optics" approximation of a linear wave equation, in the same way as ray optics is a limiting approximation of wave optics (Schrodinger [1926]). Over the following years some rigorous mathematical analogies between classical optics and quantum mechanics have been identified. One of the better known and widely exploited is based on the similarity between the time-independent Schrodinger equation and the time-independent Helmholtz equation. This analogy led to the design of multilayered optical structures with the same transmission characteristics as their quantum counterparts with OD, ID or 2D dimensions (Dragoman and Dragoman [1999]), but with the advantage of a much easier characterization due to their order-of-magnitudes larger dimensions compared to the quantum structures. The same analogy was also employed to show that the transverse modes of aspherical laser lesonators are similar to the eigenstates of the stationary Schrodinger equation with a potential well determined by the mirror profile. Although this equivalence holds only for short cavity lengths in comparison with the Rayleigh range of the ftindamental mode, higher-order corrections for longer resonator lengths can also be found (Pare, Gagnon and Belanger [1992]). Nienhuis and Allen [1993] proved that the Hermite-Gauss or the Laguerre-Gauss modes of a laser beam can be described using the operator algebra of the quantum harmonic oscillator. In particular, these modes are generated from the fundamental laser mode by applying ladder operators. In addition, displaced light beams, which are refracted by lenses according to geometrical optics, were found to be the paraxial optics analog of a coherent state. Inhomogeneous graded-index waveguides can also benefit from the quantum-theoretical formalism (Krivoshlykov [1994]). The Franck-Condon principle has found its analog in paraxial optics in the mismatch of a mode passing through two fibers with different refractive index distributions; an optical analog to the Ramsauer effect has also been identified by Man'ko [1986]. Moreover, the quantum optics squeezed states have been extended to solutions of the Helmholtz equation that contain them in the paraxial approximation (Wolf and Kurmyshev [1993]). The Helmholtz equation reduces in the paraxial approximation to the time-dependent Schrodinger equation, with the distance along the optical axis, the wavelength, and the refraction index playing the roles of time, Planck's constant and potential, respectively, in quantum mechanics. The canonically conjugate momentum describes in the optical case the direction of the ray. Ray optics and classical mechanics are then recovered in the limits A ^ 0

5, § 1]

Introduction

437

and ^ ^ 0, respectively. Even nonparaxial mappings, corresponding to aberrations in optics, can be considered. The Fresnel diffraction from a slit has found its equivalent in the two-photon process driven by a chirped pulse (Broers, Noordam and van Linden van den Heuvell [1992]). Theory and experiments show that the analog of the spectral Fresnel zone plate leads, for the case of the two-photon level, to a focusing of spectral energy in a much smaller effective bandwidth than that of the original excitation pulse. This analogy is based on the fact that the diffraction pattern is determined in both cases by the interference between different paths that lead to the same final state. The classical Malus law, which predicts an attenuation of light intensity passing through a linear polarizer with a factor depending on the angle between the polarization direction of the incoming wave and the orientation of polarizer, has also a quantum analog. In the quantum case the same attenuation occurs for spin-^ particles detected by a properly oriented Stern-Gerlach apparatus if the statistical averages involve a quasidistribution fijnction that can become negative. This analogy can be extended for arbitrary spin values s, the classical limit being obtained for 5" -^ oc (Wodkiewicz [1995]). In recent years, classical optics has been completely re-implemented using quantum particles; the newly developed "particle optics" which includes "atom optics" (Meystre [2001]) and "electron optics" (Hawkes and Kasper [1996]) is a testimony of the close relationship between quantum mechanics and classical optics. Incoherent phonons, which propagate ballistically in the crystal, can act as acoustic analogs of classical optical mirrors, lenses, filters or microscopes, generating high-resolution acoustic pictures (see Hu and Nori [1996] and the references therein). It was even demonstrated that some essential properties of quantum information and quantum computation methods are classical wave properties, the quantum nature being unquestionable only in situations where nonlocal entanglement is present (Spreeuw [1998]). The monumental work of Mandel and Wolf [1995] investigates also the relationship between classical and quantum coherence. The analogies between light propagation and atom optics can be extended beyond the paraxial approximation, to calculate for example consecutive corrections to the "optical Schrodinger equation". This generalized analogy has found applications to the harmonic motion in a graded-index fiber and to the tunneling between coupled fibers (Marte and Stenholm [1997]). The relationship between Schrodinger and classical wave propagation was applied to scattering problems. In particular, results valid for electron-impurity scattering were extended to scattering of scalar classical waves from dielectric particles (van Tiggelen and

438

Phase space correspondence between classical optics and quantum mechanics

[5, § 1

Kogan [1994]). In the latter case, the phase-destroying effects, which restrict the observation of interference in multiple electronic scattering to low temperatures or to the mesoscopic regime, are absent. On the other hand, concepts and phenomena characteristic for propagation of quantum wavefiinctions have found their analogs for classical waves. Examples include the optical crystals (Sakoda [2001]) known as photonic band gap structures, the optical Berry phase (Bhandari [1997]), tunneling (Ranfagni, Mugnai, Fabeni, Pazzi, Naletto and Sozzi [1991]), and even more exotic concepts as weak localization (van Albada and Lagendijk [1985], Akkermans, Wolf and Maynard [1986], Kaveh, Rosenbluh, Edrei and Freund [1986]), Anderson localization (Anderson [1985], Kaveh [1987]), quantized conductances (Montie, Cosman, 't Hooft, van der Mark and Beenakker [1991]), and conductance fluctuations (Kogan, Baumgartner, Berkovits and Kaveh [1993]). Last, but not least, functions that have been applied in classical optical problems have been translated in an operator language in quantum mechanics, and vice-versa. One example is the fractional Fourier transform. Although it originates in quantum mechanics (Namias [1980]), it was adopted in classical optics (Lohmann, Mendlovic and Zalevsky [1998]), and then related concepts such as the complex fractional Fourier, developed first in classical optics, have found their way back in quantum mechanics (Chountasis, Vourdas and Bendjaballah [1999]). As one would expect, despite all these fruitful analogies, there are differences between quantum and classical theories. For example, the long-wavelength classical scattering vanishes as A""*, whereas a finite (s-wave) cross-section is obtained in the same limit for Schrodinger potential scattering (van Tiggelen and Kogan [1994]). The expectation value of the orbital angular momentum of a paraxial beam of light is expressible not only in terms of an analogous angular momentum of the harmonic oscillator, but also contributions from the ellipticity of the wavefi-ontsand of the light spot are present (Nienhuis and Allen [1993]).

1.2. Why a phase space treatment of these analogies? Why do we need then a phase space (PS) comparison of quantum mechanics and classical optics? The aim of this work is to show that a PS approach to such a problem can offer a more complete answer to the question of the limit of such analogies. The previously mentioned similarities between quantum and classical phenomena are usually based on the observation of formal resemblance of mathematical equations. What about a small difference - a small, perturbation-

5, § 2]

The phase space in classical optics and quantum mechanics

439

like term - in the mathematical formulas? Would this term have any clear interpretation? Probably not. The reason is that classical mechanics and/or classical optics operates with algebraic functions, whereas quantum mechanics works with operators which are applied to different representations of quantum states. There is no obvious connection between these two mathematical theories. A better understanding of the connection between quantum mechanics and classical optics would be provided if the comparison would be made between similar mathematical languages. Since an operator approach to classical optics, or classical mechanics, would be an unnecessary complication, our attention focuses to a quantum mechanical treatment in the space of numbers. Fortunately, such a treatment has been already developed in the quantum mechanical PS.

§ 2. The phase space in classical optics and quantum mechanics 2.1. Hamiltonian formulation of the equations of motion in classical mechanics The state of a physical system is characterized by the information needed at a given time for the calculation of its future evolution. For a classical particle the state is given by the n independent components of the coordinate vector ^ ^ (^\,^2, ' • (In) and momentum vector p = (p\,p2, ... Pn)- They span the 2n dimensional PS (q,p) of classical mechanics in which the momentum and position vectors are on the same footing and interchangeable. The evolution of a system in PS under the action of a Hamiltonian Hc\(q,p) is described by a set of first-order differential equations for the conjugate variables q and p: ^i=^—^

dpi

Pi = —^-^

(2-1)

dqi

where the dot indicates the total time derivative. For a time-dependent Hamiltonian the above equations are supplemented with Hc\ = dHc\/dt. For systems of classical particles with mass m the Hamiltonian can usually be separated into a kinetic and a potential part Hd = p^/2m + V(q), with V{q) the potential energy. In this case/7/ = mqi. The PS in classical optics depends on the treatment we are using: geometrical (ray) or wave optics. Geometrical optics is an approximation to wave optics which disregards the diffraction, interference or polarization effects, valid whenever the dimensions of various apertures are very large compared to light wavelength. In geometrical optics a Hamiltonian can be defined as

440

Phase space correspondence between classical optics and quantum mechanics

[5, § 2

^opt = -[«(^)^ - p^y^^ where n(q) = n{x,y) is the local refractive index and P ^ {Px,Pv) ^ (nu:,,nuy), with u^, Uy the direction cosines made by a ray with the X, y coordinate axes. Equations (2.1) are valid also in this case, with the total time derivative replaced by the derivative with respect to z - the propagation direction of the bunch of rays. Although the Hamiltonian equations are formally similar, the PS of classical mechanics and classical optics are globally different. In classical mechanics the momentum vector/; is not restricted in value, whereas in classical optics the form of the Hamiltonian implies that \p\ ^ n. Only in the paraxial approximation, when \p\
where § = I

J is the ray vector, and / ^ I

, ^ 1 is a 2« x 2« matrix, with

/ the nxn identity matrix (« = 2 for geometrical optics). This form of the Hamiltonian equations of motion is preserved under canonical transformations, characterized by a matrix Af, which satisfies the relation MJM^ = J

(2.3)

M describes the transformation of variables from the initial set §' to the final set §^ and is defined as M,//X^') = d^l/d^]y The linear transformations that satisfy eq. (2.3) form the symplectic group. A linear transformation is symplectic in two dimensions if its determinant is 1, additional restrictive conditions being required in higher dimensions (Guillemin and Sternberg [1984]). From a geometrical point of view, Hamiltonian mechanics corresponds to transformations that preserve an antisymmetric, nondegenerate bilinear form defined on an evendimensional real vector space §, which in two dimensions can be written as

5, § 2]

The phase space in classical optics and quantum mechanics

441

cr((^i,^2) = (liP\ - qxPi- In other words, the PS area bounded by a group of trajectories is constant with time; one can directly follow the motion of a bounded region in PS rather than follow the individual trajectories that comprised that region. If instead of one particle we consider now an ensemble of non-interacting particles we can define a probability distribution function in PS/(§). Under the action of the symplectic map the probability distribution function transforms as/^(§) = / ' ( M " ' | ) . This relation is known as the Liouville's theorem. The Liouville's theorem can be explicitly expressed in terms of the Hamiltonian of the system as

f =[/,//c,,op.]p+f,

(2.4)

where the Poisson bracket is defined by r . r,-,

^r^ fdA dB

dA dB\

^-^

dA dB

The Poisson bracket expresses the symplectic structure of the classical PS. It is a binary antisymmetric relation which, applied to the elements of a commutative ring, makes it into a Lie algebra with the additional requirement that the bracket acts as a derivation of the commutative multiplication. A generalized Liouville theorem exists even for non-Hamiltonian motion, induced by forces that are not derivable from a potential, such as close-range collisions with other species of particles, synchrotron radiation or bremsstrahlung (Lichtenberg [1969]). Equation (2.4) holds not only for the probability distribution in PS, but for any function A(q,p) of canonical variables. In particular [qi^qj]? = 0, [pi,Pj]p = 0, [qi,Pj]p = ^ij and eqs. (2.1) become q = [^,//cKopt]p, ^ = [p,//ciopt]p. The expectation value of a PS function for an ensemble of non-interacting particles is defined as

{A) = JdqdpA(p,q)f(p,q).

(2.6)

In wave optics it is also possible to define a Hamiltonian, but in terms of conjugate functions instead of conjugate vectors. More precisely, expanding the vector potential of the electromagnetic field as A{Q, t) = V'^'^ J2k Zls = 12,3 ^ksQks(t) exp(i^^), where V is the volume in which the field is confined, k is the wave vector, and e^s is a unit vector along the polarization direction s, the Hamiltonian of a source-less and current-less electromagnetic field is given by

442

Phase space correspondence between classical optics and quantum mechanics

[5, § 2

^eni = \T.kAPi<^Pls + (4QksQl) with Pks = Ql and w* = kc (di Bartolo [1991]). The Hamiltonian equations are now

or v4(^, 0 = [^(^, ^),//em]p- Since the electric and magnetic fields are related to the vector potential through E = -dA/dt, B = V x A (the scalar potential of the electromagnetic field is taken as zero), their Poisson brackets are [Ei(q,tXEj(q\t)]p = 0, [Bi(q,t),B,iq'j)]p = 0, [E,(q,tlBj(q,t)]p ^ 0 (di Bartolo [1991]). 2.2. Quantization procedures and the phase space of quantum mechanics The relation between classical and quantum mechanics can be translated in group theory through the relation between the symplectic group and the metaplectic representation, the latter denoting the action of the metaplectic group - double covering of the symplectic group - on the Hilbert space. The quantization of a classical mechanical system reduces then to the association to each quadratic polynomial H^ of a self-adjoint operator H acting on the Hilbert space, such that the map from //d to -xh'^H carries the Poisson bracket into commutators (Guillemin and Sternberg [1984]). According to the Groenwald-van Hove theorem, it is not possible, however, to extend the metaplectic representation such as to include nonquadratic polynomials. More precisely, the quantization of a Hamiltonian nonrelativistic physical system proceeds by raising the classical dynamical variables q, p and any function A(q,p) of them to the category of linear operators. In doing this, the Poisson brackets [v4,5]p transform to the commutation relations -ih'^[A,B] = -ih\AB-BA). In particular, in quantum mechanics the position and momentum operators do not commute, i.e. [q,p] =" \h. This property of the position and momentum operators, although not encountered in classical mechanics, where the corresponding Poisson bracket vanishes, is not foreign to classical wave optics. In the preceding section we have assigned a nonvanishing Poisson bracket to the electric and magnetic field components of the electromagnetic radiation. Due to the non-commutativity of q and p, the quantization procedure is unambiguous only when the correspondence between A(q,p) and A(q,p) is unique, i.e. there is no ambiguity in the ordering rules of the position and momentum operators. Even in this case, however, the canonical quantization

5, § 2]

The phase space in classical optics and quantum mechanics

443

privileges the Cartesian frame. This means that, for example, a Hamiltonian expressed in terms of angle-action variables, cannot be quantized in a welldefined manner. Moreover, in the canonical quantization process the Hamiltonian must be identified with the total energy of the system in order to avoid contradictory results. Not even in the {q,p) variables is the Hamiltonian unique for a given motion (Pimpale and Razavy [1988]). However, it was recently shown that PS concepts are essential to define a general procedure of quantization of non-Hamiltonian systems (Bolivar [1998]). For mixed quantum-classical systems it is possible to define a quantumclassical bracket that reduces to the quantum commutator and the Poisson bracket in the purely quantum and classical cases, respectively. In these systems two distinct sets of variables, with their own Planck constant, correspond to the quantum and classical parts, respectively, so that the Planck constant of the classical part can approach zero leaving the quantum subsystem unchanged (Prezhdo and Kisil [1997]). In quantum mechanics all information about a quantum state is contained in the state vector, which is a vector in Hilbert space. The observables are Hermitian operators. In the Schrodinger formulation of quantum mechanics the quantum state I xp) satisfies the linear differential equation .A«=^|V>

(2.8)

with a quantum Hamiltonian H. The eigenstates \n) of the Hamiltonian are the energy eigenstates. The energy spectrum of a quantum system has usually a discrete and a continuous part. It is usually assumed that a discrete energy spectrum is a manifestation of the quantum nature of a system. But, is this really so? No. Classical optics offers the best counterexample: the propagation constants (energy levels) of classical light in a waveguide have also a discrete as well as a continuous spectrum (Snyder and Love [1983]). Energy discretization appears whenever a constraint is imposed upon the freedom of movement. Since the eigenstates \q), \p) of the position and momentum operators form complete sets of states, an arbitrary quantum state can be characterized in the position or momentum representation. In the position representation the wavefunction xp = ip(q,t) is defined as {q\ip) and the action of the operators q and p on the quantum state are described by multiplication with q and -ihVq, respectively, so that H = Hc\{q,~\hVqj). In the momentum representation the wavefunction i/; = ^(pJ) is defined as {p\ip) and H = Hc\(ihWp,p,t). Usually, only ID quantum systems are considered, so that we will restrict from now on to this case. For these systems, ipiq) and

444

Phase space correspondence betM'een classical optics and quantum mechanics

[5, § 2

xlj(p) are two representations of the same quantum state; they are related by a Fourier transform ip(q) = (iJihy^^^ J^dpip{p) Qxp(iqp/h). The squared modulus of the wavefunction in the position or momentum representation gives the corresponding probability density. We have shown in the previous section that the Hamilton's equations of motion are preserved under linear canonical transformations, which correspond to quadratic Hamiltonians. Quantum mechanical evolution equations are preserved under unitary transformations. The quadratic Hamiltonians in quantum mechanics which generate the rotation around the origin and the squeezing in PS are p^/2 + (0^x^/2 and (px-^xpyi, respectively. The squeezing operator compresses the PS along one coordinate and expands it along the other, transforming one harmonic oscillator into another with different frequency co. The other quadratic Hamiltonians, p^/2 and x^/2 describe in quantum optics the paraxial free propagation of light rays in a homogeneous medium and the action of a thin lens, respectively. Although the quantum state vector contains all information about the state, it is sometimes difficult, especially in open systems, to use this concept. Therefore, a more appropriate description, which incorporates our lack of knowledge about what pure state the system is actually in, can be given in terms of the Hermitian density operator. It is defined for pure states as p = 11/^) (V^|, or for a superposition of states It/;) = J2^^^ yj„\m) as p = E , ^ . = o H^n,il^nH{^\ = E^.n = oPninH{nl Pmn are the elements of the density matrix in the energy representation. In a mixed state we cannot describe the state by a superposition, since we only know the probability with which the component states \m) appear. The density matrix is then diagonal in the component-state representation, i.e. p„„j = 0 for m ^ n. The density operator has only non-negative eigenvalues (is non-negative) and evolves according to the von Neumann equation

f = 4[i/,p].

(2.9)

p is extremely useful in expressing the expectation values of an arbitrary operator A via the trace operation: {A)=TT(Apy

(2.10)

For a mixed state (statistical mixture) Trp^ < 1, while for pure states Trp^ = 1. Defining the variance of an operator through AA = ({A ) - {A^y^^, it can be shown that for any pairs of non-commuting operators, in particular for the position and momentum operators AqAp^h/2.

(2.11)

5, § 2]

The phase space in classical optics and quantum mechanics

445

This uncertainty relation is considered to be the most important difference between classical and quantum mechanics. It says that one cannot simultaneously measure with arbitrary precision the expectation values of two non-commuting operators. In contrast, in classical mechanics the momentum and position of a classical particle can be exactly known at any time. Well, the uncertainty relations are not restricted to the quantum realm. A similar relation (with a different meaning!) exists in wave optics between position and momentum, limited by the light wavelength, while in optical signal processing or Fourier analysis the time and frequency satisfy Aa;A^ > ^. This should be expected since, when speaking about position-momentum of photons in free space, position corresponds to time in a retarded frame and the momentum of the photon is related to the frequency via Planck's constant and the velocity of light. The development and significance of quantum mechanics is checked by the requirement that classical results are recovered in the ^ -^ 0 limit. Ehrenfest was the first to show that the equation of motion for the average values of quantum observables coincides with the corresponding classical expression, i.e. d(^)/d^ = {p)/in, d{p)/dt = F{{q)). The last equation is valid only if {F(q)) = F({q)), with the force defined as F(q) = -W(q). The validity of Ehrenfest's theorem is neither necessary nor sufficient to identify the classical regime, since the classical limit of a quantum state is not a single classical orbit, but generally an ensemble of orbits. Even when Ehrenfest's theorem fails, a quantum state may behave classically if its evolution is in agreement with the Liouville equation for a regular or chaotic classical ensemble. Thus, a more appropriate criterion for classical behavior is that quantum averages and probability distributions agree, approximately, with the respective classical quantities (Ballentine, Yang and Zibin [1994]). Potentials can, however, be found for which the quantum mechanical motion is identical to the motion of the corresponding classical ensemble. Two classes of such potentials have been found by Makowski and Konkel [1998]. The difficulty of the quantum-classical correspondence is even more emphasized by the fact that, generally, in order to obtain the correct classical limit when ^ ^ 0 the system must be mechanically connected to an infinite number of additional classical degrees of freedom. Quantum effects such as interference or tunneling originate then in the mechanical interactions between different parts of the overall infinite system (Kay [1990]). Despite all these differences there is a close relation between the classical PS variables and the corresponding quantum operators in the Hilbert space. For example, for any linear transformation in the classical PS described by a symplectic matrix M with elements A,B,C,D, a unitary operator U(M) can

446

Phase space correspondence between classical optics and quantum mechanics

[5, § 2

always be constructed such that the same relation exists between the quantum operators: q' = [U{M)YqU{M) = Aq + Bp, p' = [U{M)YpU{M) = Cq + Dp. A special case of such a linear transformation is squeezing (Fan and VanderLinde [1989]). The evolution of the system changes dramatically, however, in the case of nonlinear evolution. An example of such a situation is the interference of the wavefunction of a trapped atom with Raman-type exciting laser waves, which is the quantum analog of nonlinear optical phenomena such as parametric amplification, multimode mixing and Kerr-type nonlinearities. In this case neighboring PS zones have a different time evolution, the net result being a partitioning of the PS, which may induce strong amplitude squeezing of the motional quantum state as well as quantum interferences (Wallentowitz and Vogel [1997]). The eigenvalues of the position and momentum operators span the PS of quantum mechanics of massive particles. In quantum optics the electromagnetic field is modeled as a collection of harmonic oscillators with unit mass m = I and frequency w, characterized by bosonic annihilation and creation operators a and a^, respectively. These operators, for which [a,a^] = h, are related to the position and momentum operators of the oscillators as a = (2hmcoy^^^(m(joq + i^),

a"^ = (2hmwy^^^{mojq - ip).

(2.12)

However, in quantum optics the position and momentum operators have no clear meanings, so that q and p are called quadrature operators, and correspond to the in-phase and out-of-phase components of the electric field amplitude. The rotated quadrature operators are linear combinations of the position and momentum operators, with weights determined by the angle of rotation 6: qo = q cos 6 -\-p sin 6, po = -q sin 0 + ^ cos 6. The quantum mechanical PS, although spanned by variables with the same meaning as in classical mechanics (position and momentum), has a totally different algebraic structure than the classical PS. More precisely, the classical PS, invariant under canonical transformations, is not a metric manifold, since the separation between two points has no invariant meaning. A physically realizable state is characterized in this PS by a density d(q - qo) d(p-po), the motion of this PS point being described by the Hamiltonian equations of motion. The density d(q - qo) 8(p -po) is, however, unacceptable in the quantum PS (and in the PS of classical wave optics!), due to the position-momentum uncertainty relation. The quantum PS has thus a metric, non-Riemannian structure, which must coincide with the completely different classical PS in the limit ^ ^ 0. Also, the operator algebra of quantum mechanics must be invariant under unitary transformations.

5, § 3]

Definitions and properties of phase space distribution

fiinctions

447

To emphasize these differences the quantum PS is called mock PS (sometimes also Weyl PS) and p, q are referred to as c-numbers. Each rule of association introduces its own p, q manifold via a particular choice of the basis set. So, there is a mock PS for each rule of ordering (Balazs and Jennings [1984]). In the quantum PS it is possible to define a correspondence between a dynamical operator A(p,q) and its Weyl image A(p,q). This is essential since there is no isomorphism between the group of canonical transformations in the PS of classical dynamics and the group of unitary transformations in the Hilbert space of quantum mechanics. In the limit h = 0, p,q become the momenta and coordinates in Cartesian coordinates and the Weyl space becomes the classical PS. Only the Weyl images of linear and quadratic operator functions transform as classical dynamical quantities, the linear inhomogeneous transformations playing a preferred role in the Weyl space endowed with an equiaffine geometry (Balazs [1981]). This conclusion corresponds to that obtained from a group-theoretical point of view. The group of linear canonical transformations in PS and their applications in various branches of physics are discussed in Kim and Noz [1991]. This group of transformations for n pairs of conjugate variables, characterized by symplectic matrices which can be written as products of translation, rotation and squeezing matrices, is the inhomogeneous symplectic group ISp(2«). Its subgroup, the homogeneous symplectic group Sp(2n) (which does not include translations), is locally isomorphic to the (2 + l)-dimensional and (3 + 2)-dimensional Lorentz groups for « = 1 and n = 2, respectively. In quantum optics the SU(2) algebra can be used to calculate the electron-counting probability and the SU(1,1) Lie algebra is useful for the calculation of the photon-counting probability and for the investigation of the quantum-nondemolition measurement of photon number in four-wave-mixing (Ban [1993]). § 3. Definitions and properties of pliase space distribution functions Since it is not possible to access a (q,p) point in the quantum PS due to the uncertainty principle, we cannot define a localized probability distribution in quantum mechanics, but only quasiprobability distributions that yield correct results for observable quantities. In terms of any distribution function F(q,p, t) the expectation value of an arbitrary operator A(q,p) can be calculated similarly as in the classical PS, i.e. as {A(q,p)) = Tr ]^p(q,pj)A(q,p)\ = f dqdpA(q,p)F(q,pJX

(3.1)

where A(q,p) is the scalar function obtained by replacing the operators q,p in A with scalar variables q and p. Different scalar functions A{q,p) and hence

448

Phase space correspondence between classical optics and quantum mechanics

[5, § 3

different distribution functions are obtained for different rules of ordering of the non-commuting position and momentum operators. All distribution functions contain the same amount of information about the quantum system. The selection of a quasiprobability distribution is merely imposed by the problem to be solved, the principal requirement being usually that of simplicity. For quantum-classical correspondence problems, the most interesting quasiprobability distribution is the Wigner distribution function (WDF), which corresponds to the Weyl or symmetric rule of association. For pure and mixed states the WDF is defined as Pure : W{q,p\ t) = —— / djc exp(-i/?x/^) xp^'iq - \x\ t) ipiq + ^JC; t), Mixed : W(q,p; t) = —— / dxQxp(-ipx/h){q + ^x\p(t)\q - \x).

(3.2a) (3.2b)

The WDF is limited to | W(q,p; t)\ ^ l/jih. In the words of its inventor, the WDF "seems to be the simplest" from all bilinear expressions in the wavefunction which are linear in the expectation values of any sum of a function of coordinates and a function of momenta (Wigner [1932]). There are many books and review papers dedicated to the properties and applications of the quantum WDF, as well as to its relation to other distribution functions. We can only mention a few of them here, such as those written by Moyal [1949], Carruthers and Zachariasen [1983], Hillery, O'Connell, Scully and Wigner [1984], or more recently the excellent review of H.-W Lee [1995] and the books of Walls and Milburn [1994] and Schleich [2001]. A whole class of ^--parameterized distribution functions can be obtained from the WDF as (Cahill and Glauber [1968a,b]) Fig,p;s) = exp ( ^ - ^ ^ ^ j

exp ( " ^ ^ ^ j

n
(3.3)

where s can take also complex values (Wiinsche [1996a]). The distribution functions for ^ = -1,0, and 1, correspond to the antinormal, Wigner and normal distribution functions, respectively (m = 1 in quantum optics). For normal ordered operators, the powers ofa^ precede the powers of a, the opposite holding for antinormal ordering. The normal distribution function, also called GlauberSudarshan or P function, is mostly used in the quantum theory of optical coherence, where expectation values of normally ordered products are of interest. The distribution function corresponding to ^ = - 1 , called also Q function, gives the probability distribution for finding the coherent state | a) in the state p

5, § 3]

Definitions and properties of phase space distribution functions

449

since Q(q,p) = JT~^{a\p\a). The Q function is always positive and limited to 0 ^ Q(q,pi t) ^ )^Jth. Therefore, it is mainly employed in the PS study of chaotic systems, for which the Q function has the smoothest and simplest structure from all distribution functions. In the classical limit of large mean photon numbers the distinctions depending on the ordering of operators vanish and the expressions for Q, P become identical. The Q function is a particular case of a class of non-negative quantum distribution functions - the Husimi functions - obtained by smoothing the WDF with a minimum uncertainty squeezed Gaussian function characterized by a positive constant ^: H{q,p, t) = (Jth)-^ f dq' dp' exp [-mUq' - qf/h - {p -pf/hmt]

W{q\p\ t).

(3.4) The Q function is retrieved for t = CL), i.e. when the WDF is smoothed by a coherent state wave packet. The Husimi function is associated with the antinormal ordering of the squeezed photon annihilation and creation operators. In the coherent state representation, two distributions for different ^'s are related to one another through a convolution or smoothing operation that depends on the difference in s: F{a,s) = j

F{li,s')

exp JT(S' -S)

\

2|a-/3p s' -s

d-/3,

(3.5)

for s' > s. The quasiprobability distributions are in general singular for ^ > 0, i.e. expressed in terms of generalized functions such as delta functions and their derivatives, whereas for ^ < -1 the distribution is well defined and for 5- < 0 is always regular. These behaviors can be understood by viewing eq. (3.5) as an operation of smoothing in the direction of decreasing s. Another class of distribution functions which includes the antistandard, Wigner, and standard distributions for Z? = -1,0, and 1, respectively, is defined as (see O'Connell and Wang [1985] and the references therein): Giq,p;b) = cxp(^^^^\

W{q,p).

(3.6)

For standard ordering all powers of ^ precede those ofp, whereas for antistandard ordering all powers of^ precede those of ^. The distribution function for Z? = -1 is also called Kirkwood or Rihaczek distribution. Other association rules and corresponding distribution functions are described in Cohen [1966]. Among them are the positive P function, the

450

Phase space correspondence betM^een classical optics and quantum mechanics

[5, § 3

Rivier or Margenau-Hill ordering for which the distribution function is given by [G{q,p,-\) + G(q,p, l)]/2, the normal-antinormal ordering distribution \Q{q,p) + P{q,p)\/2, the Born and Jordan rule of ordering which gives as distribution the product of the squared modulus of the wavefunction and its Fourier transform. Some of these distributions, as for example, the last one, are not bilinear in the wavefunction. It is commonly believed that the WDF is the quantum analog of the classical PS probability distribution even for many-body problems (Shlomo [1985]). Why is the WDF a privileged quasiprobability distribution? First, it is a real distribution that can, however, take negative values over certain regions of PS. However, the negative regions of the WDF cannot extend over areas significantly wider than h/2. The realness property is also shared by the P distribution, which is highly nonsingular, and by the Q and Husimi functions, which are in addition non-negative. Then, it satisfies the marginal properties, also satisfied by a classical probability distribution, that dpW{q,p)={q\p\q),

/

DC

&qW{q,p)= {p\p\p),

(3.7)

vZ-OC

and the normalization condition j ^ dqdpW(q,p) = 1. The correct quantum mechanical marginals are not given, for example, by the P, Q and Husimi functions. The marginal distributions of WDF can be directly measured with homodyne or balanced homodyne detection schemes (see next section). For ^-parameterized distribution functions the marginal distributions are positive for s ^0, but can take negative values or even be singular for 5* > 0 (Orlowski and Wunsche [1993]). Another desirable feature of the WDF is that it is invariant with respect to time and space reflections, and it is Galilei invariant, i.e. it transforms as W(^,P) -^ W(q + q\p) and W(q,p) -> W(q + q\p) if ip(q) -^ ipiq + q') and ipiq) —^ Qxp(-ip'q/h) ip(q), respectively. Then, it is the only quasiprobability distribution that satisfies the overlap property (O'Connell and Wigner [1981]) Tr{p,p2) = 2JTh J-oc J-OC

dq

dpW^(q,p)W2(q,p).

(3.8)

JJ-OC

For all other ^-parameterized quasiprobabilities eq. (3.8) must be replaced by (Leonhardt [1997]) Tr(p,p2) = IJihJ^dqJ^dpW,(q,p; s)W2(q,p; ~ s). Moreover, the WDF was also shown to be the simplest description in nonequilibrium situations. In particular, the WDF has the simplest correspondence to

5, § 3]

Definitions and properties of phase space distribution

fiinctions

451

the Bloch equation, which has been extensively used in calculations of quantum corrections to classical distribution functions (O'Connell and Wang [1985]). Muga, Palao and Sala [1998] showed that the WDF is also the closest to the classical probability when average local values and local variances of a quantum observable are numerically compared with their classical counterpart. The WDF forms also a complete, orthonormal set, in the sense that for a pure function ^(q,t) = ^an{t)(l)n{q) with (j)n the nXh eigenstate of the system, W{q,p,t) = Y,a:{t)aM WUq.p).

(3-9)

where WnnM^P) = j ^ / dxQxp(-ipx/h) \p;,(q - {x) xpn,(q + \xl

I

dqdp W„„,{q,p) fV*,„,,{q,p) = — d,,,/(5,„„,s

Yl W„„,{q,p) KM',P') = ^Hq

- q') ^(p -p'\

The terms with n = m in eq. (3.9) are called auto-terms, the other being the cross- or interference terms. Note that the WDF for a mixed state is the weighted sum of pure WDFs (it does not contain interference terms); in contrast, the Q function cannot distinguish between statistical mixtures and macroscopic quantum superpositions. All the properties mentioned above, with the exception of the non-positiveness, are sheared by a classical probability distribution. Therefore, the WDF is called a quasiprobability. This classical-like quantum PS distribution can be identified by using only the postulate that (Bertrand and Bertrand [1987]) pr(^, 0) = J W{q cos 0 ~ ps\x\0,qsmO -\- p cos 0) dp, where pr(^, 6) is the position probability distribution after an arbitrary phase shift 6. There is no non-negative distribution, bilinear in the wavefunction, and which yields the correct quantum mechanical marginal distributions (Srinivas and Wolf [1975]). Non-negative Wigner-type distributions for all quantum states can be obtained, however, by smoothing with a Gaussian whose variance is greater than or equal to that of the minimum uncertainty wave packet, or by integrating the WDF over PS regions of the order h^", where n is the number of dimensions (Cartwright [1976]). Non-negative smoothed WDF distributions, which include as a special case the Husimi distributions, can be used to formulate quantum mechanics (Lalovic, Davidovic and Bijedic [1992]).

452

Phase space correspondence between classical optics and quantum mechanics

[5, § 3

But is the quantum PS formalism so 'quantum'? Is it related to the PS formalism of classical physics only in the ^ -^ 0 limit? Well, no. A WDF formally identical to that defined for pure and mixed quantum states has been long ago defined (and used with considerably success) for coherent (Bastiaans [1979]) and partially coherent classical light beams (Bastiaans [1986]), respectively. The only difference is that the Planck's constant h should be replaced in this case by the normalized wavelength X = X/ln and that the density matrix in eq. (3.2b) should be replaced by the coherence fiinction in classical optics. In rest, all the properties of the quantum and classical WDF defined in this way are identical; for a review of the properties and applications of the WDF in classical optics see Dragoman [1997]. Even the non-positive property of the WDF is preserved in classical optics. Not to mention that signal processing has benefited also from distribution ftinctions defined on the time-frequency PS (Cohen [1989]), in particular from the WDF (Claasen and Meklenbrauker [1980a,b]). The relationship between the quantum and classical WDFs is also supported by the work of Bialynicki-Birula [2000], who showed that for the fijU electromagnetic field the role of position and momentum is played by the magnetic and electric induction vectors and the analog of the WDF is a functional of B and Z>. Actually, similarities between the quantum and classical WDFs for particular states have been observed by many authors. Serimaa, Javanainen and Varro [1986] even defined a gauge-invariant Wigner operator and a gauge-independent Wigner function that allow for both quantized and classical electromagnetic fields. Equations (3.2a) and (3.2b) show that the WDF can be calculated from the wavefunction or the density matrix of a quantum system. However, the WDF can be obtained directly in PS by solving a system of coupled linear partial differential equations derived from the time-independent Schrodinger equation. This system is

xW{q,p) =

m dq

y.

1 /i^Y"' d'T &" W(q,p)

y>

1 f\hV

(3.10a) = 0,

iW{q,p) =

d'V &' "

^..jfr. ^y^J WW.

n'(q,p)

(3.10b) = 0,

5, § 3]

Definitions and properties of phase space distribution

fiinctions

453

where V{q) is the potential energy. Although, in general, these equations are of infinite order in /?, only a finite number of derivates contribute for a polynomial potential V{q). The simplest way to extract the WDF from these equations is to expand it into a series of products of Chebyshev polynomials depending on q or p (Hug, Menke and Schleich [1998]). How can we know that a solution of eqs. (3.10a) and (3.10b) is really a WDF, i.e. it corresponds to a certain quantum wavefianction or density matrix? A simple answer is: make sure that the density matrix obtained from the WDF through

{v,u) =/ dpQxp[ip(u-uyh]W((u j p(u,u)=

+ uy2,p)

(3.11)

has non-negative eigenvalues. More complex criteria exist, however. Narcovich and O'Connell [1986] showed that, besides satisfying the normalization condition, a function W(q,p) should have a continuous and ^-positive type symplectic Fourier transform, defined as W(u, u) = J W{q,p) exp[i(^t; - up)] dq dp, in order to be a WDF. A fijnction W is of ^-positive type if, for every choice of points a\ = (u\,u\), ai = (u2,U2), •.. a,n = (w,;;,t;,„), the mxm matrix with elements exp[i^0'(«A,^/)/2] W{aj - Uk) is non-negative. Here o{a/^,aj) = UjVk - UkVj is the symplectic form. A quantum state differs from a classical state in that it requires W{a) to be of ^-positive type instead of positive type in the sense of Bochner. The two conditions of positivity are identical in the limit ^ = 0. The ^-positive type condition assures that the uncertainty relations are respected, the opposite being not true, i.e. the uncertainty relations alone do not assure that a real PS function is a WDF. Up to now we have defined the quantum PS as being spanned by the momentum and position coordinates, or by the complex variable a for coherent states. However, it is possible to define a quantum mechanical PS starting from any two mutually incompatible, not necessarily canonically conjugate, complete sets of operators ^ = {^1,^2, • • }, B = {B\,B2, ...} with eigenvalues a = {a\,a2, •. •), b = (b\,b2, .. .)• The PS is then spanned by (a,b) = {a\,a2, ..., b\,b2, •..) with the variables taking values over the respective continuous or discrete eigenvalue spectra. When it is possible to obtain eigenfunctions of a particular complete set of operators in more than one representation, the relation of one quantum PS to the other is obtained using Dirac's transformation theory. Defining the distribution fiinction associated to a state |i/^) in the PS corresponding to the complete sets of Hermitian operators A and B as/(a,/?) = {a\ip) (iplb) {b\a) and the PS mapping of an operator O as 0{a, b) ^ {b\d\a)/{b\a), any transformation

454

Phase space correspondence between classical optics and quantum mechanics

[5, § 3

to another PS corresponding to the pair of complete sets C, D converts these quantities to

/(c,^).5:/<»,i,,(MMM).

,3.12a)

o(.,.,.5:o,»,»,(»™>).

ai2b)

In particular, the complete sets of operators can be q and H. In this case one finds that the degeneracies of the PS motion are not, in general, reflected in the degeneracies of the energy eigenvalues, and that the PS constants of motion do not always correspond to quantum constants of motion (Pimpale and Razavy [1988]). The WDF has been generalized, in particular, for relativistic spin-zero quantum particles in an external electromagnetic field (Holland, Kyprianidis, Marie and Vigier [1986]), for rotation-angle and angular-momentum variables (Bizarro [1994]), for a general angular-momentum state with applications to collections of two-level atoms (Dowling, Agarwal and Schleich [1994]), and a WDF has even been defined in the number-phase PS with analogous properties to the WDF associated to position and momentum observables (Vaccaro and Pegg [1990], Vaccaro [1995]). In order to develop in PS a mathematical formalism analogous to that of the Heisenberg equation of motion for a quantum-mechanical operator, and so to deepen the similarities between the PS and Heisenberg treatments of quantum mechanics, operators have also been defined in PS (Ghosh and Dhara [1991]). The Wigner operator A\^(q,p) = A(Q,P) is obtained from the WDF fiinction corresponding to an arbitrary quantum mechanical operator M^'>P) ^ /#^xp(i/7y/;^)(^ - y/2\A\q -\- y/2) by replacing q,p with the Bopp operators Q = q - {h/2i){d/dp\ P = p -^ (h/2i){d/dq). These Wigner operators do not act on the Hilbert space, but on fiinctions in PS; they are not needed for evaluating the expectation values, but can be used to develop time-dependent density-functional theories in PS or master-equations for open quantum systems. A quantum theory using operators can thus be derived in PS, which in the limit ^ —> 0 leads to the canonical formulation of classical mechanics. By introducing operators in the Liouville space, viewed as vectors and represented by kets, on which act superoperators, the PS formulation becomes a representation in a peculiar Liouville-space basis, transforming naturally under Galilean changes of reference frames. In this formalism the rate of change (though not the higher-order time derivatives) of the expectation value of a quadratic operator is the same as if the WDF obeyed a Liouville equation,

5, § 4]

Nonclassical states in phase space

455

whatever the Hamiltonian (Royer [1991]). In the bracket PS formalism it can be shown that the change of PS representation follows the same rules as a change of representation for a waveflinction, a much simpler transformation rule than in PS approaches based only on functions of PS variables (Wilkie and Brumer [2000]). The role of Galilean space-time symmetries in selecting a certain representation is thus more directly evidenced (Royer [1992]).

§ 4. Nonclassical states in phase space The quantum states usually encountered in quantum optics are the coherent states, the Fock states (or number states) and the squeezed states. The properties of all these quantum states are described in detail in any quantum optics textbook; we are concerned here only with their representation in PS. Any of these states can be represented in PS by any of the quasiprobability distributions, in particular by the WDF. In most cases however, they are represented by the socalled 'error-box', or contour lines of the WDF, or, in the case of the numberstates, by 'energy-bands'. The Fock states |«), eigenstates of the photon-number operator h = a^a, have a definite number of photons, but a completely random phase. The waveftinctions of the energy eigenstates \n) of a harmonic oscillator with mass m and frequency co in the position space are given by V^,?(^) "= N,jH,j(K:q) exp[-(/c^)^/2], with //„ the Hermite polynomials, K = Vmco/h, and A^,, = (K^/jty^^(2"n\y^^^. In the large-« limit (Schleich [2001]) the energy wavefunction takes the form V^«(^) = \/Z^exp(i0„) + v ^ e x p ( - i 0 „ ) where the amplitudes are given by A„ = (K/2jz)[2(n + ^) - K^q^y^^^ and the phases are (l)„(q) = Sniq) - \jT, with Sniq) the PS area enclosed by the vertical line at Kq and the circle of radius J2(n + ^). The amplitudes A,j of the right- and the left-going waves are equal because the energy wavefianction is a standing wave. Since, according to the uncertainty principle, a quantum state cannot have a PS area less than 2jiPi, the energy eigenstate is a band in PS with boundaries determined by the Planck-Bohr-Sommerfeld quantization condition: PS area 2jThn on its inner boundary and 2nh{n + 1) on its outer. Its PS area is 2jr^, as it should be for a pure quantum state (mixed states can occupy larger PS areas). The PS trajectory for the «th energy eigenstate, which encompasses the area 2Jih(n + ^), runs midway through the band. This (Kramer) PS trajectory, described in the dimensionless variables Q = Kq, P = p/PiK by the circle Q^ + P^ = 2(n -\- ^) corresponds to the classical, harmonic, PS trajectory of a particle with welldefined energy E^ = ho){n + \). The collection of Planck-Bohr-Sommerfeld

456

[l{n+\)r

Phase space correspondence between classical optics and quantum mechanics

^P

[5, § 4

AP

AP -2^^'lma

^

^e

re

2^^Rea

2"^Rea

2'"a

"0

>e

Fig. 1. PS representations of (a) the nth energy eigenstate of a harmonic oscillator, (b) a coherent state, (c) a displaced squeezed state, (d) a phase state, and (e) a superposition of two coherent states.

bands for all n values fills out the PS. The PS representation of an energy eigenstate, and the corresponding WDF have been represented in figs, la and 2a, respectively. The WDF of the energy eigenstates of the harmonic oscillator Wn(Q,P) = (-iy(Jihy^L„(2(Q^ + P^))Qxp{-Q^ - P'X where L„ are the Laguerre polynomials, depends only on the energy values and therefore is constant along the PS contours of constant energy. In fact, the WDF oscillates in the region enclosed by the classical PS trajectory and exponentially decreases away from it, ^,,(0,0) = (-ly/Jih changing its sign depending on the quantum number being even or odd. For large n numbers the energy-band in PS can be viewed as a contour of the WDF, taken at a sufficiently high value so that the crest and troughs inside the PS trajectory do not appear. The coherent state \a) is the complex eigenstate of the annihilation operator a\a) = a I a), its amplitude \a\ and phase arga corresponding to the respective quantities of the complex wave amplitude in classical optics. Due to eq. (2.12), the scaled real and complex parts of a correspond to the PS coordinates p, q (the eigenvalues of the position and momentum operators). For a number of different reasons, including the fact that the coherent state expectation value for a single-mode field operator is the same as for the classical electric field of a monochromatic wave, and the fact that the photon distributions for coherent states is Poissonian, coherent states are as close

5, §4]

Nonclassical states in phase space

O

cd

II Q

ti-

.t^

5--0

IT)

o

in

(U

s ^ S ^ ^+'-I" I o f.

> ^

C3

s:,

i

£ >

ex u to O

Q

457

458

Phase space correspondence between classical optics and quantum mechanics

[5,

to wave-like states of the electromagnetic oscillator as quantum mechanics allows. This is valid also for statistical mixtures of coherent states (like thermal fields), all other (nonclassical) states being reduced to classical ones by any kind of losses. The coherent state is represented by a minimum-uncertainty Gaussian wavefiinction i/^coh(0 "" (K^/JtY''^ exp[-2(Im a)^] Qxp[-(Q - V2af/2] displaced from the origin by V2a (displaced vacuum state, with Gaussian quadrature probability distributions with the same width as for the vacuum). Its WDF is W(Q,P) = {jzhy^ e x p K g - v ^ R e af - (P + \/2 Im af], its error box being a displaced (minimum-uncertainty) circular PS area. The coherent states have an indefinite number of photons, and so a more precisely defined phase than number states. They are generated by a highly stabilized laser operating well above threshold. The PS representation, and the WDF of a coherent state are represented in figs, lb and 2b, respectively. Analogously, the squeezed ground-state function \l)Q{Q)^{sK^/jiy^^Q~^^~^^ has the WDF W{Q,P) = {Jihy^ exp[-^2' - P^/s\ and the momentum and position uncertainties are given by A^ = {K\/2S)~\ Ap = fiKy/sTl. The PS representation is a Gaussian cigar, elongated in one direction and squeezed in the other. Actually not the state, but the fluctuations are squeezed in one variable (momentum or position) at the expense of the other whenever s ^ I. s = I corresponds to the coherent state. The squeezed quadrature offers the possibility to enhance the quantum measurement limit. Unlike in a coherent state, the quantum fluctuations in squeezed states are no longer independent of phase. Squeezed states are produced in nonlinear optical processes such as the degenerate parametric amplification. A generalized squeezed state %q(q) = (sK^/jzy^'^ exp[-25'(Im af] Qxp[-s{Kx - y/2af/2) is represented in the PS as a displaced Gaussian cigar (fig. Ic), and its WDF is shown in fig. 2c. Phase states are also encountered in quantum optics, despite the difficulty of defining a Hermitian phase operator for states of definite phase. The PS representation (Schleich [2001]) of the phase state defined as DC

|0)=(2;rr'^2^exp[i(m+i)^]|m) m=0

is shown in fig. Id. Figure le displays the PS representation of a superposition of coherent states |i/^) = (N/y/2)(\aQxp(i6)) + |aexp(-i0))) (a Schrodingercat state), for real a, where N = {\ + cos[a^ sin(2^)]exp(-2asin^ ^)}"^^^ is a normalization constant. Are these states real quantum states? Although the Fock or coherent states are defined in the frame of quantum optics, their wavefunctions in the

5, § 4]

Nonclassical states in phase space

459

position representation and the corresponding WDFs have the same form as electromagnetic fields in optical waveguides (Dragoman [2000a], Wodkiewicz and Herling [1998]) and fields produced by coherent light sources, respectively (Dragoman and Dragoman [2001], Wodkiewicz and Herling [1998]). This is true also for superpositions of these states. Moreover, rotation and squeezing of the WDF in PS can be realized in classical optics by fractional-Fourier transforming devices (Lohmann, Mendlovic and Zalevsky [1998]) and magnifier systems, respectively. Negative regions exist also for the WDF of the classical optical states, the negative regions being a consequence of the nonlocal character of the fields and not a signature of nonclassical behavior (Dragoman [2000b]). Actually, negative regions for the WDF arise for any (quantum or classical) state that occupies a PS area larger than the minimum allowed value, due to interferences in PS between the neighboring minimum-uncertainty states in which the state can be decomposed (Dragoman [2000c]). The real quantum character is manifested only in the result of measurement. How can we express in PS the nonclassicality of a quantum state? There is no single answer to this question. In Buzek and Knight [1995] PS interference between components of the macroscopically distinguishable coherent states lead to nonclassical characteristics. For nonclassical states the WDF takes negative values (this is however theoretically and experimentally invalidated for the WDF in classical optics). Nonclassical states are also defined as those that have a non-positive P distribution. Another definition, and even classification, of nonclassical states is given in terms of the P distribution function, which is related to the density operator in the diagonal representation of coherent states as p = / d ^ a P ( a ) | a ) ( a | . A state is classical if for any Hermitian operator A, Tr(pi) > 0 for every A{a) ^ 0, where {A) = Tr(pA) = J d^aP(a)A(a), and is nonclassical if Tr(p^) < 0 for some A{a) ^ 0. The definition of the nonclassical state can even be refined to include weakly and strongly nonclassical states, when the real and imaginary parts of a are specifically taken into account. In particular, for states described by Gaussian WDFs, the onset of squeezing triggers an abrupt change from the classical to the strongly nonclassical regime without passing through weakly nonclassical states (Arvind, Mukunda and Simon [1997]). For the 5-parameterized distribution functions a criterion of nonclassicity based on negative regions of quasiprobability distributions was derived in Liitkenhaus and Bamett [1995]. They showed that the s parameter associated to a given state has a critical value ^c such that distributions with s < Sc are positive definite, those with s > Sc are indefinite and for ^ = ^c are positive semidefinite. ^c is thus a measure of the degree of nonclassical behavior (^c ^ -1 since the Q function is always non-negative). The minimum-uncertainty states with Gaussian wavefijnctions,

460

Phase space correspondence between classical optics and quantum mechanics

[5,

which include the coherent and squeezed states, are the most "classical" since only for them the WDF is positive definite. For them ^c "^ 1. On the contrary, Fock states are nonclassical since their P function contains derivatives of the delta function. Another criterion defines nonclassical states through the degree of squeezing or sub-Poissonian statistics (antibunching), defined for a single mode field by S={\ (aexp(i0) + 5+ exp(-i6/))^ :) - {{a Qx^iiO) + a"^ exp(-i0)))^

g^"°"°;;;f°>'>.

(4.1)

(4.2,

respectively, where : : denotes normal ordering. However, neither squeezing nor antibunching provides a necessary condition for nonclassicality. For squeezing S is negative, while for sub-Poissonian statistics Q is negative. Fields with ig < 0 have no classical description via the P function, whereas states with g > 0 (with super-Poissonian statistics) are classical (Mandel [1979]). For the number states of light Q ^ -\. The coherent state is on the borderline between classical and nonclassical states because for it g = 0 (the photon statistics defined by |(«|V^coh)P is Poissonian). When counted, classical particles obey the same statistical law as coherent states, if taken at random from a pool with an average |ap each time. A squeezed state can have a photon number distribution broader or narrower than a Poissonian, depending on whether the reduced fluctuations occur in the phase or amplitude component of the field. In particular squeezed vacuum contains only photon pairs (its photonnumber statistics vanishes for odd photon numbers), the probability of finding a photon pair being in this case identical to the probability distribution of independently produced particle pairs. By superposing two coherent states, the photon distribution changes from Poissonian to sub-Poissonian, reflecting the nonclassical character of the superposition. A modified Q parameter can be introduced to characterize nonclassical light even if both S and Q are positive, an example of its utility being the Schrodingercat states in regions where they exhibit no sub-Poissonian statistics (Agarwal and Tara [1992]). A measure of nonclassicality of a given radiation field can also be defined as the minimum average photon number of the chaotic light that can destroy all the nonclassical properties of the field. Besides sub-Poissonian photon statistics and squeezing, there can be other nonclassical properties; the more of these there are, the smaller is the number of photons necessary to destroy only one of them (Kim [1999]). A remark should be made on the fact that squeezing of quadrature fluctuations has a classical analog. For example, for a superposition of two coherent classical

5, §5]

Measurement procedures of phase space distribution functions

461

Gaussian beams, squeezing is due to destructive interference in the A:-space (k = p/X ), which cause the reduction of the /:-vector bandwidth below the value of the original Gaussian beam (Wodkiewicz and Herling [1998]). In this case Aq,Ak are defined as statistical spreads of the light intensity and its Fourier transform, respectively (or via the respective second-order moments with WDF the weight function) and for a single Gaussian beam have the values Aq = Ak "= l/\/2, as for the uncertainty relation for a single coherent state. The degree of squeezing depends on the separation d between the Gaussians: (A^)'

d^ 1 2"^ 4 l+exp(-dV4)'

(A^) 2

_ 1

d^ exp(-dV4) Tl+exp(-dV4)"

(4.3)

An interesting object for studying the transition from classical to nonclassical behavior is the gray-body; as its absorptivity varies from 0 to 1, the gray-body changes from an extremely nonclassical to an extremely classical state (C.T. Lee [1995]).

§ 5. Measurement procedures of phase space distribution functions in quantum mechanics and classical optics Since the canonically conjugate position and momentum variables cannot be measured simultaneously in quantum mechanics, is there any chance to measure at least some of the PS distribution fianctions, or are they only mathematical constructions? Fortunately, there is not only the possibility, but there are at least three methods for measuring PS probability functions (Freyberger, Bardroff, Leichtle, Schrade and Schleich [1997]), as shown schematically in fig. 3.

Fig. 3. Schematic representation of (a) the tomographic method, (b) the simultaneous measurement method, and (c) the ring method for the determination of the WDF.

462

Phase space correspondence between classical optics and quantum mechanics

[5,

In the first method, 'sHces' through the WDF are obtained by measuring probability distributions of rotated quadratures. This tomographic method is similar to optical tomography, performed on either light beams or light pulses. In the second method the PS is sampled with areas greater than or equal to the value allowed by the uncertainty relation. Simultaneous knowledge of the conjugate variables of the WDF distribution is obtained, although an approximate, smoothed one; actually the experiments provide the Q function. The smoothing is inherently linked to the possibility of performing these approximate measurements. In the third method, called also the ring method, the WDF is obtained measuring its overlap (by photon counting) with different energy eigenstates. The tomographic method has been applied to the measurement of the WDF for quantum states of either light or matter. The method was proposed by Bertrand and Bertrand [1987] and Vogel and Risken [1989], and is excellently reviewed in the book of Leonhardt [1997]. Although heterodyne tomographic measurements have also been performed, the established method for WDF reconstruction is homodyne detection, especially balanced homodyne detection, which has the advantage of canceling technical noise and the classical instabilities of the reference field. The first practical demonstration of quantum homodyne tomography was performed by Smithey, Beck, Raymer and Faridani in 1993. In this method the quadratures of the signal beam are rotated to qo = q cos 9+ psind, po = -qsind -\- pcosd by its interference at a balanced 50/50 beam splitter with an intense coherent laser beam, called the local oscillator (LO), which provides the phase reference 0. The position quadrature, or more exactly 2^^^\(^Lo\^o, is obtained from the difference between the photocurrents measured by ideal, linear response photodetectors placed in the path of the two beams emerging from the beam splitter. The amplitude of the local oscillator | a^o \ is obtained from measurements of the sum of these photocurrents, while 6 can be varied by adjusting the LO using, for example, a movable mirror. Denoting by pr(^, 6) the set of quadrature distributions, the WDF is obtained from the measured rotated quadratures using the inverse Radon transform

•11

W(q,p) = -(2jl'r'V

/

djcd^;

n

•

{qcos 0 -{-psinS

n

^ '

-xY'

(^l)

where V denotes Cauchy's principal value. Any possible losses due to the mismatch between the LO and the quantum light field, due to non-unit detector efficiency, and so on, or any possible amplification of the signal, determine the measurement not of the WDF but of an ^--parameterized version of it. For example for detectors with a quantum efficiency rj one measures

5, § 5]

Measurement procedures of phase space distribution functions

463

^-1^(^^-1/2^^ ^-1/2^ _^ I - r]yr]) (Leonhardt and Paul [1993]). Quantum tomography measurements have been performed for classical and nonclassical states of the radiation field by Breitenbach and Schiller [1997]. For material particles the mixing mechanism between q and p is provided by the evolution of the system through free space or combinations of lenses and free spaces. Under free evolution, the WDF of the initial state suffers a shear transform in PS, the WDF being reconstructed from a set of marginal distributions (measured with atom detectors) recorded for different evolution times tcj. The rotation angle of the quadratures is related to the evolution time by 6 = t3,n~\tcih/mxl), where XQ is a scaling length, and the measured marginal distributions are in this case pr(^, /,/) = pr(^/ cos 6, mxl tan 0/h). Free evolution gives access only to rotation angles between 0 and ^Jt, additional lenses being necessary to access the whole range from -^JT to ^JT (Pfau and Kurtsiefer [1997]). Several modifications of the 'classical' balanced homodyne detection scheme have been proposed in order to measure the discrete WDF characterizing quantum states of finite-dimensional systems like atoms or spins (Leonhardt [1995]), or for the tomography of a beam of identically prepared charged particles, entering into an electric field which causes harmonic oscillations in transverse direction (Tegmark [1996]). Optical homodyne tomography was also used to measure the number-phase uncertainty relations (Beck, Smithey, Cooper and Raymer [1993]) or the ultrafast (sub-ps) time-resolved photon statistics of arbitrary single-mode weak fields from phase-averaged quadrature amplitude distributions (Munroe, Boggavarapu, Anderson and Raymer [1995]). Optical tomographic measurements were used long ago for image reconstruction from projections (Hermann [1980]) of for the reconstruction of the WDF for light beams (Lohmann and Soffer [1994]). Raymer, Beck and McAlister [1994] designed a tomographic method for the determination of the amplitude and phase of either quasimonochromatic scalar electromagnetic field or quantum wavefiinction of matter waves in a plane normal to propagation direction. The set-up is a classical optical tomography one, formed from two cylindrical lenses, oriented along different directions and at different distances from the input plane. By recording the intensity distribution in the output plane for different focal lengths and distances from the input plane of the two cylindrical lenses, it is possible to reconstruct the field (and the WDF) for coherent electromagnetic fields or pure quantum states, or to reconstruct the two-point correlation function or density matrix for partially coherent electromagnetic fields or mixed quantum states, respectively. Since time and frequency are also non-commuting variables for nonstationary signals, a time-frequency joint probability density cannot be

464

Phase space correspondence betw^een classical optics and quantum mechanics

[5,§ 5

measured directly, but can only be obtained from marginal distributions along rotated directions in the {t.o)) plane (Man'ko and Vilela Mendes [1999]). The setup of Beck, Raymer, Walmsley and Wong [1993], consists of a succession of dispersive elements and time lenses (Kolner [1994], Godil, Auld and Bloom [1994]), which mix (D and /. Quantum tomography can be simplified considerably if the density operator (and hence the WDF) is reconstructed from its normally ordered moments up to the nth. order. In this case the measurement of quadrature components for only n + 1 discrete angles are needed (Wunsche [1996b]); we would like to mention that the signal (and the WDF) can be recovered from the beam's moments also in classical optics (league [1980]). In the simultaneous method, q and p are measured at the same time with limited accuracy. The scheme consists of two entangled homodyne apparata (an eight-port homodyne detector), which measure simultaneously the q and p quadratures, respectively, of two copies of a light beam, obtained using a beam splitter. For this, the LO's of the two homodyne detectors must have a phase difference of \ji. The two homodyne detectors can measure the q and/? variables simultaneously only when the respective operators q = q, + A and p = ps + B commute, i.e. if [q,p] = 0. Since for the signal beam [qs,Ps] =" i^, extra quantum fluctuations represented by the non-commuting operators A, B must be introduced through the vacuum port of the beam splitter. These extra fluctuations, necessary in order to satisfy the uncertainty principle, double the uncertainty product for q,p and produce a fiazzy picture of the measured quadratures. The measured probability distribution is P^i^^P) '•

W{q',p') exp

R-R{p'

-pf/T

dq'dp',

(5.2)

where R and T are the reflected, and transmitted probabilities of the beam splitter, respectively. The set-up measures thus the Q function for a balanced beam splitter, for which R = T, or the Husimi function for an unbalanced one. T controls the squeezing of the resolution of the signal WDF. As for optical tomography, a more detailed description reveals that an ^--parameterized distribution is measured when detection losses appear. For non-unit detection efficiency rj, s = ~(2 - r])/r] and the minimum uncertainty product becomes (1 - s)h/2 (Leonhardt, Bohmer and Paul [1995]). The eight-port homodyne detection scheme works even in the case when the input mode contains only one photon (Jacobs and Knight [1996]). The Q function of multi-particle states, such as Bose-Einstein condensates, can be obtained from the measured atom count probabilities at the output of

5, § 5]

Measurement procedures of phase space distribution functions

465

an atomic interferometer based on Raman transitions between two hyperfine states, if both the phase and transmission parameters of the interferometer are varied (Bolda, Tan and Walls [1998]). The Q function for a light signal can also be determined from the probability of there being no photons in the amplified signal field. This method is based on the fact that in PS linear amplification is a convolution of the signal with the idler field; in this case the idler field is coherent (Kim [1997]). Cascaded optical homodyning was also proposed as a method to determine the PS distributions of optical fields in which the output photon-number statistics of an unbalanced homodyne detection scheme is measured by phase-randomized balanced homodyning (Kis, Kiss, Janszky, Adam, Wallentowitz and Vogel [1999]). The complex amplitude of the local oscillator controls in this case the PS point of interest and a sampling function maps the measured quadrature statistics onto a PS distribution. Richter [2000] has shown that a slight modification of the eight-port homodyne detection scheme allows the direct measurement of the WDF. The modified set-up consists of a 50/50 beam splitter that splits the signal, followed by a photon counter at one output beam (output mode 1), while the other output beam (output mode 2) forms the input of an eight-port balanced homodyne detection, which measures the Q function of the state. If P{q,p,n) is the joint probability to measure n photons in output mode 1 and the quadrature components q and p in output mode 2, the WDF of the input field is given by ^(^^P) "^ J2T=o(~~^y^(^^^^P^^) ^^V[HT +P^)] in normalized coordinates for which h= \. A classical equivalent of the quantum simultaneous measurement method does not really exist, since in classical optics, although the fields occupy an area in PS greater than a minimum value determined by the wavelength of light, the WDF can be measured in principle with arbitrary high resolution. However, to mimic the quantum measurement, Wodkiewicz and Herling [1998] proposed a method to determine the Q and Husimi functions in classical wave optics by simply smoothing the optical WDF by masks with Gaussian transmittance. The simultaneous method tells us that in quantum PS the results can always be expressed as a convolution of the WDF of the quantum state with a distribution of the possible states of the measurement device, always distributed over a PS area of at least the order of h; the overlaps of two WDFs is always a positive quantity, as follows from the overlap principle of WDF (in contrast, negative regions of the WDF have been directly measured in classical optics; see below). No structures of the WDF finer than h can be directly observed

466

Phase space correspondence between classical optics and quantum mechanics

[5, § 5

in real measurements, even if these exist (Zurek [2001]). The measurement, or filtering device is needed to resolve the current position and momentum of the investigated system (Wodkiewicz [1984]). The algorithm for WDF recovery is the simplest in the ring method. It is based on a method proposed by Royer [1977], who showed that the WDF at a point (qo,po) is the expectation of the displaced parity operator 11: W((io,Po) = Jt'^{ip\D(qo,po)nD+(qo,po)\ip), where D(q,p) = Qxp(ipq - iqp) is the coherent displacement operator which displaces a state across PS by an amount (q,p). The motional state of a trapped ion, for example, can be coherently displaced by applying an oscillating (resonant) field that couples to the ion's motion. The amplitude and the phase of the applied field with respect to that of the initial motional state of the ion determine the point (qo,po). For a harmonic oscillator the energy eigenstates are also parity eigenstates, so that the WDF can be obtained from the measured probability distribution of energy eigenstates Pn(qo,Po) as W(qo,po) = JT'^ E / ^ O ( ~ 0 " ^ ^ ' ( ^ O , / ? O ) . The measurement of P,j is performed indirectly, coupling the external motion to internal hyperfine levels of the ion via a resonant laser radiation applied for a duration r, and by actually measuring the population of one of the hyperfine levels after r. Measurements have been performed for different motional states of the ion, such as different harmonic oscillator states, thermal, coherent, squeezed and energy eigenstates, and even superposition of them including Schrodinger-cat states; these states are prepared by applying laser pulses and RF fields to a ion in the ground state of the harmonic oscillator (Leibfried, Meekhof, King, Monroe, Itano and Wineland [1996], Leibfried, Pfau and Monroe [1998]). Fields inside high-g microwave cavities benefited from the same method of measuring their WDF These fields can be displaced in PS by driving the cavity with a strong coherent field, and can then be probed by a two-level atom prepared initially in one of its energy levels. The measured quantity is the atomic inversion after resonant atomfield interaction for specific interaction times (Kim, Antesberger, Bodendorf and Walther [1998], Lutterbach and Davidovich [1997]). The same method was applied for determining the WDF of molecular vibrational states from measurements of fluorescence after applying a sequence of electromagnetic pulses to a molecule (Davidovich, Orszag and Zagury [1998]). A simpler experimental scheme would imply the exploitation of selection rules for Raman transitions following the coupling of the motional state with the internal levels by Raman laser pulses (Bardroff, Fontenelle and Stenholm [1999]). An endoscopic tomography of single modes of the radiation fields in the cavity, without taking them out, can be performed by coupling it through a quantum-nondemolition

5, § 5]

Measurement procedures of phase space distribution functions

467

Hamiltonian to a meter field in a highly squeezed state. Information about the signal field can be obtained fi-om balanced homodyne tomography on the meter field only in out-of-phase measurements, during which the initial state is however changed (Fortunato, Tombesi and Schleich [1999]). Banaszek, Radzewicz, Wodkiewicz and Krasinski [1999] proposed a scheme to implement the ring method for light fields. The coherent displacement is created in this case by a high-transmission beam splitter, which mixes the signal beam with a probe beam, whose amplitude and phase (and thus the point in PS) can be controlled by amplitude and phase electro-optic modulators, respectively. Ideally, the WDF should be obtained as W{li) = (2/JT) YlT=o(-^T^n(l3l where Pn(P) is the photon counting statistics of the signal transmitted by the mixing beam splitter for a PS point p. However, the ^-parameterized quasidistribution fiinction with s = -(1 - r]T)/r\T is obtained instead due to non-unit quantum efficiency r] of the detectors. T is the transmission of the mixing beam splitter. If the signal and the probe fields do not match perfectly at the beam splitter, a scaling of the PS point occurs and

T.7^,{-\rPn{li)

= (Jt/2rjT) exp[-2(l - S)|^p] W{^Wnf^~{\

- r]T)/r]n

where ^ = V/{2 - V) is the squared overlap of the two modes expressed in terms of the fringe visibility V. The principle of superposing coherently displaced replica of the signal in order to obtain the WDF is also used in classical optics. The measurement itself, however, does not rely on photon counting, so it is not directly linked with the ring method; what is measured in this case is directly the light intensity after passing through a setup that Fourier transforms the superposition of the two displaced replica of the signal. Measurements of the classical WDF for one-dimensional and two-dimensional, coherent or incoherent light beams have been performed (Bamler and Gliinder [1983], Bartelt, Brenner and Lohmann [1980], Brenner and Lohmann [1982], Conner and Li [1985], Iwai, Gupta and Asakura [1986], Weber [1992]), and even a set-up has been proposed to directly measure the WDF of light pulses (Dragoman and Dragoman [1996]). Since direct measurements of WDF in classical optics have revealed regions of negative values (see for example Brenner and Lohmann [1982]), it is sensible to conclude that their presence arises merely because of the nonlocal character of either quantum wavefunctions or classical fields. This conclusion has received unexpected backing by the measurement of the WDF of a superposition of two coherent Gaussian beams, which has the same form as the WDF of a superposition of two quantum coherent states (a Schrodinger-cat state). In dimensionless coordinates Q,P the WDF of a superposition of displaced Gaussians

468

Phase space correspondence between classical optics and quantum mechanics

[5, § 5

"tP

Fig. 4. Negative image of the experimentally determined modulus of the WDF for a superposition of two coherent and spatially separated Gaussian beams.

Fig. 5. Negative image of the experimentally determined modulus of the WDF for a superposition of two incoherent and spatially separated beams.

\1){Q) ^ exp[-(g - df/2\ classical cases by

+ exp[-(g + df/2]

W{Q, P) = Wo exp(-p2){exp[-(e -df]^

is given in both quantum and

cxp[-(Q ^df] + 2 expC-g^) cos(2Pd)l (5.3) where WQ is a normalization constant. Negative regions arise from the last, interference term. The measured WDF (its modulus) is shown in fig. 4. The interference term in the middle is only present for coherent classical light; for incoherent sources the same superposition (the fields are no longer Gaussian) looks like that fig. 5. The middle term has no longer an oscillatory behavior, but is a simple incoherent superposition of the outer, individual source terms. It should be noted that superpositions of incoherent classical sources have not the same PS representations as statistical mixtures of quantum states. In the latter case no middle interference term should exist at all (Dragoman and Dragoman [2001]). The total or partial disappearance of the interference term due to decoherence (Giulini, Joos, Kiefer, Kupsch, Stamatescu and Zeh [1996]), can be mimicked in classical optics by filtering away the middle term of the WDF. Experimental methods have been heavily employed not only for the measurement of PS distribution functions, but also for distinguishing coherent superposition of states from statistical mixtures. These can be inferred from the form of the WDF in some cases, but an unambiguous detection method

5, § 6]

Propagation of classical fields and quantum states in phase space

469

was shown to exist based on the observation of a time-dependent spectrum of spontaneous emission from the studied system. The appHcation for the case of a diatomic molecule is discussed in more detail in Walmsley and Raymer [1995]. We have insisted here only on the measurement of the WDF and Q function. Due to its highly singular character the P function cannot be generally determined experimentally, but other PS distribution functions have been determined as well. An example is the positive P distribution, measured in a four-port arrangement (Agarwal and Chaturvedi [1994]).

§ 6. Propagation of classical fields and quantum states in phase space The evolution law for the quantum WDF follows directly from the Schrodinger equation satisfied by the wavefunction, or from the von Neumann equation satisfied by the density operator. In either case, for a classical Hamiltonian H{q,p) =p^/2m+V{q) one obtains (H.-W Lee [1995]):

the left-hand side of which is identical to the classical Liouville equation. The 'nonclassical' terms in eq. (6.1) (the right-hand side) contain higher derivatives of the WDF, implying that the PS motion is no longer described by substitution of one PS point for another, as in the canonical transformation, but is analogous to a diffusion process in which a localized function spreads out. It is not a true irreversible diffusion process, which would be described by even derivatives of the WDF, but a reversible quasidiffusion involving only odd derivatives of the WDF (Bohm and Hiley [1981]). When not all derivatives in eq. (6.1) exist, an alternative, integral form can be used: dW

p dW

f^

,

with J(^,7) = i^iJTh^) / ^ dy [V{q + ^ v ) - V{q- ^y)] Qxp{-iyj/h). Equation (6.2) reveals again that the PS motion involves an integral over a set of discrete jumps (nonlocal transformations) in momentum, a concept completely different from the continuity of movement implied by classical mechanics (Wigner [1932]). The evolution law for the WDF distinguishes it again from the other PS distribution functions, which all have an additional term (or terms) even

470

Phase space correspondence between classical optics and quantum mechanics

[5, § 6

for the free particle case, and so a nonclassical evolution. These h dependent additional terms are not even of quantum origin (O'Connell, Wang and Williams [1984]). The classical-quantum correspondence with respect to PS dynamics seems again to be best studied in terms of the WDF. The classical limit of eq. (6.1) cannot be obtained by simply putting ^ = 0, because quantum contributions (and even a h dependence) may be hidden in the expression of the WDF at the initial time /Q, Wo(q,p). For a harmonic potential all ^-dependent terms in eq. (6.1) disappear and the quantum WDF satisfies an equation of motion identical to a classical probability distribution. The quantum effects are hidden in the initial conditions. Note that all quantum corrections (the terms on the right-hand side of eq. 6.1) depend only on even powers of ^. To separate the two different origins of h dependence in WDF, the /i-dependent terms in eq. (6.1) can be replaced by ah, where a is a dimensionless parameter, so that classical evolution for WDF (the causal approximation) implies a = 0, and the classical limit of the WDF is afterwards obtained by additionally taking the ^ ^ 0 limit. The first-order quantum correction to the causal approximation, called also the quasi-causal approximation (Bund, Mizrahi and Tijero [1996]), implies the consideration of the first-term on the right-hand side of eq. (6.1) (second-order in ah). The first-order quantum correction for the PS ThomasFermi and Bose distribution fiinctions for fermions and bosons, respectively, has been calculated in Smerzi [1995]. Both distribution functions reduce in the high temperature limit to the Gibbs-Boltzmann distribution. In the causal approximation each PS point of the initial WDF evolves classically (although reversed in time) as Wc\{q,p; t) = Wo{q(to - t),p(to -1)), following a trajectory according to the classical Hamiltonian equations. Wc\ is also called the semiclassical distribution ftmction. If the classical limit of the WDF exists, one obtains the classical distribution function d(q - Q(t)) d(p - P(t)), with Q(t),P(t) solutions of the Hamilton equations. It is important not to confound the classical limit of the quantum WDF with the WDF in classical optics! Wc\ depends only on the energy associated with the classical trajectory. For a bound state in a potential that vanishes at infinity Wc\{p,q) vanishes for points of positive energy, and so corresponds to a stationary distribution of particles trapped inside the potential. Studying different types of potentials, it was found that the Wc\ curves tend to move away from the regions where the quantum WDF has negative values. In particular, the region where WDF is negative is located in the domain where Wc\ vanishes (Bund and Tijero [2000]). The above results concerning the propagation law of the WDF can be reformulated in the following way: when a symplectic matrix is associated with the canonical transformation that sends the initial set of points ^ = (q,p) in the

5, § 6]

Propagation of classical fields and quantum states in phase space

471

PS to another final set of points, the corresponding quantum WDF transforms as W^{^) = W\M~^^) (as the classical probability distribution!) only when M is a linear transform (Littlejohn [1986]). Since, according to its definition, detM = 1, the area of localization of the WDF is invariant under linear transformations, although it may change its shape. This area of localization cannot be smaller than Planck's constant in quantum mechanics (Kim and Noz [1991]), and cannot be smaller than the wavelength for the WDF in wave optics. The symplectic group Sp(2«,R) technique was used to study the evolution of pure Gaussian quantum states of systems with n degrees of freedom under quadratic Hamiltonians (Simon, Sudarshan and Mukunda [1988]). This evolution was shown to be compactly described by a matrix generalization of the Mobius transformation. For nonlinear symplectic maps, such as, for example, aberrations in (ray) optics and particle dynamics (Lichtenberg [1969]), the WDF does not follow the simple relation W(^) = W(M'^^). In particular, the WDF is not exactly conserved along straight trajectories even in free space, unless the paraxial limit is considered (Wolf, Alonso and Forbes [1999]). The effect of third-order aberrations to the paraxial approximation, described by the Hamiltonians H = p'^ for the spherical aberration, H = p^x for coma, H = p^x^, H = px^ and H = x^ for astigmatism, distortion, and pocus, respectively, is described in Rivera, Atakishiyev, Chumakov and Wolf [1997]. For all these cases numerical simulations showed that the classical and the quantum Gaussian WDF evolve differently, and quantum oscillations (including negative WDF regions) appear at the concavities of an initially Gaussian WDF (the Gaussian shape is not preserved in nonlinear evolution). They are caused by self-interference in PS between different parts of the WDF, and have a smaller area than that of the vacuum state. Only the 'top' of the Gaussian Wigner function moves in agreement with classical dynamics. In both classical and quantum mechanics the uncertainty relations are not conserved at nonlinear transformations; these relations are thus a measure of nonlinearity, and not of nonclassicality, and cannot describe an element of PS volume. Such a PS volume measure can be provided by the moments 7^(0 = (k/2'''^) J W''(p,x; t)dpdx/2jt of the WDF, which are constant under classical canonical transformations, but are only preserved in the quantum case under linear transformations. Self-interference is also observed at nonlinear propagation through a Kerr medium described by H = j(p^ + w^x^) + (x/co^)(p^ + co^x^)^. In this case, standing waves along a circle in PS form at certain times x^ = LJT/M, where L,M are mutually prime integers. The interference pattern in PS at these moments, known as the M-component Schrodinger cat, is associated with pronounced peaks of the WDF moments (Rivera, Atakishiyev, Chumakov and Wolf [1997]).

472

Phase space correspondence between classical optics and quantum mechanics

[5, § 6

Note that in the quantum PS spanned by the annihilation and creation operator eigenvalues, the matrix relation satisfied by the WDF has another form. For example, an optical device which transforms linearly an input multimode field with annihilation and creation operators ai^a^ into a field with annihilation and creation operators bj,b^ can be described by b^ = ^AMaftaf] -\-LajjCi^), where the matrices M,L satisfy the relations ML - LM = 0, MM^ - LL^ = 1. For this linear transformation, an example of which is a parametric amplifier, the WDF of the output field is given by -1

Wo(z) = Wi

M-—L'

If* z •

-—M-L''

-1

z

(6.3)

where W{z) = Jt'^ Tr[p / d'a exp{-[a(z* - 5+) - a^z - a)]}]. The output WDF is thus dependent on the phase of z, even if the input WDF is phase-independent (Agarwal [1987]). Despite all these analogies, in most circumstances the dynamical time evolution of the WDF does not reduce to classical dynamics even if ^ -^ 0. Especially for highly coherent density matrices a direct ^-expansion treatment of quantum corrections is generally not possible, unless a selective re-summation of the terms in the series for the quantum PS propagation is made, in which case a revised or renormalized classical-like dynamics is obtained (Heller [1976]). The relation between classical and quantum dynamical theories for the WDF is especially interesting when chaotic systems are considered. Then, the degree of non-integrability of the system plays an important role in the quantum-classical comparison. The PS treatment of chaotic systems cannot be briefly summarized here. Therefore, we refer the readers to the review papers of Eckhardt [1988], Bohigas, Tomsovic and UUmo [1993] and Wilkie and Brumer [1997a,b] for more details. We only point out here that the PS dynamics in the quantum case is determined by the PS dynamics of the classical system. In general, the quantum eigenstates can be separated in regular and irregular groups. In integrable systems the waveftinctions peak on the invariant tori quantized by the discrete values of the action; instead, in the regular (quasi-integrable) case chaotic trajectories alternate densely with regular trajectories, exploring each a tiny fraction of the energy space, behavior which is generally valid also for the regular portions of PS for systems with soft chaos, where there is a mixture of integrable and chaotic motions. However, no simple relation exists between classical and quantum PS motion for strongly chaotic systems. Breakdown of the quantum-classical correspondence in chaotic systems is predicted in some papers (Ford and Mantica [1992]), while in others it is argued that even

5, § 6]

Propagation of classical fields and quantum states in phase space

473

in chaotic systems semiclassical methods are successful for quite long times (Tomsovic and Heller [1991], Provost and Brumer [1995]). In general, due to the uncertainty principle, the quantum WDF cannot resolve the details of the classical trajectories for long evolution times (Berry and Balazs [1979]), so that, depending on the stable or unstable character of the classical motion, the quantum PS dynamics becomes distinct from the classical PS dynamics in longer or shorter times. The rapid divergence between quantum and classical dynamics in integrable systems for initial conditions near an unstable point is caused by the coherent interference of fragments of the wave packet occurring on a short time scale, of the order of the system dynamical time. For a periodically kicked 1D particle, for example, the differences between classical and quantum motions become discernible on a time scale of the order of ^"^^^, the discrete quasienergy spectrum, and hence a structure in the quantum motion, being identifiable on longer time scales of about h~^ (Jensen [1992]). In some cases the behavior of quantum and classical chaotic systems is different. Quantum systems behave totally different than classical systems, for which the PS is divided by separatrices and stochastic webs into regions in which different types of motion are exhibited (Torres-Vega, Moller and Zuniga-Segundo [1998]). Moreover, unlike classical stationary distributions, quantum eigenstates can become localized due to the slowness in the rate with which the nonstationary PS distribution sweeps out the available PS (Heller [1987]). Ballentine and McRae [1998] showed that for chaotic and regular motions of the Henon-Heiles model the centroid of the quantum state approximately follows the classical trajectory for very narrow probability distributions, the difference between the equations of motion for quantum and classical PS moments scaling as h^. The differences between the quantum and classical dynamics grow exponentially for chaotic motion and as t^ for regular motions, the corresponding difference between the variances of the PS distributions growing also exponentially but with a larger exponent, and as t^, respectively. Quantum interference patterns in PS can be induced by the quantum oscillations of internal degrees of freedom in multicomponent systems, the chaotic dynamics destroying the coherence of the quantum oscillations (Tanaka [1998]). Habib, Shizume and Zurek [1998] showed that for initial states chosen as Gaussian packets that randomly sample the chaotic part of the PS, a smooth quantum-to-classical transition in nonlinear dynamical systems occurs via decoherence. Decoherence (Giulini, Joos, Kiefer, Kupsch, Stamatescu and Zeh [1996]) destroys the quantum interference with a degree determined by the interplay between the dynamics of the system and the nature and strength of the coupling with environment, and washes out the fine structure in the

474

Phase space correspondence between classical optics and quantum mechanics

[5, § 7

classical distribution. So, the quantum and classical predictions, in particular the expectation values of the corresponding variables, become identical.

§ 7. Interactions of classical fields and quantum states as phase space overlap The most important application of PS overlap is to calculate the transition probability between two quantum states, characterized by density matrices f)\ and p2> According to the overlap principle the transition probability can be written in terms of an overlap between the corresponding WDFs as Pn = Mp\p2)

= 2nh

'/7 /

dqdpfViiq,p)W2{q,p).

(7.1)

^-oc J-oc ^-oc

J-

A similar formula holds also in classical statistics if W\ and W2 are the PS densities for the classical states before and after the transition. In classical optics eq. (7.1) is used for defining the coupling coefficient between coherent light sources and optical waveguides, a slightly modified relation holding for the case of partially coherent light sources (see Dragoman [1997]). PS overlaps are also encountered in PS matching problems for particles passing through several set-ups (Lichtenberg [1969]). The quantum transition probability can be measured by the balanced homodyne technique discussed in the previous section, as (Leonhardt [1997])

Alternatively, the direct measurement of overlaps of two WDFs can be achieved for two single-mode light beams via photon counting, the PS overlap of the signal and probe beam being realized by a beam splitter (Banaszek and Wodkiewicz [1996]). The calculation of transition probabilities using the overlap of the WDFs has been applied, for example, in the study of Franck-Condon transitions (Schleich, Walther and Wheeler [1988], Dowling, Schleich and Wheeler [1991]), for defining transition probabilities of momentum jump after a finite-time evolution of the quantum system (Takabayashi [1954]), or, generally, for studying the transition probabilities of quantum-mechanical oscillators for either large or small perturbations (Bartlett and Moyal [1949]). The transition probabilities for coherent and squeezed states from their WDFs are given in Han, Kim and Noz

5, § 7]

Interactions of classical fields and quantum states as phase space overlap

475

[1989]. Kim and Wigner [1989] showed that the correct form-factor behavior in the harmonic-oscillator model for hadrons can also be traced to the overlap of two Lorentz-deformed PS distribution functions. Recently, a PS formulation of Fermi's golden rule have been given in Dragoman [2000d], which showed that even for time-dependent interactions described by a Hamiltonian H\nt(q,p), the transition probability between an initial (i) and a final (f) state, can be written in terms of the corresponding WDFs as ihP,f = {Wr\L,ntm,

(7.3)

where

The behavior of transition matrix elements in mixed quantum systems with a few degrees of freedom, with one or several regular islands in the ergodic sea, was studied by Boose and Main [1996]. They showed that the mean associated to the distribution of diagonal transition matrix elements is a weighted sum of classical means over the ergodic part of PS and over the stable periodic orbit. The variance characterizing distributions of non-diagonal transition matrix elements can be well approximated with the weighted sum of Fourier transforms of averaged classical autocorrelation functions along an ergodic trajectory and along the stable periodic orbit. In the formulation (7.1) of transition probabilities, the values of the two WDF over the common PS domain contribute. In some cases it is possible to identify a smaller PS domain that has a dominant contribution to the integral in eq. (7.1). This is true especially when number states are the initial or final states of the transition, since the corresponding WDF, in the large-« limit, has a prominent peak around the classical PS trajectory, a lower-amplitude oscillatory behavior inside it and an exponential decrease outside. Its WDF contribution in eq. (7.1) is therefore expected to come mainly from the neighborhood of the PS trajectory. This is indeed the case when photon distributions for arbitrary states are calculated, or when probability amplitudes of transitions (due to a sudden change in conditions) between number states are considered. In this case the PS representations of the energy states as given in fig. la and the other involved state offer an intuitive explanation of the behavior of different quantities. Due to normalization reasons, the area of intersection between the two PS representations multiplied by \/(2jTh) gives a good approximation of eq. (7.1). If the PS overlap between the quantum states consists of more than one

476

Phase space correspondence between classical optics and quantum mechanics

[5, § 7

4>e

Fig. 6. PS overlap between the «th energy eigenstate of a harmonic oscillator and (a) a coherent state, (b) a squeezed coherent state with real a, and (c) a squeezed energy eigenstate.

region, interference effects appear, as extensively discussed in Schleich [2001]. For two or more areas of overlap, the transition probability is obtained by adding the complex probability amplitudes associated to each area. The total transition probability between two states | V^) and \x) is then given by yth area of overlap \1/2

{X\H>) = Y.

IjTh

(7.4)

X exp -n ( /th area enclosed by central lines) With this area-of-overlap principle we can easily view the Poissonian energy spread of a coherent state for example, P,;.coh "" |(«|V^coh}P = (a^V«!) exp(-a^) as arising from the overlap between the PS representations of the «th eigenstate of the harmonic oscillator and of the wavefunction of the coherent state (fig. 6a). The main contribution to {n\\pcoh) arises from regions around the turning points qn = A/2(« + \)/K, and so («| V^coh) follows in its n dependence the wavefunction of the coherent state in the q variable. When two coherent states of identical mean photon number but different phases are in a quantum superposition, squeezing, as well as sub-poissonian and oscillatory photon statistics can appear (Schleich, Pernigo and Kien [1991]). This PS interpretation intuitively supports the mathematical result that the energy distribution for squeezed states gets narrower than the Poissonian (becomes sub-Poissonian) when we increase s and that for strong squeezing s —^ oo the distribution starts to oscillate with period two, since Pin +1 "^ 0, Pm ^ 0 (Schleich, Walls and Wheeler [1988], Schleich [2001]). For the overlap between a number state and a displaced strongly-squeezed state (fig. 6b) with Im a = 0, there are two areas of overlap with reduced areas of intersection A,y2jTh, disposed symmetrically above and below the coordinate axis, and eq. (7.4) reduces

5, § 8]

Classical and quantum interference in phase space

417

to Pn = \{A^/2jthy''^ Qxp{i(f„) + (An/lJThy'^ Qxp{-i%)\^ = A{An/2nh) cos^ cpn, where q)n= Sn- \n, with Sn the PS area enclosed by the vertical line at Via and the Kramers center lines of the energy band n. In classical mechanics the probability amplitudes would be given by the reduced area of overlap, no phase factors being included. The two interfering quantum probability amplitudes in PS can be viewed as the analogues of the probability amplitudes in configuration space in Young's double-slit experiment. The phase difference is now given by the PS area caught between the interfering states, whereas in Young's experiment it is determined by the difference in optical path length from the two slits to the point of detection. In the same limit of strong squeezing. An = exp[-2(« + \ - a^)/s][2ns{n + \ - a^)]~'^^. The PS overlap principle also accounts for the behavior of the energy distribution of a rotated squeezed state, which has a rapid oscillation and a slow modulation on top of it, in contrast to the nonrotated case where there is a single but large period. Examples of cases when more than two areas of overlap occur are the calculation of the photon number distribution of the squeezed number and squeezed thermal states (Kim, de Oliveira and Knight [1989]). In these cases highly structured number distributions are obtained due to interference effects from the four PS areas (see fig. 6c). In particular, if squeezed photon number states overlap photon number states with a different parity the interference is destructive and the photon number distribution vanishes. For overlaps between squeezed photon number states and photon number states with the same parity the interference is constructive and the photon number distribution has nonzero values. The area of overlap principle explains also the fact that the phase probability distribution of highly squeezed states undergoes a transition from a single- to a double-peaked shape when the product of squeeze and displacement parameters is decreased (Schleich, Horowicz and Varro [1989]). Finally, it should be noted that when the PS overlap approach is extended to the hyperbolic space, the PS overlap areas should be replaced by weighted areas, since for the hyperbolic space the PS is not represented by the same embedded sheet as the configuration space (Chaturvedi, Milburn and Zhang [1998]). The hyperbolic space can be employed to characterize active interferometers, while passive interferometers can be described in a spherical space, for which the PS has the same spherical geometry. § 8. Classical and quantum interference in phase space Classical and quantum interferences are caused by the linear superposition principle of fields and quantum wavefunctions, respectively, which follow from

478

Phase space correspondence between classical optics and quantum mechanics

[5, § 8

the linearity of the corresponding wave equations. Young's type experiments, or one-photon interference experiments, can be successfully explained by either classical or quantum theory. Differences between the two theories can only be observed in experiments that involve the interference of intensities. In intensity interferometry experiments, or two-photon interferometry, as that of HanburyBrown and Twiss, the interference/correlations between the intensities of two electric fields detected by separate photomultipliers are measured. Classical theory predicts in this case an interference of intensities, whereas quantum theory treats the interference still at the level of probability amplitudes. The predictions of the two theories are thus different. The one-photon and twophoton interference experiments have been recently reviewed by Mandel [1999]. Interference experiments have been observed also for charged and neutral particles, spins, Bose-Einstein condensates, fluxons propagating in Josephson rings, atoms, experiments even being designed to demonstrate the nonlocal nature of the quantum interference. The theoretical and experimental work in the quantum interference domain is immense; so as not to do injustice by inevitably omitting valuable papers, we specifically refer only to those works directly related to the PS approach. The correlation function between the electromagnetic field E(q, t) = i Y^(hw,,/2eoy^^[akUk(q) Qxp(-mt) -atuKq)

exp(-ia;AO]

k

= E\q,t) + E-{qj) at the space-time points x = (q,t) and x' = {q'j') is defined as G^^\x,x')=' Tx[f)E~{x)E^{x')]. This first-order correlation function is sufficient to account for classical or quantum one-photon interference experiments. Ideal detectors working on an absorption mechanism yield a signal I{q, t) ^ Tr[ pE~(q, t)E\q, t)] (Walls and Milburn [1994]). Higher-order correlation functions are necessary to describe experiments involving intensity correlations. The «th-order correlation ftinction is defined as G^^'^Xi . . . Xn,Xn+\

. . . X2n) = Tv

f)E{X\)

. . . E~(Xn)E^(Xn

+ \) . . . ^^(X2«)l ,

(8.1) while the n-fo\d delayed coincidence rate is proportional to G^"\x[... x^,x„... x\). The «th order correlation fianction satisfies the following properties: G^"\xi ... x„x„ ... xi)^0, G^"\xi

. . . X„,X,j . . . Xi)G^"\x„+i

> \G^"\xi

(8.2a) . . . X2n,X2n • • • X,; + 1)

... X„,Xn+i ... X2„)|'.

(8.2b)

5, § 8]

Classical and quantum interference in phase space

479

The odd-ordered correlation functions contain information about the phase fluctuations of the electromagnetic field, no such information being contained in the even-ordered correlation functions. The latter, including the second-order correlation function, are a measure of the fluctuations in the photon number. For fields propagating in nondispersive media no difference is made between longitudinal and temporal coherence. However, for dispersive propagation, as is the case for electrons or neutrons in vacuum, or for light propagating in a medium, one should distinguish between spatial and temporal coherence. A discussion on these two types of coherence in dispersive media can be found in Hamilton, Klein and Opat [1983]. 8.1. Classical and quantum one-photon interference In one-photon Young's type interference experiments the intensity observed on the screen is given by / = G^^\q\,q\) + G^'^(^2,^2) + 2Re[G^^^(^i,^2)], where q\, q2 are the positions of the two pinholes. The first two terms describe the intensities from the individual pinholes, the interference fringes originating from the last, interference term, which can be rewritten as 2|G^'^(^i,q2)\ cosi^{q\,qi)Note that G^^\q,q')=

f dpQxp -ipiq-q')

^/A(i±iOy

(33)

The normalized first-order coherence function, defined as g"'(^w^2)=.^.,..„ ,;^;;r ..,2{G^'Kq^,q^)&'\ql,q2)V

(«-4)

is associated with the visibility of the interference fringes, which is given by F = (/max - /min)/(^max + /min)- Morc precisely, for incoherent fields for which g'^^Kqx.qi) = 0 no interference fringes appear, whereas full coherence, corresponding to |g^^^(^i,^2)| ^ 1 is associated also with maximum fringe visibility. When the fields on each pinhole have equal intensities, V = |g^'^|; the first-order coherence function was also identified with the degree of path indistinguishability (Mandel [1991]). More generally, the fields for which the first- (higher-) order correlation fiinction factorizes are first- (higher-) order coherent. Coherent states satisfy this criterion. In particular, the wavefiinction of a coherent field incident on the pinholes 1 and 2 factorizes as |ai,a2) = \<^\)\(^2) and can therefore represent two independent light beams. Interference between independent light beams can

480

Phase space correspondence between classical optics and quantum mechanics

[5, § 8

occur if the phase relation between them varies slowly, and was observed for single-mode independent lasers by Pfleegor and Mandel [1967] even for light intensities so low that one photon is absorbed before the next is emitted by one or the other source. To have interference between independent light beams, it is necessary that the correlation function between the states of the radiation field does not vanish; this condition is not satisfied for example by the Fock states. In terms of the WDF, the 'eventuality' of interference is described by the interference term, which appears even when the two interfering beams do not yet overlap. Wolf and Rivera [1997] showed that interference terms in the WDF, called 'smile function', exist also for superpositions of coherent classical optical fields, termed by an extension of language optical Schrodinger-cat states. The marginal projection of the smile yields the transmissivity of the physical hologram obtained by superposing the two beams. Unlike classical optical experiments, where the interference pattern appears immediately, the quantum interference pattern reveals itself in time. The fringe visibility is however the same as with high-intensity light sources (see for example Franson and Potocki [1988]). Referring to a two-slit experiment, in which the coherent fields immediately after the slits have a Gaussian form, the interference term in the WDF is present even immediately behind the slits, when the coherent beams do not overlap in real space [see fig. 7a and also eq. (5.3) for the WDF of a superposition of two coherent states]. Interference in the qor p domain arises when the outer terms in the WDF, representing the interfering fields, have a common projection interval along the respective axis. So, immediately after the slits the beams interfere only along the /?-axis, interference in real space occurring after propagation through a sufficiently long distance in free space, such that the outer terms in the WDF begin to have a common projection interval over the (^r-axis also. At propagation through free space the WDF suffers a shear transform, as can be seen in fig. 7b, which does not affect the interference pattern along the /7-axis. Experimental demonstration of/7-space interference, in the absence of ^-space interference, as well as the conclusion that interference should be treated rather in PS than in the configuration space, has been already provided by Rauch [1993] and Jacobson, Werner and Rauch [1994]. The distinction between interaction and interference is best described in PS: there is interaction (transition) if the WDFs of two states overlap, and there is interference if the corresponding WDFs have common projections along q or /?, but are still well separated. Of course, interference patterns can only be observed when no knowledge is available about the slit through which the quantum particles or classical waves go. In the quantum case, the interference fringe visibility V and which-way

5, §8]

00

03 r/)

03

OX) t t ;

X)

(IJ

X3

c/2

^

C! Q


_

on

03

c/5

-tJ

O

^

TD

(U 03

!>. ^ fl

tC

^ ^ "S ^

JZ (X OU in

ao

T3

c> ^ ^3 o O) C3

Classical and quantum interference in phase space

O

cd

> o

tin

a

Q

^

03 HJ

x: t^-H

o

o X)

481

482

Phase space correspondence between classical optics and quantum mechanics

[5, § 8

knowledge K can be related through F^ + AT^ ^ 1 (Schwindt, Kwiat and Englert [1999]). Experimental results for pure, mixed and partially-mixed input states show good agreement with the theoretical prediction V^ + K^ = 2Tv{p^) - \. In PS the measurement process, including the which-path measurements, can be viewed as PS filtering, the WDF of the incoming light being filtered not only by the measurement devices, but by any part of the set-up in which they are propagating (Dragoman [2001a]). In particular, when one slit is closed, or when a which-path measurement is performed, the PS area occupied by the system is reduced to only one region (corresponding to the remaining beam), which has no other domain to interfere with. Interference is thus lost by PS filtering. The central role of the interference term in the WDF in quantum interference was also evidenced by Wallis, Rohrl, Naraschewski and Schenzle [1997]. They studied the macroscopic interference of two independent Bose-Einstein condensates and their PS dynamics, showing that the distance of interference fringes varies linearly in time with a velocity inversely proportional to the distance between the two condensates and that collisions reduce the fringe visibility. The destruction by a which-path measurement of the interference fringes in a double-slit interference, with d the slit separation, can also be interpreted as a disturbance of the particle's momentum by an amount of at least jzh/2d, in accordance with the uncertainty principle. Wiseman, Harrison, Collett, Tan, Walls and Killip [1997] showed that this momentum transfer caused by the interaction with the which-path measuring apparatus need not be local; i.e. need not act at either of the slits through which the particle passes. In a local interaction, as in Einstein's recoiling slit or Feynman's light microscope, the momentum transfer is modeled by random classical momentum kicks which have the same effect on the momentum distribution of a particle passing through a single slit, and so the single-slit diffraction pattern is also smeared, in the same way as the double-slit interference pattern. In a nonlocal momentum transfer, as in the schemes proposed by Scully, Englert and Walther [1991] or Storey, Collett and Walls [1993], the single-slit diffraction pattern does not broaden, and, unlike classical momentum kicks, the quantum momentum transfer depends in general on the initial wavefunction of the particle. The PS representation of both cases can be readily given in terms of the WDF of the (classical or quantum) momentum-transferring device, Wf(q,p), which changes the initial WDF W\{q,p) into a final one W^{q,p) = J dp' W^q.p-p') Wf(q,p'). (A 'momentum filtering' rather than a 'PS filtering' is considered). Smearing of the diffraction pattern and the destruction of the interference fringes are caused by different momentum transfers, a local P\oc(p) and a nonlocal one Pnonioc(p); only Pnoniodp) cannot be less than jzh/ld. In a double-slit experiment, with very narrow slits separated by

5, § 8]

Classical and quantum interference in phase space

483

d, the interference term in the initial WDF is ruined by a nonlocal momentum transfer with a pseudoprobability distribution PnonXodp) = ^ti^^P) (at q midway between the slits, the particle is never found!). The visibility of the interference fringes is then changed to V = /d/?Pnonioc(p) exp(i/?(^/i^) {V = |V| is the usual visibility). A which-path measurement device can change the visibility from 1 (the value in the absence of any which path measurement) to any less than unity value, up to zero for a perfect which-path measurement. In a classical momentum-transfer experiment W((q,p) is the probability distribution for a particle at position q to receive a momentum transfer p, so that the local momentum transfer when the particle is localized at the two slits is given by P\oc{p) = ^[Wr(^d,p)+ Wt{-\d,p)l It is not related to the WDF interference term and so plays no role in the destruction of interference, but determines the diffraction pattern of a particle that is in a classical mixture of being at the two slits. This interpretation of the destruction of the interference pattern by the nonlocal momentum transfer is indirectly supported by the Aharonov-Bohm effect. In this quantum nonlocal effect, the interference fringes move with the applied field within the constant diffraction envelope. A local, classical momentum transfer would shift the entire pattern, similar to the way a local which-path measurement smears the interference pattern. A direct interpretation of the Aharonov-Bohm effect in terms of the WDF is given in Dragoman [2001b]. If the two interfering beams of charged particles acquire different phases due to a non-vanishing vector potential, the interference term in the WDF shifts also, while the individual WDF terms remain the same (see fig. 7c). According to the interpretation in Dragoman [2001a], the interference pattern (given by the common projection intervals along the q ox p domains of the WDFs of the interfering beams) remains the same, while the interference fringes, defined by the interference term in the WDF, move with the vector potential. It is worth mentioning that, at least in this case, the PS interpretation has offered a new prediction: there is an Aharonov-Bohm term (a shift of the interference fringes inside the same interference pattern) in the /?-space also, not only in the ^-space. This prediction follows readily from fig. 7c. Quantum and classical interference is mostly studied using the WDF. Other PS quasidistribution functions can be used to this end. Chountasis and Vourdas [1998] advocated the advantages of the Weyl function for the study of interference. The Weyl fiinction, defined as W(Q,P) = TY[pb(Q,P)l where D{q,p) is the displacement operator, is related to the WDF through a double Fourier transform: W(Q,P) = J J dqdpW(q,p) Qxp[-i(Pq - pQ)]. This can be seen easier writing the WDF as W{q,p) = JT~^ Tr[pU(q,p)],

484

Phase space correspondence between classical optics and quantum mechanics

[5, § 8

where U(q,p) = D(q,p) Uob^{q,p) is the displaced parity operator, with Uo = Qxip(iJta^a). The Weyl function is a generahzed correlation function, the latter being a special case of the Weyl function for P = 0 or X = 0. Whereas for a superposition of states labeled by / the WDF auto-terms are located around the points ({qi), (p/)), they are located around the origin for the Weyl function. For a Schrodinger-cat state, for example, the two Gaussian auto-terms in the WDF, and the oscillating cross-term in the middle are mirrored into an oscillating term of the Weyl function around the origin representing auto-terms and two Gaussians representing cross-terms. Note that the correspondent in classical optics of the Weyl function is the ambiguity function. A more sophisticated engineering of the PS position of the auto and interference terms can be achieved using fractional Fourier operators. The fractional Fourier operator V(d) = Qxp(ida^a) rotates the momentum and position operators in phase space with an angle 6: qo = V(d)qV^(d) = qcose +psin 6,po = V(6)pV^(e) = -q sin 6^pcos 6, and can be used to define generalized WDFs W(q,p; 6) = TT[pU(q,p; 6)], where U(q,p; 0) = D(2q,2p) V(6). The WDF and Weyl functions are particular cases for 6 = Jt (for which V(jt) = X],^o(~^)'1^)("l ^^ ^^^ parity operator) and 9 = 0, respectively. For a Schrodinger-cat state, as 6 decreases from jr to 0 the autoterms in the generalized WDF change their form from Gaussians to oscillatory and move in the PS plane ending at the origin, whereas the cross-terms, which are originally oscillatory, become Gaussians. The angle 6 can even be complex, in which case its positive imaginary part describes attenuation ('filtering'), while its negative imaginary part represents amplification (Chountasis, Vourdas and Bendjaballah [1999]). Multi-dimensional interference patterns in the quantum probability or classical field intensity distribution, called intermode traces (or quantum carpets), can appear due to pair interference between individual eigenmodes of the system (Kaplan, Stifter, van Leeuwen, Lamb Jr and Schleich [1998]). The resulting interference pattern, strongly pronounced if the intermode traces are multidegenerate, can be observed in many areas of wave physics, as for example for confined quantum particles, atoms scattered at a periodic laser-induced grating, electromagnetic waveguides or light diffraction. The similitude of the quantum and classical interference pattern is based on the mathematical analogy between the Schrodinger equation for the wavefunction of a quantum particle and Maxwell's equations of classical electrodynamics under the paraxial, fixed polarization (scalar theory) approximation. This quantum-classical similarity can be exemplified by the fact that an electron in a quantum box with infinite walls is analogous to an electromagnetic wave in a waveguide with metallic walls, or by the fact that electrons moving in periodic potentials (in the conduction

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band of solids, for example) behave as almost free particles. In these two cases the similarity between the classical and the quantum case can be evidenced also using the WDF (see Marzoli, Friesch and Schleich [1998] and H.-W. Lee [1995], respectively). 8.2. Classical and quantum two-photon interference Intensity correlation experiments of the Hanbury-Brown and Twiss type measure the joint photocount probability of detecting the arrival of a photon at time t and another at time ^ + r, given by G^^\r) = {E-{t)E-{t+T)t{t+T)t{t)).

(8.5)

The two-time photon-number correlations can be measured on a sub-picosecond scale using dual-pulse, phase-averaged, balanced homodyne (McAlister and Raymer [1997]). Even the cross-correlations and mutual coherence of optical and matter fields can in principle be measured (Prataviera, Goldstein and Meystre [1999]). Second-order coherence can also be characterized by the normalized secondorder correlation fiinction g^^^(r) = G^^\T)/\G^^\0)\^. For fields which posses second-order coherence, as for laser fields, G^^^ factorizes and g^^^ = 1, independent of the delay. Coherent fields exhibit Poissonian statistics. The correlation function factorizes always for r
486

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[5, § 8

however, it is possible to have fields that can exhibit super-Poissonian statistics over some time interval, but for which g^^^(r) > g^^^(O). A review of the single and double beam experiments, which measure the degree of second-order coherence, and thus can discriminate classical against quantum theories can be found in Reid and Walls [1986]. Classical models predict for two-photon interference a maximum of 50% visibility, as does quantum theory for experiments that involve only the wave nature of radiation. However, visibilities greater than 50% are predicted by quantum theory for certain nonlocal entangled states, called Einstein-Podolsky-Rosen (EPR) states. For these states, with no classical analog, the interference visibility can ideally reach 100%. The experimental confirmation of nonclassical values for twophoton interference visibility has been provided, for example, by Shih, Sergienko and Rubin [1993]. Two-photon interference for entangled photons, obtained from type II down-conversion, have been observed in experiments in which the photons arrive at the beam splitter at much different times. This confirms the fact that for entangled photons the interference pattern cannot be viewed as produced by the interference between two individual photons (Pittman, Strekalov, Migdall, Rubin, Sergienko and Shih [1996]). Two-photon interference effects were even observed when the entangled photon pairs were generated from two laser pulses well separated in time. This effect, not expected classically, shows a visibility that depends on the delay time, reaching a maximum value of 50%. However, visibilities up to 100% can be obtained for multiple pulses delayed in time with respect to each other (Kim, Chekhova, Kulik and Shih [1999]). The applications of the WDF for the study of two-mode quantum correlations, in particular the cross-correlations between the two modes that violate classical inequalities, are discussed in Walls and Milburn [1994]. In particular. Bell's theorem (Bell [1991]) provides a test of the predictions of the whole class of local hidden variable theories against quantum mechanics. Bell's inequalities and the EPR paradox demonstrate the nonlocality of quantum mechanics as expressed by the correlations between different subsystems of an entangled quantum system, for which the eigenstate does not factorize in the eigenstates of the subsystems. Bell correlations in PS can be tested with quantum optical means (Leonhardt and Vaccaro [1995]). Banaszek and Wodkiewicz [1999] showed that the correlation fijnctions that violate Bell's inequalities for correlated two-mode quantum states of light, are equal to the joint two-mode Q function and the WDF. The connection between the nonlocality of entangled states and the Q function is surprising since the positiveness of the Q function is usually considered as a loss of quantum properties due to simultaneous measurement of canonically conjugated observables (see § 5). The non-positivity

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of the WDF, on the other hand, cannot be unequivocally related to violations of local realism, since in some cases Bell's inequalities are violated by states with a positive WDF (Banaszek and Wodkiewicz [1998]). So, the positivity or non-positivity of the WDF cannot act as criterion for the locality or nonlocality of quantum correlations. Moreover, using the WDF, the EPR correlated state of spin-^ systems can be treated analogously to a local hidden variable model if the probability distribution function is allowed to become negative (Agarwal, Home and Schleich [1992]). Negative conditional probabilities between nonorthogonal polarization components of entangled photon pairs are responsible for the violation of Bell's inequalities. These negative probabilities can be observed in finite-resolution measurements of the nonclassical polarization statistics of entangled photon pairs, or in finite-resolution measurements of the polarization of a single photon (Hofmann [2001]). Entangled photon states (in time, frequency, direction of propagation, or polarization) can be created in nonlinear optical processes such as parametric downconversion. The second-order correlation functions of two-photon wave packets entangled in polarization and space-time can be studied using the WDF formalism. For example, in quantum-beating experiments (Ben-Aryeh, Shih and Rubin [1999]) the coincidence-counting rate is proportional to the integral of the relative two-photon wave probability distribution (in relative coordinates) over the retarded time difference, or to an integral over the frequencies difference. Both momentum-frequency and position-time coordinates can be accounted for in the WDF picture, the WDF for the two-particle entangled states being a multiplication of a WDF that depends on the relative coordinates of the two-photon with a WDF that depends on the central coordinates of the twophoton. Entangled two-photon pairs, or biphotons, generated by spontaneous parametric down-conversion display some properties that are similar to those of photons generated by incoherent sources. For example, the two-particle wavefunction and the spatial pump-field distribution in the biphoton case are analogous to the second-order coherence function and the source intensity distribution in the incoherent case. Moreover, the van Cittert-Zernike theorem, valid for incoherent optical sources, and the partial-coherence theory of image formation have counterparts for biphotons. If we compare, however, the photon count rate in the incoherent case with the biphoton count rate in the entangled-photon case, a duality rather than analogy is observed, similar to the duality between single and two-photon interference of biphotons (Saleh, Abouraddy, Sergienko and Teich [2000]). This duality originates in the fact that the separability of the secondorder coherence function is associated with high-visibility ordinary interference

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[5, § 9

fringes, whereas separability in the biphoton wavefunction is associated with the absence of entanglement and so low-visibility biphoton interference fringes. The biphoton interference fringe visibility V\2 is related to the visibility Fine of single-photon fringes created by an equivalent ordinary incoherent source of the same source distribution by F12 = (1 - Vl^yi.^ + ^inc)- ^^ ^^^ ^^^^ ^^ related to the visibility V\ of the interference pattern due to signal (or idler) photons (marginal single-photon patterns) by V^ + V^j "^ ^ • § 9. Universality of the phase space treatment In the previous sections we have advocated the advantages of a PS approach to the problem of finding the correspondence between classical optics and quantum mechanics. It was shown that classical wave optics can be treated in PS via quasidistribution functions with similar properties to the quantum quasidistribution fianctions. Ray optics becomes then the 'classical' limit of wave optics in the same sense as classical mechanics is the limit of quantum mechanics. In this last section we would like to emphasize the universality of the PS treatment. It is not only employed for the study of classical mechanics, classical optics or quantum mechanical systems, but also for the study of statistical mechanics and particle optics, for example. The significance of the classical-quantum correspondence broadens considerably by realizing the universal character of the PS approach. For example, the PS concepts can be employed for the study of the propagation of electronic rays. In this case a WDF can be defined as in quantum mechanics, with the beam emittance playing the role of Planck's constant and the longitudinal coordinate of propagation playing the role of time. The electronic beams can be transported, analogous to optical rays, with combinations of quadrupoles, drift sections, bending magnets, etc. The first two devices are the electronic counterparts of optical lenses and free space sections, respectively. More complicated optical set-ups, which can realize for example squeezing along a tilted direction in PS, can be implemented in electronic optics. In particular, a set-up for measuring the Wigner angle of rotation for electron beams has been proposed by Ciocci, Dattoli, Mari and Torre [1992]. A Wigner PS representation for a reduced density operator can also be introduced in thermo-field-dynamics for the study of thermal excitation (Berman [1990]). The thermal excitation can then be viewed as a temperature-dependent radial spreading in PS. Uncertainty relations of the form AE^^AiiJ ^ ksdji can also be encountered in statistical mechanics (Gilmore [1985]), as in any physical theory that can be

5, § 10]

Conclusions

489

formulated in terms of canonically conjugated variables. In this case the variance is defined as Ax = ((x -x)^) '^^, where x is the random variable with mean value X = (x), E^ is an extensive thermodynamic variable characterizing the system, and ij] = dS/dE^^ is the intensive thermodynamic variable conjugated to E^^ in the entropy representation/For example, AUA(\/T) ^ k^, AVA(P/T) > ks, ANAii^i/T) ^ks.A similar relation A£^^A/^, ^ kQTd'il, with /^, = dU/dE'', holds in the energy representation, where AS AT ^ ksT, AVAP ^ k^T, ANAj^i ^ ksT. The factor ^ is missing in these uncertainty relations compared to those in quantum mechanics and signal processing, because the conjugation relations are obtained here taking by derivatives of a probability distribution function, not of probability amplitudes as in quantum mechanics. The uncertainty relations of statistical mechanics express the duality between probability and statistics, and are equivalent to the stability relations of equilibrium thermodynamics. As for quantum mechanics, where the classical limit corresponds to ^ ^ 0, the classical limit of statistical mechanics - thermodynamics - is obtained for ks -^ 0. In both cases the classical limits are characterized by the lack of uncertainty relations. The coherent and squeezed states, although defined and employed mostly in quantum optics, can be generated also in other domains. For example, the eigenstates of a generalized thermal annihilation operator, constructed using thermofield dynamics, are the fermionic coherent states, fermionic squeezed states, and their thermalized counterparts (Chaturvedi, Sandhya, Srinivasan and Simon [1990]). Coherent and squeezed states of phonons have also been investigated (Hu and Nori [1996]). These coherent phonon states can be excited phase coherently in Brillouin and Raman scattering experiments, or piezoelectric oscillators can generate coherent acoustic waves up to 10'^ Hz. Many more examples could be given. Since PS methods involve the same mathematical language in all these different domains: wave optics, quantum optics, statistical mechanics, thermofield dynamics, etc., it is to be expected that our understanding of the origin of the formal relations between them, and our ability to constructively speculate these similarities will improve.

§ 10. Conclusions Throughout this chapter we have revealed similarities and differences between quantum mechanics and classical optics. We have seen that the classical limit of the WDF, i.e. the classical probability distribution function, is not the same with the WDF in wave optics. The latter is an exact replica, from the

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[5, § 10

point of view of definition, properties, and even mathematical expressions for certain fields, of the quantum WDF, if the Planck's constant is replaced by the wavelength of light. There is, however, a major difference between the quantum and optical WDF from the point of view of measurement. The quantum WDF cannot be directly measured with arbitrary accuracy, whereas the optical WDF can. Therefore, it is possible to produce classical waves with the same form as quantum wavefunctions (this can be done for any superposition of coherent or Fock states), generate their WDF by classical optical means, and study its propagation through classical optical set-ups that mimic quantum systems. As regards the behavior of the WDF, the results would be identical with those obtained in quantum mechanics, with the additional advantages of easier production and accurate measurement. This is interesting conceptually, because the starting point classical fields and quantum states - have very different properties. In particular, the quantum interference principle holds for any superposition of quantum states, whereas in the classical case only overlapping fields interfere. Even the nature and significance of interference is different: for a quantum superposition of states the quantum particle is with a certain probability in only one of these states; one can measure, by repeating the experiment, only these probabilities. On the other hand, in a classical superposition the interference patterns appear as if the light is in both (or all, in general) slits at the same time, not in one or in the other. Field amplitudes are real objects in the classical world, but only probabilities in quantum mechanics. Why is it then, that the WDF washes away the difference in behavior and retains only the difference in the measurement procedure? The answer lies in its bilinear character. Due to it, interference terms in the WDF appear even if the coherent classical beams do not really overlap; the interference in PS has the same character in both the quantum and the classical world. Quantum interference in PS can be mimicked by classical interference in PS, although the corresponding classical fields behave differently than their quantum counterparts. This similitude holds for the (quantum and classical) wave theories, for which the PS can be considered as being partitioned in adjacent, interacting, finite-area cells, occupied by elementary Gaussian beams (with positive WDFs), such that whenever the occupied PS area exceeds the minimum value allowed by the uncertainty principle the PS interference between neighboring cells lead to negative values of the WDF. On the contrary, in the classical limit of quantum mechanics, or in geometrical optics the PS is continuous and a classical mechanical ensemble or light beam can be decomposed in a number of perfectly localized, non-interacting particles or rays. To get the illusion of classical mechanics the quantum uncertainties should be small on the observation scale (Royer [1991]).

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A last, but not least important remark is that, due to the probabilistic character of the quantum wavefunction, which reflects the wave-particle duality, a quantum system can be meaningful compared to a classical one only when the particle or the wave character is involved, but not both. There are no classical states similar to the EPR entangled states, although the WDF can be employed even in this case in quantum mechanics. The quantum WDF cannot always be mimicked by a classical optical WDF, despite the evident similarities between them. The quantum WDF, and the quantum PS quasidistribution functions in general, are a combination of their classical optical counterparts, photon counting, the mysterious duality, ... If we would know all the ingredients and their proportions, quantum theory would not look so beautiful.

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E. Wolf, Progress in Optics 43 © 2002 Elsevier Science B. V All rights reserved

Chapter 6

"Slow'' and "fasf' light by

Robert W. Boyd The Institute of Optics, University- of Rochester, Rochester, NY 14627, USA

and

Daniel J. Gauthier Department of Physics, Duke University, Durham, NC 27708, USA

497

Contents

Page § 1. Elementary concepts

499

§ 2.

504

Optical pulse propagation in a resonant system

§ 3. Nonlinear optics for slow light

510

§ 4.

Experimental studies of slow light

514

§5.

Experimental studies of fast light

523

§ 6.

Discussion and conclusions

528

Acknowledgements

528

References

529

498

§ 1. Elementary concepts Recent research has estabHshed that it is possible to exercise extraordinary control of the velocity of propagation of light pulses through a material system. Both extremely slow propagation (much slower than the velocity of light in vacuum) and fast propagation (exceeding the velocity of light in vacuum) have been observed. This article summarizes this recent research, placing special emphasis on the description of the underlying physical processes leading to the modification of the velocity of light. To understand these new results, it is crucial to recall the distinction between the phase velocity and the group velocity of a light field. These concepts will be defined more precisely below; for the present we note that the group velocity gives the velocity with which a pulse of light propagates through a material system. One thus speaks of "fast" or "slow" light depending on the value of the group velocity Ug in comparison to the velocity of light c in vacuum. Slow light refers to the situation u^ c or when Ug is negative. A negative group velocity corresponds to the case when the peak of the pulse transmitted through an optical material emerges before the peak of the incident light field enters the medium (Garrett and McCumber [1970]), which is indeed fast! Some of these ideas can be understood in terms of the time sequences shown in fig. 1. It is also worth noting that the transit time T through an optical medium can in general be represented as T=-,

(1)

where L is the physical length of the medium. Thus, when Vg is negative, the transit time through the medium will also be negative. The validity of the description given here and leading to fig. 1 assumes that the pulse does not undergo significant distortion in propagating through the material system. We shall comment below on the validity of this assumption. 499

500

"Slow" and "fast" light

[6, § 1 V negative

A. 1

\ M

lA

1

A

\ lA. 1

-M

1

A.

iyv^l

M

A

A_

-A^

A

iA_

A

7V

A,

Fig. 1. Schematic representation of a pulse propagating through a medium for various values of the group velocity. In each case we depict the spatial variation of the pulse intensity for increasing values of time.

We next review the basic concepts of phase and group velocity. We begin by considering a monochromatic plane wave of angular frequency co propagating through a medium of refractive index n. This wave can be described by ^ ( z , 0 = ^e'^

'''^ + c.c.,

(2)

where k = nco/c. We define the phase velocity Up to be the velocity at which points of constant phase move through the medium. Since the phase of this wave is clearly given by 0 = kz- cot,

(3)

points of constant phase move a distance Az in a time A^, which are related by kAz = (oAt

(4)

Thus Up = Az/At or w k

c n

(5)

Let us next consider the propagation of a pulse through a material system. A pulse is necessarily composed of a spread of optical frequencies, as illustrated symbolically in fig. 2. At the peak of the pulse, the various Fourier components will tend to add up in phase. If this pulse is to propagate without distortion, these

Elementaiy concepts

6, §1]

501

(a) wave packet

/

V

optical medium z

(b) t

MV \A V Mi mm

t

Fig. 2. Schematic representation of an optical pulse in terms of its various spectral components. Note that these contributions add in phase at the peak of the pulse.

components must add in phase for all values of the propagation distance z. To express this thought mathematically, we first write the phase of the wave as nwz (p = (Ot, (6) c and require that there be no change in 0 to first order in co. That is, d^/do; = 0 or dn coz nz + - - ^ = 0, (7) aco c c which can be written as z = i;g^ where the group velocity is given by c

doj

(8) ^ n + 0) dn/dco The last equality in this equation results from the use of the relation k = nco/c. Alternatively, we can express this result in terms of a group refraction index rig defined by c (9) with dn (10) ^ dw We see that the group index differs from the phase index by a term that depends on the dispersion dn/do) of the refractive index. nQ = n-\- CO -—.

502

"Slow" and "fast" light

[6, § 1

Slow and fast light effects invariably make use of the rapid variation of refractive index that occurs in the vicinity of a material resonance. Slow light can be achieved by making dn/d(0 large and positive (large normal dispersion), and fast light occurs when it is large and negative (large anomalous dispersion) ^. 1.1. Pulse distortion What is perhaps most significant about recent research in slow and fast light is not the size of the effect (that is, how fast or how slow a pulse can be made to propagate) but rather the realization that pulses can propagate through a highly dispersive medium with negligible pulse distortion. Let us examine why it is that pulse distortion effects can be rendered so small. In theoretical treatment of pulse propagation (Boyd [1992]), it is often convenient to expand the propagation constant k((D) in a power series about the central frequency o^o of the optical pulse as k((J)) = ko-\-ki(W-COo)-\- ^k2i(J)-COof -\- • • •,

(11)

where ko = k(coo) is the mean wavevector magnitude of the optical pulse, dk do;

1

Ha c

(12)

is the inverse of the group velocity, and ^d(lA^^ld«g d(o c da;

^j3^

is a measure of the dispersion in the group velocity. Since the transit time through a material medium of length L is given by T = L/Ug = Lk\, the spread in transit times is given approximately by AT^Lk2A(o,

(14)

where Aco is a measure of the frequency bandwidth of the pulse. The significance of each of the terms of the power series can be understood, for example, by considering solutions to the wave equation for a transform-

' We use the terms normal dispersion and anomalous dispersion to describe the change in the refractive index as a function of frequency (the traditional usage). In more recent texts on opticalfiber communication systems, the terms normal or anomalous dispersion refer to the change in the group index as a function of frequency. Normal (anomalous) group-velocity dispersion is the case when d«g/dco > 0 (d«g/d(x> < 0).

503

Elementary concepts

6, §1]

.

\

1

•

1

.

1

1

>

•

1

.

1

(b)

1

>

.

.

1

1

z=5Lb

1

1

: --/32 = 0

:

"-D'

• .

1

-10

.

.

.

j

'^Nd/n / .7 \ • \

I

"

/ ;/ v^-^l f J . h"-yQk . . J r < L / i 1 . V. V/i KT'^J^-^iteza

-5

0 T/To

5

10

Fig. 3. Effects of group-velocity dispersion and higher-order dispersion on a Gaussian shaped pulse, from Agrawal [1995]. (a) Dispersion-induced broadening of a Gaussian pulse propagating through glass at z = 2 I D and z = 4 I D . The dashed curve shows the incident pulse envelope, (b) Influence of higher-order dispersion. Pulse shapes at z = 5L'^ for an initially Gaussian pulse at z = 0 are shown. The solid curve is for the case when ki ^ ^ {jh in the notation of Agrawal) in the presence of higher-order dispersion; the dashed curve is the case when the characteristic length associated with group-velocity dispersion LQ and higher-order dispersion L'^ are equal. The dotted curve shows the incident pulse envelope.

limited Gaussian-shaped pulse (of characteristic pulse width To) incident upon a dispersive medium (Agrawal [1995]). When the propagation distance through the medium is much shorter than the dispersion length rr2

^D - 7 7 7 ,

(15)

the pulse remains essentially undistorted and travels at the group velocity. For longer propagation distances (or shorter To and larger Aw), the pulse broadens but retains its Gaussian shape, as shown in fig. 3a. In addition, the pulse acquires a linear frequency chirp; that is, the instantaneous frequency of the light varies linearly across the pulse about the central carrier frequency of the pulse. Red (blue) components travel faster than blue (red) components in the normal (anomalous) group-velocity dispersion regime where fe > 0 (ki < 0). For situations where ^2 ~ 0 or for large Aa;, higher-order terms in the power series expansion (11) must be considered. It is found that an incident Gaussian pulse becomes distorted significantly, as shown in fig. 3b, when the pulse propagates farther than a characteristic distance (16) associated with higher-order dispersion, where k^ ^ d^k/dco^.

504

"Slow^" and "fast" light

[6, § 2

To observe pulse propagation through a dispersive medium without significant pulse distortion, it is necessary that the spread of transit times AT given by eq. (14) be much smaller than the characteristic pulse duration TQ. AS discussed below, experiments on slow and fast light are typically conducted under conditions such that the group index rig is an extremum, so that drig/dco = 0 and hence k2 vanishes. It is for this reason that slow- and fast-light experiments are accompanied by negligible distortion so long as the propagation distance through the dispersive medium is much less than Vp (implying a narrow spectral bandwidth for the pulse). Limitations to the accuracy of the group-velocity description for propagation through an absorptive medium have been pointed out by Xiao and Oughstun [1997, 1999]. § 2. Optical pulse propagation in a resonant system Propagation of light pulses through resonant atomic systems has attracted great interest since the early 1900's because of the possibility of fast-light behavior and its implications for Einstein's Special Theory of Relativity. Sommerfeld, independently (Sommerfeld [1907, 1914]) and together with his student Brillouin (Brillouin [1914]), developed a complete theory of pulse propagation through a collection of Lorentz oscillators. Their work was published during World War I and is not widely available. For this reason, Brillouin compiled and augmented their earlier work in a beautiful treatise entitled Waue propagation and group velocity (Brillouin [I960]). They were most interested in the case in which the carrier frequency of the pulse coincides with the atomic resonance so that the pulse experiences anomalous dispersion and consequently Vg > c. They considered the case of an optical pulse that has an initial rectangular shape so that its amplitude vanishes before the beginning of the pulse - the so-called front of the pulse. They found that the speed of the front of the pulse is always equal to the speed of light in vacuum even in the anomalous-dispersion regime where Vg > c ox Ug < 0, and that the pulse experiences substantial distortion. In hindsight, the fact that the pulse experiences distortion is due to the wide bandwidth of the pulse resulting from the infinitely sharp turn on. To understand the unusual slow and fast light properties of pulse propagation through resonant systems, we review the solutions to the wave equation, paying particular attention to the manner in which the refractive index is modified in the immediate vicinity of each transition frequency. We express the refractive index as n = ^/e= yj\+4jtx,

(17)

Optical pulse propagation in a resonant system

6, § 2 ]

505

where e is the dielectric constant, and the susceptibihty is given (in Gaussian units) by Ne^/lmcoo X = (a)o-a))-iy'

(18)

for a near resonant light field. The transition frequency is denoted by COQ, ^Y is the width (FWHM) of the atomic resonance, and e (m) denote the charge (mass) of the electron. For an atomic number density N that is not too large, the refractive index n = n' + in" can be expressed as « ~ 1 + 2KX, whose real and imaginary parts are given by n'

= \+

2(a;o - w)y ^ ^ ^ ^^^^.^^ jzNe^ ImcooY (COQ - (JOY + y-

JzNe^ = bn^max) ImcDoy {O)Q- coY + y^

{WQ -

2(a>o - a))y

(19)

(o^o - (JoY + y 2 '

(jof + y2'

(20)

where 8«^"^^^^ is the maximum deviation of the phase index from unity. These functional dependences are shown in fig. 4, along with the group index + 5n(max)

i+l^^n'-^^^Vsr)

(ft)*!'"^^"*//) Fig. 4. The real {n') and imaginary {n") parts of the phase index and the real part of the group index («g) associated with an isolated atomic resonance.

506

"Slow " and "fast" light

[6, § 2

n^ = n' + oj dn'/dco. Note that the scale of the variation of the group index from unity is given by the quantities g„(max) ^ C ^

g rmin) ^ _ ( ^

(21)

Typical values for an atomic vapor are oj = lir [5 x 10^"^) s ', bn^"^^^^ = 0.1, and 7 = 2;r (l x 10^) s ^ leading to the value bnf'''^ = 5 X 10^

(22)

This is a remarkable result! Even though phase indices of atomic vapors are rarely larger than 1.5 (and the phase index is 1.1 for the numerical example just given) the group index can be of the order of 5 x 10"^. Group indices this large are not routinely measured in atomic vapors because of the large absorption that occurs at frequencies where «g is appreciable. As one can deduce from eq. (20), the linear absorption coefficient a = ln"wlc is of the order of lO'^cm' under the same conditions used to obtain result (22). 2.1. Early observations of 'slow' and fast' light propagation While there was considerable theoretical interest in pulse propagation through resonant systems over a 100 years ago, experimental investigations in the optical spectral region increased substantially with the advent of the laser. In 1966 Basov, Ambartsumyan, Zuev, Kryukov and Letokhov [1966] and Basov and Letokhov [1966] investigated the propagation of a pulse propagating through a laser amplifier (a collection of inverted atoms) for the case in which the intensity of the pulse was high enough to induce a nonlinear optical response. They found that nonlinear optical saturation of the amplifier gave rise to fast light, a surprising result since the linear dispersion is normal at the center of an amplifying resonance so that u^ < c is expected for low intensity pulses. They attributed the pulse advancement to a nonlinear pulse reshaping effect where the front edge of the pulse depletes the atomic inversion density so that the trailing edge propagates with much lower amplification. In addition, they found that the effects of dispersion give a negligible contribution to the pulse propagation velocity in comparison to the nonlinear optical saturation effects. Such pulse advancement due to amplifier saturation is now commonly referred to as superluminous propagation. Throughout this review, we are mainly concerned with propagation of pulses that are sufficiently weak so that the linear optical

6, § 2]

Optical pulse propagation in a resonant system

507

properties of the medium need only be considered, although these properties may be modified in a nonlinear fashion by the application of an intense auxiliary field. Soon after the experiment of Basov, Ambartsumyan, Zuev, Kryukov and Letokhov [1966], Icsevgi and Lamb [1969] performed a theoretical investigation of the propagation of intense laser pulses through a laser amplifier. They attempted to resolve the apparent paradox of pulses propagating "faster than the velocity of light" predicted in the work of Basov and Letokhov [1966], and it appears that Icsevgi and Lamb were unaware of the earlier work by Brillouin [1914] discussing the distinction between group velocity and front velocity and its implications for the Special Theory of Relativity. Icsevgi and Lamb distinguish between two types of pulses in their work. A pulse is said to have compact support if its amplitude is nonzero only over some finite range of times, and is said to have infinite support if the pulse is nonzero for all times. By way of example, a hyperbolic secant pulse has infinite support. Icsevgi and Lamb find in their numerical solutions of the pulse propagation equation that pulses with infinite support can propagate with group velocities exceeding that of light in vacuum c. However, there is no violation of causality because the input pulse exists for all values of time. For a pulse with compact support, they find that the region of the pulse where it first becomes nonzero cannot propagate faster than c (the front velocity in the terms of Brillouin [1914]). Their results are consistent with the work of Brillouin [1914] and extend the analysis to a nonlinear optical medium. These issues have been clarified fiarther in the work of Sherman and Oughstun [1981], who present a simple algorithm for the description of short pulse propagation through dispersive systems in the presence of loss. More recently, Diener [1996] shows that in cases in which a pulse propagates superluminally, that part of the pulse which propagates faster than c can be predicted my means of analytic continuation of that part of the pulse that lies within the "light cone", that is, the extreme leading wing of the pulse. In subsequent work, Diener [1997] introduced an energy transport velocity Cf = T^c,

(23)

which is less than or equal to c for any value of n. Subsequent experiments conducted in the late 1960s by Carruthers and Bieber [1969] and Frova, Duguay, Garrett and McCall [1969], and in early the 1970s by Faxvog, Chow, Bieber and Carruthers [1970] on weak pulses propagating through amplifying media observed slow light as expected for a linear amplifier.

508

"Slow " and "fast" light

[6, § 2

However, the effect was small because of the smallness of the available gain. Using a high-gain 3.51-(im xenon amplifier, Casperson and Yariv [1971] were able to achieve group velocities as low as c/2.5. In this same period, Garrett and McCumber [1970] made an important contribution to the field when they investigated theoretically the propagation of a weak Gaussian pulse through either an amplifier or absorber. They were the first to point out that the pulse remains substantially Gaussian and unchanged in width for many exponential absorption or gain lengths and that the location of the maximum pulse amplitude propagates at u^, even when t;g > c or 6;g < 0. For this distortion-free propagation, the spectral bandwidth of the pulse has to be narrow enough so that higher-order dispersive effects are not important, as discussed in § 1.1. Note that a Gaussian pulse is of infinite support and hence the predictions of Garrett and McCumber [ 1970] are consistent with the earlier work of Icsevgi and Lamb [1969]. Following up on the predictions of Garrett and McCumber [1970], Chu and Wong [1982a] investigated experimentally both slow and fast light for picosecond laser pulses propagating through a GaP:N crystal as the laser frequency was tuned through the absorption resonance arising from the bound ^-exciton line. Typical experimental traces are shown in fig. 5 and are summarized in fig. 6. Both positive and negative group delays are observed and the pulse shape remains essentially unchanged. The data points are found to be in good agreement with the theoretical predictions, which were obtained from a model that is a slight generalization of the model presented above. Note that the fast light observed in this experiment was obtained in the presence of a large absorptive background. This report is of significance in that it is one of the first studies to establish experimentally that the group velocity is a robust concept in the optical part of the spectrum even under conditions of significant pulse advance or delay. We note that the pulse shapes observed by Chu and Wong [1982a] and shown in fig. 5 are effected by the measurement process, as pointed out by Katz and Alfano [1982]. The pulse shapes were measured using an autocorrelation method, which is insensitive to pulse asymmetries or oscillations, but is sensitive to pulse compression. Katz and Alfano find that the pulses shown in fig. 5 experience significant compression, which may be due to true compression or due to pulse asymmetries. In response, Chu and Wong [1982b] agree that pulse compression is present in their data and can be explained theoretically by the inclusion of higher order dispersion. However, they also point out that the group velocity remains a meaningful concept even in the presence of pulse compression. Later numerical simulations by Segard and Macke [1985] of

Optical pulse propagation in a resonant system

6, §2]

509

036 meV

cd

80 120 delay (ps) Fig. 5. Experimental results of Chu and Wong [1982a] showing the transmitted pulse shapes as their laser frequency is tuned through an exciton resonance line in GaP:N.

-3.8x10*

LASER FREQ (meV)

Fig. 6. Summary of the experimental results of Chu and Wong [1982a] demonstrating that the group delay can be either positive or negative (solid line). For comparison the absorption spectrum of their sample is also shown (dashed line).

510

"Slow" and Jast" light

[6, § 3

the experiments of Chu and Wong [1982a] show that the pulses experience significant ringing, not just compression as suggested by Chu and Wong [1982b]. In the same paper, Segard and Macke [1985] also describe a fast-light experiment via a millimeter wave absorption resonance in OCS. They observe significant pulse advancement and negative group velocities with essentially no pulse distortion using a detector that directly measured the pulse shape, confirming the theoretical predictions of Garrett and McCumber [1970]. As in the previous experiments, the pulses experienced large absorption.

§ 3. Nonlinear optics for slow light The conclusion of the previous sections is that in linear optics the group refractive index can be as large as ^g

(max)

6 . = 1 + "^^ ^ 87

jj.'KT 2

where

6«^-^^^ = ^ , mwlY

(24) ^

but is accompanied by absorption of the order of 4;f§„(max)

^ -

A

'

^ ^

where A is the vacuum wavelength of the radiation. Recent demonstrations of slow light have been enabled by nonlinear optical techniques which can be used to decrease the effective linewidth 7 of the atomic transition and also to decrease the level of absorption experienced by the pulse. A typical procedure for producing slow light is to make use of electromagnetically induced transparency (EIT), a technique introduced by Harris, Field and Imamoglu [1990] to render a material system transparent to resonant laser radiation, while retaining the large and desirable optical properties associated with the resonant response of a material system. See also review articles by Harris, Yin, Jain, Xia and Merriam [1997], Harris [1997], and Lukin and Imamoglu [2001]. The possibility of modifying the linear dispersive properties of an atomic medium using an intense auxiliary electromagnetic field was first noted by Tewari and Agarwal [1986] and by Harris, Field and Imamoglu [1990]. In addition, Scully [1991] pointed out that the refractive index can be enhanced substantially in the absence of absorption using similar methods, with possible applications in magnetometry [1992]. In a later paper Harris, Field and Kasapi [1992] performed detailed calculations to estimate the size of the slow-light effect. They estimate

6, § 3]

Nonlinear optics fo?- slow light

511

b

Fig. 7. Energy-level structure utilized in a typical EIT, slow-light experiment.

that Ug = c/250 could be obtained for a 10-cm-long Pb vapor cell at an atom density of 7 x 10^^ atoms/cm^ and probed on the 283-nm resonance transition. This small group velocity is accompanied by essential zero absorption and zero group-velocity dispersion. More recently, Bennink, Boyd, Stroud and Wong [2001] have predicted that slow- and fast-light effects can be obtained in the response of a strongly driven two-level atom. Following an approach similar to that used by Harris, Field, and Kasapi, we review the relation between EIT and slow light using a density matrix calculation. We consider the situation shown in fig. 7, and for simplicity assume that in the absence of the applied laser fields all of the population resides in level a. We want to solve the density matrix equations to first order in the amplitude E of the probe wave and to all orders in amplitude ^s of the saturating wave. In this order of approximation, the only matrix elements that couple to Paa (which can be taken to be constant) are p/,,, and pea, which satisfy the equations pha

= - (^(Oha + Yha)pha

" T

{Vi^apaa + Vhcpca)

pea

= - i}(Oca + Yea)pea

" T

{Vchpha)

,

•

(26)

(27)

In the rotating-wave and electric-dipole approximations, V^a ^ -lUtaEe'^^'^^ and Vhc = -jJihc-E^e'^^'^''^ We now solve these equations in the harmonic steady state, that is, we find solutions of the form Pha - Ohae

pea " OcaC

,

(Z6)

where Oha and Oca are time-independent quantities. We readily find that

'"

(i<5-yU[i(<5-4)-y„,] + | a / 2 P '

^ ^

where 6 = 0)- co/,„, A = (o^- (O/K, and Q^ = li-ihcEs/h is the Rabi frequency associated with the strong drive field. From this equation, we determine the

512

"Slow" and "fast" light

[6, § 3

3000

250oJ"s/2^ = ^2MHz ^

2000 i

I

1500 H

1 -

/

V

(b)

0 -

[0~^

b

5. -1 C -2 -

nj2n --=

12 MHz

rl —1

-30

-20

-10

0

1—

10

20

30

5/271 (MHz)

Fig. 8. Frequency dependence of (a) the absorption coefficient and (b) the group index in the absence (dashed curves) and in the presence (solid curves) of the intense coupling field that induces the EIT effect. The parameters are estimated from the conditions of the experiments of Hau, Harris, Dutton and Behroozi [1999] and are given by 2jTN\i^f,^\~/yi,^,h = 0.013, y/,^/2;r - 5MHz, y,.,/ = 0.038MHz, and OJ/V/^^^ = 1.02x

10^.

susceptibility for the probe wave by means of the equations P = NfiahOha = X^^^^^ which, when solved for x^^\ yields /(•) = - W | M'ha\

[i{d~A)-y,.a] ( i ( 3 - y , J [ i ( 5 - Z ^ ) - n , J + |r2,/2|2

(30)

Let us recall why this result leads to the prediction of EIT. For simplicity we assume that the strong saturating wave is tuned to the co/,,. resonance so that A = 0. One finds that as the intensity of the saturating field (which is proportional to |Osp) is increased, the absorption line splits into two components separated by the Rabi frequency |Os|- Figure 8a shows a{8,A = 0) for the experimental conditions of Hau, Harris, Dutton and Behroozi [1999] for two values of Q^ to show the emergence of the EIT spectral "hole" at line center (i.e., 3 = 0). In detail, one finds that (for d = A = 0) (31)

A

YcaYha

+\^s/2\^'

6, § 3]

Nonlinear optics for slow light

513

Note that x^^^ is purely imaginary, that x^^^ is a monotonically decreasing function of |Osp, and for \Q^\^ > YcaYha that x^'^ is proportional to Yea, which under many experimental conditions has very small value. Thus, the presence of the strong saturating field leads to transparency at the frequency of the probe field, although only over a narrow range of frequencies. Let us also estimate the value of the group refractive index under EIT conditions. To good approximation, we ignore the first contribution in the expression n^ = n' + 0) dnWdco (here n' is the real part of the phase index n) and approximate the phase index by its low-density expression « ~ 1 + 2JTX^^^. We take the expression for ^^^^ in the limit of large-field amplitude |^s| and vanishing strongfield detuning (A = 0) so that n)_-iN\^ha\^

^

h

id-Yea

|Os/2|2'

.^^.

^^^^

By combining these equations we find that %jza)N\iiijci\^

/^lap

(33)

Equation (33) was used by Hau, Harris, Dutton and Behroozi [1999] in the analysis of their experimental results. They find that it gives predictions that are in reasonably good agreement with their experimental data. Note, however, from their fig. 4, that the scaling of group velocity with drive-field intensity is not accurately described by eq. (33) for a range of temperatures slightly above the Bose-Einstein transition temperature. Figure 8b shows n^{d,A = 0) for two values of Q^. For Q^ = 0, the group index is extremely large and negative, but this is accompanied by extremely large absorption (seefig.8a). The curve is dramatically different for Q^/2K = 12 MHz, taking on a large positive value of the order of 10^ with little dispersion and absorption. The group velocity aX 6 = 0 corresponds to approximately 300 m/s. For lower Q^, Ug as low as 17 m/s were observed by Hau, Harris, Dutton, and Behroozi, although with slightly increased absorption. 3.1. Kinematics of slow light While we noted above that a smooth pulse can propagate undistorted through a medium with an EIT hole, the fact that the pulse travels with such slow speed implies that the light pulse undergoes an enormous spatial compression, as pointed out by Harris, Field and Kasapi [1992] and illustrated schematically

514

"Slow^" and "fast" light

[6, § 4 nc^l

Fig. 9. Schematic illustration of pulse compression that occurs when a pulse enters a medium with a low group velocity.

infig.9. In particular, the pulse undergoes a spatial compression by the ratio of the group velocities inside and outside of the optical medium. Since the group velocity in vacuum is equal to c, this ratio is just the group index n^ of the material medium, which as we have seen can be as large as c^lO^. Since the energy density of a light wave is given (in SI units) by u=\e^n^\E\\

(34)

one sees that the energy density increases by this same factor. However, the intensity (power per unit area) of the beam remains the same as the pulse enters the medium, as one can see from the relation / = uv^.

(35)

One also sees that the electric field strength remains (essentially) constant as the pulse enters the material medium, as can be seen from the relation /=^eoc«|£'|^

(36)

and there is little if any discontinuity in n at the boundary of the medium. These results have been discussed in greater detail by Harris and Hau [1999]. Their report also notes that large nonlinear optical effects often accompany the creation of slow light. One sees from the discussion just given that the linear response tends to be large not because the electric field is enhanced within the optical medium but rather because the conditions that produce slow light also tend to produce a large nonlinear optical susceptibility.

§ 4. Experimental studies of slow light One of the first experiments to measure the dispersive properties of an EIT system was performed by Xiao, Li, Jin and Gea-Banacloche [1995] using a gas of hot rubidium atoms and using a slightly different energy

6, § 4]

Experimental studies of slow light

515

level configuration than that considered in the previous section. They directly measured the phase imparted on a wave propagating through the rubidium vapor and tuned near the ^Si/2 -^ ^^3/2 transition using a Mach-Zehnder interferometer. A strong continuous wave laser beam tuned near the ^^3/2 -^ ^D5/2 transition (the so-called iadder' configuration) and counterpropagating with the probe beam created a Doppler-free EIT feature, thereby reducing a and increasing «g. While they did not directly measure the delay of pulses propagating through the vapor, they indirectly determined that Ug = c/13.2 for their experiment via the measurement of the phase shift of the wave. Soon thereafter Kasapi, Jain, Yin and Harris [1995] measured the temporal and spatial dynamics of nanosecond pulses propagating through a hot, dense 10-cm-long Pb vapor cell in an EIT configuration similar to that described in the previous section. In the absence of a coupling field, they inferred a probebeam absorption coefficient of 600 cm ^ With the coupling field applied, they measured a probe-beam transmission of 55% (corresponding to a = 0.026 cm"^) and Ug = c/165. These initial experiments demonstrated that it is possible to achieve slow light with dramatically reduced absorption, and they set the stage for later experiments on ultraslow light where the group velocities are extremely small. The key to achieving lower group velocities was to reduce significantly the dephasing rate Yea of the ground-state coherence, thereby narrowing the width of the EIT feature and increasing dn/dco. As mentioned in § 1.1, narrowing the EIT feature requires the use of significantly longer pulses in comparison to the nanosecond pulsed used by Kasapi, Jain, Yin and Harris [1995].

4.1. Ultraslow light in a ultracold atomic gas Hau, Harris, Dutton and Behroozi [1999] performed an experiment in 1999 that is largely responsible for the recent flurry of interest in slow light. This experiment made use of a laser-cooled sodium atomic vapor at a temperature of 450 nK near that of the transition to a Bose-Einstein condensation. The experimental setup for this study is shown in fig. 10. Briefly, they laser-cool and trap a cloud of atoms, spin-polarize the atoms by optically pumping them into the |F = 1, w/^ = -1) ^Si/2 ground state, and load the atoms into a magnetic trap at an approximate temperature of 50 |iK and a density of ~6 x 10^ ^ cm"^. At such low temperatures, the Doppler width of the optical transitions is less than the natural (spontaneous) width of the transition and hence the stationary-atom theory presented in § 3 is applicable. The temperature is further decreased via

"Slow" and "fast" light

516

(a)

[6, §4

^

i

" Pinhole

Imaging beam F=2-»3, linear

(b) t

60 MHz

14)= IF =3, M^=-2) 13)= \F=2,M='2)

12)= \F=2,M^=-2) 11)= |F=1,/W^=-1)

Fig. 10. (a) Experimental setup and (b) energy levels and laser frequencies used in the slow-light experiment of Hau, Harris, Dutton and Behroozi [1999].

evaporative cooling of the cloud, resulting in fewer trapped atoms but slightly higher atomic number densities and hence lower Ug. We note that the magnetic trap is asymmetric, leading to an oblong cloud of cold atoms. In the slow-light phase of the experiment, a strong coupling laser at frequency cOc drives the |2) -^ |3) transition of the sodium D2 resonance line (see fig. 10b) and propagates along one of the short axes of the cloud, as shown in fig. 10a. The group velocity of a pulse of light of center-frequency cOp is then determined as it propagates along the long axis of the cloud. The group velocity is monitored as probe beam frequency is scanned through the |1) —> |3) transition.

6, §4]

517

Experimental studies of slow light

E

u> c

s I-

1.006 ^

1.004

b

; | 1.002 I

1.000

I 0.998 C 0.996

0.994 -30

-20

-10

0

10

20

30

Probe detuning (MHz)

Fig. 11. (a) Theoretically predicted probe absorption spectrum, and (b) resulting modification of the phase refractive index under the experimental conditions of Hau, Harris, Dutton and Behroozi [1999].

The conceptual understanding of this method is illustrated in the theoretical simulations of the experiment shown in fig. 11. The upper part of this figure shows that a narrow transparency feature has been induced by the coupling field into the broad absorption profile of the gas. Note that this induced feature is of the order of 2 MHz in spectral width. Under their experimental conditions, the width of this feature is determined by power broadening effects (that is, the (Ds/2)^ term in eq. (38), although fiandamentally the narrowness of this feature is limited by the relaxation rate between the |1) and |2) levels). The lower part of this figure shows the resulting modification of the refractive index of the vapor. Note the steep, nearly linear increase of refractive index with frequency near the transition frequency. It is this behavior that leads to the large group index of this system. In fact, Hau, Harris, Dutton and Behroozi [1999] shows that the group index is given (in the power-broadened limit) by the expression he

lOc,

SjTCOp ||l(i3p7V'

(37)

Note that the group velocity decreases with decreasing control field intensity so long as this expression is valid. Some of the results of this experiment are shown in fig. 12. Here the open circles show a transmitted pulse propagating at the velocity of light in vacuum and the solid circles show the pulse induced to propagate slowly. Note that the induced pulse delay is considerably

518

[6, § 4

"Slow " and "fast" light

Ts 450 nK t o ^ = 7.05 ± 0.05 \iS L = 229 ± 3 urn

>

V. = 32.5 ± 0.5 m s"'

g

-2

0

2

.•A..J 4

6

8

10

12

Time (us)

Fig. 12. Some of the experimental results of Hau, Harris, Dutton and Behroozi [1999] demonstrating ultra-slow propagation of a light pulse. The open circles show the input pulse; the solid circles show the transmitted, delayed pulse.

greater than the duration of the pulse. In this example, the group velocity was measured to be 32.5 m/s corresponding to a group index of the order of 10^. In other measurements, these researchers observed group velocities as low as 17 m/s.

4.2. Slow light in hot vapors One might incorrectly deduce that the experiment of Hau, Harris, Dutton and Behroozi was enabled through use of a cold atomic gas. In fact, very similar experimental results have been obtained by Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully [1999] in a coherently driven hot ( r = 360K) gas of rubidium atoms using the apparatus shown in fig. 13. The key idea is that a narrow EIT resonance can be obtained by suppressing linebroadening mechanisms arising from the motion of the atoms and Zeeman splitting of the magnetic sublevels arising from stray magnetic fields. The dominant broadening mechanism in a hot gas is the Doppler effect. The EIT resonance can be rendered Doppler-free by making the strong continuous-

6, §4]

519

Experimental studies of slow' light

(b)

(a) F=2'

Time-delay Measurement

Spectrum Analyzer

T - ^f

A

A

F=r

a BCLD 1 l^ (probe) F=2

Amplitude Modulator

F=I Fig. 13. Experimental set-up of Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully [1999] for creating HIT features in a dense hot atomic vapor of rubidium. Note that their notation for the atomic energy levels (part a of the figure) is different from that of § 3 of the present chapter.

wave coupling beam copropagate precisely with the probe beam. To see why this is the case, recall that the susceptibility for a hot gas is given by ,.(i)_-i^lM/>aP A

*

{i[d-A ^ [i(d + k-v)-

+

(38)

(k-k,)^u)]-y,,}

YtaMd -A + {k-h)'v)\-

n , J + \Qs/l\

where k (ks) is the propagation vector for the probe (saturating) beam, u is the velocity of an atom, and (• • •)D denote an average over the thermal velocity distribution. It is seen that the term in the numerator, primarily responsible for the EIT resonance, contains the difference of the two propagation vectors. A narrow EIT resonance can thus be obtained for copropagating, nearly equal frequency probe and saturating waves so that (k - ks) essentially vanishes. For this configuration, the condition for the formation of a well-defined EIT hole is given approximately by |OsP > Yca^c^o, where AO^D is the Doppler width of the transition. Therefore, it is imperative to reduce Yea as much as possible. For a single stationary atom, YaJ^^ can be of the order of 1 Hz or less since transitions between the ground state of alkali-metal atoms are electricdipole forbidden. In a hot dense gas, the observed widths are much greater, due primarily to the finite time an atom spends in the laser beam as it moves through

520

"Slow" and "fast" light

[6,

the vapor cell and, to a lesser extent, due to collisions with surrounding atoms that can induce transitions between the states. The transit-time broadening can be reduced significantly by introducing a buffer gas to the vapor cell that reduces the mean-fi-ee-path of the alkali-metal atoms. Noble gas elements are preferred because there is little interaction between the buffer gas atoms and the alkalimetal atoms, thereby minimizing collision-induced transitions. Typical buffer gas pressures are of the order of 10 Torr for a 1 mm diameter laser beam. The final step in achieving narrow EIT resonances involves magnetic shielding. The energy level structure of an alkali-metal atom is more complex than that shown in fig. 13a; for each level there are (2F + 1) degenerate quantum states in zero magnetic field, where F is the total angular momentum quantum number. Because of the Zeeman effect, these states experience a shift in energy of the order of 1 MHz/Gauss. Therefore, to realize an approximation to the idealized three-level atomic system considered in § 3, stray magnetic fields must be reduced to better than 1 mGauss for y^a of the order of 1 kHz. Well-designed containers for the vapor cell constructed from high-permeability metals can achieve such low ambient fields. Using all of these techniques, Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully [1999] were able to attain y^a — 1 kHz in the laser-pumped rubidium vapor with a 30 Torr neon buffer gas and magnetic shielding. They measured directly the dispersive properties of the vapor using a modulation technique and from this data inferred a group velocity as low as 90m/s. They did not directly launch pulses of light through the vapor and hence did not address issues related to possible pulse distortion discussed in § 1.1. Some of their results are summarized in fig. 14 where it is seen that the group velocity 1

1

,

1

,

1

,

, —

250

1000

5 u i 200 1-°^ >. 150 _o a> Q .^ 100

•\\

\

cT \

•^^

—^"

r^

< o o

- 400

><" <

200

50 .

0

1

.

1

2

1

.

1

3

.

1

4

'

c

T3

- 600

X

\

Q. 3 O

O

X— X

- 800

3

n u

Drive Laser Power [mW] Fig. 14. Experimental results of Kash, Sautenkov, Zibrov, Hollberg, Welch, Lukin, Rostovtsev, Fry and Scully [1999] demonstrating slow-light propagation in a hot atomic vapor.

6, § 4]

Experimental studies of slow light

521

decreases with decreasing laser power for reasons mentioned above. We note that group velocities as low as 8 m/s have been inferred in a rubidium experiment by Budker, Kimball, Rochester and Yashchuk [1999] using a similar technique.

43. "Stopped" light Liu, Dutton, Behroozi and Hau [2001] have provided experimental evidence that a light pulse can effectively be brought to a halt in a material medium by proper control of the coupling field in an HIT configuration. Such processes hold considerable promise for applications such as coherent optical storage of information. The coupling configuration used in this work is shown in fig. 15. The propagation of a probe beam tuned near the |1) - |3) transition is monitored in the presence of a coupling beam tuned to the |2) - |3) transition. This experiment - | ^ - — |3> = |F=2, MF=0>

192 MHz

J_X

|4> = |F=1, MF=0>

Na D1 line X= 589.6 nm

\Coupling I Probe

1.8 GHz

|2> = |F=2.Mp=+1> .|1> = |F=1,/Wp=-1>

Fig. 15. Energy levels and laser frequencies used in the stopped-light experiment of Liu, Dutton, Behroozi and Hau [2001].

can be understood by noting that the probe beam would be very quickly absorbed were it not for the presence of the coupling beam. This experiment was performed in a laser-cooled atomic sodium vapor near the temperature for Bose-Einstein condensation. Some of the experimental resuhs of Liu, Dutton, Behroozi and Hau are shown in fig. 16. The upper panel shows three traces. The sharp peak centered at ^ = 0 (dotted line) shows a time reference obtained from the transmission of an input pulse so far detuned from the atomic resonance that it propagates essentially at the velocity of light in vacuum. The smaller peak centered at 12 [is is the transmitted, delayed pulse obtained under EIT conditions (solid line). The dashed curve shows the time evolution of the saturation field (referred to as the coupling field in the figure). The lower panels of fig. 16 shows data illustrating the storage of the probe pulse. In this experiment, the coupling field is turned on before the arrival of incident probe pulse. However, at time t = 10 |is after the pulse has fially entered

[6, §4

"Slow" and "fast" light

522

f CL

.y

1

^

0.8

1

/'f^-^^^^^'^-^v A-^^'"^-'V-'--^--

E o

I

0.6

' : '

0.4

'

:

•

1

/^

0.2 0 -20

- 1 20

60

40 Time (jts)

o O

80

1 0.8

/ 2

0.6 0.4 4, :

0.2

'. 0

*,

0 -20

20

/

40

V

^w >i

60

80

Time(jjLS)

e o

I o z

o O 20

820

840

Time (^,s)

Fig. 16. Experimental results of Liu, Dutton, Behroozi and Hau [2001] showing the stopping of light in an ultra-cold atomic medium. See the text for details.

the interaction region but before it has emerged from the exit side, the coupling field is abruptly turned off and is left off until / = 45 |is, at which point it is turned back on. During the time interval in which the coupling pulse is turned off, the probe pulse cannot propagate and remains stored in the medium. We see from the graph that in this case the pulse has been delayed by 25 |is, the time that the coupling beam has been turned off. In other experiments Liu, Dutton, Behroozi and Hau [2001] have observed pulse delays as long as 1 ms. The interpretation of this experiment is that when the coupling field is turned

6, § 5]

Experimental studies of fast light

523

off the probe beam is almost immediately absorbed. The excitation associated with the incident probe beam is not however thermalized; phase and amplitude information regarding the pulse is stored as a coherent superposition of states |1) and |2), that is, knowledge of the incident probe pulse is stored as a groundstate spin coherence. The energy of the probe pulse is coherently scattered into the direction of the coupling field. When the coupling field is later turned on again, light from the coupling field scatters coherently from the ground-state spin coherence to re-create the probe pulse. Note that the spatial compression of the light pulse as it enters the material medium (as described in § 3.1 above) is crucial to the process of light storage, as it is necessary that the entire pulse be contained within the interaction region. It is largely a matter of semantics whether the light has been temporarily "stopped" within the medium (the wording of the original workers) or whether the light pulse has temporarily been transformed to a material degree of freedom and later turned back into an optical field. It is also worth noting that the physics of the process of light storage is quite similar to that of the generation of Raman echos (Hartmann [1968], Hu, Geschwind and Jedju [1976]). As in the case of the creation of slow light, one might mistakenly assume that the use of a cold atomic vapor was crucial to the success of the temporary storage of light pulses. In fact, Phillips, Fleischhauer, Mair, Walsworth and Lukin [2001] have demonstrated very similar results through use of a hot Rb vapor. An additional physical mechanism for stopping the propagation of light has recently been proposed by Kocharovskaya, Rostovtsev and Scully [2001]. This mechanism is based on the spatial dispersion of the refractive index in a Dopplerbroadened atomic medium.

§ 5. Experimental studies of fast light As described above in the discussion of slow light, a practical requirement for the production of slow light is the attainment of a very large normal dispersion in the absence of higher-order dispersion and absorption. The natural question arises as to whether it is possible to obtain large anomalous dispersion, also with low absorption and low higher-order dispersion, and thereby produce fast (superluminal) light. Recall the work of Chu and Wong [1982a] described above where they observed large anomalous dispersion but in the presence of very large absorption. This work has been extended recently by Akulshin, Barreiro and Lezama [1999] who used electromagnetically induced absorption in a driven two-level atomic system to obtain very large anomalous dispersion (with an

524

"Slow^" and "fast" light

[6, § 5

inferred Vg of -c/23 000), but still in the presence of large absorption. Another demonstration of superluminal effects, also in the presence of large absorption, has been observed by Steinberg, Kwiat and Chiao [1993] in the context of singlephoton tunnelling through a potential barrier. One possible approach for avoiding absorption is to use the nonlinear (saturating) optical response of an amplifier as in the work of Basov, Ambartsumyan, Zuev, Kryukov and Letokhov [1966] describe above. Alternatively, one can make use of the cooperative (superfluorescence-like) response of a collection of inverted two-level atoms to produce superluminal propagation (Chiao, Kozhekin and Kurizki [1996]). Both of these approaches necessarily require the use of intense pulses. Another approach, described by Bolda, Garrison and Chiao [1994], is to make use of a nearby gain line to create a region of negative group velocity. In the present section, we describe a related scheme that has recently been realized experimentally based on the use of a pair of gain lines.

5.1. Gain-assisted superluminal light propagation We have seen above how EIT can be used to eliminate probe wave absorption, and in doing so produces slow light. An alternative procedure, proposed initially by Steinberg and Chiao [1994] and recently demonstrated by Wang, Kuzmich and Dogariu [2000] makes use of a pair of Raman gain features to induce transparency and to induce a large dispersion of the refractive index. The sign of d«/dco in this circumstance is opposite to that induced by EIT, with the result that the group velocity is negative in the present case. The details of this procedure are shown in the accompanying figures. Figure 17 shows the energy level description of the experiment. Two pump fields E\ 6P3/2

10)

|F=4,m=-3) A

|F=4.m=-4> Fig. 17. Energy levels and laser frequencies used in the superluminal pulse propagation experiment of Wang, Kuzmich and Dogariu [2000].

6, §5]

Experimental studies offast light

525

Is O

- 4 - 3 - 2 - 1 0 1 2 3 Probe detuning (MHz) Fig. 18. Theoretically predicted gain spectrum and associated variation of the phase refractive index under the experimental conditions of Wang, Kuzmich and Dogariu [2000].

and E2 with a frequency separation of 2 MHz are sufficiently detuned from a particular Zeeman component of the cesium resonance line that the dominant interaction is the creation of two Raman gain features. These gain features and the resulting modification of the refractive index are shown in fig. 18. The probe wave is turned midway between these gain features to make use of the maximum dispersion of the refractive index. Some experimental results are shown in fig. 19. Here the solid curve shows the time evolution of the probe pulse in the absence of the pump beams, and the dashed curve shows the time evolution in the presence of the pump beams. One sees that in the presence of the pump beams the probe pulse is advanced by 62 ns, corresponding to Vg = -c/310. The ratio of the pulse advancement to pulse width in this case is of the order of 1.5%. The fractional size of the effect clearly is not large. One of the motivations for performing this experiment was to demonstrate that superluminal light propagation can occur under conditions such that the incident laser pulse undergoes negligible reshaping. Indeed, it is remarkable how closely the input and output pulse shapes track one another. At one time, it had been believed that severe pulse reshaping necessarily accompanies superluminal propagation. While these experimental results are consistent with semi-classical theories of pulse propagation through an anomalous-dispersion media, there is continued discussion about the propagation of pulses containing only a few photons where quantum fluctuations in the photon number are important. Aharonov, Reznik and Stem [1998] argue that quantum noise will prevent the observation of a superluminal group velocity when the pulse consists of a few photons. In a subsequent analyses, Segev, Milonni, Babb and Chiao [2000] find that a superluminal signal will be dominated by quantum noise so that the signal-to-

526

"Slow" and "fast" light

[6, §5

> E c

0)

Time (jis) Fig. 19. Experimental results of Wang, Kuzmich and Dogariu [2000], demonstrating superluminal propagation without absorption or pulse distortion. The solid curve shows the pulse propagating through vacuum; the dashed curve shows the transmitted pulse. The insets are blow-ups of the leading and falling edges of the pulse.

noise ratio will be very small, and Kuzmich, Dogariu, Wang, Milonni and Chiao [2001] have introduced a "signal" velocity defined in terms of the signal-tonoise velocity that should be useful for describing the propagation of few-photon pulses. More recently, Milonni, Furuya and Chiao [2001] predict that the peak probability for producing a "click" at a detector can occur sooner than it could if there were no material medium between it and the single-photon source. We await experimental verification of these concepts and predictions. 5.2. Causality One might fear that the existence of negative group velocities would lead to a violation of the nearly universally accepted notion of causality. Considerable discussion of this point has been presented in the scientific literature, with the unambiguous conclusion that there is no violation of causality, as discussed by Chiao [1993] and by Peatross, Glasgow and Ware [2000]. Thorough reviews of the extended scientific discussion have been published by Chiao [1996] and Chiao and Steinberg [1997]. One can reach this conclusion by noting that the prediction of negative group velocity follows from a frequency-dependent (linear, for simplicity)

6, § 5]

Experimental studies offast light

527

susceptibility that is the Fourier transform of a causal time-domain response function. Thus, there is no way that the predictions of such a theory could possibly violate causality. But this argument does not explain how causality can be preserved, for instance, for situations in which the (peak of a) pulse emerges from a material medium before the (peak of the) same pulse enters the medium. The explanation seems to be that any physical pulse will have leading and trailing wings. The distant leading wing contains information about the entire pulse shape, and this information travelling at normal velocities such as c will allow the output pulse to be fully reconstructed long before the peak of the input pulse enters the material medium. For any physical pulse that has a non-compact support, the front velocity is limited to c while the group velocity, signal velocity, etc. can exceed c. For the case in which the front is located close to but before the peak of the pulse and t;g > c or d;g < 0, pulse distortion will occur leading to a "pile-up" of the pulse at the front as discussed by Icsevgi and Lamb [1969]. The nature of superluminal velocities can also be understood from a frequency domain description of pulse propagation. In such a description, each frequency component is present at all times; the coherent superposition of these frequency components constitute a pulse that is localized in time. When such a pulse enters a dispersive medium, the various components propagate with different phase velocities, leading to pulse distortion and/or propagation with a modified group velocity. While these ideas have not been tested experimentally for propagation of electromagnetic waves, Mitchell and Chiao [1997] have studied the propagation of voltage pulses through a very low frequency bandpass electronic amplifier. They show that the amplifier transmits Gaussian-shaped pulses with a negative group delay as large as several milliseconds with little distortion, as shown in fig. 20a. They also created an abrupt discontinuity (a front) on the waveform and found that it propagates in a causal manner, as shown in fig. 20b. It is seen that the peak of the output is produced in response to earlier input, which does not include the input peak. This result is expected for a causal system where the output depends on the input at past and present, but not on fiiture times. For a front at the beginning of the pulse, they observe that the front reaches the output no earlier than it reaches the input and that no signal precedes the front, as expected. In summary, even though t;g > c or t;g < 0, relativistic causality is not expected to be violated in electromagnetic wave propagation experiments. Specifically, the front of any physical pulse of compact extent should travel at a speed less than c, and it should distort to avoid overtaking the front, consistent with the dispersion properties of the medium.

528

"Slow" and "fast" light L t

f

t

>

1

t

t

*

l

|

l

f

l

T

|

[6 *

l

T - r - i

H 12.1 ms > 3 B 2 r output o > 1 pu b r

/ ^

A

input 1

t

i

1

1

1

1

•1

/NV / Y^

1

1

1

>

1

«

1

^^ 1

1

1

1 t 1

50

100 150 time (ms)

200

50

100 150 time (ms)

200

Fig. 20. Experimental results of Mitchell and Chiao [ 1997] demonstrating negative group delays, but causal propagation, (a) Input/output characteristics of a chain of low-frequency bandpass amplifiers. (b) Input/output characteristics for a pulse with a sharp "back".

§ 6. Discussion and conclusions This very recent research on slow and fast Hght demonstrates that our understanding of atom-field interactions has truly developed to a high degree. It is now possible to tailor the absorption, amplification, and dispersion of multi-level atoms using intense electromagnetic fields. The developments are of fundamental interest, and they hold promise for advances in practical areas from optical communications and devices to quantum computing. Fundamentally, they challenge our understanding of century-old physical laws.

Acknowledgements RWB wishes to thank C.R. Stroud Jr. and R. Epstein for their encouragement in preparing a review of this sort. The authors also wish to thank L. V Hau, E. Cornell, S.E. Harris, M. Fleischhauer, G.R. Welch, F.A. Narducci, M.O. Scully,

6]

References

529

M.D. Lukin, L.J. Wang, and S.L. Olsen for fruitful discussions regarding the content of this review. RWB was supported in part by ONR grant NOOO14-991-0539, and DJG was supported in part by NSF grant PHY-9876988.

References Agrawal, G.P., 1995, Nonlinear Fiber Optics, 2nd Ed. (Academic Press, San Diego, CA). Aharonov, Y, B. Reznik and A. Stem, 1998, Phys. Rev. Lett. 81, 2190. Akulshin, A.M., S. Barreiro and A. Lezama, 1999, Phys. Rev. Lett. 83, 4277. Basov, N.G., R.V. Ambartsumyan, VS. Zuev, PG. Kryukov and VS. Letokhov, 1966, Sov. Phys. Dokl. 10, 1039 [Sov. Phys. JETP 23, 16]. Basov, N.G., and VS. Letokhov, 1966, Sov. Phys. Dokl. 11, 222. Bennink, R.S., R.W. Boyd, C.R. Stroud Jr and V Wong, 2001, Phys. Rev. A 63, 033804. Bolda, E.A., J.C. Garrison and R.Y. Chiao, 1994, Phys. Rev. A, 49, 2938. Boyd, R.W, 1992, Nonlinear Optics (Academic Press, San Diego, CA); see, for instance, eq. (6.5.20). Brillouin, L., 1914, Ann. Physik 44, 203. Brillouin, L., 1960, Wave Propagation and Group Velocity (Academic Press, New York). Budker, D., D.F. Kimball, S.M. Rochester and VV Yashchuk, 1999, Phys. Rev. Lett. 83, 1767. Carruthers, J.A., and T. Bieber, 1969, J. Appl. Phys. 40, 426. Casperson, L., and A. Yariv, 1971, Phys. Rev. Lett. 26, 293. Chiao, R.Y, 1993, Phys. Rev. A 48, R34. Chiao, R.Y, 1996, in: Amazing Light: A volume dedicated to Charles Hard Townes on his 80th birthday, ed. R.Y. Chiao (Springer, Berlin) p. 91. Chiao, R.Y, A.E. Kozhekin and G. Kurizki, 1996, Phys. Rev. Lett. 77, 1254. Chiao, R.Y, and A.M. Steinberg, 1997, in: Progress in Optics, Vol. 37, ed. E. Wolf (Elsevier, Amsterdam) p. 345. Chu, S., and S. Wong, 1982a, Phys. Rev. Lett. 48, 738. Chu, S., and S. Wong, 1982b, Phys. Rev. Lett. 49, 1293. Diener, G., 1996, Phys. Lett. A 223, 327. Diener, G., 1997, Phys. Lett. A 235, 118. Faxvog, FR., C.N.Y Chow, T. Bieber and J.A. Carruthers, 1970, Appl. Phys. Lett. 17, 192. Frova, A., M.A. Duguay, C.G.B. Garrett and S.L. McCall, 1969, J. Appl. Phys. 40, 3969. Garrett, C.G.B., and D.E. McCumber, 1970, Phys. Rev A 1, 305. Harris, S.E., 1997, Phys. Today (July), p. 36. Harris, S.E., J.E. Field and A. Imamoglu, 1990, Phys. Rev. Lett. 64, 1107. Harris, S.E., J.E. Field and A. Kasapi, 1992, Phys. Rev. A 46, R29. Harris, S.E., and L.V Hau, 1999, Phys. Rev. Lett. 82, 4611. Harris, S.E., G.Y Yin, M. Jain, H. Xia and A.J. Merriam, 1997, Philos. Trans. R. Soc. London A 355,2291. Hartmann, S.R., 1968, IEEE J. Quantum Electron. 4, 802. Hau, L.V, S.E. Harris, Z. Dutton and C.H. Behroozi, 1999, Nature 397, 594. Hu, P, S. Geschwind and T.M. Jedju, 1976, Phys. Rev. Lett. 37, 1357. Icsevgi, A., and WE. Lamb Jr, 1969, Phys. Rev. 185, 517. Kasapi, A., M. Jain, G.Y. Yin and S.E. Harris, 1995, Phys. Rev. Lett. 74, 2447. Kash, M.M., VA. Sautenkov, A.S. Zibrov, L. Hollberg, G.R. Welch, M.D. Lukin, Y Rostovtsev, E.S. Fry and M.O. Scully, 1999, Phys. Rev. Lett. 82, 5229. Katz, A., and R.R. Alfano, 1982, Phys. Rev. Lett. 49, 1292.

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"Slow " and "fast" light

[6

Kocharovskaya, O., Y. Rostovtsev and M.O. Scully, 2001, Phys. Rev. Lett. 86, 628. Kuzmich, A., A. Dogariu, L.J. Wang, RW. Milonni and R.Y. Chiao, 2001, Phys. Rev. Lett. 86, 3925. Liu, C , Z. Dutton, C.H. Behroozi and L.V Hau, 2001, Nature 409, 490. Lukin, M.D., and A. Imamoglu, 2001, Nature 413, 273. Milonni, RW., K. Furuya and R.Y. Chiao, 2001, Opt. Express 8, 59. Mitchell, M.W, and R.Y. Chiao, 1997, Phys. Lett. A 230, 133. Peatross, J., S.A. Glasgow and M. Ware, 2000, Phys. Rev. Lett. 84, 2370. Phillips, D.R, M. Fleischhauer, A. Mair, R.L. Walsworth and M.D. Lukin, 2001, Phys. Rev. Lett. 86, 783. Scully, M.O., 1991, Phys. Rev. Lett. 67, 1855. Segard, B., and B. Macke, 1985, Phys. Lett. 109A, 213. Segev, B., RW Milonni, J.F. Babb and R.Y. Chiao, 2000, Phys. Rev. A 62, 022144. Sherman, G.C., and K.E. Oughstun, 1981, Phys. Rev. Lett. 47, 1451. Sommerfeld, A., 1907, Physik. Z. 8, 841. Sommerfeld, A., 1914, Ann. Physik 44, 177. Steinberg, A.M., and R.Y. Chiao, 1994, Phys. Rev. A 49, 2071. Steinberg, A.M., PG. Kwiat and R.Y. Chiao, 1993, Phys. Rev. Lett. 71, 708. Tewari, S.P, and G.S. Agarwal, 1986, Phys. Rev. Lett. 56, 1811. Wang, L.J., A. Kuzmich and A. Dogariu, 2000, Nature 406, 277. Xiao, H., and K.E. Oughstun, 1997, Phys. Rev. Lett. 78, 642. Xiao, H., and K.E. Oughstun, 1999, L Opt. Soc. Am. B 16, 1773. Xiao, M., Y.-q. Li, S.-z. Jin and J. Gea-Banacloche, 1995, Phys. Rev Lett. 74, 666.

E. Wolf, Progress in Optics 43 © 2002 Elsevier Science B. V All rights reserved

Chapter 7

The fractional Fourier transform and some of its applications to optics by

A. Torre ENEA, UTS - Tecnologie Fisiche Avanzate, via E. Fermi 45, 00044 Frascati (Rome), Italy; E-mail: [email protected]

531

Contents

Page § 1.

Introduction

533

§ 2.

The fractional Fourier transform

534

§ 3.

The optical fractional Fourier transform

541

§ 4.

Fractional Fourier transform and lens optics

550

§ 5.

Fractional Fourier transform and Wigner optics

561

§ 6.

Fractional Fourier transform and Fourier optics

569

§ 7.

Fractional Fourier transform and wave-propagation optics

. .

579

§ 8.

Operational properties of the fractional Fourier transform

. .

584

§ 9.

Conclusions

591

§ 10.

Acknowledgments

593

References

593

532

A mia madre Ai suoi occhi nei miei Al suo volto fra le mie mani

§ 1. Introduction The fractional Fourier transform (FrFT) has been invented and reinvented several times within different contexts of both mathematics and physics. The group of transformations and the associated representations on appropriate Hilbert spaces of functions have provided the early framework, where the fractionalization of the Fourier transform has originally been conceived (Weyl [1927], Wiener [1929], Condon [1937], Bargmann [1961]). In contrast, the quantum mechanics of dynamical states describable in terms of Hermite-Gauss eigenstates (Namias [1980a], McBride and Kerr [1987]) and the optics of light propagation through continuous media and lens systems (Mendlovic and Ozaktas [1993], Ozaktas and Mendlovic [1993a,b], Lohmann [1993]) as well as signal information processing (Almeida [1994]) have benefited from the more recent reinvention of the FrFT. The tQvm fractional evokes the charming vision of the extension to continuum of a process that occurs through finite-size steps. Thus, Condon [1937] investigated a continuous group of functional transformations isomorphic with the group of rotations of a plane about a fixed point by multiples of an arbitrary angle, thus generalizing the property of the Fourier transform group, which corresponds to rotations by multiples of the right angle. Likewise, Namias [1980a] elaborated the functional form of Fourier-like operators, admitting, as the Fourier transform, the Hermite-Gauss eigenfunctions, but with eigenvalues evenly separated by 2i fraction of the imaginary unit. Also, the optical definition of the FrFT (Mendlovic and Ozaktas [1993], Ozaktas and Mendlovic [1993a,b]) stems from the performance of an optical system perceived as the limit of a larger and larger number of shorter and shorter free-space sections interleaved with weaker and weaker lenses. In parallel, the extension to arbitrary angles of the property of the Fourier transform to rotate by a right angle the Wigner chart in the inherent Wigner phase plane is basic to the implicit definition of the fractional Fourier transform, proposed by Lohmann [1993]. The equivalence of all these definitions stems from the fact that the FrFT belongs to the family of the linear canonical transforms generated by linear canonical transformations (Wolf [1974a,b, 1979]). The rotation by a right angle in the plane of the Fourier conjugate variables is the underlying 533

534

The fractional Fourier transform

[7, § 2

canonical transformation of the Fourier operation. Letting the rotation angle vary continuously leads to the fractional Fourier operation. Thus, the finite-step transformations marked by multiples of the right angle become specific events within the smooth evolution of the process governed by the continuously varying angle over the In range. After its introduction into the field of optics, the concept of FrFT and the relevant formalism gave rise to a huge variety of applications, investigations and new formulations in an increasingly enriched optics scenario. This chapter is organized as follows. In §2 we introduce the FrFT within a purely formal context and illustrate its basic properties in the case of both real and complex order. In § 3 we situate the problem within the framework of optics. Sections 4-7 are devoted to a review of the relation of the FrFT to various aspects of optics, such as lens optics (§ 4), Wigner optics (§ 5), Fourier optics (§ 6) and wave-propagation optics (§ 7). Finally, in § 8 we recover the formal context of § 2 to describe the operational calculus associated with the fractional Fourier operator. Section 9 concludes. The topics are treated at an introductory level; each of them has evolved into a multifarious universe, for which we suggest the reading of the review papers by Lohmann, Mendlovic and Zalevsky [1998] and by Ozaktas, Kutay and Mendlovic [1999] and of the book by Ozaktas, Zalevsky and Kutay [2000]. § 2. The fractional Fourier transform The Fourier transform pair is defined by the integrals (Bracewell [1978]) + CXD

q}{K) = ^

+OC

/ e-"^>(p(x)dx,

(fix) = - ^

f e''q){K)AK.

(2.1)

In the present context, we will regard the Fourier transform q) of the function cp as resulting from the action of the linear operator T on (p: q) = J^ cp. Thus, eqs. (2.1) can be recast in operator form to define the Fourier operator T as + OC

[T (p]{x) = q>ix) = ^

J e~"^' (p(x') dx',

(2.2)

-OC

and accordingly the inverse operator T~^ through + OC

[J^' (p]{x) = ^ L 2ji

/ e"'^' (fix') dx'.

(2.3)

7, § 2]

The fractional Fourier transform

535

Evidently, the operators T and T~^ satisfy the inversion theorem TT~^ = T~^ T = I, with I the identity operator, and are complex conjugate to each other: T^^ ^ T~^. Accordingly, T is unitary, which expresses in operator form the Parseval theorem, stating the equality of the Dirac integrals of (f) and its transform ^:

l\(pix)\'dx= -oc

l\^(x)f
(2.4)

-oc

Furthermore, the following table of integer powers of T can be traced: ^ (p{x) = q)(xl P(p(x) = cp{-x\

P (fix) = cp {-x\ Pcp{x) = (p(x),

where P means that J^ is applied y times in a row. The above manifest the wellknown cyclic behaviour of the Fourier transform, the original function being retrieved after applying the transform four times. In eqs. (2.2), the symbol x is used to denote the arguments of both the function q) and the Fourier transform ^. In contrast, conventional practice assigns different symbols to the arguments of the ftinction and its transform, as in eq. (2.1), where x and K may denote the space variable and the spatial frequency. We will deal here with suitably scaled variables and functions. The physical context, to which the transforms (2.2) and (2.7) are pertinent, will give specific meaning, role and dimensions to the variables x and x^ and correspondingly to q) and q). The Fourier transform of fractional order arises from a fractionalization of the one-step transformation (2.1). In a sense, the process of fractionalization is complementary to that expressed in eq. (2.5), where Fourier blocks are joined one to the other. The FrFT realizes the continuous fragmentation of the ordinary transform; it interpolates between the ftinction (p(x) and its Fourier transform (p(x), introducing between the space and frequency domains a continuum of fractional Fourier domains. Fourier transforms become apparent therefore as forming a subclass of the more general and wider class of FrFTs. We will now introduce the mathematical definition of the FrFT and then discuss some of its properties, separately for real and for complex order. 2.1. Real order The FrFT of order a is conceived to define the operator P^ with the following basic properties:

536

The fractional Fourier transform

[7, § 2

(i)

J^^^ is continuous with respect to the order parameter a: P"^ tends to J^^ as fi tends to a, (ii) J^^ obeys the semigroup property, so that composing two fractional Fourier transforms of orders a\ and ai yields the fractional transform of order «! +a2* pM^a, ^ pt.pM

^ ^^^. +^^:^

(2.6)

(iii) T^^ reduces to the identity operator when a = 0 and to the ordinary Fourier transform when a= \\ J^ = \ and JT' = ^ . The mathematical definition of the FrFT can be obtained, within the purely formal context of the linear canonical transforms, as unitary representation of the canonical transformation, which produces a clockwise rotation, governed by the order parameter a, in the plane (x, K) of the Fourier-conjugate variables (Wolf [1974a,b, 1979], Kramer, Moshinsky and Seligman [1975], Abe and Sheridan [1994a,b]). Accordingly, the required properties (i)-(iii) emerge from the general properties of the linear canonical transforms with respect to the parameters of the generating canonical transformations. On the other hand, the procedure we describe in the next section, which resorts to the optical interpretation of the ordinary Fourier transform, provides a physical account of the mathematical formalism, thus yielding the formal definition of the fractional transform as well as its optical interpretation in terms of graded-index media and thin lenses. Here, we will focus on the essential properties of the transformation j^^^ described by the linear integral transform (Namias [1980a], McBride and Kerr [1987]) [r'(p](x) = tp,{x) = J K„(.r,.r') (f{.x') dr'.

(2.7)

-DC

The kernel Ka(x,x^) is given by the explicit expression Ur ,

r.

/l-icot0

Kaix.x ) = W

—

exp 2sin0 [x'" cos 0 + x^ COS 0 - 2xx')

n

0 = a^,

(2.8) where the argument of the square root is taken in the interval {-\ji,\n]. Evidently, the value a = 1 corresponds to the usual Fourier transform (2.2). The kernel Ka{x,x') is periodic with respect to the angle (j) and hence to the order a. Any increment of a by 4/', withy an integer, leaves Ka{x,x') unchanged. The operator J^^^ is therefore cyclic in the sense that T''^^'(p{x) = T"cp(x)

V 7 G N,

(2.9)

7, § 2]

The fractional Fourier transform

537

which generaUzes to any a the cyclic behaviour of the ordinary transform and accordingly allows to limit the values of a within the interval a G (-2,2]. In this respect, since the kernel (2.8) is not strictly defined when a is an even integer, the values a = 0 and a = 2 deserve some comments. We recast eq. (2.8) as Ka{x,x) = A/1 + i tan 0 exp[-Lx.x^ tan(0/2)]

exp

/\2

2i tan

-Ax-x'y

(2.10)

\J1JX\ tan ^

which, according to the well-known limit b(x) = lim/, _ o e''-^'"'^'^V\/2j7riZ?, can be seen to approach the delta function d{x -x') for vanishing values of a. The operator J^ turns therefore into the identity operator: P(p(x)=\(p{x)=(p{x).

(2.11)

Likewise, to deal with the value a = 2, it is convenient to rewrite eq. (2.8) as exp Ka(x,x')= >/l + i tan 0 Qxp[-bcx' cot(0/2)] -

L ( x + y)'

2i tan

y/2jTi tan (j)

(2.12)

which for a = 2 approaches the delta function d{x-\-x'), thus giving T^(p(x) = (p(-x),

(2.13)

in accordance with eq. (2.5). The relations (2.11) and (2.13) estabhsh the continuity of the definition (2.7) with respect to the order a. In addition, the order additiuity property, expressed in eq. (2.6), is of crucial relevance within the context of the FrFT formalism. It implies, for instance, that transforms of different orders commute with each other and also that the inversion theorem for T^'^ can be formulated in the form ?''T~''q){x) = ^-''P(p(x)

= ^(f{x)

= (fix),

(2.14)

which extends the theorem holding for the ordinary Fourier transform to As a fiirther consequence of the additivity property (2.6), the FrFT of order a can be mathematically understood as the ath power of the Fourier transform: ^ ^ = [^]".

(2.15)

To close the present list of properties of ^^^ we note that the kernel Ka(x,x') is symmetric with respect to its arguments: K(({x,x') = K^ix^x), thus implying that

538

[7, §2

The fractional Fourier transform Table 1 Fractional Fourier transform for some common functions^

Function

Fractional Fourier transform

ay/\ + itan0exp(-^x tan^ / 1-ico

d{x)

V - 2 F^^ ^ exp(2^^ cot^

step(j<:)

y/\ + i t a n 0 exp(- 2x^ tan 0) F

rect(jc)

A/1 + i t a n 0 exp(- ^.v^ tan 0) <^ F f I'-a - cos ^

V!sin(20)| y/\s\n(20)\ J

-F

.v+a cos (p \^|sin{20)

)}

V^l+itan0exp(-^^2 tan0)(^ tan0)"//,, ( ^ ^ n ^ ^ ) V T T T t ^ e x p ( ^^2 cot(20))(-i tan(pfDy

[y/^^^x^

exp(ifljc)

y/\ + itan0 exp[(-2^,QgA (JC^ sin 0 + «^ sin ^ - 2ax)]

exp(iZ)jc^)

/ l+itan0 . 1 2( 2/7-tan 0 . V l+2/)tan0 c^PV2-^ ^ l+2/)tan0^

^ a denotes the semiaperture of the rectangle function rect(x); F{y) is the complex Fresnel integral: Fiy) = - 7 ^ / exp(si/2)d^ s = sign(sin(20)).

the roles of the variables x and x' in eq. (2.7) can be interchanged. Furthermore, the complex conjugate relates to the inverse according to /<*(x,xO == K_a(x,%0The fractional transform operator J^^'^ is therefore unitary: [^«]t=^-«^[^«]-i^

(2.16)

(the symbol f denotes the adjoint operator) and accordingly the Parseval theorem applies to the fractional transform q)^ as well + OC

+OC

J \(p(x)\'dx=

J

\q^a(x)\'dx.

(2.17)

It can be easily proved that ^ " is not hermitian: j ^ " ^ [J^^^]K In table 1 we list the FrFTs of some simple functions. Thus, the transform of the rectangle function involves the complex Fresnel integral F(y), and the transforms of the monomials x'\ n integer, and x^\ v complex, relate respectively to

7, § 2]

The fractional Fourier transform

539

the Hermite polynomials //„ and to the parabolic cylinder functions Dy. Notably, the chirp function e'^^ can be fractionally transformed into a delta function if the order of the transform is chosen to be a = - ^ tan"'(^); this property is used to remove chirp-type noise from signals (Ozaktas, Barshan, Mendlovic and Onural [1994], Dorsch, Lohmann, Bitran, Mendlovic and Ozaktas [1994]). Finally, we notice that the FrFT of (p(x) is related to the Fourier transform of the function 0(x) = y^l - i cot0 (p(x) exp[i(x^/2) cot0] through [jr«(^](jc) = exp(i(xV2)cot0)[J^0] — - . \ sm 0 /

(2.18)

Thus, if q)(x) belongs to the Hilbert space C^(R) of the square integrable functions over the real axis M, the transform q)aix) is square integrable as well. In fact, the transformation in eq. (2.7) defines a unitary mapping of C^(W) onto £^(R). However, a remarkable property of the FrFT is that it can also be calculated for functions which are not square integrable, such as the monomialsx", n = 0,1,2,... (see table 1). Interestingly, the interpretation (2.18) of the FrFT in terms of simpler operations, such as chirp multiplication, scaling and ordinary Fourier transformation, is conveniently exploited in the digital computation of the fractional transform (Ozaktas, Arikan, Kutay and Bozdagi [1996], Marinho and Bernardo [1998]). Also, with the change of variable x —> ;c sec 0 and some algebraic manipulations, the basic form (2.7) can be recast in terms of the Fresnel transform as ^^ \ - i cot 0 \ x'^ ~\ f .(x'-x)^ \T q)\{xcos0) = Y — 1 ^ — exp -\— sm(20) / exp 2 tan 0

(p(x)dx\

(2.19) The above expression permits also the interpretation of the FrFT as a wavelet transform (Daubechies [1990]), whose mother wavelet /?(?) is provided by the quadratic phase function, h{^) = e'^'^^, and the daughter wavelets hah(^) are generated by translation and dilatation from h(^), i.e., hahi^) = 7^^(^7^)' with the scale and shift parameters a and b being identified ^s a = v^tan0 and b= y. 2.2. Complex order The extension of the definition (2.7) from the real to the complex domain for the order parameter a is naturally suggested and straightforwardly accounted for by the correspondence between the real-order transforms (2.7) and the real

540

The fractional Fourier transform

[7, § 2

canonical transformations, representing rotations in the Fourier plane (x, K). In fact, the real nature of the order a, and hence of the angle 0, amounts to realangle rotations, from which the transforms (2.7) arise as unitary representations on the Hilbert space £^(R). Correspondingly, complex values of the order a, and hence of the angle 0, correspond to complex-angle rotations, from which the associated integral transforms emerge in the base form (2.7) and with the basic properties (i) and (ii), but with substantial differences with respect to the real-order transforms as far as the variable domains, the function spaces and the operator properties, such as boundness and unitarity, are concerned. A detailed formal account of the FrFT of complex order within the general context of the linear canonical transforms generated by complex canonical transformations is beyond the purpose of this chapter. The interested reader is referred to Bargmann [1961, 1967], Wolf [1974a,b, 1979] and Kramer, Moshinsky and Seligman [1975]. We briefly synthesize the ruling features of the transform (2.7), when the quantities identified as sin0 and cos0 are allowed to be in general complex. Let us use the notation 0 = (aR + i a , ) | = 0 R + i0i,

(2.20)

where aR,0R denote the real parts of a and 0, and a\,(j)\ represent the corresponding imaginary parts; the limits for aR and a\ are then -2 < aR < 2 and -oo < a\ < +00; accordingly
(2.22)

7, § 3]

The optical fractional Fourier t?-ansform

541

The measure d^ above is explicitly given by

^i^W " \/:;r~T7^X^ ^^^\ " T ~ T 7 ^ X ^ W^ cosh(20i) + x* ^ cosh(2(^i) - 2xx*] I y :/rsinh(20i) |^ 2sinh(20i) "^ ^J X d(Rex)d(Imx), (2.23) where Rex and Imx denote the real and imaginary parts of x, respectively. When (/)[ = 0, the Bargmann-Hilbert space 3^ collapses to the ordinary £^-space, the scalar product (2.22) turning into the usual Dirac integral

which defines the metric in the ordinary Hilbert space £^(M). Besides, when the complex-order transform (2.7) is bounded, i.e., when sinh(20i) < 0, the associated mapping between £^(IR) and 5^ is unitary, and therefore &, ^h = (
(2-25)

In contrast, when considered as a mapping from £^(E) to £^(R), the integral transform (2.7), corresponding to complex orders, is nonunitary, even if bounded. As a final remark, we note that the order additivity property allows one to write ^«R + i«, ^ ^ « R ^ m = pmp^K

^

(2.26)

according to which the complex-order operator T^^"^ ^'"' can be seen as arising from the combined operation of the real and purely imaginary order operators T^^"^ and j ^ ' ^ ' , the specific sequence being unessential due to commutativity.

§ 3. The optical fractional Fourier transform We introduce the FrFT resorting to the optical realization of the ordinary Fourier transform, to which the fractional transform closely relates through a direct procedure of fractionalization. The formal definition given in eq. (2.7) is immediately recovered. Also, the optical implementation of the FrFT in terms of refractive lenses and graded-index media is naturally established, thus allowing

542

[7, §3

The fractional Fourier transform

n,

(a)

(b)

Fig. 3.1. The 2/-system (a) and the Fourier tube (b) are the basic setups to implement the Fourier transform of the signal.

to place the fractional transform within the context of the imaging and optical wave propagation in the paraxial regime. The ordinary Fourier transform can easily be implemented by lenses (Goodman [1968]). The single- and double-lens setups of fig. 3.1 are the basic geometries of the optical devices performing the Fourier transform of the input signal, as a result of a suitable interplay between the propagation through a homogeneous and isotropic medium (e.g., free space) and the focusing by positive or negative thin lenses. Likewise, single- and double-lens geometries have been proposed as optical realizations of the FrFT (Lohmann [1993], Bernardo and Soares [1994a,b]). Both the single- and the double-lens Fourier transform systems are characterized by only one parameter, i.e., the lens focal distance /s, which operates as a size parameter for the system and as a space scale and amplitude amplification parameter for the imaging of the object into the Fourier transform. Let us briefly recall the conventional rules adopted when dealing with optical systems. In general, the various optical elements are supposed to be aligned along the optical axis, and the light is assumed to propagate from left to right. The object and image distributions are referred to two planes, ilj and /Jo, transversal to the optical axis. In particular, in the single-lens geometry (fig. 3.1a), the input and output reference planes, /Jj and /7o, are placed, respectively, at the entrance to the first free-space section and at the exit from the second section. Correspondingly, in the double-lens configuration (fig. 3.1b), the input and output planes are located immediately to the left of the first lens and immediately to

7, § 3]

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543

the right of the second lens. In both geometries one detects on the exit plane the Fourier transform of the optical distribution on the input plane, apart from a multiplicative factor and a scale transformation, depending on the optical wavelength and the focal length/s. The object and the transform distributions are both real or virtual according to whether the lens focal distance/s is positive or negative. Throughout the chapter we assume that the light distributions are monochromatic and confined to the proximity of the optical axis (paraxial propagation). 3.1. Input-output relations for linear optical systems The complex amplitude distributions xl)\{q) and V^o(^) of the signals respectively entering into and emerging from a first-order optical system are related through the diffraction integral (Siegman [1986], Saleh and Teich [1991]),

%{q) = j g{q.q'\ z„z,)Uq)dq'

= M{z,,z,)Uql

(3-1)

involving the impulse response function g{q,q'\ ZQ,Z\). The z-coordinate is commonly measured along the optical axis with respect to a fixed origin; thus, z\ and ZQ specify the locations on the axis of the input and output planes, /7i and ilo, where cartesian coordinates are taken with the origin on the axis. The one-dimensional picture of the signal transfer implied in eq. (3.1) and below in eq. (3.2), is allowed for separable astigmatic systems, which we consider here for simplicity. As expressed in the right-hand side of eq. (3.1), the transformation performed by the optical system on the incoming signal can be represented through the linear operator M{ZQ,Z\), which transfers the amplitude distribution over the input plane /7i to the corresponding distribution over the output plane /TQ. The transfer of the signal from /7i to /7o can also be expressed in matrix form,

involving the 2x2 ray transfer matrix M = I

j . The above establishes

a linear relation between the coordinates (q,p) of the light ray at the input and output planes. Specifically, q denotes the transverse coordinate of the intersection of the ray with the reference plane, as the paraxial optical momentum p is given

544

The fractional Fourier transform

[7, § 3

by the product of the angle between the ray and the optical axis at the intersection point and the on-axis value of the refractive index at the reference plane. The real or complex ray matrices M describing first-order real or complex optical systems belong to the symplectic group of the 2x2 matrices, hence they are unimodular (Gilmore [1974)'. The link between the two formulations (3.1) and (3.2) of the light-signal transfer by optical systems is established by expressing the response function g in terms of the entries of the ray matrix M as follows (Nazarathy and Shamir [1980, 1982a,b], Siegman [1986]): g{q,q\z^,z,)=

V^—^exp ^^[Aq'^^Dq'-lqq')

(3.3)

where k = In/X is the wavenumber in the vacuum, A being the optical wavelength. The above expresses within the formalism of first-order optics the general relation holding between linear canonical transformations, of which eq. (3.2) is an example, and the associated representations on the Hilbert space >C^(M), which in general take the form of linear integral transforms of the type (3.1) with the kernel (3.3) (Wolf [1974a,b, 1979], Nazarathy and Shamir [1982a,b], Abe and Sheridan [1994a,b]). In particular, as noted in § 2.1, the representations associated with real canonical transformations are unitary; unitary is therefore the transfer operator /W(zo,Zi) describing real, i.e., lossless and gainless, optical systems.

5.2. The optical Fourier transform The optical ray matrix F ( / ) between planes connected by a Fourier-transforming relation, such as those in the setups depicted in fig. 3.1, reads

^(/s)=(Ji ^ ) -

(3-4)

' In general, (real or complex) symplectic matrices represent (real or complex) canonical transformations, which map canonically conjugate variables into canonically conjugate variables, thus preserving the inherent Poisson brackets or Lie products, according to whether classical variables or quantum-mechanical observables are concerned. The property of being symplectic manifest itself in explicit relations between the entries of the matrix, yielding the unimodularity relation, detM = AD - BC = 1, in the simple case of 2x2 matrices, both real and complex.

7, § 3]

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545

Thus, the output signal, whose ampHtude distribution writes as

-oc

can be recognized as the Fourier transform of the input signal apart from multiplicative and space-scale factors \/k/\f^ and k/f^. Correspondingly, in the optical phase plane ^ the ray-representative point is transferred from {q\,p\) to {qo,Po), with qo=fsPu

Po = -jqx-

(3.6)

In particular, with /s = 1 the motion (3.6) is geometrically identified with , clockwise rotation by the angle \ji, the Fourier matrix F{\) equaling the 2'1\Krotation matrix: F( 1) = /?(^JT) ^. Evidently, by arranging a sequence of identical Fourier transformers one obtains the optical implementation of the power table (2.5). In fact, if \p\{q) is the input signal, one consecutively detects at the exit from each F system, apart from multiplicative and scale factors, the Fourier transform of \l>\{q), the inverted image ipii-q), the Fourier transform of \l)\{-q), and finally again the input signal \i)\{q). In other words, cascading optical Fourier-transforming configurations, we realize physical systems which perform integer powers of the Fourier transform. It is natural to inquire whether it is possible to realize fractional Fourier transformers, i.e. optical systems which, when arranged in sequence, produce

^ Let us recall that the phase space is understood as the cartesian space (or plane, for systems with one degree of freedom) formed by the conjugate variables {q,p) suited to the problem under study, obeying the Poisson-bracket relation {q,p} = 1The light-ray coordinates {q,p) are canonically conjugate variables. Thus, they can span the phase plane, where light rays are represented by points, and accordingly the ray transfer through optical systems corresponds to the transfer of the representative point {q\,Px) to the point {qo,Po)- For instance, free-space propagation and focusing produce translations of the ray representative point parallel to the q- and to the /?-axis, respectively. ^ More properly, the {q,p) pair is conveniently scaled to the new pair (q, r]), with rj ^fp having the dimension of length. In the {q, r]) plane, the Fourier matrix F{f) corresponds to clockwise rotation by ^JT, as the fractional Fourier matrix F"(/^), defined by eq. (3.19), corresponds to clockwise rotation by ait/2.

546

The fractional Fourier transform

[7, § 3

a Fourier transform. This is a particular aspect of the more general problem of decomposing an optical system M into a cascade of a certain number m of identical optical systems Af''", which can be expressed through the ray-matrix relation M^/'^'M'/"' • •

-M'/'"

= [M'/'"]'" = M

(3.7)

The problem (3.7) has been examined by Shamir and Cohen [1995], where the FrFT arises as one of the m root systems of F.

3.3. The square root of the optical Fourier transform We firstly ask for an optical system such that a sequence of two of them produces the Fourier-transforming configuration of focal lengthy^. Denoting by F^^^ the ray matrix of such a system, we write for F'^^ as a particular case of eq. (3.7) the relation F^^^ F^^^ = F(f).

(3.8)

Evidently, as different configurations can be envisaged to implement the Fourier system F, different geometries can be designed to realize F^^^ as well. Taking as a model the single-lens geometry of the Fourier transform, we may think of the F'^^ system as formed by a thin lens of focal length/1/2 placed midway between two reference planes separated by 2d\/2 (Mendlovic and Ozaktas [1993]). The relation (3.8) yields for the system parameters/1/2 and d[/2 the expressions fw2 = V2f,

du2 = iV2-\)f,

(3.9)

where/s takes the meaning of the focal length of the overall system 77^^^ F^^^. The square root of the optical Fourier transform is therefore represented by the ray matrix FV2.r.( P (M

cos(i;r) V-isin(i;r)

/ssin(i;r)\ cos(i;r) j '

^^"^^^

which with f = I is immediately identified to produce a clockwise rotation by ^Jt of the ray-representative point in the optical phase plane.

7, § 3]

The optical fractional Fourier transform

547

In terms of the diffraction integral, we obtain

^o(^) - J^/^

I ^^P[i^ {q"^r-2V2qq')'^ Uq)^q- (3.11)

Rewriting the above integral in terms of the scaled variable x = \fTdf^q and the function q)(x) = ^(y/fs/kq), we are naturally led to the formal expression (3.12) where the integral transform

[^^'VlW = W ^

/ exp [ i ( y ^ ^x' -

iVlxx') (p(x')dx',

(3.13)

is squared to the Fourier integral (2.2), and hence can be viewed as the mathematical definition of the square root T^'^^ of the Fourier operator T, i.e., of the Fourier transform of fractional order a = ^. ^.4. The ath power of the optical Fourier transform As a further step in the process of fragmenting Fourier transform units, we think of optical fractional Fourier transformers of order \/n, as obtained by cutting ordinary Fourier transform systems into n identical subsystems. Accordingly, the ray matrix of such systems, denoted as F ' " , obeys the relation \/n wr\/n /T'/^TT

r^\/n _

[wrl/nl" [F''^]=F(f).

(3.14)

Assuming for F^^" the same optical setup as for F ' ^ with relative parameters f/n and d\/„, on account of eq. (3.14), we obtain the expressions f\/n -

/s

sin(jr/2«)

^1/,,-/stan(;r/4«).

(3.15)

which reproduce eqs. (3.9) for « = 2. Then, the ray matrix F^ " takes the form cos(jr/2«) /s sin(jT/2n) \ -jr sm(jz/2n) cos{Jt/2n) J '

(3.16)

548

The fractional Fourier transform

[7, § 3

according to which the transfer of the hght signal V^i(^) through the nth root F^^" of the optical Fourier transform is described by the diffraction integral

%(q) =

iJiifs sin(;r/2«) ^ / e x p L ^ . \ ,- , [q'-cosiJT/2n) + J {2fs sm{ji/2n)

q^cos(Jt/2n)-2qq']\il^,iq')dq'. ')

-OO

(3.17) As expected, when applied n times, the above yields the Fourier integral (3.5). Evidently, the key parameter to continuously fragment the Fourier transform is just the angle, entering the circular functions in eqs. (3.16) and (3.17), which will then be adjusted to continuous values, say aji/l. Thus, we define the ath power of the optical Fourier transform, with a an arbitrary real number, through the ray matrix

F\f.)=( r . ^ ^^''"/V

0 = «?,

(3.18)

^•^^ \ ^ - ^ s m 0 c o s 0 y ^ 2 and correspondingly through the Huygens integral

y 2;ri/s sm0 y

|^ 2/s sm0 ^

')

= [FMqy (3.19) The tuning parameter a measures the degree of fractionalization, and/s relates to the focal length of the reconstructed Fourier system, F(/s) = [F"(fs)V^^^The expression (3.19), assigned to the optical FrFT operator F^^, whose dependence on the order parameter a and the arbitrary fixed focal length/s is explicitly evidenced, corresponds to the mathematical definition (2.7) of the FrFT operator J^^ through the scaling of the optical coordinate q to the dimensionless variable x= Jjq and through the factor e~'^^^:

FI -^ e-'^/2 ^« ^-^j^ q -^ x= J J q.

(3.20)

Also, resorting to the phase-plane picture introduced above, it is immediately apparent that the matrix (3.18) with/s = 1 corresponds to clockwise rotation by

7, § 3]

The optical fractional Fourier transform

549

an angle 0 = an/2 of the ray-representative point. Accordingly, the continuous fractionalization of the Fourier transform parallels the continuity of rotations in the phase plane, the Fourier transform corresponding to the particular case (j) =

\7l.

3.5. Fractional Fourier transform and quadratic graded-index media In order to explore the physical content of the matrix (3.18) and integral transform (3.19), following Ozaktas and Mendlovic [1993a,b] and Mendlovic and Ozaktas [1993], we note that F^^ resembles the ray matrix relevant to a quadratic graded-index medium, whose refractive index n(q, z) shows a quadratic dependence on the distance from the optical axis according to n{q,z) = nQ-\n2q^,

(3.21)

where n^ and ni are the medium parameters, which for the purposes of the present discussion we suppose to be «o > 1 and ni > ^ (focusing medium). The ray variables {q,p) are propagated along the medium (3.21) by the matrix , , / cos(a;r/2) M(z,Zi)= ^ \-y7Zo«2 sm(a;r/2)

-7=!= sin(a;r/2) \ v/^^ ^ M , cos(ajr/2) /

^^ ^^, (3.22)

where z\ specifies the location of the input reference plane, and z that of the observation plane inside the medium. Correspondingly, the diffraction integral for the propagation of the light signal \l)\{q) from z\ to z takes the form (3.19), with/s = \/y/n^. Interestingly, the parameter a, given by a-^^,

L='-J

—,

(3.23)

measures, in units of L, the distance from the input plane at which the image is detected. In turn, L specifies the proper length required for performing a Fourier transform of the object; propagation over the distance L from the input plane results in the ordinary Fourier transformation, being M(z\ + L, z\) = F ( l / y « o ^ ) . At distances aL from Zi, the system behaves as a Fourier transformer of fractional order a. Accordingly, we can imagine the graded-index medium to be formed by a continuum of planes from z\ through the optical axis; on each plane at z, we can observe the FrFT of order a = (z - z\)/L of the signal at z\. At the

550

The fractional Fourier transform

[7, § 4

planes at regular distances from Zj, L,2L,3L,..., we may detect the successive powers of the Fourier transform of the input signal. Also, the cascading property of the ray matrix (3.22) and of the corresponding diffraction integral, according to which the propagation from z\ to zi and then from z\ to Z2 is equivalent to the propagation from Zj directly to Z2, allows us to state the additivity of the optical fractional integral (3.19) with respect to the order a, since the sequence of two fractional Fourier transformers of order a\ = (z\ - z\)/L and ai = {zj - z\)/L turns into the fractional transform of order a = a\-\- a2 = {z2- z\)/L. The quadratic graded-index medium offers the possibility of physically defining T^cp in terms of the scalar light distribution at the plane z = Zj + aL, resulting from the propagation of the input signal through the medium from z\ to z. In particular, the ordinary Fourier transform of the signal can be observed at the distance L = ^Jt^/n^7n^ from the input plane. Hence, L fixes the z-size of the Fourier transform units of which the medium may be thought as composed. The direct correspondence between the order parameter a and the propagation variable z suggests the relevance of the FrFT for wave-propagation optics (§ 7). The expression (3.19) for the optical FrFT, as well as the mathematical definition (2.7), has been straightforwardly generalized to the two-dimensional case (Mendlovic and Ozaktas [1993], Ozaktas and Mendlovic [1993a,b]). Also, the two-dimensional FrFT with different orders in the two dimensions has been introduced; benefitting for its optical implementation from both anamorphic optics (Sahin, Ozaktas and Mendlovic [1995], Mendlovic, Bitran, Dorsch, Ferreira, Garcia and Ozaktas [1995], Erden, Ozaktas, Sahin and Mendlovic [1997]) and elliptic graded-index media (Yu, M. Huang, Wu, Lu, W. Huang, Chen and Zhu [1998]), it has found successful applications in pattern recognition and fikering (Mendlovic, Bitran, Dorsch, Ferreira, Garcia and Ozaktas [1995], Garcia, Mendlovic, Zalevsky and Lohmann [1996]). Finally, we note that the rigorous mathematical procedure of fractionalization has been applied to the twodimensional Fourier transform by Simon and Wolf [2000], paralleling the onedimensional problem dealt with by Shamir and Cohen [1995], and suggesting generalization to more than two dimensions.

§ 4. Fractional Fourier transform and lens optics A lens is able to perform two basic operations in optical processing: Fourier transform and imaging. The concept of FrFT permits consideration of both operations within the same context, where the imaging can be seen as a fractional

7, § 4]

Fractional Fourier transform and lens optics

551

Fourier transformation, and correspondingly the ordinary Fourier transform can be seen as a particular realization of the fractional transformation (Lohmann [1993], Bernardo and Soares [1994a,b], Ozaktas and Mendlovic [1995], Liu, Xu, Zhang, Chen and Li [1995], Sahin, Ozaktas and Mendlovic [1995], Bernardo [1996], Erden, Ozaktas, Sahin and Mendlovic [1997], Sahin, Ozaktas and Mendlovic [1998]). The FrFT can be obtained optically by interleaving free-space sections and lenses. In fact, the well-known single and double-lens realizations of the ordinary Fourier transform have been extended to the FrFT as well. Interestingly, the interpretation of the integral transform (2.7) in terms of elementary optical operations, i.e., free propagation and focusing, can be conveniently used to design appropriate algorithms for the numerical calculation of the FrFT. Basically, free-space propagation is numerically performed by using the fast Fourier transform, whereas lens operation is accounted for by multiplication by a quadratic phase factor (Marinho and Bernardo [1998]). 4.1. Type-I and type-II optical setups A single-lens configuration can be designed to correspond to the fractional transform ray matrix (3.18), consisting of a thin lens of focal length/^^ placed midway between two reference planes spaced by Ida (fig- 4.1a). For assigned values of a and/s in (3.18), the parameters/^ and da are explicitly given by f a = ^ .

da=fst^n(0/2l

(4.1)

reproducing the ordinary Fourier transform setup with a= \, i.e.,/i = d\ =fs. For the order a ranging within the interval (-2,2], we obtain SL family of optical systems with a geometry like in fig. 4.1a, first introduced by Lohmann [1993] as the type-I geometry. The family is uniquely marked by the real parameter/s which, denoted as the standard focal length, represents the scale of the ordinary Fourier transform F\ which results from appropriately cascading fractional transforms of lower order belonging to the same family/s. The concept of a family grouping fractional transforms with the same standard focal length together arises naturally from the rule by which the systems (3.18) are cascaded. It is evident that only a cascade of fractional units having the same absolute value off yields an overall system which is an FrFT again (Bernardo [1996]). The double-lens realization of the fractional transform, modeled on the doublelens geometry of the ordinary transform, is composed of two thin lenses with

552

[7, §4

The fractional Fourier transform

n„

n,

Hi

n.

fa

A

ii

\L

(a)

SJ (b)

Fig. 4.1. Optical implementation of the fractional Fourier transform of order a. (a) Single-lens (type I) and (b) two-lens (type II) configurations.

the same focal distance/« separated by the free-space length da (fig. 4.1b). The parameters/cf and da relate to a and/s through ^=/sCot(0/2),

da =fs sin (l>,

(4.2)

thus realizing Lohmann's type-II geometry (Lohmann [1993]), which with a = 1 turns into the two-lens Fourier transform, described by/i == d\ =f^. Both eqs. (4.1) and eqs. (4.2) can be recast in terms of the lens focal distance/a, thus yielding for the free-space length da and the family parameter ^^ the expressions Type I

da =/«(l - cos0),

f, =fa sin0,

Type II

da =fa(\ - COS0), /s =fa tan(0/2).

(4.3) (4.4)

For a given focal length/^, the single- and double-lens configurations group into different families, the corresponding values of the standard focal length/s being indeed different. Also, the ray matrix (3.18) can be properly rewritten in term of the lens focal distance/« for both type-I and type-II configurations. On account of eqs. (4.3) and (4.4), we obtain, respectively: t70.f.

_ /cos0

F\fa)

=

/asin^0

COS0 -^ cos^(0/2)

(4.5)

2/^^sin'^(0/2)\ COS0 J'

(4.6)

The transfer of optical information by a lens between planes, symmetrically placed on the two sides of the lens, can always be regarded as a fractional Fourier

7, §4]

Fractional Fourier transform and lens optics

553

Fig. 4.2. Optical setup for a fractional Fourier transform with a non-collimated input signal.

transformation, the inherent order a being determined by the lens focal length/^ and by the object-to-lens distance d(( through cos(ajr/2) = 1 - y^. Real values of the order parameter a are obtained for d^ ^ 2/;^. Accordingly, Fourier transformations by lenses belong to the wider class of fractional Fourier transformations by lenses, which are characterized by the basic parameters a and /s, both operating as design parameters. In addition, the standard focal length/s is the scaling parameter, which relates the spatial coordinates and the spatial frequencies in the fractional Fourier domain, and also identifies the family of fractional Fourier transformers, within which the claimed semigroup property of the mathematical definition (2.7) finds an optical implementation. In the type-I and type-II configurations discussed above, the elementary optical operations, i.e., focusing and free-space propagation, are arranged into a symmetrical geometry: two identical free-space lengths or two identical lenses. This straightforwardly reflects the symmetry of the F^^-matrix, for which A= D. The A and D entries of the ray matrix have complementary roles within the diffraction integral (3.1); in fact, according to the response function (3.3), they are responsible for the quadratic phase factors involving the spatial coordinates on the input and output planes, respectively, i.e. the ^' and q variables. Thus, in a FrFT the input and output planes are mirror-symmetric. Finally, we note that also the transfer of spherical waves by lenses can be described in terms of FrFTs (Bernardo and Soares [1994a], Bernardo [1996]). We refer to fig. 4.2, where the basic single-lens setup is depicted with a spherical wave incident on the input plane. Once assigned the object-to-lens distance zi, the lens focal length / and the curvature radius p of the incident wave, the

554

[7, §4

The fractional Fourier transform

parameters/« and (j) of the equivalent fractional transform system, and the lensto-image distance Z2, where the FrFT of the object can be observed, are fixed to fa =

Pf

COS0 = 1

p-f^-zx'

Z2 = Z i

7'

p-f

+ zx'

(4.7)

In the case of collimated illumination, the above configuration becomes the symmetrical geometry (4.3), since, for p -^ oo.fa ^ f and z^ -^ zx = / ( l -cos<;^). 4.2. The fractional Fourier transform and imaging systems Fourier transform systems can be arranged to form afocal configurations for perfect imaging"^. Figure 4.3 shows the basic setup of an afocal system, formed

ni

ri„ Z

.

/:

A

A ^--^ •---..^^ •

•

•

•

/

\

^^<^-'' ^ *^^

^\i

-

. 1 ./l

r'

./l +./2

/

•

>

Fig. 4.3. Afocal system producing a perfect image of the object obtained by cascading two conventional Fourier transformers. The dashed lines show the ray tracing for plane and spherical waves.

by cascading Xv^o perfect Fourier transforms F^ with different focal lengths/j and 72. The system produces a perfect image with magnification m = -fi^fx • ^^ shown in the figure, ray tracing confirms that for both plane and spherical waves the imaged distributions have the same curvature as the input distributions.

We recall that in a perfect imaging system the output amplitude distribution is identical to the input distribution, except for a possible scaling, whereas in an imperfect imaging system the image distribution displays quadratic phase terms, which are not present in the object distribution. Similarly, a lens produces ?i perfect Fourier transform on its back focal plane if the object is placed at the front focal plane. If the object is out of the front focal plane, an imperfect Fourier transform, displaying a quadratic phase term, is observed on the back focal plane.

7, § 4]

Fractional Fourier transform and lens optics

555

Evidently, perfect imaging systems can be obtained by properly cascading fractional transform units to an overall system with a = 2. In fig. 4.4, some geometries involving fractional transformations of different orders are depicted, all composing into an afocal system with magnification m = -\, equivalent to the optical configuration of fig. 4.3 with/, =f2 (Bernardo and Soares [1994a,b]). Generally speaking, a lens system for perfect image formation can be realized by arranging in tandem a different number of fractional units belonging to the same family (Dorsch and Lohmann [1995], Dorsch [1995]). As a basic rule, the order a = 2 of the overall imaging system must be achieved in summing up the orders of the various fractional units, according to

Y,njaj = 2,

(4.8)

where rij denotes the number of component units of order a,, the total number of components being A^. As an example, we may consider an imaging system formed by A^ lenses with the same focal length Z^, placed at equal distance from each other (Dorsch and Lohmann [1995]). In that case, the tandem rule (4.8) becomes Na = 2,

(4.9)

the orders a, of all component units being equal: GJ = a (fig. 4.5). Accordingly, the lens-to-lens distance d turns out to be d = 2fa[\-C0S(jT/N)]

^fa(Jt/N)\

(4.10)

the approximation above being suited to a large number N of lenses. The overall length D of the system is therefore D = Nd^fa'^.

(4.11)

The above formulae can be used in various ways, according to the specific design problem at hand. Equation (4.11), for instance, may provide the number N of lenses with assigned focal length fa needed to carry the signal over the distance D. Perfect imaging can also be implemented by cascading FrFT units with different family parameters. In that case, the fractional units belonging to the same family must be arranged to form integer-order modules. The conventional

556

[7, §4

The fractional Fourier transform

'/2y,,3

Vi/,,

viL,

'/j/,,

vj:_.

Vif,^,,

A

A

(b) '/2/,„,

'/2./,„

v^f,^

yiy,,

./:..

A

(c)

%/,

' ^ . / , , Vl/,3 V^./,, VS./,,

Fig. 4.4. Imaging systems producing a perfect inverted image of the object obtained by cascading fractional Fourier units: (a) three units with a = ^; (b) two units with « = j and « = j (/2/3 = f^/^)\ (c) three units with a = | , « = - j and « = 4 (/4 3 = -/-2 3)-

7, §4]

Fractional Fourier transform and lens optics

A

A-

A

557

A

D = Nd

Fig. 4.5. Design of a perfect imaging system composed of A^ identical lenses placed at equidistant locations.

afocal system of fig. 4.3 with/, ^f2 is the simplest example of such an optical system: it is formed by two modules of integer order, F\f\) and F ' ( / 2 ) , which do not constitute a fractional tiansform, because/s, = / , ^ / 2 =/s.. It is evident that the same system can be realized by arbitrarily decomposing each a = 1 unit into a certain number of fractional transformers of lower order. Three identical units with Oi= \ a n d / = / , , for instance, can be cascaded to compose the first F^(/i)-system, whereas two cascaded units with Oi\ = \ and «2 = | , having the same standard focal length/ = / 2 , are equivalent to the F^fj) system. More general optical systems, not necessarily symmetrical, have been demonstrated to be capable of implementing the FrFT under specific geometric conditions (Bitran, Mendlovic, Dorsch, Lohmann and Ozaktas [1995], Liu, Xu, Zhang, Chen and Li [1995], Lohmann [1995], Dorsch [1995], Bernardo [1996], Erden, Ozaktas, Sahin and Mendlovic [1997], Sahin, Ozaktas and Mendlovic [1998]). Various multiple-lens architectures, also in modular arrangements, have been suggested to realize the optical FrFT with the further feature that the order a and/or the standard focal length/ can be adjusted (Liu, Xu, Zhang, Chen and Li [1995], Lohmann [1995], Dorsch [1995]). For instance, Lohmann [1995] suggested optical designs for an FrFT with the order continuously adjustable from -1 to +1. The proposed geometries are based on type-I and type-II FrFT systems with fake zoom lenses enabling the order of the transform to be varied continuously; the zoom systems basically consist of two optical Fourier transform units, i.e., a 2 / system or a Fourier tube or also a piece of gradedindex fiber, with a free-space section of variable length between them. Also, Liu, Xu, Zhang, Chen and Li [1995] considered a base architecture composed of two lenses/J and/2 separated by the distance d, which image the input

558

The fractional Fourier transform

[7, § 4

plane at distance d\ from/j into the output plane at distance dj from/2. A suitable parameter involving the various free-space lengths of the system has been envisaged as an additional degree of freedom, allowing one to tune both the order parameter and the standard focal length within well-defined ranges. A comprehensive and systematic treatment of a great variety of multiple-lens configurations, capable of realizing the FrFT with the desired parameters, was presented by Erden, Ozaktas, Sahin and Mendlovic [1997], and fiarther developed by Sahin, Ozaktas and Mendlovic [1998], within the general context of the anamorphic transform. Thus, optical systems with two, four and six cylindrical lenses were considered. In each case, the flexibility of the system, as regards the control of the various parameters of the transform, was discussed and quantitatively expressed by the relevant design equations, relating the inherent free-space lengths and lens focal distances to the transform parameters. In §6 we will discuss the relation of the FrFT to the general optical transform (3.1), and we will establish the conditions under which flat and spherical surfaces may be related by a FrFT under propagation through a general real ABCD system. A wider extension of the fractional Fourier transforming relation to optical systems is achieved by letting the order parameter range within the complex domain (§ 4.3) and modifying the basic definition of the fractional transform to appropriate forms for including additional adjustable parameters besides a and/s (§6.3).

4.3. Optical interpretation of the fractional Fourier transform of complex orders It is evident from the relations (4.3), (4.4) and (4.7) that the real or the complex nature of the order a of the fractional Fourier transformations, performed by the corresponding optical systems, is closely related to the relevant design parameters. In fact, if the geometry of the optical configuration yields values of cos 0 and sin 0 in these relations that are outside of their trigonometric domain, the angle 0, and then the order a, becomes complex. Also, complex orders naturally arise when complex optical systems are implemented. Gaussian lenses, characterized by complex focal lengths, and Gaussian apertures, modelled by imaginary focal lengths, are examples of complex optical systems, described by complex ABCD matrices (Siegman [1986]). In that case, the angle 0 is complex because the quantities cos0 and sin0 are complex.

7, §4]

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559

Hi

Fig. 4.6. Imaging system composed of two fractional units of complex order a\ = 2 - iO.84 and «2 = - 2 + i0.84.

Evidently, the extension of the order domain from the real line to the complex plane allows one to apply the description in terms of FrFT to optical systems with more general geometries than those discussed in the previous subsection. A further extension is obtained by other generalized forms of the base transform (2.7), as will be discussed in §6.3. Figure 4.6, for instance, shows an imaging system composed of two fractional units of complex orders a\ =2- iO.84 and a2 = -2 + iO.84 (Bernardo and Soares [1996]). The first is implemented by a symmetrical single-lens setup, the second by a symmetrical two-lens geometry; in both cases the above-mentioned restriction on the freespace length da ^ 2fa, required to have real orders, has not been applied; indeed, /«i "/«2 ^ / ^^^ ^(i\ "^ ^(i^ "" ^f- Bernardo and Soares [1996] performed a detailed study of the domain of the order a in the complex plane (a^, a\) for the case that the optical transform is implemented by bulk-lens systems, both real and complex. It is also interesting to approach the problem by considering the representative ray matrix (3.18) for an assigned real value of the standard focal length/s. Letting the angle 0 be complex, with the notation (2.20), it is straightforward to write

r^(/s) = r^H/s)/^'"'(/s),

(4.12)

which reproduces the additivity property obeyed by the associated integral transform, already stated in eq. (2.26). Accordingly, the fractional transform of complex order can be optically implemented by cascading a real-order and a purely imaginary-order unit, the respective optical matrices being F^'-^{fs) and F""(fs)-

560

The fractional Fourier transform

[7, § 4

Evidently, the real-order units can be realized by bulk-lens systems or gradedindex media. On the other hand, the matrix F'^^'(/s), corresponding to the purely imaginary angle i0i, is explicitly given as 7'm(f\=. I

cosh0,

i/ssinh0i

Following Shih [1995a], we separate F'"' into the product of lower and upper triangular matrices according to ,uiuf^^[ 1 ^^'^ \^_itanh(0,/2)

{)\(\ l^V^

i/ssmh0,>^ ( \ ^ J \-jrtanh{(l)i/2)

0 1 (4.14) The complementary sequence of upper, lower and upper triangular matrices is also allowed, with the off-diagonal entry being i/s tanh(0|/2) for the upper triangular matrices and -(i/f^) sinh (/)] for the lower triangular matrix. Notably, the above applies to F'^^' the general decomposition of an ABCD matrix in terms of lens-like and free-space-length-like matrices. The optical realizations of the real-order fractional transform, basically the type-I and type-II configurations, are formally based on this kind of decomposition, as confirmed by the design relations (4.1) and (4.2); in fact, the corresponding lower and upper triangular real matrices describe real lenses and free-space lengths. The imaginary nature of the off-diagonal elements in eq. (4.14) lead to different optical interpretations of the inherent matrices and hence of the overall optical system F'^^'(/s). The lower triangular matrices in eq. (4.14) account for the transmission through Gaussian apertures, whereas the upper triangular matrix synthesizes the convolution of the signal amplitude distribution with a Gaussian function. In general, the convolution can be performed by applying the inverse ordinary Fourier transform to the product of the ordinary Fourier transforms of the convolved fiinctions. The inner matrix in the product (4.14) may therefore be realized by cascading two ordinary Fourier-transforming configurations and inserting a Gaussian aperture at the common focus. Accordingly, the optical interpretation of the imaginary-order Fourier transform amounts to a two-lens self-imaging optical system with three Gaussian apertures placed at both ends and at the focus (fig. 4.7). The Gaussian apertures at both ends have the same form but are different from that at the focus. The combined operation of a realorder transform and an imaginary-order transform according to eq. (4.12) yields the optical realization of the complex-order FrFT (Shih [1995a]). Shih [1995a] also studied the evolution of Gaussian beams under propagation through optical systems equivalent to complex-order Fourier transforms.

Fractional Fourier transform and Wigner optics

7, §5] Hi

561

n.

V /

/

/

/

Fig. 4.7. Optical interpretation of an imaginary-order fractional Fourier transform T^^^^ as a selfimaging system with three Gaussian apertures at both ends and at the focus.

§ 5. Fractional Fourier transform and Wigner optics The FrFT bridges the gap between classical optics and "Wigner optics", i.e., optics described in terms of the Wigner distribution function (WDF) (Wigner [1932], de Bruijn [1973], Hillery, O'Connell, Scully and Wigner [1984], Cohen [1989, 1995]). Both the WDF and the FrFT yield hybrid signal representations, displaying space and spatial-frequency features together. Interestingly, performing the FrFT of a signal manifests into a clockwise rotation of the corresponding Wigner distribution in the Wigner phase plane with an angle an/I, where a is the order of the transform; this correspondence between the WDF and the FrFT is formally expressed through the Radon transform. The relation of the FrFT to the WDF was firstly asserted by Lohmann [1993] who, as a supposedly natural extension of the behavior of the ordinary Fourier transform, defined the FrFT of order a as the transformation displayed by the function when the associated Wigner distribution is clockwise rotated by the angle an/I in the inherent Wigner phase space. Evidently, this definition is particularly appealing since the WDF has many applications in the characterization of beam quality and sharpness for both completely and partially coherent light signals. It also has applicative advantages, suggesting, for instance, optical designs for efficient signal multiplexing or for optimal filtering of signals from degradating noise terms (Ozaktas, Barshan, Mendlovic and Onural [1994], Kutay, Ozaktas, Arikan and Onural [1997]). As shown in fig. 5.1, it may happen that the Wigner distributions of the signal and the noise overlap, when projected,

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The fractional Fourier transform

[7, § 5

signal Fig. 5.1. Wigner charts for signal and noise.

on both the space and spatial-frequency axes, but have little or no overlap on some other direction, rotated by the angle (j) with respect to the Fourier axes. Removal of the undesired signal components can therefore be accomplished by using suitable amplitude masks after performing the appropriate FrFT of the signal, i.e., the appropriate rotation in the Wigner plane. Moreover, the Wigner phase space represents a particularly useful tool in quantum optics when dealing with coherent and squeezed states, which indeed can be generated from the vacuum state by phase-space rotation, translation and squeezing operators (Kim andNoz [1991]). In addition, the established link between the FrFT and the WDF suggested the optical implementation of the fractional Fourier operator by lens optics, simply modeled on the optical realizations of the ordinary Fourier operator, as previously described. Thus, as a straightforward generalization of the lens realizations of the FrFT relevant to the WDF of one-dimensional signals, an effective strategy to perform rotations of WDFs of two-dimensional signals under very general conditions has been envisaged. Furthermore, translating the optical implementations into photonic implementations, where optoelectronic modulators and optical fibers with suitable dispersion replace thin lenses and free-space sections, the FrFT of time signals can be performed (Lohmann and Mendlovic [1994]). Also, as a simple restatement of the Wigner rotation property, the connection between the Radon-Wigner transform and the FrFT was demonstrated by Lohmann and Soffer [1994], making it possible to transpose the signal processing capabilities of the Radon-Wigner transform into the fractional Fourier field, which benefits from easy digital computation and optical realization.

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Further investigations have been performed to explore the role of the FrFT within the wide realm of the phase-space representations (Ozaktas, Erkaya and Kutay [1996]) as well as to introduce new YQlaiQd fractional distributions and transforms such as, for instance, the fractional WDF (Dragoman [1996]) and the fractional Radon transform (Zalevsky and Mendlovic [1996b]), which extend and enrich the field of applications of the corresponding ordinary distributions and transforms.

5.1. Basics of the Wigner distribution function The WDF provides a ID representation of the optical signal simultaneously in the space domain and in the spatial frequency domain (Walther [1968], Bastiaans [1978, 1979], Dragoman [1997]). An optical signal can be described by its complex amplitude V^(^,z), or by its spatial ft-equency spectrum \1)(K,Z). Both descriptions are complete and equivalent, being related through the Fourier integrals (2.1). However, the representation of the signal by the Fourier pair yields separate bodies of information about the behavior of the signal in the q domain and in the K domains, failing to display the correlation between the q and K patterns. The WDF W{q, K, Z) provides a hybrid representation of the signal, displaying space and ft-equency information simultaneously through the convolution integral in ^-space

nq,'C) =^ J ^(1+ y)r{q-

W)^-""' M',

(5.1)

-OC

or, equivalently, in /c-space

W{q,K)=~

f \p(K+^K')\p*{K-^K')e"'''^dK'.

(5.2)

-OO

The complete symmetry of the two definitions indicates that the spatial coordinate and the spatial frequency have equal weights in the hybrid ^-representation. For notational convenience, the dependence on z of the complex amplitude xjj, the Fourier transform ip and the WDF is not displayed explicitly in the above formulae, where the integrals are intended to be performed over z = const, planes.

564

The fractional Fourier transform

[7, § 5

The positional and angular power spectrum as well as the total energy of the signal can be retrieved from the Wigner representation by the integral projections onto the ^-axis, the /f-axis and the q- and ;c-axes, according to

IV(?)P = \ j W(q,K)AK,

jv/Wl' = i y" W{q,K)dq, -CXD

-l-OC +OC

+00

(5.3)

j \n>{qf dq=^ J J W(q,K)dKdq. The reconstruction of the two representations ipiq) and \p(K) is based on the inverse formulae + 00

"^oo

^(K) = -J—-

A:i/;*(0) J

(5-4)

f W(q, \K) exp-''^^ dq,

-oc

from which both ^(q) and \IJ(K) are recovered apart from a constant phase factor, the magnitudes of t/;*(0) and t/;*(0) being known from the squared moduli in eq. (5.3) with q = 0 and K = 0. Remarkably, the WDF is propagated through a linear optical system according to the ray-matrix formalism of geometrical optics. Thus, the transformation experienced by the optical signal, when propagating through an optical system, manifests into the WDF, which is transformed from Wi(q,K) to (Bastiaans [1978]) Wo(q,K)=WXM-'r),

(5.5)

where the matrix M describes the transformation experienced by the Fourierconjugate variables (q, K), gathered into the column vector r = (^). The conjugate variables suited to Hamiltonian optics are the cartesian coordinate q and the optical momentum p. The ray-transfer matrix M describes the transformation of the pair (q,p) under propagation along the relevant optical system according to eq. (3.2). The Fourier-conjugate variables q and K are naturally introduced in Wigner optics. Due to the correspondence between the spatial frequency and the ray momentum as K = kp, any transformation of the

7, § 5]

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565

ray variables {q,p) turns into a transformation of the Wigner pair {q, K) through a matrix M, obtained from M with the replacements B -^ B/k and C ^^ kC', A B/k^ thusM . ^ ^ ^

5.2. Fourier transform and Wigner distribution function: optical analog of the ^Jt rotation in the Wigner phase plane It is convenient, as noted in § 3.2, to scale the K variable by the factor/s/fe to yield r] = (f/k) K, where f is an arbitrary but fixed focal length. Evidently, the transformation of the ray variables performed by an ABCD system is described in the (q, r/) plane by the matrix -M = I ^^

n

]' Then, the Fourier transform

matrix F{f) straightforwardly corresponds to the rotation matrix with angle ^Jt, and accordingly the transformation of the WDF from W\(q, rj) to Wo(q,r])=m(-rj,q)

(5.6)

manifests itself in a clockwise rotation by ^;r. Interestingly, the bulk-optics implementations of the Fourier transform operation (fig. 3.1) suggest an optical procedure to perform the ^Jt rotation of the Wigner chart in the (q, rf) plane. In this respect, we note that the free-space propagation along the distance d, described by the ray-optics matrix

nd)=(l

\ \

is.i)

produces a shearing of the Wigner distribution along the ^-direction in the {q, rj) plane, as formally expressed by Wo(q,V)=W,(q-j:r],r]\.

(5.8)

Likewise, the passage through a thin lens with focal length / , described by

corresponds to a shearing of the Wigner distribution along the //-direction, the transformation law (5.5) giving

Wo(q,V)=W,Ur] + j:qY

(5.10)

Typical examples of the deformations (5.8) and (5.10) experienced by the Wigner chart in the (q, t]) plane under the T and L transformations are shown in fig. 5.2 in the plain case of a rectangular Wigner chart W\(q, t]).

566

The fractional Fourier transform V k

r] A

(a)

[7, §5

(b)

(c)

Fig. 5.2. (a) Wigner chart in the {q, r]) plane and examples of deformations due to (b) free-space propagation and (c) passage through a lens.

Translating into (q, r/)-plane operations the sequences of free-space propagation and focusing relevant to the single- and double-lens optical setups of the Fourier transform (fig. 3.1), it is evident that the ^ic rotation of the Wigner distribution in the (q, rj) plane results from the sequence of three shearing operations, i.e., q, rj,q shearings or rj,q, rj shearings, with the relevant d a n d / parameters appropriately sized to d =f =fs. 5.3. Fractional Fourier transform and Wigner distribution function: optical analog of the (j) rotation in the Wigner phase plane The FrFT operation manifests itself as a clockwise rotation by 0 of the Wigner chart in the {q, rj) plane. In fact, the optical FrFT matrix F^(f) corresponds to the cos (/> sin 0 rotation matrix l?(0) = in the (q, rj) plane. Thereby, as the sig- sin 0 cos 0 nal xp\(q) is acted on by the optical FrFT operator F^^: ij^iiq) -^ %(q) = [F/^ V^i](^)» the corresponding WDF changes from W\(q, rj) to

wi"\q,n)=wA[Rm

I I = WiiqCOS
(5.11) which is obtained by rotating W[(q, rj) clockwise by the angle (j) = an/l. This has two basic consequences, concerning the possibility of an alternative and equivalent definition of the FrFT as well as its bulk-optics implementation (Lohmann [1993]). According to eq. (5.11), the FrFT of order a of the signal ^^xiq) can be implicitly defined as the transformation that is experienced by ^\{q) when the associated WDF is rotated in the {q, rj) plane by the ath fraction of ^JT. Furthermore, as the rotation by ^JT can be achieved

7, § 5]

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567

by optical means, which reaUze consecutively the proper q, r], q shearings or r/, q, T] shearings, similarly the rotation by the angle 0 can be optically realized by arranging appropriately sized free-space lengths and thin lenses, which rotate the WDF in the (q, r/) plane by the same type of three-stage transformations, i.e., q, r},q shearings or r],q, r] shearings. Specifically, the d a n d / parameters of the q and r] shears (eqs. 5.8 and 5.10) are given as ^=/stan(i0), ^

f-—^

(5.12) sm0

for the single-lens realization, and as ^=/sSin0,

/=/,cot(l0)

(5.13)

for the two-lens implementation. Apparently, the above correspond to the optical setups of types I and II, respectively, of the FrFT, discussed in §4.1. As already noted, the relation of the FrFT to the WDF, stated here as a consequence of the transformation law (5.5) and the fractional transform ray matrix (3.19), was introduced by Lohmann [1993] as an implicit definition of the FrFT. Lohmann's definition parallels that given by Mendlovic and Ozaktas [1993], modeled on the variation of the signal amplitude when propagating along a quadratic graded-index medium by a length proportional to the order of the transform (§3.5). The equivalence of the two definitions was proved by Mendlovic, Ozaktas and Lohmann [1994]. It relies on the fact that linear canonical transformations, which are represented by symplectic matrices (2x2 for one-dimensional systems), manifest in the signal-space representation through integral transforms having the mathematical form (2.7), and in the space-frequency representation through the transformation law (5.5) for the WDF. The linear canonical transformation underlying the ath-order FrFT is just the clockwise rotation with angle ajz/l in the phase space of the appropriate conjugate variables.

5.4. Fractional Fourier transform and Radon transform The two-dimensional Radon transform (Radon [1917]) is a set of ID frmctions obtained by integrating a 2D function over lines. It is convenient to think of the Radon transform as a projection of the 2D input function v{x',y') onto an axis y, which is rotated by an angle (j)fromthe y' axis. Accordingly, the Radon

568

The fractional Fourier transform

[7, §5

line of integration

Fig. 5.3. Geometry for the 2D Radon transform.

transform consists of two steps: rotation followed by projection, as expressed in the mathematical definition (Deans [1983], Barrett [1984])

^0Mx^y)}-|K[^(0)]-'(;))d7,

(5.14)

where R{(j)) denotes as usual the clockwise rotation matrix and (x,y) represent a coordinate system rotated by (j) with respect to the original set {x',y') (fig. 5.3). For a fixed angle 0, the above yields a ID function. Likewise, the 3D Radon transform is a set of ID functions obtained by integrating a 3D function over planes. The reconstruction of the 2D or 3D object from the ID projections is performed via the 2D or 3D inverse Radon transform (Clack and Defrise [1994]). The structure of the inverse transform is different in the 2D and 3D cases, or, more generally, in spaces of odd and even dimensionality. Apparently, when applied to the WDF the above definition straightforwardly leads to the squared modulus of the FrFT Explicitly, applying the definition (5.14) to the WDF of a given signal V^i(^), i.e..

7^^ {W{q.r])}= j

W,{[Rm-'{:i))dm

(5.15)

-oc

we obtain the Radon-Wigner transform (Wood and Barry [1992, 1994b]).

7, § 6]

Fractional Fourier transform and Fourier optics

569

On account of the Wigner rotation property, the direct correspondence between the Radon-Wigner transform (5.15) and the squared modulus of the FrFT of the signal V^i(^) becomes evident (Lohmann and Sofifer [1994]), i.e., ^ 0 {^(^, r])] =fs \F2IP]' (ql

(p = an/2.

(5.16)

By calculating and displaying the FrFT for all possible angles 0, one obtains the {q, a) display of the signal (Mendlovic, Zalevsky, Dorsch, Bitran, Lohmann and Ozaktas [1995]), also reported as the Radon-Wigner display, for which both optical and digital applications have been discussed by Mendlovic, Dorsch, Lohmann, Zalevsky and Ferreira [1996]. Although it is undoubtedly medical computed tomography (Wood and Barry [1994a]) that has attracted most attention, the Radon transform has been usefully applied also in pattern recognition (Gindi and Gmitro [1984]) and signal processing (Barrett [1982], Easton, Ticknor and Barrett [1984], Wood and Barry [1992, 1994b], Woolven, Ristic and Chevrette [1993]). Basic to all the applications of the Radon transform is the central-slice or projection-slice theorem (Deans [1983], Barrett [1984]). This theorem states that the ID Fourier transform of a projection of a 2D function is directly one line through the 2D Fourier transform of the function itself This line passes through the origin of the 2D Fourier frequency space (hence the term central), which accordingly can be sampled on a set of lines through the origin by transforming projections at different angles 0. The property of the Radon transform asserted by the central-slice theorem has been rephrased in terms of the FrFT, thus leading to the definition of what has been called the fractional Radon transform (Zalevsky and Mendlovic [1996b]). Application of this new transform has been demonstrated, for instance, in connection with the performance of the fractional Wiener filter (Zalevsky and Mendlovic [1996a]), being the minimization of the mean-square error, obtained after filtering non stationary signals, directly related to the fractional Radon transform.

§ 6. Fractional Fourier transform and Fourier optics The discussion in the previous sections has shown that the FrFT may arise as a mathematical model for specific diffraction and image formation problems, such as those related to light propagation in quadratic graded-index media and to the design and the analysis of lens systems.

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The fractional Fourier transform

[7, § 6

We now discuss the relation of the FrFT to Fourier optics, investigating its role within the general context of diffraction and image formation by arbitrary firstorder optical systems. We examine the conditions under which planar or spherical surfaces may be fractionally transformed into each other under propagation along an arbitrary ABCD system. The Fourier-transforming relation between privileged planes in an optical system turns out to be a particular case of the more general fractional Fourier-transforming relation between intermediate surfaces. The order of the transform, which is closely related to the entries of the transfer matrix between the given surfaces, provides a measure of how far the system is from performing the Fourier transform of the object (Pellat-Finet [1994], Pellat-Finet and Bonnet [1994], Abe and Sheridan [1995], Ozaktas and Mendlovic [1995, 1996], Ozaktas and Erden [1997], Hua, Liu and Li [1997], Mas, Ferreira, Garcia and Bernardo [2000]). Fourier optics is based on the formalism of the ABCD matrix M (3.2) and the associated ABCD integral (3.1), which provides the functional form of the transfer operator M relevant to the optical system described by M. Due to the isomorphic correspondence, expressed through the relation (3.3), between the optical matrix M and the corresponding operator M, we describe the problem within the matrix representation of Fourier optics, the expression (3.3) providing the tool to translate relations between ray matrices into relations between transfer operators whenever we are interested in the transformations of wave fields. 6.1. The ABCD integral and the fractional Fourier transform Any 2x2 real unimodular matrix M = f ^

1 can be decomposed into

different forms (Nazarathy and Shamir [1980, 1982a,b], Dattoli and Torre [1991]). In the present context, we consider the parametrization (Ozaktas and Erden [1997]) 'A C

B\^f\ Dj [-V

0\fm l)[o

O\/cos0 :.. l) 1^-11 .sin0

/ssin0 .... COS0

/.

(6.1)

which gives M as the product of the lower triangular matrix LiV), the diagonal matrix S(m) and the fractional Fourier-transforming matrix F^(f). The realvalued parameters m, V and 0, defining the product above, are given explicitly as 2

.2

^^

.

^

^

AC + DB/f^

,^^^

7, § 6]

Fractional Fourier transform and Fourier optics

571

The focal length /s works as a scale parameter needed to turn from the optical coordinate q to the dimensionless variable JC = y/k/Jlq, thus adapting the definition (3.19) of the optical fractional transform to the mathematical definition (2.7). Each matrix in the product (6.1) can be given an optical interpretation. The diagonal matrix S{m) is implementable as a magnifier with magnification factor m, while the lower triangular matrix L{J^) is properly accounted for by a thin lens with focusing power V = \/f. Correspondingly, the lens systems previously discussed may represent appropriate optical realizations of F^(/s). Thus, while F^(fs) generates the diffraction integral (3.19), S(m) and L(V) manifest themselves in the complex amplitude distribution \p\(q) of the incoming signal through, respectively, the scaling operation ^p^(q)=-L^^i^g/m)

(6.3)

and the chirp modulation %(q) = exp(-i^P q^/2) xp^iq).

(6.4)

Accordingly, any Fourier optical system can be synthesized by the input-output relation %(q) = - — exp(-i/:P q^/2)[Flxp,]{q/ml

(6.5)

which expresses the amplitude distribution %(q) over the output plane as the scaled FrFT of the amplitude distribution \p\(q) over the input plane with a frirther quadratic phase term, i.e., as the imperfect scaled FrFT of il^\(q)^. The above offers an equivalent representation of the ABCD integral, which then can be alternatively characterized by the scale factor m, the curvature parameter V and the angle 0, and hence the order a, of the FrFT. Evidently, V specifies the curvature radius R = -\/V of the spherical surface on which the perfect (scaled) FrFT can be observed^. When P = 0, the quadratic phase factor

^ One may talk of an imperfect FrFT, in the same sense as one talks of an imperfect image or an imperfect Fourier transform. An imperfect FrFT displays a residual quadratic phase factor in the output variable with respect to the basic definition (3.19). ^ We note that the sign of R is taken according to the sign convention used to describe beam wavefi-onts, and hence is positive for diverging beams and negative for converging beams.

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The fractional Fourier transform

[7, § 6

disappears and the perfect (scaled) FrFT is observed on a planar surface {R = oo). The pure optical FrFT corresponds to m = 1 and V = 0. As an example, we discuss the Fresnel diffraction between two surfaces. 6.2. Fresnel diffraction between two planar surfaces The free propagation between two planar surfaces, say TIj and UQ, is described by the Fresnel integral

""'"'"^'alw/"'' .i(,-,')'

Uci')dq\

(6.6)

where d = Zo-Z[ specifies the distance between the two planes. It has been proved that the expression (6.6) for the Fresnel integral can be adapted to the expression (3.19) of the FrFT, thus paralleling the relation of the Fraunhofer integral to the standard Fourier transform (Pellat-Finet [1994], Pellat-Finet and Bonnet [1994], Gori, Santarsiero and Bagini [1994], Abe and Sheridan [1995], Ozaktas and Mendlovic [1995], Ozaktas and Erden [1997]). In fact, according to eq. (6.5), the Fresnel transform can be interpreted as a magnified FrFT with a residual phase curvature. Explicitly, the upper triangular matrix (5.7), representing the fi-ee-space propagation by d, can be decomposed as prescribed by eq. (6.1) into T{d) = L{V)S{m)F\f\

(6.7)

the relevant parameters m, V and 0 being

m^ = x + dl r~

rr

^ /s^ci+rfv//)'

tan 0 = -

(6.8)

Here, the scale parameter can be conveniently set to y^ = A:. Then, according to eq. (6.5), the Fresnel diffraction pattern can be interpreted as the scaled FrFT of the diffracting signal. The transform is observed on the spherical surface 1^ with curvature radius

R, = d(\Jf\

(6.9)

at a distance d from ili. The order of the transform increases monotonically 1. with the distance d. As d -^ oo, the angle (p approaches ^Jt and hence the

7, § 6]

Fractional Fourier transform and Fourier optics

573

fractional transform turns into the ordinary Fourier transform. Correspondingly, since m —^ d/fs and R -^ d, one recovers the far-field diffraction pattern, which is the Fourier transform of the signal at /Jj, observed on the spherical surface with curvature radius d. A unifying mathematical description of the diffraction in a free medium is therefore provided by the fractional Fourier formalism, which embraces the ordinary Fourier formulation as a limiting case (see also § 7.1). Interestingly, the product (6.7), inverted for F^(fs), corresponds to the functional expression (2.19) giving the FrFT in terms of the Fresnel integral. We emphasize also that the diffraction integral has been demonstrated to have the typical structure of a wavelet transform (Onural [1993]), thus supporting the considerations developed in connection with the representation (2.19). A noteworthy fact is that the amplitude distributions on the plane /Ij and on the spherical surface IQ of curvature radius RQ are directly connected by a fractional Fourier transformation. The expression (6.9) for RQ closely resembles that for the curvature radius R of the wavefront at ^ of a Gaussian beam having the plane 77i as waist, i.e., R(d) = d(l ^z^/d^) (Siegman [1986], Saleh and Teich [1991]). The scale parameter/s then takes the meaning of the Rayleigh length ZR of the beam: ZR = ^kwQ, with Wo the waist radius'^. Besides, the magnification factor m has the form of the normalized beam width w at the distance d from the waist which, according to the law w^(z) = w^izYwl = 1 -\-z^/zl, yields m^ as in eq. (6.8) with z = d and ZR=f^. Finally, the order a of the fractional transform is proportional to the Gouy phase ^ of the beam which, expressed as C(z) = arctan(z/z/?),

(6.10)

turns into the angle 0 of the transform with properly setting z = J and ZR^f^. Thus, we can say that the propagation of a Gaussian beam in free space naturally occurs through scaled fractional Fourier transformations, whose order is directly linked to the Gouy phase shift as the scale factor accounts for the variation of the beam width with propagation. 6.3. Fresnel diffraction between two spherical surfaces Let us consider two spherical surfaces 1\ and li, centered on the common optical z-axis, with the vertices located respectively at z\ and zj (z\ < zi), the

We recall that when dealing with the propagation laws of Gaussian beam parameters, the z-coordinate is commonly measured with respect to the beam waist.

574

The fractional Fourier transform

[7, § 6

origin of the propagation variable being arbitrarily assigned^. We denote the separation distance hy d = zi- z\, and the curvature radii of the surfaces by Rx andi?2. Evidently, the diffraction integral for the free propagation from 1\ to I2 can be interpreted as a FrFT with a further suitable scaling factor, if the geometric quantities R\, R2 and d satisfy appropriate relations. The ray-transfer matrix for the free propagation from I\ to I2 can be cast as g\

d

where the g-parameters, defined as (Siegman [1986], Saleh and Teich [1991]) gi = l - | - ,

g2 = l - | - ,

K\

K2

(6.12)

are customary within optical resonator theory^. In order to examine the condition under which the two surfaces are connected by a perfect magnified FrFT, we write G as G(gug2) = S(m)F'(U

(6.13)

and solve for the parameters w,/s and 0. Thus, we obtain 0 = arccos(di v ^ ^ ) ,

(6.14)

which provides the inequality 0 < g i g 2 ^ 1,

(6.15)

as a necessary and sufficient condition for a fractional Fourier-transforming relation of real order to hold between I\ and 12- It is needless to say that if complex orders are allowed the condition (6.15) can be relaxed (see §§2.1 and 4.3). Interestingly, the above inequality is known in the theory of optical resonators as the stability (or confinement) condition; it characterizes

^ We speak quite improperly of spherical surfaces; actually, circular or cylindrical surfaces are more proper terms in connection with the one-dimensional systems we are dealing with. ^ The sign of the curvature radii R\ and R2 is now taken in accordance with the convention adopted to describe resonator mirrors, and hence is positive or negative according to whether the corresponding surface is concave or convex as seen from the space in between.

7, § 6]

Fractional Fourier transform and Fourier optics

575

stable resonators, while the complementary inequality, i.e., g\g2 > 1, is distinctive of unstable resonators. The ordinary Fourier transform relation is obtained between I\ and I2 if gi = 0 or g2 = 0 or also if gi = gi = 0, which are known to define, respectively, the confocality and symmetrical confocality relation between the two surfaces. The scale factor m and the parameter/§ are obtained in the form m' = '-^,

f^=d'

g2

''

..

(6.16)

gl(l-glg2)

On account of the well-known expressions for the parameters of the Gaussian beam that properly fits between the two surfaces I\ and I2, it is easy to recognize m^ as the magnification factor of the beam width for propagation from I\ to I2, i.e., m^ = wj/wj, where w\ and W2 denote the beam radii at I\ and I2. Correspondingly, the parameter/s is linked to the beam width at I\ through Then, according to eqs. (6.14) and (6.10), the angle 0 is recognized as the Gouy phase shift associated with the propagation of the fitting beam from 2'i to 0-t(^2)-?(zi),

(6.17)

where z\ and Z2 fix the location of I\ and I2 with respect to the waist. In particular, with g] = 1, the geometry turns into a plane and a spherical surface and accordingly we regain the results of the previous subsection. The Fresnel diffraction between two surfaces can therefore be expressed as an appropriately magnified fractional Fourier transformation of real order if the g-parameters relevant to the two surfaces obey the stability condition (6.15). Evidently, the above considerations establish also the relevance of the fractional transform formalism to the theory of optical resonators; fractional transforms of real order directly relate to stable resonators, while complex-order transforms are appropriate for describing unstable resonators (Ozaktas and Mendlovic [1994]). Similarly, the symmetrical decomposition (Ozaktas and Mendlovic [1995]) G(gu gi) = 5(m2)F«(/s)5(mi),

(6.18)

which explicitly involves two scaling factors, m\ and ^2, yields the relation (6.14) for the angle 0, and the expressions

for the scale parameters m\ and ^2. Within the picture of a Gaussian beam having I\ and I2 as wavefronts, the above reproduce the expressions for the

576

The fractional Fourier transform

[7, § 6

normalized Gaussian beam radii at the wavefronts Il\ and Hi, i-e., m\^ = wf/wg and ^2 = H^I^^O' where the parameter^ plays the role of the Rayleigh length. The two representations (6.13) and (6.18) are totally equivalent. In both cases, the resulting integral transform takes the basic form (2.7), with the input and output optical variables being scaled to dimensionless variables through the top hat beam radii w\/\fl and W2/V2, respectively.

6.4. The ABCD integral as an extended FrFT Let us apply the representations (6.13) or (6.18) to an arbitrary optical system, described by the ray matrix M. We write, for instance, M = S{m2)F\f,)S{mx\

(6.20)

which, once solved for the inherent parameters, provides the general conditions under which the optical system M can be viewed as performing the FrFT of the input signal with two scale factors, one for the input and the other for the output. Thus, we obtain the relation cos^(l)=AD,

(6.21)

which yields the inequality 0 ^ ^D ^ 1

(6.22)

as a necessary and sufficient condition to interpret the optical transform by M of the input into the output plane as a FrFT of real order. Correspondingly, the scale factors mi and m2 write as _4

B^

D

4

B^

A

which are the obvious generalizations of expressions (6.19). In the more general case of complex-order transforms, the condition (6.22) can be relaxed, and accordingly the scale parameters mi and m2 are allowed to be complex.

7, § 6]

Fractional Fourier transform and Fourier optics

577

If the condition 0 ^ AD ^ 1 is not satisfied, we may assign spherical surfaces, instead of planar ones, as input and output reference surfaces for the system. Thus, if M refers to planar reference surfaces, the optical matrix

accounts for spherical reference surfaces. The g-parameters, defined as h=A-^,

g2=D-^,

(6.25)

where R\ and R2 denote the curvature radii of the reference surfaces 1\ and I2, can be regarded as a kind of g-parameters of the optical system with respect to 1\ and 2*2, on account of the fact that the entry B is commonly interpreted as the effective length of the optical system. Evidently, the condition for an FrFT of real order relating the input and output signals as observed on 1\ and I2 is 0^gig2^1,

(6.26)

which then can be satisfied by an appropriate choice of the curvature radii R\ and R2 of the reference surfaces. Practically, the configuration (6.24) can be realized by appending lenses to the input and output reference planes of the system, with the focusing powers V\ and V2 of the lenses suitably fixed according to the relation V = -l/R and the condition (6.26) (Ozaktas and Mendlovic [1995]). It is interesting to inspect the expression for the ABCD integral corresponding to the sequence of optical transformations generated by eq. (6.20), namely.

[M%-\{q) =

m

^k

2/sSin0

f

J ,j

q

,

COS0H

1

2

. ^^1

xq COS0-2—< ^2 rn2

Uq)Aq.

(6.27) which suggests a generalized, also called extended (Hua, Liu and Li [1997]), form of FrFT. In fact, scaling the optical coordinates to dimensionless variables.

578

The fractional Fourier transform

[7, § 6

the above integral naturally suggests the generalized form of the transform (2.7) as [^.,«,.^](^)=..l-'COt0 IJT + OC

/exp

2sin0

(six' ^ COS (j) + ^^JC^ COS (j) -

2SoS\Xx')q){x') dx\

-(X)

(6.28) which depends on the angle (j) and on the two scale parameters s\ and 5*0 for the input and the output. The multiplication factor above has been adjusted in accordance with the definition (2.7). The basic fractional transform T^ is recovered from the generalized form ^^«'^''^' with ^o ^ -^i ^ 1Evidently, the generalized fractional transform (6.28) expresses in mathematical terms the three-step optical operation represented by (6.20), and hence is equivalent to scaling the object by the factor s\, then performing the basic FrFT on it, and finally scaling the result by ^-^^ The matrix correspondence r-''^'^^S{sl')R{(l>)S{s,)

(6.29)

can therefore be stated, whereas for the basic transform T^ we proved J"^ <^ i?(^).

(6.30)

In addition, the parameters ^i, a and SQ are in general allowed to be complex, thus enabling one to also take account of field curvatures. On account of the equivalence stated by eq. (6.20), the extended fractional transform can describe more general (real and complex) optical systems, and hence physical phenomena, than the basic fractional transform. Thus, for instance, whilst the basic FrFT relates symmetrical planes on the two sides of a lens, the extended FrFT connects arbitrary planes. The condition for the additive operation of the generalized transform (6.28) with respect to the order a is immediately inferred from the matrix correspondence (6.29). Since indeed S{s';')R{(l)2)S{s[)S{sl')R{(t>^)S{s,) = S{C')R{0^

+02)5(^i),

(6.31)

only if ^j'^-"^ = 1, the transform (6.28) is seen to be additive in the sense that ^Cci2,s[^So,ax,s,

^ ^s'^.a\+a2,s, ^

(6.32)

only if the output scale parameter ^o of the prior cascade is equal to the reciprocal of the input scale parameter s[ of the next one (Hua, Liu and Li [1997]).

7, § 7]

Fractional Fourier transform and wave-propagation optics

579

However, even when the order additivity is not satisfied, cascading two or more generalized fi-actional transforms of the type (6.28) always composes into an overall generalized transform of the same type, whose parameters s\, (/) and 5-0 are obtained through the matrix relation S(s':-')R((l)2)S(s[')S(s','')R((l>0S(sl)

= S(s-')R((l>)S(s,).

(6.33)

In the language of § 7.2 below, the tandem rule (6.33) relies on the fact that the integral transform (6.28) is the general representation of the generic element in the group, generated by second-order differential operators; thus, eq. (6.28) simply expresses the closure property of the group. § 7. Fractional Fourier transform and wave-propagation optics Namias [1980a] proved that the FrFT and its associated operational calculus provide a convenient and systematic technique for solving certain classes of ordinary and partial differential equations of the second order containing quadratic terms in the variables involved. In fact, applications to the timeindependent Schrodinger equation for the ft-ee and forced one-dimensional quantum-harmonic oscillator have been illustrated, and then extended to the three-dimensional problem of the motion of electrons in constant, as well as time-varying, magnetic fields. Once introduced in optics, the concept of FrFT and the relevant formalism revealed their potential as a mathematical tool to solve partial differential equations, ruled by second-order differential operators, in connection with the specific problem of the paraxial wave propagation in passive optical media (Alieva, Lopez, Agullo-Lopez and Almeida [1994]). The problem fits naturally in the general context of the linear canonical transforms and the closely related evolution operator formalism (Agarwal and Simon [1994], Dattoli, Torre and Mazzacurati [1998], Torre [2001]), thus suggesting a ftirther extension of the field of applications of the FrFT from classical to quantum optics (Aytur and Ozaktas [1995]). 7.1. Wave propagation in free and graded-index media The scalar wave equation for the paraxial propagation of the monochromatic wave t/^(^,z)e~^^^'^^~^"^ in a homogeneous and isotropic medium is written in the well-known parabolic form as:

380

The fractional Fourier transform

[/, § /

where HQ denotes the refractive index of the medium and, as usual, \l)\{q) is the wave ampHtude at the input reference plane. The solution to the above equation is expressed by the Fresnel diffraction integral (6.6), with the fixed length d being properly replaced by the varying distance (z-Zi)/«o- Consequently, the analysis developed in §6.2 allows us to state the relevance of the FrFT for the wave propagation in a free medium, with the order of the transform given by tan (p = k{z- z\)/n(). Likewise, the propagation in a graded-index medium, modeled by a quadratic refractive index profile such as eq. (3.21), where the gradient parameter «2 niay generally vary with the propagation variable z, i.e., n{q,z) = no - ^n2(z)q^, is described by the following equation for the wave amplitude ^^: il^2F;^$-5«^(^)^')'^(^'^)=«'

'/'(^'-.)='^.(^)'

(7.2)

The solution to eq. (7.2) has been proved to have the form of the Huygens integral (3.1) with the relevant response fiinction g given according to eq. (3.3) (Dragt, Forest and Wolf [1986], Dattoli, Solimeno and Torre [1987], Dattoli and Torre [1991]). In that case, the coefficients A, B and D are understood as fiinctions of the propagation variable z. They arise as solutions of the first-order differential equations

A{z,,zO = 1,

C(zi,Zi) = 0,

5(zi,Zi) = 0,

D(zi,Zi) = 1,

(7.3)

where the primes denote z-derivatives. In other words, the wave propagation in a graded-index medium is equivalent to the propagation through an optical system '^(z,Zi) B(z,z,) described by the z-dependent ABCD matrix M(z,Zi) . ^. \ n/ ^ \ C (Z, Z[)

Lf(Z, Z[)

whose entries are obtained from eqs. (7.3). The general analysis of §6.1 can be applied to M(z,Zi), thus enabling a solution to the wave-propagation problem (7.2) in the form of the integral (6.5). The relevant parameters m, (j) and V are understood as depending on the propagation variable z and related to the medium parameters «o and n2 through eqs. (6.2) and (7.3). The scale parameter/s is naturally suggested as/s = l/y/nonj. '^ The one-dimensional picture, inherent to both eqs. (7.1) and (7.2), is suitable for the propagation of a factored wave amplitude through a separable astigmatic medium, whose index profile may be thereby modelled as n(qy,qy,z) = «o - 2«2.v(^)^.v - ^"2vi^)^y- In the general case, the transversal Laplacian operator V^ takes the place of its one-dimensional counterpart

7, § 7]

Fractional Fourier transform and wave-propagation optics

581

by the established link between the wave propagation in a constant-parameter graded-index medium and the FrFT. Evidently, for constant medium parameter niiz) = «2, we obtain m= \,V = 0 and (f) = \/n^7rio(z - z^) = ^jr(z -Zi)/L, where L is the characteristic distance of the medium (§3.5). Therefore, the formalism of the FrFT can be adequately used to describe the wave propagation both in a homogeneous isotropic medium and in a quadraticrefractive-index optical fiber, which then turn out to be connected through a unifying and elegant formulation closely related to the Fourier-transform formalism (Alieva, Lopez, Agullo-Lopez and Almeida [1994]). Also, since the FrFT is well suited for numerical computation by the use of the fast Fourier transform algorithm (Garcia, Mas and Dorsch [1996], Marinho and Bernardo [1998]), the formulation of the optical wave-propagation problem in terms of the FrFT may have practical, in addition to conceptual, advantages. From a purely mathematical viewpoint we established the relation between the FrFT and the solution of parabolic differential equations, ruled by second-order differential operators, of which eqs. (7.1) and (7.2) are basic examples (Namias [1980a], Dattoli, Torre and Mazzacurati [1998]). In this respect, it is worth stressing the formal similarity of eq. (7.2) to the time-dependent Schrodinger equation for the quantum-harmonic oscillator, whose classical frequency depends on time. It has been demonstrated by Agarwal and Simon [1994] that the Green function for the quantum-harmonic oscillator, when expressed through appropriate dimensionless quantities, is the same as the kernel (2.8) of the FrFT. The order parameter of the transform corresponds in that case to time, whereas, as proved above, in the case of the wave-propagation problem it relates to the coordinate z along the direction of propagation. Both time and z play the role of the evolution variable for their respective problems. The correspondence established above relies on the general property of the linear canonical transforms to arise as evolution operators associated with parabolic differential equations (Wolf [1976]), as will be briefly reviewed in the next subsection. 7.2. Canonical transforms and parabolic differential equations Equations (7.1) and (7.2) belong to the class of differential equations of the form d i—(p(x, r) = H(p{x, r),

(p{x, T,) = cpXx),

(7.4)

where H is understood as a linear operator acting on the function (p(x, r), and the parameter r plays the role of the evolution variable. Evidently, in the case of the

582

The fractional Fourier transform

[7, § 7

wave equations (7.1) and (7.2), r corresponds to the real propagation variable z, while H takes the specific realizations Homogeneous isotropic medium:

H ••

Quadratic-index-profile medium:

H = -— - ^ - -riiiz) a . 2kno dq^ 2

2kno dq^' \

d^

k

J

It is a well-established approach, which resorts to the time-evolution operator formalism of quantum mechanics (Messiah [1961]), to express the solution of eq. (7.4) as the mapping of the fiinction at T\ to the fiinction at r, by exponentiating the operator H through {T -T\) according to cp{x, T) = exp[-i(r - r,) H] (pi(x).

(7.5)

Specifically, when H can be expressed as a linear combination of the secondorder expressions of the Schrodinger observables x and -id/dx, i.e., H = ax^ + bx—- + C-—r, dx

(7.6)

djc^

the transfer operator exp[-i(r - T\) H] takes the form of the integral transform I

+CXD

(p,(x)dx'.

(7.7)

The functions A, B and V entering the triangle function in the integral kernel Air,T,) 6(r,ri) above are entries of a 2x2 symplectic matrix M(r,r\) . ^. . ^, . and obey in general a set of first-order differential equations involving the coefficients a, b and c in the linear operator (7.6)' ^ The representation of the evolution operator as the single exponentiated form exp[-i(r - ri)H] is strictly correct only in the case when the operator H does

^ * In order to give a flavour of the Lie-algebra-related context in which parabolic differential equations of the type (7.4) are commonly considered, we note that the second-order expressions for the position- and momentum-like operators x and -i d/djc close into a finite-dimensional Lie algebra under the corresponding Lie products. In fact, the operators

''^=2^'

''- = - 2 5 ? '

''3=-2(-J^ + 2)

W

7, § 7]

Fractional Fourier transform and wave-propagation optics

583

not depend explicitly on the evolution parameter r. Hence it is not appropriate to the wave equation (7.2) with a z-dependent medium parameter riiiz). In the case of a r-dependent operator H, suitable techniques allow one to express the relevant transfer operator as the product of proper exponential forms, however with the resulting functional transformation having the form of the integral transform (7.7) (Wilcox [1967], Dattoli, Gallardo and Torre [1988]). A direct correspondence between the dynamical problem (7.4), ruled by the second-order differential operator (7.6), and the linear canonical transforms is therefore estabUshed (Wolf [1976]). The diffraction integral (3.1), with the kernel (3.3), and the associated ray-transfer problem (7.3) provide an example of such a correspondence in the case of the wave-propagation problem (7.2). Basically, this correspondence is based on the matrix realization of any exponential form exp(-irH) as a 2x2 symplectic, generally complex, matrix. In particular, for the exponentiated forms of the single second-order expressions \x^, \ d^/dx^ and xd/dx the following matrix realizations can be recognized

exp(-ipy]

^ L(p) =

-P(i3^)

-- ni) =

" 5 /

exp

d

IM

J('^^0.

U ?)•

(so-

^. . / m ^ 5(m) =

0\

lo I)

(7.8) . ,.. ra = exp(5/2)

The similarity with the optical ray matrices representing chirp modulation, freespace propagation and scale operation is apparent, and accordingly the functional forms of the relevant exponential operators, which directly follow from the general strategy (7.7), write as those of the corresponding optical operations.

are easily proved to produce the commutator brackets [K+,

KJ =2iK3,

[K±,K3J

=±iK±.

They represent the basic reaUzation of the generators of the dynamical algebra su(l,l) of the quantum-harmonic oscillator, whose isomorphism with other Lie algebras as the algebra of the unimodular 2x2 matrices is well known (Gilmore [1974]). When the operator H is of the form (7.6), it belongs to the algebra (*) and hence its exponentiated forms through real or complex parameters belong to the corresponding group SU(1,1). From the viewpoint of optics, the second-order expressions K+, K_ and K3 correspond directly, by exponentiation to the group, to lens operation, free-space propagation and scale operation.

584

The fractional Fourier transform

[7, § 8

Particular attention deserves the simple linear operator

-^^''-h)

W= IU^-:rT

>

(7-9)

having the base structure of the Hamiltonian of the quantum-harmonic oscillator. The corresponding evolution operator can be seen to have as representative matrix the rotation matrix /?(0):

Thus, comparing the integral transform (7.7), generated by the canonical transformation /?(^), with the mathematical expression (2.7) for the FrFT operator, we may infer the so-called hyperdififerential representation of T^ according to ^« = e x p ( . 0 e x p ( - . 0 7 ^ ) = e x p [ - . ^ ( x ^ - ^ - l )

(7.11)

Differentiating the above with respect to (j) enables one to represent the dynamical problem associated with the FrFT operator in the form of the quantum-harmonic oscillator-like differential equation,

where the angle (j) plays the role of the evolution variable. Apparently, the generalized form ^''-^'-'' of the FrFT arises from the dynamical problem ruled by the general second-order differential operator (7.6).

§ 8. Operational properties of the fractional Fourier transform We briefly describe some of the operational properties of the FrFT, which generalize the well-known properties of the ordinary Fourier transform (Namias [1980a], McBride and Kerr [1987], Mendlovic and Ozaktas [1993]). Basic to the present analysis is the hyperdifferential representation (7.11) of the FrFT operator ^ ^ , from which the basic properties listed in §2 follow straightforwardly.

7, § 8]

Operational properties of the fractional Fourier transform

585

8.1. The similarity rule We deduce the similarity rule for the FrFT, recalling that for the Fourier transform it is expressed by the well-known relation T{cp{ax)] = ^q){-\ \a\ a

(8.1)

with a an arbitrary real parameter. The function q){ax) can be regarded as resulting from the scale transformation of q){x) performed by the operator exp[-|5(jc^ + ^)], with a = exp(-^5). Then, we can write f''{q){ax)]

= —= exp(i0/2)exp(-i07^)exp

^M-^^i

cp{x). (8.2)

Evidently, on account of the matrix realizations (7.8) and (7.10), the operator exp(-i0W)exp[-^5(x^ + -)] can be seen to correspond to the matrix /?(0)5'(l/a), which, according to the general rule (6.1), is decomposed into Ri<^)S f l )

= ( ^rt,

' ' " " t ) =LiV)Sim)Ri,t>').

(8.3)

The relevant parameters m, V and 0' are identified through tan0' tan 0

7

^

,A \

cos^0'\ cos- 0 y

sin0 sm 0'

.

,JT 2

(8.4) Exploiting again the correspondence between the matrices in eq. (8.3) and, respectively, the lens, scale and fractional Fourier operations, we can manipulate the right-hand side of eq. (8.2) to establish the similarity rule for the FrFT in the form X sind)^ COS-^ 0^ ;^n r . .^ 1 /I -icot0 ^''{(Piax)} = —J . I exp i ^ ( l-TTTTT lcot0 (pee ' \a\ y 1 - i c o t 0 ' 2 V^ cos^0 ^a sin0 (8.5) Notably, the ath-order FrFT of q){ax) cannot be expressed as the scaled version of ^o(x), as in eq. (8.1); indeed, it turns out to be the scaled and chirp-modulated version of ^o'(^) for a different order a', determined by the order a and the scale factor a as prescribed by the first of equations (8.4).

586

The fractional Fourier transform

[7, §8

8.2. The multiplication rule The FrFT of the product xcp{x) is a rotational mixture of the x-products in the normal and Fourier sense. We write ^''{xcpix)} = [^^x^-«]^^{(p(jc)} = X(-a) q)a{xl

(8.6)

where the operator X(a), formally defined through the similarity transformation X(a) = ^ - ^ x ^ « ,

X(0) = X,

(8.7)

expresses the evolution of the position-like operators under T^. Exploiting the exponential representation (7.11) of ^ , we can give X(a) the explicit form ^^ X(a) = exp(i^W)x exp(-i0H) = cos0x-isin0-—,

(8.8)

QX

thus providing for J^"{xq)(x)} the expression ^«{^
cos0x + i s i n 0 - - Cpaix), ax

(8.9)

which generaUzes to a ^ I the relation holding for the Fourier operator: (8.10)

T{x(pix)} = i— (fix).

8.3. The derivative rule The FrFT of the derivative £ cp{x) is a rotational mixture of the derivatives in the normal and Fourier sense. We write

f'{iii')]-

Ax

r'{(p{x))

= iP(-a)(?)„(x),

'^ We recall that operators of the type e"'^ Be "^ are defined by the series ^aA 3 ^-aP. ^ 3 ^ ^^^^ 3 j ^

j ^ ^ j ^ ^ 3JJ _^

[^j^^ j ^ ^ 3JJ] ^

(8.11)

7, § 8]

Operational properties of the fractional Fourier transform

587

where the operator P(a) expresses, through the formal definition

PC^)-^"!-^)^",

?(0) = ->^,

(8.12)

the evolution of the momentum-like operator - i ^ under T^. It is given explicitly by

P(a) = exp(i0W) I - i — | exp(-i0W) = - s i n ^ x - i c o s ^ — , ^ ox I ax

(8.13)

thus leading to

f'{m-

isin0x + cos0

dx

cpa(x).

(8.14)

With a = 1, we regain the well-known result for T:

H—(p{x)\=\x(f){x).

(8.15)

Interestingly, according to eqs. (8.8) and (8.13) the operators X(a) and P(a) evolve with a from the values at a = 0, i.e., X(0) = x and P(0) = - i ^ , through the rotation matrix R{^). Thereby, it is easy to infer the commutation product [X(a),X(a')] = - i s i n ( 0 - 0 ' ) ,

(8.16)

on account of the well-known commutator [X(0), P(0)] = i. This commutation relation implies the uncertainty relation

(X{af)(x{a'f)^\sm\(l>~(l>')

(8.17)

for the variances of X(a) and X(a') (Lohmaim [1993], Aytur and Ozaktas [1995]).

588

[7, §8

The fractional Fourier transform

8.4. The fractional Fourier transform and the Weyl group We will now derive the shift theorem for the FrFT. Let us apply the operator i^^ to the shifted ftinction q){x -b). On account of the fact that (8.18)

q){x - Z?) = exp ( -Z?— ) cp{x\ where exp(-Z)^) is the shift operator, we can write ^«{<j9(x-6)} = ^ « e x p ( - Z , - ) ^ - «

J^^^(p{x) = exp iZ)P(-a)

(8.19) with P(a) given by eq. (8.12). Applying the decomposition formula for the Weyl group to the operator exp[-iZ7P(-a)] (Wilcox [1967]), we end up with

f b^

\

J^^{q){x - b)} = exp I i— sin 0 cos (j) j exp(-iZ?x sin (p) q)a(x - b cos 0), (8.20) from which the relation for the Fourier transform follows in the well-known form T{(p(x -b)} = exp(-iZ?;c) q)(x).

(8.21)

According to eq. (8.20), a shift of the input causes a shift of the output, controlled by the order of the FrFT. This order-dependent shift variance characterizes all FrFT-based operations, and has proved to be desirable in some applications, such as, for instance, in pattern recognition processes, since the position of the object may provide an additional encoding feature (Lohmann, Zalevsky and Mendlovic [1996]). Likewise, let the operator ^ ^ act on the ftinction exp(iZ>x) q){x). Writing ^'^{exp(iZ?x) cp{x)} = [^« exp(iZ>jc)^-'^] P'cpix) = exp[iZ?X(-a)] q)a{x\ (8.22) with X(a) given by eq. (8.7), and decomposing the operator exp[iZ?X(a)], we obtain

f b^

\

^

.

^^{exp(iZ?x) (p{x)} = exp I -i — sin (j) cos 0 I exp(iZ?x cos (j)) (pa(x - b sin 0), (8.23) which with a = 1 turns into the familiar result for the ordinary Fourier transform: J^{exip(ibx) (f(x)} = (p(x - b).

(8.24)

7, § 8]

Operational properties of the fractional Fourier transform

589

As a final comment, we emphasize that the relations exp[iZ>X(a)] = exp{iZ?[cos0X(0) + sin # ( 0 ) ] } ,

,^ _._

exp[iZ?P(a)] = exp{iZ?[- sin 0 X(0) + cos 0P(O)]} confirm that the Weyl group evolves into itself under T^, i.e., under quantum-harmonic oscillator-like dynamics. 8.5. The transform of a product Here we compute the FrFT of the product 0{x) of two functions, 0{x) = (p,{x)cp2{x).

(8.26)

Expressing (p\(x) in terms of its fractional transform, T^(f)\, we get'^

^n
exp(iy cot0)

J

IK

sin^ 0

+ OC

X

(8.27) Then, the FrFT of the product 0= q)\q)2 is obtained by multiplying the fractional transform of (p\ by a chirp function, convolving with the scaled Fourier transform of q)2, and finally multiplying by a chirp function and a scale factor. Evidently, the roles of the two functions q)\ and (p2 can be interchanged. The relation for the ordinary Fourier transform is well known: 1 /" _ _ T{(pi(x)(p2(x)} = -1= / (pi(x)(f^(x-x)dx' V2jr J

1 _ _ = -j= {(pi * (P2) (x), y/ZJT

-oc

(8.28) where the symbol * denotes the convolution operation. The Fourier transform of the product of two functions is basically the convolution of their Fourier transforms. ^^ To avoid any confusion, we stress that in this and the following subsections ^1 and ^2 denote the ordinary Fourier transforms of the functions q)] and (P2.

590

The fractional Fourier transform

[7, § 8

8.6. The transform of a convolution We now consider the convolution of the functions q)\ and cp2'. ^{x) = ((39, * (P2){x) = f (ft (x')
(8.29)

-OC

Acting on 0(x) by the FrFT operator i^", we have +00

^"{4>ix)} = j

(ptix')T"{(p2{x-x')}dx',

(8.30)

-oo

which, on account of eq. (8.20), after a simple algebraic manipulation becomes +OC

2

-OC

(8.31) The FrFT of the convolution 0 = q)\ * (p2 is therefore obtained by chirpmodulating the FrFT of one of the ftinctions, convolving with the scaled version of the other function, and finally multiplying by a chirp fijnction and a scale factor. With a= 1, we recover the convolution theorem for the Fourier transform, T{0(x)}

= q),ix)q)2{xl

(8.32)

which gives the Fourier transform of the convolution of two functions as the product of their Fourier transforms. Other expressions for the FrFT of both a product and a convolution can be found in Almeida [1997]. Notably, the product and the convolution are conjugate operations with respect to the Fourier transform. This duality is lost in the case of the FrFT. Also, the convolution formula (8.31) does not generalize the nice result (8.32) for the Fourier transform. Zayed [1998a] proposed an alternative convolution structure for the FrFT, which preserves the Fourier-transform property (8.32). 8.7. Eigenfunctions and eigenvalues We conclude the review of the properties of the FrFT by briefly commenting on the problem of the eigenfiinctions and corresponding eigenvalues.

7, § 9]

Conclusions

591

As is well known, the eigenfunctions of the Fourier operator T are the Hermite-Gauss functions Unix), which form an orthonormal and complete set. In symbols:

^Unix)

= (-if

Unix),

W^(x) = J -~^Hnix)

^Xpf-y j ,

(8.33)

where //«(x) is the Hermite polynomial of order n. Since T^ is the ath power of T, it admits the same eigenfunctions as T, with corresponding eigenvalues (-i)"^; explicitly: ^"^ Unix) = i-ir^

Unix).

(8.34)

Remarkably, Namias [1980a] presented the above relation as the defining equation of the FrFT operator J^^\ whose integral representation (2.7) was indeed derived from the eigenvalue equation (8.34). Likewise, the optical definition of the FrFT, based on the gedanken-experiment with graded-index media, was given in terms of the w„'s, which basically are the eigenmodes of quadratic graded-index media (Mendlovic and Ozaktas [1993], Ozaktas and Mendlovic [1993a,b]). The Hermite-Gauss eigenfunctions are also employed for the numerical implementation of the FrFT (Ozaktas and Mendlovic [1993b]) as well as for the reconstruction of an image from its mode content (Alieva and Bastiaans [1999]). Moreover, the eigenvalue equation (8.34) is of great significance in a purely mathematical framework. It is exploited, for instance, to infer from the spectral decomposition of the kernel (2.8) generating-function-like relations for the w„'s (Dattoli, Torre and Mazzacurati [1998]). As well, useful relations involving special functions, such as the Hermite and the parabolic cylinder functions, can be worked out from appropriate decompositions of the operator T^ within the inherent group, when applied to functions whose transform can be analytically calculated (Wolf [1974a], Torre [2001]).

§ 9. Conclusions The FrFT synthesizes a new conceptual and mathematical approach to a great variety of physical processes and mathematical problems. Its formal structure enables the presentation of the process not in terms of space (time) or frequency purely but in terms of both space (time) and frequency with a continuous degree

592

The fractional Fourier transform

[7, § 9

of emphasis on space (time) or on frequency features. Accordingly, the FrFT improves and extends the potentiahties and utihties of the ordinary Fourier transform, with the further advantage that it can be implemented numerically with fast digital algorithms, and optically with bulk optics much like the ordinary Fourier transform. Fourier mathematics has been naturally replaced by fractional Fourier mathematics. Thus, several fractional operations, such as the fractional convolution (Ozaktas, Barshan, Mendlovic and Onural [1994], Almeida [1997], Zayed [1998a]), the fractional correlation (Ozaktas, Barshan, Mendlovic and Onural [1994], Mendlovic, Ozaktas and Lohmann [1995], Mendlovic, Bitran, Dorsch and Lohmann [1995], Lohmann, Zalevsky and Mendlovic [1996], Lohmann and Mendlovic [1997], Granieri, Arizaga and Sicre [1997]) and the fractional filtering (Ozaktas, Barshan, Mendlovic and Onural [1994], Mendlovic, Zalevsky, Lohmann and Dorsch [1996], Zalevsky and Mendlovic [1996a], Kutay, Ozaktas, Arikan and Onural [1997]), which employ the FrFT as building block in place of the ordinary Fourier transform, have been implemented and proved in some cases to be superior to the conventional operations. Similarly, just replacing the Fourier transform by the FrFT in the inherent definitions, the fractional Radon transform (Zalevsky and Mendlovic [1996b]), Hankel transform (Namias [1980b], Yu, Lu, Zeng, M. Huang, Chen, W. Huang and Zhu [1998]), Hilbert transform (Lohmann, Mendlovic and Zalevsky [1996], Zayed [1998b]) and wavelet transform (Mendlovic, Zalevsky, Mas, Garcia and Ferreira [1997]) have been introduced and successfully applied to signal analysis and optical propagation problems. Moreover, a different scheme of fractionalization has been proposed, which can in general be applied to any periodic mathematical operation (Shih [1995b], Alieva and Calvo [2000]). Many investigations have also been concerned with the discrete FrFT, for which a satisfactory definition is desired, having the same relation with the continuous FrFT as the discrete Fourier transform has with the ordinary continuous Fourier transform (Pei and Yeh [1997], Atakishiyev and Wolf [1997], Candan, Kutay and Ozaktas [2000], Pei and Ding [2000a]). It has recently been recognized that the FrFT belongs to the wider class of linear canonical transforms (LCT), thus animating new investigations and new opportunities (see also §7.2). In fact, some canonical, i.e., LCT-based, operations, such as the canonical convolution, correlation and Hilbert transform, have been introduced; although lacking in some of the properties of the corresponding fractional operations, these offer the advantage of a greater flexibility and simpler implementation, both digitally and optically (Barshan, Kutay and Ozaktas [1997], Almanasreh and Abushagur [1998], Sahin, Ozaktas

7]

References

593

and Mendlovic [1998], Pei and Ding [2000b]). The relations between the LCTs and the space-(time-)frequency distributions have also been investigated (Pei and Ding [2001]), thus paralleling the similar analysis concerning the FrFT. The potential of LCTs within the signal-processing world is still to be explored.

§ 10. Acknowledgments The author wishes to express her appreciation and gratitude to Professor E. Wolf for his helpful comments, to Professor W.A.B. Evans for his invaluable suggestions, to Professor V. Lakshminarayanan and Dr. A. De Angelis for their stimulating discussions, and to Dr. F. Mucci and Mrs. G. Gili for their warm encouragement. The benefit of interactions with Dr. G. Dattoli and Dr. L. Mezi is acknowledged. An especial thanks is reserved to zia Aida for her generous sympathy at all times. Finally, the author apologizes to all those whose contributions have not been properly mentioned in this finite-length chapter.

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Author index for Volume 43

Anderson, D. 77, 83, 86, 87, 89-92, 94, 96, 99, 101, 110, 122, 124, 127, 129, 148 Anderson, M.E. 463 Anderson, RW. 438 Andrekson, P.A. 99, 101 Andreoni, G. 55 Ankiewicz, A. 165, 167 Antesberger, G. 466 Arikan, O. 539, 561, 592 Arizaga, R. 592 Artoni, M. 300 Arvind 459 Asakura, T. 467 Aspect, A. 386 Atakishiyev, N.M. 471, 592 Auld, B.A. 464 Aytiir, O. 579, 587

Aakjer, T. 101 Abdullaev, F.Kh. 88, 89, 143, 146-148, 153, 165 Abe, S. 536, 544, 570, 572 Ablowitz, M.J. 75, 123, 127, 149, 158, 160 Abouraddy, A.F. 487 Abraham, M. 339 Abram, I. 299, 307, 308, 310, 312, 323, 324, 326, 327, 329-334, 338, 343, 345, 362, 374, 375, 378, 382, 383, 385 Abrarov, R.M. 165 Abushagur, M.A.G. 592 Aceves, A.B. 148, 175 Adachi, S. 346 Adam, P. 465 Afanasjev, V.V. 89, 94, 117, 119, 121, 122 Agarwal, G.S. 402, 454, 460, 469, 472, 487, 510, 579, 581 Agranovich, VM. 247, 248 Agrawal, G.P. 73, 76, 77, 127, 128, 153, 375, 503 AguUo-Lopez, F. 579, 581 Aharonov, Y. 525 Aiello, A. 362 Aitchison, J.S. 76, 104, 110 Akhmediev, N. 84, 119, 165-167 Akiba, S. 148 Akkermans, E. 438 Akulshin, A.M. 523 Alexeeva, N.V. 84 Alfano, R.R. 508 Alieva, T. 579, 581, 591, 592 Allen, L. 436, 438 Almanasreh, A. 592 Almeida, L.B. 533, 579, 581, 590, 592 Alonso, M.A. 471 Ambartsumyan, R.V. 506, 507, 524

B Babb, J.F 525 Babiker, M. 328 Bagini, V 572 Baizakov, B.B. 153 Balazs, N.L. 447, 473 Ballentine, L.E. 445, 473 Bamler, R. 467 Ban, M. 341,447 Banaszek, K. 467, 474, 486, 487 Banyai, W.C. 128 Barashenkov, I.V. 82, 84, 122, 176, 177, 179 BardrofiF, P 461 Bardroff, PJ. 466 Bargmann, V. 533, 540 Bamett, S.M. 299-302, 307, 360, 385-387, 389, 392-394, 396-398, 402, 403, 405, 421, 459 Barreiro, S. 523 597

598

Author index for Volume 43

Barrett, H.H. 568, 569 Barry, D.T. 568, 569 Barshan, B. 539, 561, 592 Bartelt, H.O. 467 Barthelemy, A. 107, 110 Bartlett, M.S. 474 Basov, N.G. 506, 507, 524 Bastiaans, M.J. 452, 563, 564, 591 Baumberg, J.J. 385 Baumgartner, R. 438 Bechler, A. 302 Beck, M. 462-464 Beckwitt, K. 86 Beenakker, C.W.J. 438 Behroozi, C.H. 499, 512, 513, 515-518, 521, 522 Belanger, P.-A. 84, 149, 436 Bell, J.S. 486 Ben-Aryeh, Y. 299, 300, 303, 323-325, 336345, 358, 386, 487 Bendjaballah, C. 438, 484 Bennink, R.S. 511 Bennion, I. 148 Berge, L. 91, 183 Berkovits, R. 438 Berman, M. 488 Bernardo, L.M. 539, 542, 551, 553, 555, 557, 559, 570, 581 Bemtson, A. 86, 92, 96, 99, 148-150, 152, 153, 156 Berry, M.V 473 Bertilsson, K. 101 Bertolotti, M. 303, 342-344 Bertrand, J. 451,462 Bertrand, R 451,462 Bethe, H.A. 211 Bhandari, R. 438 Bialynicka-Birula, Z. 298, 312-314, 346 Bialynicki-Birula, I. 298, 303, 312-315, 346, 452 Bieber, T. 507 Bijedic, N. 451 Biondini, G. 127, 149, 158, 160 Bitran, Y. 539, 550, 557, 569, 592 Bizarro, J.R 454 Bjorken, J.D. 306 Blanchard, R 83, 85 Bleany, B. 349 Bleany, B.I. 349 Bloembergen, N. 297, 346, 349, 350

Bloom, D.M. 464 Blow, K.J. 148, 299, 319, 322, 375, 377, 382 Bodendorf, C.T. 466 Boffeto, G. 123 Bogdan, M.M. 82, 84 Boggavarapu, D. 463 Bohigas, O. 472 Bohm, D. 469 Bohmer, B. 464 Boivin, L. 300 Bolda, E.A. 524 Bolda, E.L. 465 Bolivar, A.O. 443 Bondeson, M. 77, 83 Bonnet, G. 570, 572 Boose, D. 475 Bom, M. 208, 209, 242, 246, 280, 284, 288, 297, 346 Bouasse, H. 206, 214, 222 Bouwkamp, C.J. 211 Boyd, A.R. 104 Boyd, R.W. 297, 502, 511 Bozdagi, G. 539 Braat, J. 15 Bracewell, R.N. 534 Breitenbach, G. 463 Brenner, K.-H. 467 Brillouin, L. 504, 507 Brodhun, E. 5 Broers, B. 437 Bronski, J. 153 Brown, S.A. 302 Brumer, R 455, 472, 473 Brun, T.A. 300 Briining, E. 83, 85 Budker, D. 521 Bullough, R.K. 81 Bund, G.W. 470 Buzek, V 459 C Caglioti, E. 84 Cahill, K.E. 303, 448 Caldeira, A.O. 385 Calvo, M.L. 592 Campbell, D.K. 105 Candan, C- 592 Caputo, J.G. 88, 89, 143, 146-148 Carmichael, H.J. 326, 359, 361 Carruthers, J.A. 507

Author index for Volume 43 Carruthers, P. 448 Carter, S.J. 326, 327, 346, 375 Cartwright, N.D. 451 Casperson, L. 508 Cauchy, A.L. 206, 214, 275 Caves, CM. 323, 326, 336, 340, 361, 364, 386 Chakravarty, S. 127, 158 Chanan, G. 20, 21, 37 Chaturvedi, S. 469, 477, 489 Chekhova, M.V 486 Chen, H.H. 92 Chen, L. 551, 557 Chen, M. 550, 592 Chemikov, S.V 140 Chevrette, P. 569 Chew, W.C. 413 Chiao, R.Y. 182, 524-528 Chizhov, A.V 302, 418 Chountasis, S. 438, 483, 484 Chow, C.N.Y. 507 Christiansen, PL. 183 Christodoulides, D.N. 175 Chu, PL. 89, 94, 99, 105, 107, 109, 117, 121, 164-166, 168, 169, 179 Chu, S. 508-510, 523 Chumakov, S.M. 471 Ciocci, F. 488 Cirac, J.L 86 Claasen, T.A.C.M. 452 Clack, R. 568 Clarke, S.R. 141 Clausius, R. 274 Clebsch, A. 285 Cohen, E. 299, 326, 327, 329-334, 338, 343, 345, 362, 374, 375, 378, 382, 383, 385 Cohen, G. 165 Cohen, L. 449, 452, 561 Cohen, L.G. 74, 148 Cohen, N. 546, 550 Cohen, R. 41 Cohen-Tannoudji, C. 226, 328, 390, 417 Collett, M.J. 316, 326, 359, 361, 482 Condon, E.U. 533 Conner, M. 467 Conti, C. 176 Cooper, F. 91 Cooper, J. 463 Cosman, B.C. 438 Couder, A. 30

599

Coullet, P 115 Cowley, D. 6 Creedon, J.F 11 Crossignani, B. 84 Crouch, D.D. 323, 326, 336, 340, 386 Cui, X. 45, 46, 67 Cullum, M. 56 D Dalton, B.J. 300-302 Darmanyan, S.A. 165 Dattoli, G. 488, 570, 579-581, 583, 591 Daubechies, I. 539 Davidovic, D.M. 451 Davidovich, L. 466 De Angelis, C. 91 de Bruijn, N.G. 561 deGroot, S.R. 418 de Oliveira, FA.M. 477 De Rossi, A. 176 de Sterke, CM. 136, 175 Deans, S.R. 568, 569 Dechoum, K. 345 Defrise, M. 568 Dekens, F. 21 Delabre, B. 67 Desaix, M. 91, 92, 94, 124 Desyatnikov, A. 86, 92, 114 Deutsch, I.H. 299, 362-365, 367-371, 374 Dhara, A.K. 454 di Bartolo, B. 442 Di Stefano, O. 303, 412 Dianov, E.M. 122, 140 Diener, G. 507 Dierickx, Ph. 3, 67 Dimitrevski, K. 92 Ding, J.-J. 592, 593 DiPorto, P 84 Dirac, PA.M. 297, 306 DiVittorio, M. 12 Dodd, R.J. 86 Dogariu, A. 524-526 Doran, N.J. 148, 150, 152, 157 Dorsch, R.G. 539, 550, 555, 557, 569, 581, 592 Doty, S.L. 165 Dowling, J.P 454, 474 Dragoman, D. 436, 452, 459, 467, 468, 474, 475, 482, 483, 563 Dragoman, M. 436, 459, 467, 468

600

Author index for Volume 43

Dragt, A.J. 580 Drell, S.D. 306 Drude, P. 222 Drummond, P. 86 Drummond, PD. 299, 300, 302, 304, 320, 322, 326, 327, 337, 345, 346, 349-355, 357, 359, 375, 382, 396 Duguay, M.A. 507 Dung, H.-T. 301, 405-409, 413 Dupont-Roc, J. 226, 328, 390, 417 Dutra, S.M. 301, 362, 385, 416-418, 420, 421, 423 Dutton, Z. 499, 512, 513, 515-518, 521, 522

Easton Jr, R.L. 569 Eckhardt, E. 472 Edagawa, N. 148 Edrei, I. 438 Efremidis, N. 149 Eisenberg, H.S. 104 El-Reedy, J. 124 Enard, D. 45, 46 Englert, B.-G. 482 Erden, M.R 550, 551, 557, 558, 570, 572 Erkaya, N. 563 Etrich, C. 85 Evangelides, S.G. 148 Ewald, PR 242 F Fabeni, P 438 Falinejad, H. 413 Fan, H.-Y. 446 Fano, U 385, 391 Faridani, A. 462 Faxvog, ER. 507 Ferreira, C. 550, 569, 570, 592 Field, J.E. 510, 513 Finlayson, N. 128 Fleischhauer, M. 523 Florjanczyk, M. 165 Flytzanis, N. 143 Fontenelle, M.T. 466 Forbes, G.W. 471 Ford, J. 472 Fordy, A.P 81 Forest, E. 580 Fork, R.L. 165 Fortunato, M. 467

Forysiak, W. 148, 150, 152, 157 Fox, A.G. 360 Francis, M.R. 303 Franson, J.D. 480 Frantzeskakis, D.J. 149 Franza, F 11, 55 Fresnel, A. 206 Freund, I. 438 Freyberger, M. 461 Friberg, S.R. 164, 346 Friedland, L. 82 Friesch, O.M. 485 Frova, A. 507 Fry, E.S. 518-520 Furuya, K. 301, 416-418, 420, 421, 423, 526 G Gabitov, I. 148 Gagnon, L. 436 Gaididei, Yu.B. 183 Gallardo, J.C. 583 Garabedian, PA. 410 Garcia, J. 550, 570, 581, 592 Garcia-Ripoll, J.J. 86 Gardavsky, G. 359 Gardiner, C.W. 316, 326, 359, 361 Garmire, E. 182 Garrett, C.G.B. 499, 507, 508, 510 Garrison, J.C. 299, 362-365, 367-371, 374, 524 Gatti, A. 300 Gea-Banacloche, J. 299, 316, 360-362, 514 Georges, T. 160 Gerjuoy, E. 84 Geschwind, S. 523 Ghatak, A.K. 91 Ghosh, S.K. 454 Gilmore, R. 488, 583 Gilson, C.R. 300 Gindi, G.R. 569 Ginzburg, VL. 247, 248 Girlanda, R. 303, 412 Giuliani, G. 122 Giulini, D. 468, 473 Glasgow, S.A. 526 Glauber, R.J. 299, 303, 327, 346, 396, 418, 448 Gloag, A.J. 157 Glunder, H. 467

Author index for Volume 43 Gmitro, A.F. 569 Gmitter, T. 346 Gocheva, A.D. 84 Godil, A.A. 464 Goldstein, E.V 485 Golles, M. 165 Golovchenko, E.A. 160 Goodman, J. 542 Gordon, J.G. 148 Gordon, J.P. 76,110 Gori, F. 572 Gorshkov, K.A. 83 Graham, R. 367 Grangier, P. 326, 336, 346 Granieri, S. 592 Grimshaw, R. 145, 146 Grimshaw, R.H.J. 141 Grischkowsky, D. 122 Griinbaum, F.A. 96, 123 Gruner, T. 300, 302, 396, 399-401, 403, 405, 409, 423 Grynberg, G. 226, 328, 390, 417 Guerra, E.S. 300 Guerreiro, A. 315 Guillemin, V 440, 442 Guisard, S. 36, 57, 60 Gupta, A.K. 467 H Habib, S. 473 Haelterman, M. 132 Haken, H. 367 Halas, N.J. 122 Hamilton, M.W. 359 Hamilton, W.A. 479 Han, D. 474 Harris, S.E. 499, 510, 512-518 Harrison, RE. 482 Hartmann, S.R. 523 Hasegawa, A. 74, 117, 160 Hatami-Hanza, H. 164 Hau, L.V 499, 512-518, 521, 522 Haus, H.A. 300, 346, 375, 377, 382, 385, 386 Haus, J.W. 165 Hawkes, RW. 437 Hawton, M. 303 He, H. 86 He, J. 145, 146 Healey, W.R 353

601

Hedekvist, P.-O. 101 Heller, E.J. 472, 473 Heriing, G.H. 459,461,465 Hermann, G.T. 463 Hiley, B.J. 469 Hillery, M. 298, 302, 304-306, 327, 328, 336, 346, 364, 376, 396, 448, 561 Hinds, E.A. 346 Hizanidis, K. 149 Hmurcik, L.V 123 Ho, K.C. 301 Ho, S.-T. 299, 355 Hofmann, H.F 487 Holland, PR. 454 Hollberg, L. 518-520 Holmes, P 148 Home, D. 487 Honold, A. 346 Hook, A. 122 Hopfield, J.J. 299, 385 Hori, H. 413 Home, R.L. 127, 158 Horowicz, R.J. 477 Hortmanns, M. 49 Hradil, Z. 300 Hu, P 523 Hu, X. 437, 489 Hua, J. 570, 577, 578 Huang, E. 26, 66 Huang, M. 550, 592 Huang, W. 550, 592 Hubin, N. 4 Hug, M. 453 Huttner, B. 299, 300, 307, 323-325, 340, 385-387, 389, 393, 394, 396-398, 403, 421

lannone, E. 77 Icsevgi, A. 507, 508, 527 Ikeda, K. 346 Imamoglu, A. 510 Imoto, N. 300, 316, 340, 341 Inagaki, T. 303 Infeld, L. 297, 346 Inoue, T. 413 looss, G. 167, 182 Islam, M.K. 89, 94 Itano, W.M. 466 Itzykson, C. 332, 375, 377, 384

602

Author index for Volume 43

Iwai, T. 467 lye, M. 34, 65

Jackel, J.L. 76, 110 Jacobs, K. 464 Jacobson, D.L. 480 Jain, M. 510, 515 Jamin, J. 206, 221 Janszky, J. 465 Jauch, J.M. 297, 327 Javanainen, J. 303, 452 Jedju, T.M. 523 Jefifers, J.R. 300, 301, 341, 402, 403, 405 Jennings, B.K. 447 Jensen, J.H. 473 Jin, S.-z. 514 Jones, C.K.R.T. 148 Joos, E. 468, 473 Joseph, D.D. 167, 182 Joseph, R.I. 175 Juul Rasmussen, J. 183 K Kaplan, A.E. 484 Karlsson, M. 94, 148 Karpman, VI. 83, 110, 161 Kartner, EX. 300, 377 Kasapi, A. 510, 513, 515 Kash, M.M. 518-520 Kasper, E. 437 Kath, W.L. 90, 93, 98, 129, 135, 148, 166 Katz, A. 508 Kaup, D.J. 81, 83, 123, 124, 127, 129-131, 135, 148, 149, 157-159, 169, 171-173, 175, 179 Kaveh, M. 438 Kay, K.G. 445 Keller, O. 201, 211, 212, 225, 226, 239, 249, 281, 283 Kennedy, T.A.B. 367 Kerker, M. 286 Kerr, EH. 533, 536, 584 Keshavarz, A. 301 Khosravi, H. 402 Kiefer, C. 468, 473 Kien, EL. 476 Killip, R.B. 482 Kim, K. 460 Kim, M.S. 465, 466, 477

Kim, Y.-H. 486 Kim, Y.S. 447, 471, 474, 475, 562 Kimball, D.E 521 King, B.E. 466 Kirchhoff, G. 207, 210, 257 Kirkman, D. 21 Kis, Z. 465 Kisil, VV 443 Kiss, T. 465 Kitagawa, M. 375, 376 Kittel, C. 385 Kivshar, Y.S. 83, 105, 122, 129, 165, 169 Kivshar, Yu.S. 165 Klein, A.G. 479 Klingshim, C.E 420 Knight, P.L. 300-302, 359, 459, 464, 477 Knoll, L. 298, 299, 301-303, 316, 318, 346, 396, 402, 403, 405^14 Knox, EM. 148 Kobe, D.H. 303 Kocharovskaya, O. 523 Kodama, Y. 74, 117 Kogan, E. 437, 438 Kogelnik, H. 74, 148 Kolner, B.H. 464 Kolobov, M.I. 300 Kolokolov, A.A. 106, 171 Konkel, S. 445 Konotop, VV 86 Kooi, P.S. 413 Korobov, VI. 82, 84 Korolkova, N. 359 Kottler, E 210 Kozhekin, A.E. 524 Kragh, H. 200, 276, 288 Kramer, P. 536, 540 Krasinski, J.S. 467 Kiepelka, J. 303, 342-344 Krivoshlykov, S.G. 436 Krokel, D. 122 Krolikowski, W. 122 Kryukov, PG. 506, 507, 524 Kubo, R. 330 Kubota, H. 148 Kuehl, H.H. 140 Kulik, S.P 486 Kumar, A. 91 Kumar, P 299, 355 Kumar, S. 160 Kupsch, J. 468, 473

Author index for Volume 43 Kurizki, G. 524 Kurmyshev, E.V 436 Kurtsiefer, Ch. 463 Kutay, M.A. 534, 539, 561, 563, 592 Kutz, J.N. 148 Kuzmich, A. 524-526 Kuznetsov, E.A. 93 Kwiat, P.G. 482, 524 Kyprianidis, A. 454

Laedke, E.W. 148 Lagendijk, A. 438 Lai, Y. 346, 375, 386 Lakoba, T.I. 148, 149, 169, 172, 173, 175, 179 Lalovic, D. 451 Lamb Jr, W.E. 299, 301, 359, 360, 484, 507, 508, 527 Landau, L.D. 84, 88, 103, 124, 349 Lang, R. 299, 301, 359, 360 LaPorta, A. 346 Lassell, W. 7 Lawrence, B.L 92 Lax, M. 367 Leaird, D.E. 76, 110 Lederer, R 75, 85, 160, 163, 165 Ledinegg, E. 381 Lee, C.T. 461 Lee, H.-W. 448, 469, 485 Lee, Y.C. 92 Leffert, E. 37 Lega, J. 115 Leibfried, D. 466 Leichtle, C. 461 Leong, M.S. 413 Leonhardt, U. 299, 339, 396, 410, 414, 450, 462^64, 474, 486 Lesage, F. 149 Letokhov, VS. 506, 507, 524 Leuchs, G. 359 Leung, P.T. 301 Levenson, M.D. 298, 315, 381, 386 Lewenstein, M. 86, 299, 327, 346, 396, 418 Lewis, Z.W. 123 Ley, M. 359 Lezama, A. 523 Li, C. 551, 557 Li, G. 570, 577, 578 Li, L.W. 413

603

Li, T. 360 Li, Y. 467 Li, Y-q. 514 Lichtenberg, A.J. 441, 471, 474 Lifshitz, E.M. 84, 88, 103, 124, 349 Lin, C. 74, 148 Lindgren, A.G. 11 Lisak, M. 77, 83, 86, 87, 89-92, 94, 96, 99, 101, 110, 122, 124, 127, 129, 148 Littlejohn, R.G. 471 Liu, C. 521, 522 Liu, L. 570, 577, 578 Liu, S. 551,557 Liu, X. 86, 183 Liu Wong, D. 375 Logan, N.A. 200, 285, 286, 289 Lohmann, A. 550 Lohmann, A.W. 438, 459, 463, 467, 533, 534, 539, 542, 551, 552, 555, 557, 561, 562, 566, 567, 569, 587, 588, 592 Lopez, V 579, 581 Lorentz, H.A. 274 Lorenz, L. 197, 198, 200, 203, 204, 206, 207, 209, 210, 214, 221, 227, 228, 230-232, 234, 237, 241, 248, 251, 252, 256, 257, 260, 262, 265, 268, 269, 273-276, 281, 283-285, 289-291 Loudon, R. 299, 300, 303, 319, 320, 322, 328,341,359,375, 377,382,402 Louisell, W.H. 367 Love, J.D. 443 Lu, N. 299, 316, 360-362 Lu, Y 550, 592 Lucheroni, C. 91 Luis, A. 303, 344 Lukin, M.D. 510, 518-520, 523 Luks, A. 303, 323, 341-344 Luther-Davies, B. 122 Liitkenhaus, N. 459 Lutterbach, L.G. 466 M Maassen van den Brink, A. 301 Machida, S. 316, 346 Macke, B. 508,510 Maimistov, A.I. 84, 86, 92, 96, 114, 123, 165 Main, J. 475 Mair, A. 523 Mak, W.C.K. 179

604

Author index for Volume 43

Makhankov, VG. 84 Makowski, A.J. 445 Malomed, B.A. 83-87, 89, 92, 94, 99, 101, 103, 105, 107, 109, 110, 113-117, 121, 124, 127, 129-131, 135-137, 140-142, 145, 146, 148, 149, 153, 156-161, 163-166, 168-172, 175, 176, 179, 183 Manakov, S.V 75, 76, 81, 82, 123, 125, 129 Mandel, L. 356, 437, 460, 478^80 Maneuf, S. 76 Man'ko, VI. 436, 440, 464 Mantica, G. 472 Mail, C. 488 Marie, Z. 454 Marinho, F.J. 539, 551, 581 Marshall, T.W. 345 Marte, M.A.M. 437 Martins, A.M. 315 Marzoli, I. 485 Mas, D. 570, 581, 592 Mast, T. 12,20,21,41, 51,66 Matera, F. 77 Mathieu, P. 149 Matloob, R. 300, 301, 413, 414 Matsumoto, M. 148, 160 Maxwell, J.C. 273 Maynard, R. 438 Mazilu, D. 75 Mazzaeurati, G. 579, 581, 591 MeAlister, D.F 463, 485 MeBride, A.C. 533, 536, 584 MeCall, S.L. 507 MeCumber, D.E. 499, 508, 510 MeDonald, K.T. 323 MeKnight, W.B. 367 MeLeod, B.A. 35 MeRae, S.M. 473 Meeozzi, A. 77 Meekhof, D.M. 466 Meklenbrauker, W.FG. 452 Mendlovie, D. 438, 459, 533, 534, 539, 546, 549-551, 557, 558, 561-563, 567, 569, 570, 572, 575, 577, 584, 588, 591, 592 Mendonga, J.T. 315 Menke, C. 453 Menyuk, C.R. 92, 128, 160 Merriam, A.J. 510 Messiah, A. 582 Meyer, K. 204 Meystre, R 437, 485

Mezentsev, VK. 84, 148, 183 Michaelis, D. 75 Michaels, S. 21 Michinel, H. 86, 92, 104 Mie, G. 200, 285 Migdall, A. 486 Mihalache, D. 75 Mikhailov, A.V 93 Milbum, G.J. 298, 315, 359, 381, 448, 477, 478, 486 Milonni, PW. 419, 420, 525, 526 Misner, C.W. 313 Mitchell, M.W. 527, 528 Mitschke, FM. 110 Mizrahi, S.S. 470 Mlodinow, L.D. 298, 302, 304-306, 327, 328, 336, 346, 364, 376 Mollenauer, L.F 76, 110 Moller, K.B. 473 Moloney, J.V 73 Monroe, Ch. 466 Montie, E.A. 438 Morandotti, R. 104 Morita, I. 148 Moshinsky, M. 536, 540 Mossotti, O.F 274 Mostofi, A. 169 Moyal, J.E. 448, 474 Mrowczyhski, S. 313 Muga, J.G. 451 Mugnai, D. 438 Mukunda, N. 459, 471 MuUer, B. 313 Munroe, M. 463 Muraki, D. 129 Muschall, R. 165 Musher, S.L. 148 N Nakano, H. 346 Nakazawa, M. 148 Naletto, G. 438 Namias, V 438, 533, 536, 579, 581, 584, 591, 592 Naraschewski, M. 482 Narcovich, F.J. 453 Nazarathy, M. 544, 570 Nazmitdinov, R.G. 418 Nelson, J. 12,20,21,41, 51, 66 Nepomnyashchy, A.A. 84, 115, 116

Author index for Volume 43 Neumann, F. 206 Newell, A.C. 73, 75, 81, 83, 123 Niculae, A.N. 157 Nienhuis, G. 362, 436, 438 Nijhof, J.H.B. 148, 150, 152, 157 Nilsson, O. 316 Nistazakis, H.E. 149 Noethe, L. 4, 11, 18, 21, 31-33, 36, 45, 46, 48, 49, 54, 55, 57, 60, 67 Noll, J.N. 59 Noordam, L.D. 437 Nori, F. 437, 489 Novikov, S.P. 75, 76, 82 Noz, M.E. 447, 471, 474, 562 O O'Connell, R.F 448^51, 453, 470, 561 Oh, Y. 165 Ohara, C. 20 Ohgren, A. 92 Oliver, M.K. 76,110 Onishchukov, G. 127 Onural, L. 539, 561, 573, 592 Opat, G.I. 479 Orlowski, A. 450 Orszag, M. 466 Osborne, A.R. 123 Ostrovsky, L.A. 83 Oughstun, K.E. 504, 507 Ozaktas, H.M. 533, 534, 539, 546, 549-551, 557, 558, 561, 563, 567, 569, 570, 572, 575, 577, 579, 584, 587, 591, 592

Palao, J.P. 451 Panova, E.Yu. 122 Papanicolaou, G.C. 153 Pare, C. 84, 149, 165, 436 Parker, D.F 142 Partington, J.R. 280, 284 Paul, H. 463, 464 Payne, D.N. 140 Pazzi, G.R 438 Peatross, J. 526 Pedrotti, L.M. 299, 316, 360-362 Pegg, D.T. 454 Pei, S.-C. 592, 593 Peierls, R. 339 Pelinovsky, D.E. 152, 176, 177, 179 Pelinovsky, E.N. 83

605

Pellat-Finet, P 570, 572 Peng, G.D. 99, 105, 107, 109, 164-166, 168, 169 Penrose, R. 264 Perez-Garcia, VM. 86 Pefina, J. 303, 344, 349 Pefina Jr, J. 303 Pefinova, V. 303, 323, 341-344 Pemigo, M. 476 Peschel, T. 85 Peschel, U. 75, 85 Petviashvili, VI. 117 Pfau, T. 463, 466 Pfleegor, R.L. 480 Phillips, D.F 523 Phoenix, S.J.D. 299, 319, 322, 375, 377, 382 Pihl, M. 200, 210, 222, 228, 237 Pilipetskii, A.N. 160 Pimpale, A. 443, 454 Pinsker, Z.G. 242 Pitaevskii, L.P 75, 76, 82 Pittman, T.B. 486 Piwnicki, P. 414 Piatt, B.C. 14 Pomeau, Y. 115 Pooseh, G. 413 Postema, H. 45, 46 Potasek, M.J. 326, 336, 346, 386 Potocki, K.A. 480 Power, E.A. 328, 353 Prasad, S. 299,316, 360-362 Prataviera, G.A. 485 Prezhdo, O.V 443 Provost, D. 473 Prytz, PK. 276 Pushkarov, D.I. 91 Pushkarov, Kh.I. 91 Puzynin, I.V 84

Qian, L.J. 86, 183 Quiroga-Teixeiro, M. 86, 92, 96, 99, 101 Quiroga-Teixeiro, M.L. 86, 94, 124, 148, 169 R Racine, R. 6 Radmore, PM. 299, 360, 385, 392 Radon, J. 567 Radzewicz, C. 467

606

Author index for Volume 43

Ranfagni, A. 438 Raszillier, H. 440 Rau, A.R.R 84 Rauch, H. 480 Ray, F.B. 4 Rayleigh, Lord (J.W. Strutt) 208, 280, 286, 288, 289 Raymer, M.G. 462-464, 469, 485 Razavy, M. 443, 454 Reichel, T. 77, 87, 89, 90 Reid, M.D. 346, 386, 486 Reimhult, E. 92 Reynaud, F. 76, 110 Reznik, B. 525 Richardson, D.J. 140 Richter, Th. 465 Riemann, B. 260, 273 Rindler, W. 264 Risken, H. 462 Ristic, VM. 569 Rivera, A.L. 471, 480 Robinson, F.N.H. 418, 423 Rochester, S.M. 521 Roddier, C. 14 Roddier, F 14 Rohrl, A. 482 Romagnoli, M. 84 Roman, R 340 Rosenau da Costa, M. 385 Rosenbluh, M. 346, 375, 438 Rosenfeld, L. 200, 292 Rosewame, D.M. 418,419 Rostovtsev, Y. 518-520, 523 Roy, V 149 Royer, A. 455, 466, 490 Rubin, M.H. 486, 487 Ruostekoski, J. 303, 424

Sahin, A. 550, 551, 557, 558, 592 Sakoda, K. 438 Sala, R. 451 Saleh, B.E.A. 487, 543, 573, 574 Salmon, D. 6 Samir, W. 136 Sammut, R.A. 136 Sandhya, R. 489 Santarsiero, M. 572 Santos, E. 345 Sarkar, S.N. 91

Satsuma, J. 90 Sautenkov, VA. 518-520 Savasta, S. 303, 412 Scacca, L.R. 124, 127 Schaefer, B. 20 Schaefer, T. 84 Schafer, T. 148 Scheel, S. 301-303,409-411 Schempp, W. 440 Schenzle, A. 482 Schiller, S. 463 Schleich, W.P. 448, 453^55, 458, 461, 467, 474, 476, 477, 484, 485, 487 Schmidt, E. 301, 302, 402, 403, 405 Schneermann, M. 45, 46 Schrade, G. 461 Schram, K. 418 Schrodinger, E. 436 Schroeder, D.J. 35 Schubert, M. 346 Schumaker, B.L. 361, 364 Schwesinger, G. 30, 33, 44, 49, 55 Schwindt, P.D.D. 482 Scully, M.O. 299, 301, 316, 359-362, 448, 482, 510, 518-520,523,561 Sczaniecki, L. 333 Seaton, C.T. 128 Sedighi, D 301 Segard, B. 508,510 Segev, B. 525 Segur, H. 75, 123 Seligman, T.H. 536, 540 Sergeev, A.M. 117 Sergienko, A.V 486, 487 Serimaa, O.T. 452 Serkin, VN. 122 Serulnik, S. 299, 323-325, 336-340, 386 Settembre, M. 77 Sfez, B.G. 164 Shabat, A.B. 123 Shack, R.V 14, 35 Shalaby, M. 107 Shamir, J. 544, 546, 550, 570 Shapiro, E.G. 148 Shelby, R.M. 346, 375, 386 Shen, Y.R. 297, 304, 305, 327, 332 Shepard, H. 91 Sheperd, T.J. 299, 319, 322, 382 Sheppard, A.R 132 Sheridan, J.T. 536, 544, 570, 572

Author index for Volume 43 Sherman, G.C. 507 Shih, C.-C. 560, 592 Shih, Y. 486 Shih, Y.H. 486, 487 Shimokhin, LA. 93 Shipuhn, A. 127 Shirasaki, M. 375, 382, 385 Shizume, K. 473 Shlomo, S. 450 Shumovsky, A.S. 418 Sibilia, C. 303, 342-344 Sicre, E.E. 592 Siegman, A.E. 543, 544, 558, 573, 574 Silberberg, Y. 76, 104, 110, 164 Simon, R. 459, 471, 489, 550, 579, 581 Sipe, J.E. 136, 175, 254, 272, 280, 285, 418 Sirko, E. 20, 37 Skinner, I. 165, 169 Skinner, I.M. 168 Sklyarov, Yu.M. 96, 123 Slusher, R.E. 326, 336, 346 Smerzi, A. 470 Smimov, Yu.S. 84 Smith, N.J. 148 Smith, RS. 164 Smith, RW.E. 76, 110 Smithey, D.T. 462, 463 Smyth, N.R 90, 93, 98, 137, 142, 148, 166 Snyder, A.W. 132, 443 Soares, O.D.D. 542, 551, 553, 555, 559 Sodano, R 91 Soffer, B.H. 463, 562, 569 Solimeno, S. 580 Solov'ev, VV 83, 110, 161 Sommerfeld, A. 211, 504 Soto-Crespo, J.M. 84, 166, 167 Sovka, J. 6 Sozzi, C. 438 Spatschek, K.H. 148 Spreeuw, R.J.C. 437 Spruch, L. 84 Spyromilio, J. 56, 57, 60 Srinivas, M.D. 451 Srinivasan, V 489 Stamatescu, I.-O. 468, 473 Stegeman, G. 92 Stegeman, G.I. 122, 128, 129, 164, 167 Steinberg, A.M. 524, 526 Steinberg, S. 337 Stenflo, L. 86

607

Stenholm, S. 377, 437, 466 Stepp, L. 26,29, 31, 66 Stem, A. 525 Sternberg, S. 440, 442 Stifter, R 484 Stokes, G.G. 207, 208 Stolen, R.H. 76, 128 Storey, R 482 Strekalov, D.V 486 Stroud Jr, C.R. 511 Su, D. 67 Sudarshan, E.C.G. 471 Suttorp, L.G. 396 Suzuki, M. 148 Svensson, E. 92

't Hooft, G.W. 438 Taga, H. 148 Takabayashi, T. 474 Tan, S.M. 465, 482 Tanaka, A. 473 Tanev, S. 91 Tara, K. 460 Tarasov, VE. 345 Tasgal, R.S. 129-131, 135, 136, 175, 176 league, M.R. 464 Tegmark, M. 463 Teich, M.C. 487, 543, 573, 574 Tewari, S.P. 510 Thompson, K. 35 Thome, K.C. 313 Ticknor, A.J. 569 Tijero, M.C. 470 Tip, A. 302, 303 Toda, M. 346 Tomas, M.S. 408, 409 Tombesi, P. 467 Tomov, I.M. 91 Tomsovic, S. 472, 473 Toren, M. 300, 339-345 Tomer, L. 75 Torre, A. 488, 570, 579-581, 583, 591 Torres-Vega, G. 473 Torruellas, W. 92 Towers, I. 183 Townes, C.H. 182 Tran, H.T 136 Trillo, S. 84, 122, 128, 129, 164, 176 Troy, M. 20, 21, 37

608

Author index for Volume 43

Tucker, J. 304, 326, 336 Turitsyn, S.K. 84, 148 U Ueda, T. 129, 135 Ujihara, K. 359 Ullmo, D. 472 Uzunov, I.M. 165

Vaccaro, J.A. 454, 486 Vakhitov, M.G. 106, 171 Valentiner, H. 199, 285 van Albada, M.P. 438 van Bladel, J. 264 van der Mark, M.B. 438 van Kranendonk, J. 254, 272, 280, 285, 418 van Leeuwen, K.A.H. 484 van Linden van den Heuvell, H.B. 437 van Tiggelen, B.A. 437, 438 VanderLinde, J. 446 Varro, S. 477 Varro, S. 452 Vestergaard, B. 303 Vigier, J.P. 454 Vilela Mendes, R. 464 Vogel, E.M. 76 Vogel, K. 462 Vogel, W. 298, 316, 318, 346, 396, 412, 414, 446, 465 von Helmholtz, H. 207 Vourdas, A. 438, 483, 484 W Wabnitz, S. 84, 122, 128, 129, 164, 167, 175 Wai, P.K.A. 92 Wald, M. 160, 163 Walker, G.W. 286 Wallentowitz, S. 446, 465 Wallis, H. 482 Walls, D.F. 298, 304, 315, 326, 336, 359, 381, 386, 448, 465, 476, 478, 482, 486 Walmsley, LA. 464, 469 Walsworth, R.L. 523 Walther, A. 563 Walther, H. 466, 474, 482 Wang, L. 122,449,451,470 Wang, L.J. 524-526 Wang, Y. 67

Wang, Z.H. 105, 107, 109 Ware, M. 526 Watanabe, K. 346 Watson, K.M. 297, 327 Weber, H. 467 Weber, W 258, 262 Weiner, A.M. 76, 110, 164 Weinstein, M.I. 86 Welch, G.R. 518-520 Welsch, D.-G. 298, 300-303, 316, 318, 346, 396, 399^03, 405-^14, 423 Werner, S.A. 480 Westfahl Jr, H. 385 Wetthauer, A. 5 Weyl, H. 533 Wheeler, J.A. 313,474,476 Whitham, G.B. 82 Whittaker, E.T. 223, 240, 242, 264, 265 Wiener, N. 533 Wigner, E. 561 Wigner, E.R 448, 450, 469, 475, 561 Wilcox, R.M. 583, 588 Wilhelmi, B. 346 Wilkie, J. 455, 472 Williams, H.A. 470 Wilson, R.N. 4, 11, 35, 36, 39, 41, 55, 66, 67 Wineland, D.L 466 Wise, F. 86 Wise, F.W 86, 183 Wiseman, H.M. 482 Wizinowich, R 12, 20 Wodkiewicz, K. 299, 316, 360-362, 437, 459, 461, 465^67, 474, 486, 487 Wolf, E. 208, 209, 242, 246, 280, 284, 288, 356,437,451 Wolf, K.B. 436, 440, 471, 480, 533, 536, 540, 544, 550, 580, 581, 583, 591, 592 Wolf, RE. 438 Wong, S. 508-510, 523 Wong, V 464,511 Wood, J.C. 568, 569 Woolley, R.G. 328 Woolven, S. 569 Worthy, A.L. 166 Wright, E.M. 92, 122, 129, 164, 167, 367, 375 Wu, L. 550 Wubs, M. 396 Wimsche, A. 448, 450, 464

609 X Xia, H. 510 Xiao, H. 504 Xiao, M. 514 Xu, J. 551, 557

Yablonovitch, E. 346 Yajima, N. 90 Yamamoto, S. 148 Yamamoto, Y 316, 346, 375, 376 Yang, C.C. 122 Yang, C.N. 358 Yang, J. 136, 148, 149, 157-159 Yang, T.S. 148 Yang, X. 122 Yang, Y 445 Yao, Z. 67 Yariv, A. 339, 342-345, 365, 508 Yashchuk, VV 521 Yeh, M.H. 592 Yeh, P. 339, 342-345 Yeo, T.S. 413 Yin, G.Y 510, 515 Yoder, P.R. 7 Young, K. 301 Yu, L. 550, 592 Yu, T. 160 Yurke, B. 326, 336, 346, 359, 361, 386

Zachariasen, F. 448 Zagury, N. 466 Zakharov, V.E. 75, 76, 82, 123 Zaks, M.A. 84 Zalesny, J. 315 Zalevsky, Z. 438, 459, 534, 550, 563, 569, 588, 592 Zayed, A.I. 590, 592 Zeh, H.D. 468, 473 Zemlyanaya, E.V 84, 176, 177, 179 Zeng, X. 592 Zhang, Y 551, 557 Zhang, Z. 477 Zhamitsky, V. 148 Zhu, Z. 550, 592 Zibin, J.P. 445 Zibrov, A.S. 518-520 Zienau, S. 328, 353 Zinn-Justin, J. 375, 377 Zoller, P 86 Zubairy, M.S. 360 Zuber, J.B. 332, 375, 377, 384 Zuev, VS. 506, 507, 524 Zuniga-Segundo, A. 473 Zurek, W.H. 466, 473

Subject index for Volume 43

D Dirac's transformation theory directional coupler 164 dispersive medium 503 displacement operator 299

action-at-distance 197 active optics, principles of 4-12 adaptive optics 6, 13 aether 197 - vibrations in polarized light 204-213 Aharonov-Bohm effect 483 Anderson approximation 93 anomalous-dispersion wave 122 antinormal distribution function 448 Avogadro's number 289

453

Ehrenfest's theorem 445 Einstein-Podolsky-Rosen state 486 electric dipole approximation 207 electromagnetically induced absorption — transparency 510 energy transport velocity 507 energy-momentum tensor 299 entangled state 486, 491 ether, see aether

B Bargmann-Hilbert space 540, 541 Bell's inequalities 486 Berry phase 438 Bessel fiinction 288 bistability 103 Bogoliubov transformation 311,325 Bose-Einstein condensate 86, 464 Bragg grating 179 bremsstrahlung 441 Brewster angle 214,221

523

F Fabry-Perot resonator 300 fast light 511 — , experimental studies of 523-527 Fock state 458 Fokker-Planck equation 128 four-wave mixing 447 fractional Fourier transform (FrFT) 438, 459, 533-541 , operational properties 584 , optical 541-550 and Fourier optics 569-579 imaging system 554—558 lens optics 550-560 wave-propagation optics 579 Wigner optics 561-569 of complex order 558-560 Franck-Condon principle 436 — transition 474

Casimir effect 301 Cauchy dispersion formula 275 Cauchy-Riemann equations 410 central intensity ratio (CIR) 3 chaos, dynamical 86 -, quantum 346 Chebyshev polynomial 453 coherent state 458, 489 correlation function 478 Coulomb gauge 225, 327, 367, 407 cross-phase modulation (XPM) 107, 122, 128, 157 611

612

Subject index for Volume 43

Fresnel diffraction 437, 572, 575, 580 - formulae 214, 218, 228, 245, 246, 260 - integral 538 - transform 539 fringe visibility 480

Galerkin truncation 83, 84 Galilean transformation 75, 204 geometrical optics 439 Gibbs-Boltzmann distribution 470 Ginzburg-Landau equation 115,116 Gladstone-Dale formula 275 Glauber-Sudarshan function 448 graded-index fiber 437 — media 541 Groenwald-van Hove theorem 442 Gross-Pitaevskii equation 86 group velocity 499, 500, 507 H Hankel transform 592 Heisenberg equation 318, 330 — picture 297, 304, 324, 373, 408, 424 Heisenberg-Langevin equation 341 Heisenberg-Weyl algebra 440 Helmholtz equation 398, 407, 436 Helmholtz-Kirchhoff integral 207, 210 Henon-Heiles model 473 Hermite-Gauss eigenfunction 533, 591 — mode 436 homodyne detection, balanced 463, 465 — , eight-port 464 Hopfield model 299 Husimi ftmction 449^51, 465 Huttner-Bamett model 307 Huygens integral 580 Huygens-Fresnel principle 210,211 I impulse response fiinction 543 inverse scattering transform 75 Jost function

123

K Kepler problem 88, 89, 96 Kerr coefficient 74, 183 - effect 74, 128, 375

- medium 182, 299, 375, 383, 404, 425 - nonlinearity 75, 91, 301 Kirchhoff", quasistatic theory of 257 - boundary conditions 207 - diffraction theory 207 - equations 257, 262 Kirkwood distribution 449 KolmogorofF statistics 61 - turbulence 61 Korteweg-de Vries equation 82 Kramers-Kronig dielectric 301, 420 - - relations 386, 398, 425

Lagrangian 78, 79, 82, 87, 96, 97, 109, 110, 161, 305 -density 78, 81, 128, 337 Laguerre-Gauss mode 436 Liouville equation 445, 454, 469 Liouville's theorem 441 Lorentz oscillator 504 - - model 420 - transformation 205 Lorentz-Lorenz formula 198, 201 Lorenz, electrodynamic theory 256-272 -gauge 201, 262-264 - law 199 - number 199 Lorenz-Lorentz relation, discovery of 272280 Lorenz-Mie scattering theory 285 Loschmidt number 289 Lyapunov frmctional 115,122 M Mach-Zehnder interferometer 515 magnetometry 510 Malus law 437 Margenau-Hill ordering 450 Maxwell equations 73, 198, 200, 205, 206, 211, 212, 223-225, 239-241, 249, 271, 276, 297, 298, 300, 308, 316, 337, 352, 397, 405, 424, 484 - stress tensor 309, 324 Maxwell-Lorentz equations 276 - - theory 240 Minkowski vector 324 mirror, meniscus 29, 32 -, monolithic 27, 28

Subject index for Volume 43 N Neumann function 288 nonlinear optics, quantum theory of normal distribution function 448

297

O Ohm's law 258 optical fiber, nonlinear 87, 122, 128 - resonator theory 574 P P ftmction 448, 450, 459, 460 — , positive 449 paraxial approximation 440 Parseval theorem 538 Pauli equation 212 Peierls-Nabarro potential 105 phase operator 458 - space, classical and quantum interference in 477^88 - in classical optics 439, 461 quantum mechanics 439, 461 - velocity 500 phasing camera system 20 Poisson bracket 329, 442, 544 - equation 260 Poynting vector 308, 310 Q Q function 449, 450, 459, 460, 464, 465, 469 quantum computation 437 - information 437 - interference 480, 490 - nondemolition measurement 447 -well 212

Radon transform 462, 567, 592 Radon-Wigner transform 562 Raman scattering 359 Ramsauer effect 436 ray transfer matrix 543 Rayleigh diffraction formula 208, 209 - length 576 Rayleigh-Ritz optimization procedure 84 retarded potential 197 - -, Lorenz 259-262 Riemann-Silberstein-Kramers vector 312 Rihaczek distribution 449 Rivier ordering 450

613

Schrodinger-cat state 460, 466, 467, 480 - equation, nonhnear 73, 358 — , time-independent 436 - picture 297, 304 Seidel-Lie coma map 440 self-focusing 75 - phase modulation 157 Sellmeir equations 302 Shack-Hartmann device 44 - lenslets 18 - - m e t h o d 14, 16,20 - parameters 17 - sensor 19 slowHght 499,511 — , experimental studies of 514-523 — , kinematics of 513, 514 — , nonlinear optics for 510-514 slowly-varying-envelope approximation 367 soHton, Bragg-grating 174-182 -, bright 122 -, dark 121 -, gap 169-171 -, higher-order 141 -, optical 73 -, quantum 425 -, spatial 76 -, temporal 76 -, vector 127, 128, 135, 137 -, walking 75 -, zero-velocity 175, 179 - in bimodal birefringent fiber 130-136 - dual-core optical fiber 164-173 - - o p t i c a l fiber 87-104 source-field operator 316, 318 squeezed light, propagation of 300 - state 359, 489 squeezing, pulsed 346 -, spectrum of 299, 361 Stem-Gerlach apparatus 437 stopped light 521 Strehl ratio 3 superluminal propagation 524 synchrotron radiation 441

telescope, active 7 -, Gemini 26 -, Keck 12, 20, 21, 38, 40, 41, 51-53, 64 -, Ritchey-Chretien 11, 12

614

The fractional Fourier transform

Thirring model 175 tomography, optical homodyne

463

Vakhitov-Kolokolov criterion 106 van Cittert-Zemike theorem 487 variational methods 77 von Neumann equation 444, 469 W wavefront sensing 14-21 wavelength-division multiplexing wavelet transform 539, 592

156

Weber constant 258 Weyl function 483 - group 588 Wigner distribution function 448, 449, 452, 561, 563, 565, 566 - functional 314 Wigner-Weisskopf approximation 341,415 WKB approximation 123

Zakharov system 86 Zakharov-Shabat linear equations 123 Zemike polynomial 11, 15, 24, 25, 38

Contents of previous volumes'^

VOLUME 1 (1961) 1 The modem development of Hamiltonian optics, RJ. Pegis 2 Wave optics and geometrical optics in optical design, K. Miyamoto 3 The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat 4 Light and information, D. Gabor 5 On basic analogies and principal differences between optical and electronic information, H. Wolter 6 Interference color,//. AjMZ>o/a 7 Dynamic characteristics of visual processes, A. Fiorentini 8 Modem alignment devices, ^.C5'. Van Heel

1-29 3 1 - 66 67-108 109-153 155-210 211-251 253-288 289-329

VOLUME 2 (1963) 1

Ruling, testing and use of optical gratings for high-resolution spectroscopy, G.W. Stroke 2 The metrological applications of diffraction gratings, J.M. Burch 3 Diffusion through non-uniform media, R.G. Giovanelli 4 Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi 5 Fluctuations of light beams, L. Mandel 6 Methods for determining optical parameters of thin films, F. Abeles

1-72 73-108 109-129 131-180 181-248 249-288

VOLUME 3 (1964) 1 The elements of radiative transfer, F. Kottler 2 Apodisation, P. Jacquinot, B. Roizen-Dossier 3 Matrix treatment of partial coherence, H. Gamo

1- 28 29-186 187-332

VOLUME 4 (1965) 1 Higher order aberration theory, J. Focke 2 Applications of shearing interferometry, O. Bryngdahl 3 Surface deterioration of optical glasses, K. Kinosita 4 Optical constants of thin films, P. Rouard, P. Bousquet

" Volumes I-XL were previously distinguished by roman rather than by arabic numerals. 615

1- 36 37- 83 85-143 145-197

616 5 6 7

Contents of previous volumes The Miyamoto-Wolf diffraction wave, A. Rubinowicz Aberration theory of gratings and grating mountings, WT. Welford Diffraction at a black screen, Part I: Kirchhofif's theory, E Kottler

199-240 241-280 281-314

VOLUME 5 (1966) 1 2 3 4 5 6

Optical pumping, C. Cohen-Tannoudji, A. Kastler Non-linear optics, P.S. Pershan Two-beam interferometry, W.H. Steel Instruments for the measuring of optical transfer fiinctions, K. Murata Light reflection from films of continuously varying refractive index, R. Jacobsson X-ray crystal-structure determination as a branch of physical optics, H. Lipson, C.A. Taylor 7 The wave of a moving classical electron, ^ P/c/z/

1- 81 83-144 145-197 199-245 247-286 287-350 351-370

VOLUME 6 (1967) 1 2 3 4 5 6 7 8

Recent advances in holography, E.N. Leith, J. Upatnieks Scattering of light by rough surfaces, R Beckmann Measurement of the second order degree of coherence, M. Erangon, S. Mallick Design of zoom lenses,/;r. Yamaji Some applications of lasers to interferometry, D.R. Herriot Experimental studies of intensity fluctuations in lasers, J.A. Armstrong, A. W. Smith Fourier spectroscopy, G.A. Vanasse, H. Sakai Diffraction at a black screen. Part IL electromagnetic theory, E Kottler

1- 52 53- 69 71-104 105-170 171-209 211-257 259-330 331-377

VOLUME 7 (1969) 1 2 3 4 5 6 7

Multiple-beam interference and natural modes in open resonators, G. Koppelman Methods of synthesis for dielectric multilayer filters, E. Delano, R.J. Regis Echoes at optical frequencies, I.D. Abella Image formation with partially coherent light, B.J. Thompson Quasi-classical theory of laser radiation, A.L. Mikaelian, M.L. Ter-Mikaelian The photographic image, iS". Ooue Interaction of very intense light with free electrons, J.H. Eberly

1- 66 67-137 139-168 169-230 231-297 299-358 359-415

VOLUME 8 (1970) 1 2 3 4 5 6

Synthetic-aperture optics, J. W. Goodman The optical performance of the human eye, G.A. Ery Light beating spectroscopy, H.Z. Cummins, H.L. Swinney Multilayer antireflection coatings, A. Musset, A. Thelen Statistical properties of laser light, H. Risken Coherence theory of source-size compensation in interference microscopy, T Yamamoto 7 Vision in communication, L. Levi 8 Theory of photoelectron counting, CL. Mehta

1- 50 51-131 133-200 201-237 239-294 295-341 343-372 373-440

Contents of previous volumes

617

VOLUME 9 (1971) 1 2 3 4 5 6 7

Gas lasers and their application to precise length measurements, A.L. Bloom Picosecond laser pulses, A.J. Demaria Optical propagation through the turbulent atmosphere, J.W. Strohbehn Synthesis of optical birefringent networks, E.O. Ammann Mode locking in gas lasers, L. Allen, D.G.C. Jones Crystal optics with spatial dispersion, VM. Agranovich, V.L. Ginzburg Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, JPetykiewicz 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden

1- 30 31-71 73-122 123-177 179-234 235-280 281-310 311^07

VOLUME 10 (1972) 1 2 3 4 5 6 7

Bandwidth compression of optical images, T.S. Huang The use of image tubes as shutters, R. W. Smith Tools of theoretical quantum optics, M.O. Scully, K.G. Whitney Field correctors for astronomical telescopes, C.G. Wynne Optical absorption strength of defects in insulators, D.Y. Smith, D.L. Dexter Elastooptic light modulation and deflection, E.K. Sittig Quantum detection theory, C. W Helstrom

1- 44 4 5 - 87 89-135 137-164 165-228 229-288 289-369

VOLUME 11 (1973) 1 2 3 4 5 6 7

Master equation methods in quantum optics, G.S. Agarwal Recent developments in far infrared spectroscopic techniques, H. Yoshinaga Interaction of hght and acoustic surface waves, E.G. Lean Evanescent waves in optical imaging, O. Bryngdahl Production of electron probes using a field emission source, A. V. Crewe Hamiltonian theory of beam mode propagation, JA. Arnaud Gradient index lenses, E.W Marchand

1- 76 77-122 123-166 167-221 223-246 247-304 305-337

VOLUME 12(1974) 1 2 3 4 5 6

Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto Self-induced transparency, R.E. Slusher Modulation techniques in spectrometry, M. Harwit, JA. Decker Jr Interaction of light with monomolecular dye layers, K.H. Drexhage The phase transition concept and coherence in atomic emission, R. Graham Beam-foil spectroscopy, S. Bashkin

1- 51 53-100 101-162 163-232 233-286 287-344

VOLUME 13 (1976) 1

On the validity of Kirchhofif's law of heat radiation for a body in a nonequilibrium environment, H.R Baltes 2 The case for and against semiclassical radiation theory, L. Mandel 3 Objective and subjective spherical aberration measurements of the human eye, W.M. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G Schulz, J. Schwider

1- 25 27-68 69- 91 93-167

618

Contents of previous volumes

Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, VK. Tripathi Aplanatism and isoplanatism, W.T. Welford

169-265 267-292

VOLUME 14(1976) 1 2 3 4 5 6 7

The statistics of speckle patterns, J.C. Dainty High-resolution techniques in optical astronomy, A. Labeyrie Relaxation phenomena in rare-earth luminescence, LA. Riseberg, M.J. Weber The ultrafast optical Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, RJ. Vernier Optical fibre waveguides - a review, P.J.B. Clarricoats

1 2 3 4 5

Theory of optical parametric amplification and oscillation, W. Brunner, H. Paul Optical properties of thin metal films, P. Rouard, A. Meessen Projection-type holography, T. Okoshi Quasi-optical techniques of radio astronomy, r ^ Co/e Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, IE. Sipe

1- 46 47- 87 89-159 161-193 195-244 245-325 327^02

VOLUME 15 (1977) 1- 75 11-\31 139-185 187-244 245-350

VOLUME 16(1978) 1 2 3 4 5

Laser selective photophysics and photochemistry, VS. Letokhov Recent advances in phase profiles generation, J.J. Clair, CI. Abitbol Computer-generated holograms: techniques and applications, W.-H. Lee Speckle interferometry, A.E. Ennos Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis 6 Light emission fi-om high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanical framework, I.R. Senitzky

1- 69 71-117 119-232 233-288 289-356 357-411 413^48

VOLUME 17 (1980) 1 Heterodyne holographic interferometry, R. Ddndliker 2 Doppler-free multiphoton spectroscopy, E. Giacobino, B. Cagnac 3 The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wilhelmi 4 Michelson stellar interferometry, W.J. Tango, R.Q. Twiss 5 Self-focusing media with variable index of refraction, A.L. Mikaelian

1-84 85-161 163-238 239-277 279-345

VOLUME 18 (1980) 1 Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan 2 Photocount statistics of radiation propagating through random and nonlinear media, J Perina

1-126 127-203

Contents of previous volumes 3 4

619

Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, V.U. Zavorotnyi 204^256 Catastrophe optics: morphologies of caustics and their diffraction patterns, M V. Berry, C. Upstill 257-346

VOLUME 19 (1981) 1 2 3 4 5

Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow 1- 43 Surface and size effects on the light scattering spectra of solids, D.L Mills, K.R. Subbaswamy 45-137 Light scattering spectroscopy of surface electromagnetic waves in solids, 5. Ushioda 139-210 Principles of optical data-processing,//.J ^M/r^A^'^cA: 211-280 The effects of atmospheric turbulence in optical astronomy, F. Roddier 281-376

VOLUME 20 (1983) 1 2 3 4 5

Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtes, P. Cruvellier, M. Detaille, M. Saisse 1—61 Shaping and analysis of picosecond light pulses, C Froehly B. Colombeau, M. Vampouille 63-153 Multi-photon scattering molecular spectroscopy, S. Kielich 155-261 Colour holography, P. Hariharan 263-324 Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P Stoicheff 325-380

VOLUME 21 (1984) 1 2 3 4 5

Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, L..4. Lt/g/^/o The Radon transform and its applications, H.H. Barrett Zone plate coded imaging: theory and applications, N.M. Ceglio, D.W. Sweeney Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J.C. Englund, R.R. Snapp, WC. Schieve

1- 67 69-216 217-286 287-354 355-428

VOLUME 22 (1985) 1 2 3 4 5

Optical and electronic processing of medical images, D. Malacara Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema Spectral and temporal fluctuations of broad-band laser radiation, A. V. Masalov Holographic methods of plasma diagnostics, G. Pf 05/wt;5^ya, Yu.I. Ostrovsky Fringe formations in deformation and vibration measurements using laser light, /. Yamaguchi 6 Wave propagation in random media: a systems approach, R.L. Fante

1- 76 11-\AA 145-196 197-270 271-340 341-398

VOLUME 23 (1986) 1

Analytical techniques for multiple scattering ft-om rough surfaces, J.A. DeSanto, G.S. Brown 2 Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka 3 Optical films produced by ion-based techniques, P.J. Martin, R.P Netterfield

1- 62 63-111 113-182

620 4 5

Contents of previous volumes Electron holography, A. Tonomura Principles of optical processing with partially coherent light, F.T.S. Yu

183-220 221-275

VOLUME 24 (1987) 1 2 3 4 5

Micro Fresnel lenses, H. Nishihara, T. Suhara Dephasing-induced coherent phenomena, L. Rothberg Interferometry with lasers, P. Hahharan Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, /. Glaser

1- 37 39-101 103-164 165-387 389-509

VOLUME 25 (1988) Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, LM. Narducci Coherence in semiconductor lasers, M Ohtsu, T. Tako Principles and design of optical arrays, Wang Shaomin, L. Ronchi Aspheric surfaces, G. Schulz

1-190 191-278 279-348 349-^15

VOLUME 26 (1988) 1 2 3 4 5

Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals, /. C Khoo Single-longitudinal-mode semiconductor lasers, G.P Agrawal Rays and caustics as physical objects, Yu.A. Kravtsov Phase-measurement interferometry techniques, K. Creath

1-104 105-161 163-225 227-348 349-393

VOLUME 27 (1989) 1 The self-imaging phenomenon and its applications, K. Patorski 2 Axicons and meso-optical imaging devices, L.M. Soroko 3 Nonimaging optics for flux concentration, LM. Bassett, W.T. Welford, R. Winston 4 Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C Sibilia 5 Generalized holography with application to inverse scattering and inverse source problems, R.P Porter

1-108 109-160 161-226 227-313 315-397

VOLUME 28 (1990) 1 Digital holography - computer-generated holograms, O. Bryngdahl, F. Wyrowski 2 Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa, G. Bjork 3 The quantum coherence properties of stimulated Raman scattering, M. G. Raymer, LA. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, i?../Coo^

1- 86 87-179 181-270 271-359 361^16

Contents of previous volumes

621

VOLUME 29 (1991) 1 Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall 2 Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, VD. Ozrin, A.I. Saichev 3 Generation and propagation of ultrashort optical pulses, LP. Christov 4 Triple-correlation imaging in optical astronomy, G. Weigelt 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol

1- 63 65-197 199-291 293-319 321^11

VOLUME 30 (1992) 1 2 3 4 5

Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C Fabre 1- 85 Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P. Shchepinov 87-135 Localization of waves in media with one-dimensional disorder, V.D. Freilikher, S.A. Gredeskul 137-203 Theoretical foundation of optical-soliton concept in fibers, Y. Kodama, A. Hasegawa 205-259 Cavity quantum optics and the quantum measurement process, P. Meystre 261-355 VOLUME 31 (1993)

1 2 3 4 5 6

Atoms in strong fields: photoionization and chaos, PW. Milonni, B. Sundamm Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov Optical amplifiers, N.K. Dutta, J.R. Simpson Adaptive multilayer optical networks, D. Psoitis, Y Qiao Optical atoms, R.J.C. Spreeuw, J.P. Woerdman Theory of Compton fi-ee electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre

1-137 139-187 189-226 227-261 263-319 321-412

VOLUME 32 (1993) 1 Guided-wave optics on silicon: physics, technology and status, B.P Pal 1- 59 2 Optical neural networks: architecture, design and models, F.T.S. Yu 61-144 3 The theory of optimal methods for localization of objects in pictures, L.P Yaroslavsky 145-201 4 Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, V.U. Zavorotny 203-266 5 Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, VL. Ginzburg 267-312 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C.Manus 313-361 VOLUME 33 (1994) 1 The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin 2 Quantum statistics of dissipative nonlinear oscillators, V Perinovd, A. Luks 3 Gap solitons, CM. De Sterke, J.E. Sipe 4 Direct spatial reconstruction of optical phaseft-omphase-modulated images, VI Vlad, D. Malacara 5 Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt 6 Digital halftoning: synthesis of binary images, O. Bryngdahl, T. Scheermesser, F Wyrowski

1-127 129-202 203-260 261-317 319-388 389-^63

622

Contents of previous volumes VOLUME 34 (1995)

1 2 3 4 5

Quantum interference, superposition states of light, and nonclassical effects, V Buzek, P.L Knight 1-158 Wave propagation in inhomogeneous media: phase-shift approach, LP. Presnyakov 159-181 The statistics of dynamic speckles, T. Okamoto, T. Asakura 183-248 Scattering of light from multilayer systems with rough boundaries, /. Ohlidal, K. Navrdtil, M. Ohlidal 249-331 Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss 333^02 VOLUME 35 (1996)

1 Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov 2 Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis 3 Interferometric multispectral imaging, K. Itoh 4 Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo 5 Coherent population trapping in laser spectroscopy, E. Arimondo 6 Quantum phase properties of nonlinear optical phenomena, R. Tanas, A. Miranowicz, Ts. Gantsog

1-60 61-144 145-196 197-255 257-354 355^46

VOLUME 36 (1996) 1 Nonlinear propagation of strong laser pulses in chalcogenide glass films, V Chumash, I. Cojocaru, E. Fazio, E Michelotti, M. Bertolotti 2 Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders 3 Super-resolution by data inversion, M. Bertero, C De Mol 4 Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, LA. Apresyan 5 Photon wave fianction, /. Bialynicki-Birula

1- 47 49-128 129-178 179-244 245-294

VOLUME 37 (1997) 1 The Wigner distribution fiinction in optics and optoelectronics, D. Dragoman 2 Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura 3 Spectra of molecular scattering of light, I.L Fabelinskii 4 Soliton communication systems, R.-J. Essiambre, GP Agrawal 5 Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller 6 Tunneling times and superluminality, R.Y. Chiao, A.M. Steinberg

1- 56 57- 94 95-184 185-256 257-343 345^05

VOLUME 38 (1998) 1 Nonlinear optics of stratified media, S. Dutta Gupta 2 Optical aspects of interferometric gravitational-wave detectors, P Hello 3 Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W. Nakwaski, M. Osinski 4 Fractional transformations in optics, A. W. Lohmann, D. Mendlouic, Z. Zalevsky 5 Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L Horner 6 Free-space optical digital computing and interconnection, J. Jahns

1- 84 85-164 165-262 263-342 343^18 419-513

Contents of previous volumes

623

VOLUME 39 (1999) 1 Theory and applications of complex rays, Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan 1- 62 2 Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T. Opatrny 63-211 3 Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford 213-290 4 The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker 291-372 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs 373-469 VOLUME 40 (2000) 1 Polarimetric optical fibers and sensors, T.R. Wolinski 2 Digital optical computing, J. Tanida, Y. Ichioka 3 Continuous measurements in quantum optics, V. Pefinovd, A. Luks 4 Optical systems with improved resolving power, Z Zalevsky, D. Mendlouic, A.W.Lohmann 5 Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, F. Wyrowski 6 Spectroscopy in polychromatic fields, Z Ficek and H.S. Freedhoff

1- 75 11~\ 14 115-269 271-341 343-388 389-441

VOLUME 41 (2000) 1 Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang 2 Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur 3 Ellipsometry of thin film systems, /. Ohlidal, D. Franta 4 Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu 5 Quantum statistics of nonlinear optical couplers, J. Perina Jr, J. Pefina 6 Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. Sdnchez-Soto 7 Optical solitons in media with a quadratic nonlinearity, C Etrich, F Lederer, B.A. Malomed, T. Peschel, U. Peschel

1- 95 97-179 181-282 283-358 359-417 419^79 483-567

VOLUME 42 (2001) 1 Quanta and information, S.Ya. Kilin 2 Optical solitons in periodic media with resonant and off-resonant nonlinearities, G. Kurizki, A.E. Kozhekin, T. Opatrny, B.A. Malomed 3 Quantum Zeno and inverse quantum Zeno effects, P. Facchi, S. Pascazio 4 Singular optics, M.S. Soskin, M.V Vasnetsov 5 Multi-photon quantum interferometry, G Jaeger A. V. Sergienko 6 Transverse mode shaping and selection in laser resonators, R. Oron, N. Davidson, A.A. Friesem, E. Hasman

1-91 93-146 147-217 219-276 277-324 325-386

Cumulative index - Volumes 1-43*

Abeles, K: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical frequencies Abitbol, C.I., see Clair, J.J. Abraham, N.B., P. Mandel, L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, VM., V.L. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.P., see Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett, M. Babiker: The orbital angular momentum of light Ammann, E.G.: Synthesis of optical birefiingent networks Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A., A.W. Smith: Experimental studies of intensity fluctuations in lasers Amaud, J.A.: Hamiltonian theory of beam mode propagation Asakura, T, see Okamoto, T. Asakura, T., see Peiponen, K.-E. Asatryan, A.A., see Kravtsov, Yu.A. Babiker, M., see Allen, L. Baltes, H.P: On the validity of KirchhoflF's law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, VD. Ozrin, A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-fi"ee diffraction images Barrett, H.H.: The Radon transform and its applications Bashkin, S.: Beam-foil spectroscopy Bassett, I.M., W.T. Welford, R. Winston: Nonimaging optics for flux concentration Beckmann, P.: Scattering of light by rough surfaces Beran, M.J., J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M.

* Volumes I-XL were previously distinguished by roman rather than by arabic numerals. 625

2, 249 7, 139 16, 71 25, 11, 9, 26, 37, 9, 39, 9, 41, 36, 35, 6, 11, 34, 37, 39,

1 1 235 163 185 179 291 123 97 179 257 211 247 183 57 1

39, 291

13,

1

29, 65 1, 67 21, 217 12, 287 27, 161 6, 53 33, 319 35, 61

626

Cumulative index - Volumes 1-43

Berry, M.V, C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M., C. De Mol: Super-resolution by data inversion Bertolotti, M., see Mihalache, D. Bertolotti, M., see Chumash, V Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Bjork, G., see Yamamoto, Y. Bloom, A.L.: Gas lasers and their application to precise length measurements Bouman, M.A., W.A. Van De Grind, P. Zuidema: Quantum fluctuations in vision Bousquet, P., see Rouard, P. Boyd, R.W. and D.J. Gauthier: "Slow" and "fast" light Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W., H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: Applications of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., F. Wyrowski: Digital holography - computer-generated holograms Bryngdahl, O., T. Scheermesser, F. Wyrowski: Digital halftoning: synthesis of binary images Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Buzek, V, PL. Knight: Quantum interference, superposition states of light, and nonclassical effects Cagnac, B., see Giacobino, E. Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur: Principles of optical disk data storage Casasent, D , D. Psaltis: Deformation invariant, space-variant optical pattern recognition Ceglio, N.M., D.W. Sweeney: Zone plate coded imaging: theory and applications Chang, R.K., see Fields, M.H. Chamotskii, M.L, J. Gozani, VI. Tatarskii, VU. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T., Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y, A.M. Steinberg: Tunneling times and superluminality Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, WM. Christov, I.P: Generation and propagation of ultrashort optical pulses Chumash, V, I. Cojocaru, E. Fazio, F Michelotti, M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.J., C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.J.B.: Optical fibre waveguides - a review Cohen-Tannoudji, C , A. Kastler: Optical pumping Cojocaru, I., see Chumash, V Cole, T.W: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.X: Quantum jumps Courtes, G., P. Cruvellier, M. Detaille, M. Saisse: Some new optical designs for ultra-violet bidimensional detection of astronomical objects Creath, K.: Phase-measurement interferometry techniques

18, 257 36, 129 27, 227 36, 1 16, 357 36, 245 28, 87 9, 1 22, 77 4, 145 43, 497 23, 1 35, 61 15, 1 4, 37 11, 167 28, 1 33, 389 2, 73 19, 211 34,

1

17, 85 41, 97 16, 289 21, 287 41, 1 32, 203 41, 37, 41, 13, 29,

283 345 97 69 199

36, 16, 14, 5, 36, 15, 20, 28,

1 71 327 1 1 187 63 361

20, 1 26, 349

627

Cumulative index - Volumes 1-43 Crewe, A.V: Production of electron probes using a field emission source Cruvellier, P., see Courtes, G. Cummins, H.Z., H.L. Swinney: Light beating spectroscopy

11, 223 20, 1 8, 133

Dainty, J.C.: The statistics of speckle patterns Dandliker, R.: Heterodyne holographic interferometry Dattoli, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton fi-ee electron lasers Davidson, N., see Oron, R. De Mol, C , see Bertero, M. De Sterke, CM., J.E. Sipe: Gap solitons Decker Jr, J.A., see Harwit, M. Delano, E., R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses DeSanto, J.A., G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Detaille, M., see Courtes, G. Dexter, D.L., see Smith, D.Y. Dragoman, D.: The Wigner distribution fiinction in optics and optoelectronics Dragoman, D.: Phase space correspondence between classical optics and quantum mechanics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dutta, N.K., J.R. Simpson: Optical amplifiers Dutta Gupta, S.: Nonlinear optics of stratified media

14, 17, 31, 42, 36, 33, 12, 7, 9,

Eberly, J.H.: Interaction of very intense light with fi-ee electrons Englund, J.C, R.R. Snapp, W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity Ennos, A.E.: Speckle interferometry Essiambre, R.-J., G.P Agrawal: Soliton communication systems Etrich, C , F. Lederer, B.A. Malomed, T Peschel, U. Peschel: Optical solitons in media with a quadratic nonlinearity Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C, see Reynaud, S. Facchi, P., S. Pascazio: Quantum Zeno and inverse quantum Zeno effects Fante, R.L.: Wave propagation in random media: a systems approach Fazio, E., see Chumash, V Ficek, Z. and H.S. FreedhofiF: Spectroscopy in polychromatic fields Fields, M.H., J. Popp, R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Flytzanis, C , F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics Focke, J.: Higher order aberration theory Forbes, G.W., see Kravtsov, Yu.A. Frangon, M., S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlidal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, VD., S.A. Gredeskul: Localization of waves in media with one-dimensional disorder

1 1 321 325 129 203 101 67 31

23, 1 20, 1 10, 165 37, 1 43, 12, 14, 31, 38,

433 163 161 189 1

7, 359 21, 355 16, 233 37, 185 41, 483 37, 95 in

1

1

42, 22, 36, 40, 41, 1,

147 341 1 389 1 253

29, 4, 39, 6, 41, 40,

321 1 1 71 181 389

30, 137

628

Cumulative index - Volumes 1-43

Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions Friesem, A.A., see Oron, R. Froehly, C , B. Colombeau, M. Vampouille: Shaping and analysis of picosecond light pulses Fry, G.A.: The optical performance of the human eye Fu, Z., see Chen, R.T. Gabor, D.: Light and information Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H., G.H. Weiss: Random walk and diifusion-like models of photon migration in turbid media Gantsog, Ts., see Tanas, R. Gauthier, D.J., see Boyd, R.W. Ghatak, A., K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E., B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., see Reynaud, S. Giannessi, L., see Dattoli, G. Ginzburg, V.L., see Agranovich, VM. Ginzburg, VL.: Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Giovanelli, R.G.: Diffusion through non-uniform media Glaser, I.: Information processing with spatially incoherent light Gniadek, K., J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, X, see Chamotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, YD.

9, 311 42, 325 20, 63 8, 51 41, 283 1, 109 3, 187 34, 35, 43, 18, 13, 17, 30, 31, 9,

333 355 497 1 169 85 1 321 235

32, 267 2, 109 24, 389 9, 8, 32, 12, 30,

281 1 203 233 137

Hache, F, see Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, R: Colour holography Hariharan, R: Interferometry with lasers Hariharan, P., B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M., J.A. Decker Jr: Modulation techniques in spectrometry Hasegawa, A., see Kodama, Y. Hasman, E., see Oron, R. Heidmann, A., see Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Herriot, D.R.: Some applications of lasers to interferometry Homer, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images

29, 29, 20, 24, 36, 12, 30, 42, 30, 38, 10, 6, 38, 10,

321 1 263 103 49 101 205 325 1 85 289 171 343 1

Ichioka, Y, see Tanida, J. Imoto, N., see Yamamoto, Y Itoh, K.: Interferometric multispectral imaging

40, 77 28, 87 35, 145

Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, P., B. Roizen-Dossier: Apodisation

5, 247 3, 29

629

Cumulative index - Volumes 1-43 Jaeger, G., A.V Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection Jamroz, W., B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation Javidi, B., J.L. Homer: Pattern recognition with nonlinear techniques in the Fourier domain Jones, D.G.C., see Allen, L.

42, 277 38, 419 20, 325

Kastler, A., see Cohen-Tannoudji, C. Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Keller, O.: Optical works of L.V Lorenz Khoo, I.e.: Nonlinear optics of liquid crystals Khulbe, P., see Carriere, J. Kielich, S.: Multi-photon scattering molecular spectroscopy Kilin, S.Ya.: Quanta and information Kinosita, K.: Surface deterioration of optical glasses Kitagawa, M., see Yamamoto, Y. Klein, M.C., see Flytzanis, C. Klyatskin, VI.: The imbedding method in statistical boundary-value wave problems Knight, PL., see Buzek, V Kodama, Y, A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, F: The elements of radiative transfer Kottler, F: Diffraction at a black screen. Part I: Kirchhoff's theory Kottler, F: Diffraction at a black screen, Part II: electromagnetic theory Kozhekin, A.E., see Kurizki, G. Kravtsov, Yu.A.: Rays and caustics as physical objects Kravtsov, Yu.A., see Barabanenkov, Yu.N. Kravtsov, Yu.A., L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Forbes, A.A. Asatryan: Theory and applications of complex rays Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatmy, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities

5, 37, 43, 26, 41, 20, 42, 4, 28, 29, 33, 34, 30, 7, 3, 4, 6, 42, 26, 29, 36, 39, 1, 40,

Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves Lederer, F, see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N., J. Upatnieks: Recent advances in holography Letokhov, VS.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H., C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Lohmann, A.W., D. Mendlovic, Z. Zalevsky: Fractional transformations in optics Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L.A.: Theory of optical bistability Luis, A., L.L. Sanchez-Soto: Quantum phase difference, phase measurements and Stokes operators

14, 11, 41, 16, 6, 16, 39, 8, 41,

47 123 483 119 1 1 373 343 97

5, 38, 40, 35, 21,

287 263 271 61 69

38, 343 9, 179 1 257 195 105 97 155 1 85 87 321 1 1 205 1 1 281 331 93 227 65 179 1 211 343

42, 93

41, 419

630

Cumulative index - Volumes 1-43

Luks, A., see Pefinova, V Luks, A., 5ee Pefinova, V Luks, A. and V Pefinova: Canonical quantum description of light propagation in dielectric media

Machida, S., see Yamamoto, Y. Mainfray, G., C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas Malacara, D.: Optical and electronic processing of medical images Malacara, D., see Vlad, VI. Mallick, S., see Fran9on, M. Malomed, B.A., see Etrich, C. Malomed, B.A., see Kurizki, G. Malomed, B.A.: Variational methods in nonlinear fiber optics and related fields Mandel, L.: Fluctuations of light beams Mandel, L.: The case for and against semiclassical radiation theory Mandel, P., see Abraham, N.B. Mansuripur, M., see Carriere, J. Manus, C , see Mainfray, G. Marchand, E.W.: Gradient index lenses Martin, P.J., R.P. Netterfield: Optical films produced by ion-based techniques Masalov, A.V: Spectral and temporal fluctuations of broad-band laser radiation Maystre, D.: Rigorous vector theories of diffraction gratings Meessen, A., see Rouard, P. Mehta, C.L.: Theory of photoelectron counting Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Michelotti, K, see Chumash, V Mihalache, D., M. Bertolotti, C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L., M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mikaelian, A.L.: Self-focusing media with variable index of refraction Mills, D.L., K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, PW., B. Sundaram: Atoms in strong fields: photoionization and chaos Miranowicz, A., see Tanas, R. Miyamoto, K.: Wave optics and geometrical optics in optical design Mollow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence Murata, K.: Instruments for the measuring of optical transfer functions Musset, A., A. Thelen: Multilayer antireflection coatings

Nakwaski, W., M. Osiriski: Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., see Carriere, J. Narducci, L.M., see Abraham, N.B. Navratil, K., see Ohlidal, I. Netterfield, R.P, see Martin, PJ. Nishihara, H., T. Suhara: Micro Fresnel lenses Noethe, L.: Active optics in modem large optical telescopes

33, 129 40, 115 43, 295

28, 32, 22, 33, 6, 41, 42, 43, 2, 13, 25, 41, 32, 11, 23, 22, 21, 15, 8, 38, 40, 30, 36, 27, 7, 17,

87 313 1 261 71 483 93 71 181 27 1 97 313 305 113 145 1 77 373 263 271 261 1 227 231 279

19, 45 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201

38, 41, 25, 34, 23, 24, 43,

165 97 1 249 113 1 1

631

Cumulative index - Volumes 1-43 Ohlidal, I., K. Navratil, M. Ohlidal: Scattering of light from multilayer systems with rough boundaries Ohlidal, I., D. Franta: Ellipsometry of thin film systems Ohlidal, M., see Ohlidal, I. Ohtsu, M., T. Tako: Coherence in semiconductor lasers Okamoto, T, T. Asakura: The statistics of dynamic speckles Okoshi, T.: Projection-type holography Ooue, S.: The photographic image Opatmy, T, see Welsch, D.-G. Opatmy, T, see Kurizki, G. Oron, R., N. Davidson, A.A. Friesem, E. Hasman: Transverse mode shaping and selection in laser resonators Orrit, M., J. Bernard, R. Brown, B. Lounis: Optical spectroscopy of single molecules in solids Osinski, M., see Nakwaski, W. Ostrovskaya, G.V, Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.I., see Ostrovskaya, G.V Ostrovsky, Yu.I., V.P Shchepinov: Correlation holographic and speckle interferometry Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, YD., see Barabanenkov, Yu.N. Padgett, M.J., see Allen, L. Pal, B.P: Guided-wave optics on silicon: physics, technology and status Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., see Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., see Brunner, W. Pegis, R.J.: The modem development of Hamiltonian optics Pegis, R.J., see Delano, E. Peiponen, K.-E., E.M. Vartiainen, T Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C , see Carriere, J. Penna, J.: Photocount statistics of radiation propagating through random and nonlinear media Perina, J., see Perina Jr, J. Perina Jr, J., J. Penna: Quantum statistics of nonlinear optical couplers Perinova, V, A. Luks: Quantum statistics of dissipative nonlinear oscillators Pennova, V, A. Luks: Continuous measurements in quantum optics Perinova, V, see Luks, A. Pershan, P.S.: Non-linear optics Peschel, T, see Etrich, C. Peschel, U, see Etrich, C. Petykiewicz, J., see Gniadek, K. Picht, J.: The wave of a moving classical electron Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view Popp, J., see Fields, M.H. Porter, R.P.: Generalized holography with application to inverse scattering and inverse source problems Presnyakov, L.P: Wave propagation in inhomogeneous media: phase-shift approach Psaltis, D., see Casasent, D

34, 41, 34, 25, 34, 15, 7, 39, 42,

249 181 249 191 183 139 299 63 93

42, 325 35, 38, 22, 22, 30, 24, 33, 29,

61 165 197 197 87 165 319 65

39, 32, 35, 42, 27, 15, 1, 7,

291 1 197 147 1 1 1 67

37, 57 41, 97 18, 41, 41, 33, 40, 43, 5, 41, 41, 9, 5, 31, 41,

127 359 359 129 115 295 83 483 483 281 351 139 1

27, 315 34, 159 16, 289

632

Cumulative index - Volumes 1-43

Psaltis, D., Y. Qiao: Adaptive multilayer optical networks

31, 227

Qiao, Y, see Psaltis, D.

31, 227

Raymer, M.G., LA. Walmsley: The quantum coherence properties of stimulated Raman scattering Renieri, A., see Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino, C. Fabre: Quantum fluctuations in optical systems Ricard, D., see Flytzanis, C. Riseberg, L.A., M.J. Weber: Relaxation phenomena in rare-earth luminescence Risken, H.: Statistical properties of laser light Roddier, P.: The effects of atmospheric turbulence in optical astronomy Roizen-Dossier, B., see Jacquinot, P. Ronchi, L., see Wang Shaomin Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems Rosenblum, W.M., XL. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P., P. Bousquet: Optical constants of thin films Rouard, P., A. Meessen: Optical properties of thin metal films Roussignol, Ph., see Flytzanis, C. Rubinowicz, A.: The Miyamoto-Wolf diffraction wave Rudolph, D., see Schmahl, G. Saichev, A.L, see Barabanenkov, Yu.N. Saisse, M., see Courtes, G. Saito, S., see Yamamoto, Y Sakai, H., see Vanasse, G.A. Saleh, B.E.A., see Teich, M.C. Sanchez-Soto, L.L., see Luis, A. Sanders, B.C., see Hariharan, P Scheermesser, T, see Bryngdahl, O. Schieve, W.C, see Englund, J.C. Schirripa Spagnolo, G., see Paoletti, D. Schmahl, G., D. Rudolph: Holographic diffraction gratings Schubert, M., B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G., J. Schwider: Interferometric testing of smooth surfaces Schulz, G.: Aspheric surfaces Schwider, J., see Schulz, G. Schwider, J.: Advanced evaluation techniques in interferometry Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics Senitzky, LR.: Semiclassical radiation theory within a quantum-mechanical fi-amework Sergienko, A.V, see Jaeger, G. Sharma, S.K., D.J. Somerford: Scattering of light in the eikonal approximation Shchepinov, VP, see Ostrovsky, Yu.L Sibilia, C , see Mihalache, D. Simpson, J.R., 5e^ Dutta, N.K. Sipe, J.E., see Van Kranendonk, J. Sipe, J.E., see De Sterke, CM.

28, 181 31, 321 30, 29, 14, 8, 19, 3, 25, 35,

1 321 89 239 281 29 279 1

13, 24, 4, 15, 29, 4, 14,

69 39 145 77 321 199 195

29, 20, 28, 6, 26, 41, 36, 33, 21, 35, 14,

65 1 87 259 1 419 49 389 355 197 195

17, 163 13, 93 25, 349 13, 93 28, 271 10, 89 16, 413 42, 277 39, 213 30, 87 27, 227 31,189 15, 245 33, 203

Cumulative index - Volumes 1-43

633

Sittig, E.K.: Elastooptic light modulation and deflection Sizmann, A., G. Leuchs: The optical Kerr effect and quantum optics in fibers Slusher, R.E.: Self-induced transparency Smith, A.W., see Armstrong, J.A. Smith, D.Y., D.L. Dexter: Optical absorption strength of defects in insulators Smith, R.W.: The use of image tubes as shutters Snapp, R.R., see Englund, J.C. Sodha, M.S., A.K. Ghatak, VK. Tripathi: Self-focusing of laser beams in plasmas and semiconductors Somerford, D.J., see Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices Soskin, M.S., M.V Vasnetsov: Singular optics Spreeuw, R.J.C., J.R Woerdman: Optical atoms Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Stoicheff, B.R, see Jamroz, W. Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy Subbaswamy, K.R., see Mills, D.L. Suhara, T, see Nishihara, H. Sundaram, B., see Milonni, RW. Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams Sweeney, D.W., see Ceglio, N.M. Swinney, H.L., see Cummins, H.Z.

10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355

Tako, T, see Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets Tanas, R., A. Miranowicz, Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J., R.Q. Twiss: Michelson stellar interferometry Tanida, J., Y. Ichioka: Digital optical computing Tatarskii, VI., VU. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium Tatarskii, VI., see Chamotskii, M.I. Taylor, C.A., see Lipson, H. Teich, M.C., B.E.A. Saleh: Photon bunching and antibunching Ter-Mikaelian, M.L., see Mikaehan, A.L. Thelen, A., see Musset, A. Thompson, B.J.: Image formation with partially coherent light Thyagarajan, K., see Ghatak, A. Tonomura, A.: Electron holography Torre, A., see Dattoli, G. Torre, A,: The fi-actional Fourier transform and some of its applications to optics Tripathi, V.K., see Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatialfi*equencyfiltering Turunen, J., M. Kuittinen, F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.

25, 191 23, 63

Upatnieks, J., see Leith, E.N. Upstill, C , see Berry, M.V

6, 1 18, 257

13, 169 39, 213 27, 109 42, 219 31, 263 5, 145 37, 345 20, 325 9, 73 2, 1 19, 45 24, 1 31, 1 12, 1 21, 287 8, 133

35, 355 17, 239 40, 77 18, 204 32, 203 5, 287 26, 1 7, 231 8, 201 7, 169 18, 1 23, 183 31, 321 43, 531 13, 169 2, 131 40, 343 17, 239

634

Cumulative index - Volumes 1-43

Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids

19, 139

Vampouille, M., see Froehly, C. Van De Grind, W.A., see Bouman, M.A. Van Heel, A.C.S.: Modem alignment devices Van Kranendonk, J., J.E. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vanasse, G.A., H. Sakai: Fourier spectroscopy Vartiainen, E.M., see Peiponen, K.-E. Vasnetsov, M.V, see Soskin, M.S. Vernier, P.J.: Photoemission Vlad, VI., D. Malacara: Direct spatial reconstruction of optical phase from phase-modulated images Vogel, W., see Welsch, D.-G.

20, 63 22, 77 1, 289

Walmsley, I.A., see Raymer, M.G. Wang Shaomin, L. Ronchi: Principles and design of optical arrays Weber, M.J., see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weiss, G.H., see Gandjbakhche, A.H. Welford, W.T.: Aberration theory of gratings and grating mountings Welford, WT.: Aplanatism and isoplanatism Welford, W.T., see Bassett, I.M. Welsch, D.-G., W Vogel, T. Opatmy: Homodyne detection and quantum-state reconstruction Whitney, K.G., see Scully, M.O. Wilhelmi, B., see Schubert, M. Winston, R., see Bassett, I.M. Woerdman, J.P, see Spreeuw, R.J.C. Wolinski, T.R.: Polarimetric optical fibers and sensors Wolter, H.: On basic analogies and principal differences between optical and electronic information Wynne, C.G.: Field correctors for astronomical telescopes Wyrowski, F, see Bryngdahl, O. Wyrowski, F, see Bryngdahl, O. Wyrowski, F, see Turunen, J. Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa, G. Bjork: Quantum mechanical limit in optical precision measurement and communication Yanagawa, T, see Yamamoto, Y Yaroslavsky, L.P: The theory of optimal methods for localization of objects in pictures Yeh, W.-H., 56^ Carriere, J. Yoshinaga, H.: Recent developments in far infrared spectroscopic techniques Yu, F.T.S.: Principles of optical processing with partially coherent light Yu, F.T.S.: Optical neural networks: architecture, design and models

15, 6, 37, 42, 14,

245 259 57 219 245

33, 261 39, 63 28, 25, 14, 29, 34, 4, 13, 27,

181 279 89 293 333 241 267 161

39, 10, 17, 27, 31, 40,

63 89 163 161 263 1

1, 10, 28, 33, 40,

155 137 1 389 343

22, 271 6, 105 8, 295 28, 28, 32, 41, 11, 23, 32,

87 87 145 97 77 221 61

Cumulative index - Volumes 1-43 Zalevsky, Z., see Lohmann, A.W. Zalevsky, Z., D. Mendlovic, A.W. Lohmann: Optical systems with improved resolving power Zavorotny, VU., see Chamotskii, M.I. Zavorotnyi, VU, see Tatarskii, V.I. Zuidema, P., see Bouman, M.A.

635 38, 263 40, 32, 18, 22,

271 203 204 77