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
F. Gori
Rome, Italy
L.M. Narducci
Philadelphia, USA
J. Pefina
Olomouc, Czech Republic
R.M. Sillitto
Edinburgh, Scotland
H. Walther
Garching, Germany
Preface This volume contains six review articles on diverse topics that have become of particular interest to optical scientists and optical engineers in recent years. The first article, by N. Davidson and N. Bokor, reviews researches on the anamorphic shaping of laser beams and of diffuse light whose input or output is substantially elongated along one direction. Such elongated beams have come to be of special interest in recent years with the appearance of high-power laser diode bars, whose excellent properties are offset by their poor beam quality and by the fact that the output beam profile is highly anamorphic (with typical aspect ratio 1000:1) and hence unsuitable for many applications. Several techniques have been developed to collimate and shape the output beams of such laser diode bars into symmetric spots. Anamorphic beam shaping has also been used to concentrate symmetrical fields such as solar radiation into very narrow lines, for use in heating water pipes, for side-pumping solar lasers, and in optical metrology (e.g. for improving resolution in surface profile measurement and high-resolution spectrometry). The article presents a review of the main reflective, refractive, diffractive, and adiabatic techniques for anamorphic beam characterization. The second article, by I. Glesk, B.C. Wang, L. Xu, V Baby and P.R. Prucnal, presents a review of recent progress in the development of ultra-fast alloptical switching devices with various applications for future optical networks. The operation principle and performance of different all-optical switches based on nonlinearities in optical fiber semiconductor optical amplifiers (SOA) and passive waveguides are discussed. Special attention is paid to interferometric SOA-based all-optical switches. Several testbed demonstrations are described. The next article, by J. Yin, W. Gao and Y. Zhu, is concerned with the generation of dark hollow beams and their applications. Such beams have been used to form optical pipes, optical tweezers, atomic pipes, atomic tweezers, atomic refrigerators, and atomic motors. They can be applied in the accurate, non-contact manipulation and control of microscopic particles, such as biological cells, neutral atoms and molecules. The principles and experimental methods for generating various types of dark hollow beams are discussed. Applications of such beams in optical traps for microscopic particles including biological cells
vi
Preface
are also discussed, as are recent studies of dark hollow beams in atom optics and coherent matter-wave optics. The fourth article, by D.J. Gauthier, presents a review of a new type of quantum oscillator known as two-photon laser. Such devices are based on twophoton stimulated emission processes whereby two photons incident on an atom induce it to drop to a lower energy state and four photons are scattered. This kind of laser has been realized only relatively recently by combining cavity quantum electrodynamics experimental techniques with novel nonlinear optical interactions. Research on two-photon quantum processes, leading to the development and characterization of two-photon masers and lasers, is discussed. The unusual quantum-statistical and nonlinear dynamical properties predicted for the device are also reviewed. The subsequent article, by G. Gbur, discusses a rather old but poorly understood subject, the so-called non-radiating sources and the related concept of non-visible objects. These are certain extended charge-current distributions that may oscillate without generating radiation. Such sources have many intriguing mathematical and physical properties whose existence is intimately related to nonuniqueness of the solution of the so-called inverse source problem. The current state of understanding of such sources is discussed and they are compared with other classes of "invisible objects". The concluding article, by H. Cao, is concerned with random lasers. These are unconventional lasers whose feedback is provided by disorder-induced scattering. Random lasers may be separated into two categories: those with coherent feedback and those with incoherent feedback. In this article both types are discussed, as well as measurements of a variety of properties of such devices, such as the lasing threshold, lasing spectra, emission pattern, dynamical response, photon statistics and speckle patterns. Furthermore, investigations regarding the relevant length scales are described. Large disorder in the lasing material leads to spatial confinement of the lasing modes, which is the foundation for the micro-random laser. Some theoretical models of random lasers with coherent feedback are briefly introduced. Such research helps the understanding of the interplay between light localization and coherent amplification. In view of the broad coverage presented in this volume, it is hoped that many readers will find in it something that is of particular interest to them. Emil Wolf Department of Physics and Astronomy and the Institute of Optics University of Rochester Rochester, NY 14627, USA April 2003
E. Wolf, Progress in Optics 45 © 2003 Elsevier Science B. V All rights reserved
Chapter 1
Anamorphic beam shaping for laser and diffuse light by
Nir Davidson Dept. of Physics of Complex Systems, Weizmann Institute of Science, 76100 Rehovot, Israel
and
Nandor Bokor Dept. of Physics, Budapest University of Technology and Economics, 1111 Budapest, Hungary
Contents
Page § 1. Introduction
3
§ 2. Reflective techniques
7
§ 3. Refractive/diffractive techniques
20
§ 4. Adiabatic techniques
36
§ 5. Applications
44
§ 6.
50
Summary
References
50
§ 1. Introduction Beam shaping constitutes a large and important field in optics. For many applications the beam must be shaped in one transverse direction substantially differently than in the other transverse direction. In what follows, the term anamorphic beam shaping designates techniques in which the one-dimensional beam quality in one transverse direction is improved at the expense of reducing the one-dimensional beam quality in the orthogonal direction. In a broader sense, techniques for one-dimensional beam shaping would include onedimensional concentration of diffuse light with a curved diffractive element (Bokor, Shechter, Friesem and Davidson [2001], Bokor and Davidson [2001c, 2002b]) and one-dimensional diffuse beam shaping using a single reflection on a curved step mirror (Bokor and Davidson [2001a]). The simplest element capable of one-dimensional beam shaping is a cylindrical lens. In the examples listed above the one-dimensional beam-quality factors in the two orthogonal transverse directions remain the same. In this article we will not consider such techniques. Comprehensive reviews on concentration of diffuse beams, with special emphasis on non-imaging concentrators and solar energy applications, have been provided by Winston and Welford [1989] and Bassett, Welford and Winston [1989].
1.1. Diffuse light and its phase-space representation We first define a number of concepts related to diffuse light. The term diffiise light refers to beams for which the diffraction-limited angles and spot sizes are much smaller than the diffusive ones. This means that the transverse (spatial) coherence length of the beam is much smaller than its size, and that we can use the geometrical optics approximation and ray-tracing techniques to describe the beam propagation. The longitudinal (temporal) coherence is less relevant to the scope of this chapter, except when a broad wavelength range may cause considerable chromatic aberrations, in particular for diffractive optical elements. An effectively spatially incoherent, diffiise light can be formed even for laser experiments, by sending the laser beam through a rotating diffiiser.
4
Anamorphic beam shaping for laser and diffuse light
[1, § 1
Following Winston and Welford [1989], we define the four-dimensional phasespace volume (PSV) (often referred to as "etendue") of a diffuse beam as ?SY=
f f
f
j doc'Ay'd(sina'^)d(sin«;),
(1.1)
X y sin a^ sin Qy
where x and y are the sizes, and a^ and ay the diffusive angles, of the beam in two orthogonal directions (jc and y are perpendicular to the direction of beam propagation). For a beam with uniform and space-invariant diffiisivity (explained below) we define the phase-space areas (PSAs) in the x- and ^-directions as follows: PSA;r = y/jix sin ajc,
PSA^ = y/Hy sin Qy.
(1.2)
Note that, in general, a diffuse beam is represented by a non-uniform distribution function I{x,y, o^, ay) in four-dimensional phase-space. Space-invariant diffusivity means that the distribution function can be written as a product of two distributions: I\{x,y) • hia^, ay). In this chapter we will mostly assume uniform distributions with Cartesian symmetry that are thus characterized by the four quantities x, y, ax and ay. This largely simplifies the notation and captures most of the basic effects and ideas that we will describe. For the extension to nonuniform distributions (which are quite common in practice), the distribution as well as the conservation laws should be described in different terms than in the uniform case, e.g., by the RMS sizes of the distribution function. Optical brightness is defined as: B = ^ ,
(1.3)
where P is the optical power transmitted by the beam. A fiindamental conservation law - closely related to the second Law of thermodynamics - states that for spatially incoherent light and passive optical transformations B cannot increase. In the optimal case brightness is conserved. For lossless transformations P is constant and the conservation of brightness implies a conservation of PSV This so-called etendue invariance (Winston and Welford [1989]) is the main guiding principle in the design of diffuse beam-shaping techniques. In the paraxial approximation [sin a^^ ^ 1; sin o^ <^ 1] the PSV can be written as FSY^AQ,
(1.4)
where Q is the solid angle of divergence and A is the cross-sectional area of the beam. Recently it was proposed that the notation M^ should be used to describe
1, § 1]
Introduction
Fig. 1.1. Phase-space representation of a space-invariant, uniform, rectangular diffuse beam: the size of the rectangle represents the spatial dimensions of the beam, and the length of the arrow for each direction represents the diffuse angle.
the two-dimensional beam-quality factor, defined in the paraxial case as (Graf and Balmer [1996]): M'=M^M^ = ^ ,
(1.5)
where - in the practical case of quasi-monochromatic beams - A is the center wavelength, and M^ and My are the one-dimensional beam-quality values in the X- and ^-directions, respectively, defined as: Mi = ^ - f " - , A
M^ = ' - ^ .
(1.6)
A
It is obvious fi-om eqs. (1.2) and (1.4)-(1.6) that in the paraxial case of a monochromatic beam the beam-quality factor and the phase-space volume are equivalent descriptions of the beam. Even in the non-paraxial case the symmetry condition M^ = My is equivalent to the symmetry condition PSA^^ = PSA^. Since all the techniques described below contain these symmetry requirements either at the input or at the output, we will often use the concepts of M^ and My in the text - because of their wide acceptance - even for beams with large diffusive angles. For diffraction-limited beams - e.g., in the x-direction - M^ takes its optimum value: M^ = \. For diffuse beams M^^ > 1. Note that the beam quality M^ is related to the focusability of the beam, i.e. it gives the minimum spot size to which it can be focused, in units of the diffraction-limited area A^. Following Davidson and Khaykovich [1999], we now introduce a graphical method to depict the four-dimensional PSV of a diffuse light distribution. A beam with a space-invariant diffusivity can be represented with a simple diagram, as shown in fig. 1.1. The spatial dimensions are represented by the size of the rectangle, and the length of the arrow for each direction represents the diffuse angle. The beam represented in fig. 1.1, for example, has larger dimensions in the ^-direction than in the x-direction, however, its diffusivity is larger in the x-direction than in the y-direction. We will use this type of phasespace representations - called phase-space diagrams - throughout the text.
6
Anamorphic beam shaping for laser and diffuse light
[1, § 1
1.2. Anamorphic beam shaping We now return to anamorphic beam shaping, i.e. when only the 4-dimensional PSV is conserved (in the optimal case), while PSA^c and PSA^ are not conserved individually during the beam transformation. Anamorphic beam shaping gained special significance in recent years with the appearance of high-power laser diode arrays (LDAs), whose excellent properties are offset by their poor beam quality (M^ > 1), and especially by their highly asymmetric M^ and My values (the typical aspect ratio of an LDA is y/x ^ 1000, while the divergence angles are comparable in the jc- and ^-directions), making the output beam of an LDA unsuitable for many applications. Several techniques have been developed to collimate and shape the output beam of such LDAs into symmetric spots, e.g., for end pumping of solid-state lasers, for printing and for materials processing. Here the task is to produce an output beam where the orthogonal beam-quality factors are as much equalized as possible (M^ ^ My), without increasing M^ significantly. Alternatively, initially symmetrical beams are often concentrated primarily in one transverse direction for applications such as concentration of solar energy into narrow water pipes for water heating systems and for excitation of photochemical reactions, side pumping of solar-pumped lasers, formation of narrow light lines for optical scanners, faxes, copy-machines and high-resolution grating-based spectrometers. Here the task is to transform the initially symmetric M^ and M^ values at the input into highly asymmetric ones at the output in order to achieve extremely high concentration in one of the transverse directions. For uniform notation, we define the JC- and >^-directions so that for each technique presented throughout the text the input/output obeys M^
1, § 2]
Reflective techniques
7
beam quality in one transverse direction, corresponding to a large reduction in the PSA in that direction. Of course, this must be accompanied by (at least) a corresponding increase in the PSA in the orthogonal direction so as to conserve the total (4-dimensional) PSV of the beam. Such transformations of beam quality from one transverse direction to the other must involve coupling between the two directions. Based on the type of optical operation utilized to achieve coupling between orthogonal directions, anamorphic beam shaping techniques can be divided into the following categories: • Reflective techniques (§ 2). • Refi-active and diffractive techniques (§ 3). • Adiabatic techniques (§ 4). We will review the different methods according to the groups listed above, focusing on the basic principles of operation. Note that techniques based on refractive optics and techniques based on diffractive optics are frequently interchangeable, i.e. the same basic idea can be implemented using both refractive and diffractive optics. For this reason, we will treat these two fields in the same section (§ 3). In § 5 we will review the most important application areas of anamorphic beam shaping. We conclude with a summary in § 6.
§ 2. Reflective techniques The basic idea of all reflective techniques is to divide the input beam into sub-beams and then - using specular reflections from metallic surfaces or total internal reflections inside prisms - to apply an appropriate optical transformation for each sub-beam, so that the transformed (modified) sub-beams are recombined at the output to form a beam with symmetrical beam-quality factors in the xand ^-directions. The main advantage of any reflective technique is its general applicability to any light source, including diffuse incoherent sources, e.g., sunlight. However, most of the techniques described below were intended for laser-diode applications. 2.1. The two-mirror method Clarkson and Hanna [1996] proposed and demonstrated a reflective beamshaping method for high-power laser diode bars. The purpose was to equalize the beam-quality factors in orthogonal directions without significantly decreasing the brightness. The method is based on two parallel plane mirrors, a small distance d
Anamorphic beam shaping for laser and diffuse light
[1, § 2
(a) •3^0;
(b)
Fig. 2.1. Optical setup of the two-mirror method of Clarkson and Hamia [1996]: (a) view from the side; (b) view from below.
apart (fig. 2.1). The incident beam - which is highly elongated in the jv-direction, i.e. My > M^ - can be considered to consist of A^ adjacent sub-beams (N = 3 in fig. 2.1, the necessary value for N is calculated below). Sub-beam 1 is not incident upon either mirror: it passes by the side of mirror A (fig. 2.1a) and passes over mirror B (fig. 2.1b), emerging with no change in its direction. Sub-beam 2 also passes by the side of mirror A, but then hits mirror B and is reflected back onto mirror A inmiediately next to sub-beam 1. Then it is reflected at mirror A and emergesfiromthe two-mirror system parallel to sub-beam 1, but displaced next to it. Similarly, sub-beam 3 undergoes a reflection at mirror B and is directed onto mirror A, then it is reflected back onto mirror B, where it is reflected again at mirror A, subsequently emerging parallel to sub-beams 1 and 2, but displaced next to sub-beam 2. The basic idea in the two-mirror method is thus to chop the incident beam into a specific number of sub-beams and then direct the sub-beams in the proper directions so that they emerge firom the beam shaper stacked next to each other. The input and output light distributions are represented in the phase-space
Reflective techniques
1,§2] Ctyl
X p<2 \ (a)
(b)
Fig. 2.2. Phase-space diagrams at (a) the input and (b) the output of the two-mirror method.
diagrams of fig. 2.2. As is obvious fi-om figs. 2.1 and 2.2, the diffuse angles are unchanged: ^X2 = sin a^Xi, sin ar,
Sin ay, = sin ay^,
(2.1)
but the beam sizes change considerably in the x- and j^-directions: (2.2)
X2 = iV • Xi,
hence M^ is increased by N and My is decreased by N. Since at the input My^ > M^^, the beam qualities can be made equal at the output, i.e.: X2 sm ax2 = yi sin ay^.
(2.3)
The necessary number of sub-beams N needed to achieve M^ = My at the output is thus: N=
yi sin ayi
x\ sin a,'XI
PSA.'yi PSA,
(2.4)
By choice of appropriate values for mirror separation d and inclination angles Oi and 6)2, it is possible to set N so that it satisfies eq. (2.4). If the light source whose beam is to be shaped is a discrete laser-diode array - consisting of individual emitters with dead (or "dark") space between them - then it may in certain cases be better to choose N to be equal to the number of individual emitters. In this way it is possible to remove the dead space
10
Anamorphic beam shaping for laser and diffuse light
[1, §2
X
input LDA
beam shaper Fig. 2.3. Optical setup for the method of Villareal, Baker, Abram, Jones and Hall [1999] (LDA: laser diode array).
between adjacent emitters and obtain an output beam which - although not symmetrical - has larger brightness than the input beam. This effective increase in brightness is not in contradiction with the brightness conservation law, which applies only to continuous sources with no dead space between emitters. Note that in eqs. (2.1)-{2.2) we neglected spreading due to diffraction effects at the sharp boundaries of the mirrors. We also neglected the spreading due to multiple reflections between mirrors A and B. This latter spreading effect makes the output beam slightly asymmetric, since the last sub-beam has the largest optical path length in the system - having undergone 2(7V - 1) reflections, while the first one has the smallest path length - not having undergone any reflection. Moreover, in case of large N care must be taken to ensure very high reflection coefficients R, since the intensity of the last sub-beam is attenuated by B?-^^-^\ Also note that imperfections of either the mirrors' flatness or their parallelism will also be "amplified" by ~2A/^. However, since this technique uses only simple plane mirrors, extremely high optical quality and reflection coefficients are readily available. A modified version of the two-mirror method of Clarkson and Hanna [1996] was proposed and demonstrated by Villareal, Baker, Abram, Jones and Hall [1999] to increase the brightness of two-dimensional laser-diode arrays. The setup of their method is shown in fig. 2.3. The problem with a two-dimensional laser-diode array - that consists of a series of linear diode bars - is the poor fill factor between the diode bars, caused by the need to interleave water-cooled electrodes between them. The beam shaper consists of two mirrors as with Clarkson and Hanna [1996], but here the first mirror contains transparent parallel slots, the number of which is equal to the number of linear laser diode bars in
Reflective techniques
1,§2]
11
(a)
(V
(b) Fig. 2.4. Phase-space diagrams at (a) the input and (b) the output of the method of Villareal, Baker, Abram, Jones and Hall [1999].
the two-dimensional array. As seen in fig. 2.3, the lower half of every diode bar is redirected and repositioned next to the upper half As is obvious fi-om the phase-space diagram (fig. 2.4), the effect of the beam shaper in this case is to decrease the effective x-size of the beam by a factor of two, while leaving the j^-size approximately the same. Since the diffusive angles are not modified either, the overall effect is that the brightness approximately doubles.
2.2. The micro step-mirror technique Another reflective technique for the shaping of laser-diode bars was proposed and demonstrated by Ehlers, Du, Baumann, Treusch, Loosen and Poprawe [1997]. Here, the beam-shaping device consists of two identical micro step mirrors, as depicted in fig. 2.5. Each micro step mirror is made oiN reflective microfacets which are tilted 45'' relative to the direction of propagation. The horizontal size of the micro-mirrors is equal to the separation of adjacent micro-mirrors along the direction of propagation. The phase-space diagram for an input beam with a highly asymmetric beam quality is shown in fig. 2.6a. Just like in the two-mirror method of Clarkson and Hanna [1996], the input beam, which is highly elongated in the >^-direction, can be considered to consist of A^ sub-beams, where N is the number
12
Anamorphic beam shaping for laser and diffuse light
[1,§2
Fig. 2.5. Optical setup of the micro step-mirror technique of Ehlers, Du, Baumami, Treusch, Loosen and Poprawe [1997].
(a)
(b)
aU
a^-*
r^ (c)
Fig. 2.6. Phase-space diagrams of the micro step-mirror method: (a) at the input; (b) at the output (ideal case); (c) at the output (realistic case).
of microfacets on each micro step mirror (A^ = 3 in fig. 2.6a). The first sub-beam hits the left micro-facet on the first micro step mirror. It is then directed upwards, toward the left microfacet of the second micro step mirror. There it is reflected again, and directed perpendicularly to its original propagation direction. The second sub-beam undergoes similar reflections, but only at the second-fi"om-left microfacet of each micro step mirror. The overall action of the micro step-mirror method is thus to divide the input beam into sub-beams, and then rotate each sub-beam by 90°, without displacing them relative to each other. Unlike the twomirror method, the micro step-mirror technique gives each sub-beam an equal optical path length, thereby making the output beam spread more symmetrical.
1, § 2]
Reflective techniques
13
Also, reflection losses are less pronounced, since each sub-beam undergoes only two reflections. The phase-space diagram of the output beam in the ideal case (no loss in brightness) is depicted in fig. 2.6b. Each sub-beam is rotated, and hence their x and j sizes, and their diffusive angles a^ and o^, are interchanged. This leads to the following equations: ^2 = - ^ ,
sin a^^ = sin o^j,
y2=N'Xu
sin o^^ ^ sin a^^.
(2.5)
(2.6)
To obtain M^ = My at the output we get the same expression for N as in eq. (2.4), obtained for the two-mirror technique. Indeed, comparison of figs. 2.2 and 2.6 indicates that the phase-space diagrams of the input are exactly the same for the two methods, whereas the phase-space diagrams of the output are simply rotated versions of each other. Note that fig. 2.6b represents an idealistic case when the sub-beams are closely stacked next to each other at the output, and no loss in brightness occurs. A more realistic representation of the output can be seen in fig. 2.6c. Because of the given geometry of the micro step mirrors, the sub-beams are separated by the same distance in the output as in the input. This leads to a decrease in the fill factor of the output beam, and thus the relation y2>N-xi
(2.7)
applies [instead of the second of equations (2.5)], leading to a considerable reduction in brightness. A way to overcome this problem is to reduce the longitudinal separation of microfacets on the first micro step mirror, and simultaneously reduce the horizontal size of the microfacets on the second micro step mirror. However, this would lead to shading losses at the second micro step mirror, due to the relatively large diffusive angle a^i- The shading losses can be eliminated if one makes use of the fact that the input beam of a laserdiode bar is near diffraction-limited in one of the transverse directions (in our case, the x-direction). The modified phase-space diagrams can be seen in fig. 2.7. The beam of the laser-diode array (fig. 2.7a) is first collimated in the x-direction with a cylindrical lens, so that , xi sin ar, sma; = ; ' <1.
(2.8)
The collimated beam (fig. 2.7b) is the input for the beam shaper. In this case, since no shading losses occur at the second micro step mirror, the fill factor of the
14
Anamorphic beam shaping for laser and diffuse light
[1,§2
(a) tti
(b)
a,2
(c) Fig. 2.7. Phase-space diagrams of the modified version of the micro step-mirror method: (a) at the input; (b) after colhmation in the x direction; (c) at the output.
output beam can be made 1 (fig. 2.7c), and hence brightness can be conserved. Equations (2.5)-(2.6) can still be applied, with the substitutions xi -^ x[, al . Since the one-dimensional collimation conserves y\ - ^ > ^ i , « v , PSAv and PSAy separately, condition (2.4) still holds. 2.3. The microprism-array technique An alternative way to divide the input beam into sub-beams and rotate the sub-beams by 90° while retaining their relative position is by use of a single microprism array, as was proposed and demonstrated by Yamaguchi, Kobayashi, Saito and Chiba [1995]. Figure 2.8 shows a schematic view of the microprism array, together with the input and output beams. Each sub-beam that enters a microprism undergoes three internal reflections at different facets of the microprism, as shown in fig. 2.9. The exact geometry of the microprism, specified by Yamaguchi, Kobayashi, Saito and Chiba [1995], ensures that each
1,§2]
15
Reflective techniques
microprism array
output beam
Fig. 2.8. Optical setup of the microprism array technique of Yamaguchi, Kobayashi, Saito and Chiba [1995].
Fig. 2.9. Schematic view of one microprism.
sub-beam enters the microprism perpendicularly at facet SI, then undergoes internal reflections at facets S2, S3 and S4, and exits perpendicularly at facet S5. Unlike in the micro step mirror method, here the optical axes of the input and output beams are parallel. This is achieved by making the beam undergo three reflections instead of two. Since all three reflections occur within the same microprism, this beam-shaping technique yields a very compact setup. Note that although the input and output optical axes are parallel, they are slightly displaced with respect to each other. The displacement is p/2 in the horizontal direction and 5p/2 in the vertical direction, where p is the width of a microprism as shown in fig. 2.9. The phase-space diagrams for the microprism-array technique can be seen in fig. 2.10 for an array consisting of three microprisms. First, the beam of the laser-diode array (fig. 2.10a) is collimated in the x-direction (fig. 2.10b). This
16
Anamorphic beam shaping for laser and diffuse light
[1, §2
Otxl
ttv l^~l
_L "T .
^ Yi
(a)
H
o;,^
1t 1y; ^ (b)
(c) Fig. 2.10. Phase-space diagrams of the microprism array technique: (a) at the input; (b) after collimation in the jc-direction; (c) at the output.
is necessary in order to avoid overlap of adjacent sub-beams at the output. The coUimated beam of fig. 2.10b is the input for the beam shaper. The output beam (fig. 2.10c) consists of the original sub-beams rotated and stacked next to each other. Note that the phase-space diagrams for the microprism-array technique are almost identical to those for the micro step-mirror method, since the basic idea of beam shaping is the same. One difference is that in the case of the microprism array the output fill factor can made 1 only if the sub-beams have a square cross-section. This imposes a condition for the focal length / of the collimating cylindrical lens. For one-dimensional collimation, x[ =/sina;,j,
y[ =yx. sin a = sin ay,,
(2.9) (2.10)
where/ is the focal length of the collimating cylindrical lens. The condition for the sub-beams to have a square cross-section is y, =x',-AT,
(2.11)
1, § 2]
Reflective techniques
17
where N can be obtained from eq. (2.4). This leads to the expression
/=
/_3Z1
.
(2.12)
y sin Qx^ sin Qy^ If f is chosen according to eq. (2.12) then beam shaping with no loss in brightness can be achieved with the microprism technique. Note that technological problems impose severe limitations on both the micro step-mirror and the microprism techniques. For example, the usual length of a laser-diode array is y\ ^ 10 mm, and typically My/M^^ ^ 2000, leading to A/^ « 45. Thus the size of one micromirror-facet or microprism has to be yi/N ^ 200 |im. The manufacture of such small-size micro-optical elements is very difficult, so often N is chosen somewhat smaller than the value obtained from eq. (2.4). This eases the requirements for manufacture, but makes the output beam-quality factors somewhat asymmetric. 2.4. The retroreflector technique A technique similar to the micro step-mirror method and the microprism-array method was proposed and demonstrated by Davidson, Khaykovich and Hasman [1999], to improve the spectral resolution of grating-based spectrometers (see § 5.3.). The task of the beam shaper here is exactly the opposite of what has been discussed so far. Here the input light comes from a diffuse source that is symmetric in transverse beam-quality factors, (M^^ = My^), and the goal is to obtain an output beam which is nearly diffraction-limited in one direction (M^^ ~ 1), while conserving M^. Naturally - just like in the previous cases the same beam-shaping setup can be used for the inverse task too, by simply reversing the beam directions and exchanging input and output. The anamorphic beam concentrator operates in three stages: (1) focus in the x-direction with a cylindrical lens of focal length/i; (2) exchange the divergence angles in the x- and ^-directions with an array of one-dimensional Porro prisms; and (3) focus again in the x-direction with an additional cylindrical lens of focal length/2. The optical setup of the retroreflector technique is shown in fig. 2.11. Note that the prism array is rotated 45"^ relative to the optical axis in order to separate the reflected wave from the incident wave. The phase-space diagrams of different cross-sections along the beam shaper are presented in fig. 2.12. Figure 2.12a represents the input beam, with symmetrical beam-quality factors in the x- and y-directions. The input beam is extended, so that its size is large, while its divergence angles are relatively small.
Anamorphic beam shaping for laser and diffuse light fi
[1, § 2
fi
x.t prism array y,6 input ^2^
output ^X4
Fig. 2.11. Optical setup of the retroreflector technique of Davidson, Khaykovich and Hasman [1999].
a,,<-
(C)
(d)
Fig. 2.12. Phase-space diagrams of the retroreflector technique: (a) at the input; (b) after the first concentration in the x-direction, with the Porro prism array shown schematically; (c) immediately after the retroreflection of the sub-beams on the Porro prism array; (d) at the output.
1, § 2]
Reflective techniques
19
The light distribution at the focal plane of the first cylindrical lens is presented in fig. 2.12b. Now the x-size is much smaller than the j-size, but with a larger divergence angle, according to the relations X2=f\sma^^, X]_
sma^, = -^.
(2.13) (2.14)
/i
Note that this transformation is not yet anamorphic, since M^^ = M^^, The concentration ratio Xin/x^^jt = x\/x2 so far is smaller than 1/sinGX^, as expressed by the conservation of PSA;^. The value l/sina^ci is the one-dimensional thermodynamic limit of light concentration, in cases when no anamorphic beam transformations are used (Bokor, Shechter, Friesem and Davidson [2001]). Next, the divergence angles in the x- and j;-directions are interchanged by use of an array of one-dimensional retroreflectors located at the focal plane of the first cylindrical lens and oriented at 45'' relative to the x- and j-axes. Each retroreflector is a 90''-reflecting Porro prism, as shown schematically in fig. 2.12b. The light distribution immediately after reflection from the prism array is shown in fig. 2.12c. As seen, for each sub-beam both size and divergence are exchanged between the x- and ^-directions. However, the overall effect of the retroreflection is that the total x- and j-sizes are nearly unchanged, while the X- and ^-divergences are interchanged. Therefore, after the retroreflection, the x-direction is much smaller than the j-direction and also has smaller divergence angles. Finally, the beam is concentrated again in the x-direction with the second cylindrical lens, according to the relations X4=/2sina^3,
(2.15)
sina^4 = ^ .
(2.16)
The final concentration ratio Xin/xout = X1/X4 is proportional to l/(sin a^^ sin o^j), an improvement of 1/ sin o^j over the one-dimensional thermodynamic limit. Davidson, Khaykovich and Hasman [2000] described and demonstrated a modification of this technique to eliminate the step-like structure shown in fig. 2.12c, forming a better output beam shape. In their modified method the Porro prisms were substituted by specially designed metallic retroreflector elements. The modified phase-space diagrams are shown in fig. 2.13. Here, the size of the local square retroreflector - consisting of reflecting facets A and B - is chosen to match X2 exactly, so the x- and >^-sizes of the sub-beams are
20
Anamorphic beam shaping for laser and diffuse light
[1,§3
(b)
a,4-*-
(c)
(d)
Fig. 2.13. Phase-space diagrams of the metaUic retroreflector array technique of Davidson, Khaykovich and Hasman [2000]: (a) at the input; (b) after the first concentration in the jc-direction; (c) immediately after the retroreflection of the sub-beams on the metallic retroreflector array (shown schematically in fig. 2.14); (d) at the output.
Fig. 2.14. Schematic view of a metallic retroreflector.
actually unchanged on retroreflection. Each square retroreflector is composed of two reflecting triangular planes making an angle of 90° with each other, as shown in the schematic view of fig. 2.14 (their projections in the xy plane are marked as triangles A and B in fig. 2.13c). § 3. Refractive/diffractive techniques In both refi-active and diffractive techniques for beam shaping, light rays are deflected into the desired directions by appropriately designed phase elements,
1, § 3]
Refractive/diffractiue techniques
21
either refractive or diffractive. In general, the use of diffractive optics, e.g., holographic optical elements (HOEs) is desirable whenever a complicated aspheric wavefront is to be produced. The disadvantages of diffractive techniques are: (a) their large chromatic aberration, which restricts them to quasi-monochromatic applications, and (b) their low diffraction efficiency. High diffraction efficiencies can be achieved using Bragg volume holograms or blazed gratings. Due to the angular selectivity of Bragg holograms, the diffusive angles they can handle with high diffraction efficiencies (>95%) are typically restricted to approximately <5°. Finally, Fresnel zone plates may be a promising alternative to refractive and diffractive techniques. They are versatile, flexible, very cheap and easily mass produced. Their typical poor optical quality may be adequate for many applications concerning diffuse beams with poor optical quality factors (high M"^). Although Fresnel zone plates have not been used for anamorphic beam shaping yet, they were successfully implemented (Kritchman, Friesem and Yekutieh [1979], Kritchman [1980, 1981]) for solar energy concentration, where the ability to bend them to any desired shape was exploited to achieve improved performance over conventional parabolic concentrators.
3.1. The Fourier transform technique Leger and Goltsos [1992] and Davidson and Friesem [1993a] proposed and demonstrated an anamorphic beam-shaping method based on three essential steps: (1) the input beam is divided into sub-beams, and each sub-beam is directed in the proper direction by a refractive/diffractive element, (2) a twodimensional Fourier transformation is performed on the resulting wavefront, and (3) a second refractive/diffractive element redirects sub-beams in the direction of the optical axis. The setup of the Fourier transform technique is presented in fig. 3.1. First the input - which is highly elongated in the j-direction undergoes a one-dimensional Fourier transform in the x-direction. The Fourier transform is performed by a cylindrical lens (Li) of focal length/i. The aspect ratio of the beam at the one-dimensional Fourier plane is N, the value of which is determined below. A linear array of N square prisms is located at the onedimensional Fourier plane. Each prism deflects the light impinging on it at a different angle in the x-direction. After being Fourier-transformed by a spherical lens (L2) of focal length^, the light emerging from each prism reaches a different zone in the (two-dimensional) Fourier plane. Each sub-beam that impinges on such a zone has a different off-axis angle which must be compensated for. This is done with suitable deflection prisms located at the different zones in the two-
22
Anamorphic beam shaping for laser and diffuse light
<
•—•
<
input •^^
•
[1, §3
•
prism array
prism array
Fig. 3.1. Optical setup of the Fourier transform technique used by Leger and Goltsos [1992] and by Davidson and Friesem [1993a].
dimensional Fourier plane. At the output the beam has a square cross-section and equal diffusive angles in the x- and >^-directions, and hence symmetric beamquality factors in the two lateral directions. Figure 3.2 shows the phase-space diagrams of the beam at the input plane, at the one-dimensional Fourier plane and at the two-dimensional Fourier plane. For the one-dimensional Fourier transform: X2 =f\ sin a,j,
y2=y\-^ 2/i sin a^, ^ yi,
(3.1)
sm a^, = TT,
sm a^, = sin av,,
(3.2)
where we have neglected the diffusive spread in the j^-direction (a good approximation only if 2/i sinav, <^y\). The beam is then divided into A^ sub-beams. Each sub-beam is deflected at a different angle by a prism. Next, a twodimensional Fourier transformation is performed. The Fourier transform of individual sub-beams are formed in different zones in the two-dimensional Fourier plane, as represented by different shadings in figs. 3.2b and 3.2c. Accordingly, the beam dimensions and diffuse angles in the two-dimensional Fourier plane are: X3 = Nf2 sin a^.,
y^=f2 sin a,,,
(3.3)
y2_ (3.4) smav3 = smav3 = —, h Note that besides the diffusive angles shown in fig. 3.2c and expressed in eqs. (3.4), there is also an off-axis angle which is different for each zone. This is ^2
1, §3]
Refractive/dijfractive techniques
23
0Cy2
T (a)
(b)
{ (c) Fig. 3.2. Phase-space diagrams of the Fourier transform technique: (a) at the input; (b) at the first prism array; (c) at the output.
compensated for by a second prism array placed in the two-dimensional Fourier plane. From the equality of the orthogonal output beam-quality factors we get condition (2.4) for N. The focal length of the cylindrical lens must be chosen so that the aspect ratio of the one-dimensional Fourier transform is N (see fig. 3.2). Accordingly,
A=
xiyi
sm Ocj sin Gy^
(3.5)
The two prism arrays can be substituted by other beam-deflecting elements, as was demonstrated experimentally by Davidson and Friesem [1993 a]. Here optically recorded holographic elements (HOEs) were used for steps (1) and (3). The first HOE consisted of N linear gratings, each of which was recorded as the interference pattern between two plane waves of appropriate offset angles. The second HOE was fabricated in a simple, single-exposure technique: the interference between a plane wave and the entire wavefront emerging fi-om the first HOE and Fourier-transformed by the spherical lens was recorded at the back focal plane of the spherical lens.
24
Anamorphic beam shaping for laser and diffuse light
[1,§3
output
Fig. 3.3. Optical setup of the modified Fourier transform technique of Davidson and Friesem [1993b].
3.2. The modified Fourier transfiyrm technique The problem of loss in brightness caused by the diffusive spread in the };-direction [expressed by the second of equations (3.1)] can be solved by slightly modifying the geometry, as was proposed and demonstrated by Davidson and Friesem [1993b]. This technique - illustrated in fig. 3.3 for anamorphic concentration of a uniform diffuse beam - consists of similar steps as the previously described Fourier transform technique, but now two two-dimensional Fourier transforms are performed, as opposed to a one-dimensional and a twodimensional Fourier transform. First, the input beam is incident on a linear array of A^ prisms (the value of A'^ is determined below). The deflection angles from the prisms in the array are so arranged as to transform the light from the input to an array of \/N by \/N zones in the Fourier plane. A second prism array placed in the Fourier plane and made up of v ^ by y/N prisms (each covering a zone) redirects each sub-beam parallel to the optical axis. The beam is then concentrated with a second lens. The phase-space diagram of this method is presented in fig. 3.4. Considering that the input diffusive angles are equal in the X- and j^-directions (a^, = ay, = a\), we have the following equations for the sizes and diffusive angles of the beam at the Fourier plane: JC2 =
VN/
sin ai, Xx
sm ax2 = TTT,
Nf
72 =
sin a\,
VN/
y\ sm ay, = —,
(3.6) (3.7)
f
where/ is the focal length of the Fourier transform lens. Note that the isotropic shape of each zone in the Fourier plane corresponds to the isotropy of diffusive angles in the input plane. On the other hand, the diffusive angles in the Fourier
1, §3]
Refractive/dijfractive techniques
25 ay2
T (a)
(b)
ax3-*-
(C) Fig. 3.4. Phase-space diagrams of the modified Fourier transform technique: (a) at the input; (b) at the second prism array; (c) at the output.
plane are anisotropic because of the anamorphic shape of each prism in the input plane. After the second Fourier transform we have at the output plane: X3 =fsma^^ = sma^cs = — =
VN
sm «!, Gi, sin
73 =f sin Oy, =71,
(3.8)
sm Oiy^ ( sin = Y ^ ^^^^^ ^^'
(3.9)
For maximum concentration, we require that sin a^^ and sin Oy^ approach unity, so we can extract N fi-om eqs. (3.9) to be N
1 sin^ «!
(3.10)
If condition (3.10) is satisfied then the concentration is done at the theoretical (two-dimensional) thermodynamic limit, as indicated by the first of equations (3.8). For the experimental implementation of the two refractive/diffractive elements the following simplifications can be used: (1) every \/A^ prisms of the input
26
Anamorphic beam shaping for laser and diffuse light
[1, § 3
prism array (corresponding to different rows of zones in the Fourier plane) can be combined to form an essentially equivalent off-axis cylindrical lens. The resulting reduction of the number of zones in the input plane (from N to \/5V) largely simplifies the optical setup, as well as reduces the undesired diffraction that mainly arises from the discontinuities between zones. Each of the y/N cylindrical lenses can be formed as holographic elements, by recording the interference between a wavefront emerging from a refractive cylindrical lens and an off-axis plane wave; (2) the array of y/N by y/N redirecting elements at the Fourier plane can also be replaced by \/N off-axis cylindrical lenses. However, an even simpler approach is to record in a single exposure the interference between a plane wave and the entire wavefront emerging from the input HOE array and Fourier-transformed by the first lens. 3.3. The tilted cylindrical lens method All clipping losses and undesired diffraction effects can be eliminated if only continuous-phase elements are used in the beam-shaping process. One such technique was proposed and demonstrated by Davidson and Khaykovich [1999]. It is based on a one-dimensional Fourier transform, a tilted cylindrical lens, and a holographic redirecting element. The setup is presented in fig. 3.5. The input fi
2t
MDUtpUt
input
HOE
Yt
Fig. 3.5. Optical setup of the continuous-phase elements technique of Davidson and Khaykovich [1999].
beam has highly asymmetric beam-quality factors in the x- and j;-directions. We define the ratio of input phase-space areas as PSA^ PSA,
yi sm a,y\
x\ sin Gx
(3.11)
27
Refractive/diffractive techniques
1, §3]
OCyS
a.!^^
r (a)
(c)
(b)
(d)
Fig. 3.6. Phase-space diagrams of the continuous-elements technique: (a) at the input; (b) at a distance/i after lens L\; (c) at a distance If2 after lens L2, neglecting diffusive spread; (d) at a distance 2/2 after lens L2, including diffusive spread.
For simplicity, we consider equal diffusive angles in the input beam in the xand j-directions: a^,=ay^=ax.
(3.12)
In this case n is simply the aspect ratio of the input beam cross-section: .y\ (3.13) Xx
The first stage of the optical arrangement depicted in fig. 3.5 consists of a cylindrical lens Li that performs a one-dimensional Fourier transform (in the x-direction) of the input. The phase-space diagrams of the light distribution at different planes of the setup are shown in fig. 3.6. Figure 3.6a presents the input light distribution, and fig. 3.6b shows the light distribution at the back focal plane of Li. In the x-direction we have the following relations: X2=/isinai, Xx_
sma.'X2 7i'
(3.14) (3.15)
28
Anamorphic beam shaping for laser and diffuse light
[1» § 3
where/i is the focal length of lens Li. In the j-direction: 72 = >^i + 2/i sinai ^yx,
(3.16)
sin 0^2 "= sin o^j.
(3.17)
Note that eq. (3.16) is valid only if 2/1 sin a\ ^y\. to yield a linear magnification of M = - = V^.
The focal length/i is chosen (3.18)
Hence, fi-om eqs. (3.13), (3.14) and (3.18) we get
/. = J^^-
(3.19)
V sin «! With this choice of/i, after the one-dimensional Fourier transform the light field is M = v ^ times larger in size, and M = y/n times more diffusive in the j^-direction than in the x-direction. Therefore, if the x and>^ spatial dimensions are interchanged, whereas the diffusive angle is not (or vice versa), then the x and y phase-space areas will be exactly the same, and hence the beam-quality factors will be symmetrical in the two orthogonal directions. Such an interchange which, as described in §§ 2.2 and 2.3, can be performed with a microprism array or with two micro step mirrors - can be achieved with a single cylindrical lens (second stage, L2 in fig. 3.5) placed in the Fourier plane and rotated 45° relative to the X- and >'-directions around the optical axis. To see how this occurs, let us first consider a simplified case in which the diffuse angles are neglected and the light beam impinging on L2 is assumed to be collimated {QX^ = a^, ~ 0). The refi'action angles given to the beam by L2 at point (X2, Y2) within the paraxial approximation are . -(X2 + 72) . , -(X2 + Y2) ,._, sm % = — , sm Gy = — , (3.20) 4/2 2/2 where ^2 is the focal length of lens L2. Therefore at a distance 2/2 behind the lens, the transformation is X^ =X2+ 2/2 sin a'j^ = -Y2, Y^ = ^2 + 2/2 sin a'y = -X2. (3.21) Figure 3.6c shows the spatial distribution at a distance 2/2 behind L2 in this simple case. As seen, the operation of L2 is merely an interchange between the X- and >^-directions: A^yi.
y'^=x2'
(3-22)
Note that the lens L2 also changes the angular distribution of the beam in a spacevariant way (i.e., different angle change for each location across the beam) as illustrated by the thick dashed arrows in fig. 3.6c.
1, § 3]
Refractive/diffractive techniques
29
Coming back to the real situation of a diffuse beam {a^^ = Oy^ ^ 0), the following complication exists: while traveling a distance 2^ in free space each point at the beam expands to a rectangle of area 2^ sin a^^ x 2^ sin ay^. Since sin a^2 "" ^ sin a^^ these rectangles are M times larger in the >^-direction. The focal length^ is chosen such that the x-expansion If2 sin a^^^ is equal to X2 (and hence also lf2 sin Oy^ = yi). Each of these tall rectangles is shifted differently by the lens L2 and hence is centered around a different point on the wide rectangle of fig. 3.6c. The total size of the output light distribution is isotropic (see fig. 3.6d): ^ 3 - 7 3 ~ 71-
(3.23)
Careful analysis (Davidson and Khaykovich [1999]) indicates isotropic diffuse output angles: sinai sm a^^ = sm Oy^ = -^TJ--
(3.24)
Next, a collimating element must be applied to correct the angles given to the beam by the lens L2. The location of each beam on the collimating element is affected by its diffuse angle, and thus a more complicated phase element than L2 itself is required. As in the previous cases when the collimating element was a HOE, it can simply be recorded in situ, by recording the interference between a plane wave and the entire wavefront emerging from L2. After collimation, the output beam is isotropic in its spatial dimensions as well as in its diffuse angles, while the four-dimensional phase-space volume is conserved. A similar technique, based on a slightly tilted cylindrical lens, was proposed and demonstrated by Goring, Schreiber and Possner [1997] for laser-diode arrays, i.e. light sources that consist of typically 10-20 individual emitters. The setup of this technique is shown in fig. 3.7. The laser-diode array (LDA) - the sides of which are parallel to the x'- and j'-axes - is put adjacent to a slightly tilted cylindrical lens (TCL). The cylindrical lens performs a one-dimensional Fourier transform on the input beam which is thus slightly tilted relative to the lens axes x and j . The beam then undergoes a two-dimensional Fourier transform performed by the spherical lens (SL), and is finally recollimated by a grating array (GA) parallel to the optical axis. As seen in fig. 3.7, the effect of the tilt in the TCL is that the beams of the individual emitters, i.e. the sub-beams, propagate in different directions between the TCL and the SL. A more detailed discussion is possible with the help of the phase-space diagrams of fig. 3.8, in the {X', Y') coordinate system. The input beam is presented in fig. 3.8a. In the onedimensional Fourier plane (fig. 3.8b) the beam cross-section is approximately
30
Anamorphic beam shaping for laser and diffuse light
LDA
[1, § 3
GA
TCL (a)
iY'
(b)
Fig. 3.7. Optical setup of the tilted cylindrical lens method of Goring, Schreiber and Possner [1997] (LDA: laser-diode array, TCL: tilted cylindrical lens, SL: spherical lens, GA: grating array).
ttxr
(a)
(b)
(c) Fig. 3.8. Phase-space diagrams of the tilted cylindrical lens method: (a) at the input; (b) in the back focal plane of the TCL; (c) at the GA.
1, § 3]
Refractive/diffractive techniques
31
rectangular, with sides parallel to the X and Y axes. (The TCL is slightly tilted with respect to the LDA, hence the phase-space diagram of the one-dimensional Fourier transform is also slightly tilted in the {X', Y') frame.) However, the TCL adds a space-variant tilt to the beams of the individual emitters, as shown by the dashed arrows in fig. 3.8b. The beam can thus be considered to consist of sub-beams, each having a square cross-section, the same diffusive angles, and an additional tilt angle which is different for each sub-beam. The SL performs a Fourier transform on the beam of fig. 3.8b. Because of the additional tilt associated with each sub-beam, the Fourier transform of each sub-beam is formed in different regions in the two-dimensional Fourier plane, as shown in fig. 3.8c. The shaded square sub-beams of fig. 3.8b correspond to the shaded rectangular regions of fig. 3.8c. For the two-dimensional Fourier transform we have the following relations: X3 = Nf2 sin a^^, y^ =f2 sin o^^. sina;,3 = -^,
smay^ = ^ ,
0-25) (3.26)
where ^ is the focal length of the SL, and A/^ is the number of sub-beams. If the sub-beams in the two-dimensional Fourier plane are packed closely adjacent to each other (as shown in fig. 3.8c) then no loss in brightness occurs. This corresponds to the relation ax„
(3.27)
where a^ is the tilt angle of the extreme sub-beams, as shown in fig. 3.8b. The goal is to make the output beam-quality factors equal, hence we get fi-om eqs. (3.25)-(3.26):
^ = yi='h^, X2
(3.28)
sm a^^
In order to get an output beam with isotropic beam-quality factors and no loss in brightness, the focal length and the tilt angle of the TCL are chosen to satisfy eqs. (3.27) and (3.28). Since N - the number of sub-beams - here equals the number of individual emitters in the LDA, eq. (3.28) can in general be satisfied only approximately for a laser-diode array. Finally, since each sub-beam in the two-dimensional Fourier plane has a different tilt in the ^-direction, a refractive or diffractive element must be used to redirect the sub-beams parallel to the
32
Anamorphic beam shaping for laser and diffuse light
[1,§3
optical axis. Goring, Schreiber and Possner [1997] proposed and experimentally implemented an array of blazed gratings for this redirecting element. 3.4. The method of two grating arrays Although the beam-quality factors in the two orthogonal directions cannot be made equal with one lens, the diffusive angles alone can be equalized quite easily with a single cylindrical lens. In this case the ratio of beam-quality factors in the X' and ^-directions of the beam after the cylindrical lens equals the aspect ratio of the beam. If the beam is then divided into sub-beams and the sub-beams are rearranged to form a square beam cross-section then the output beam-quality factors will be equal in the two orthogonal directions. Such a technique was described and demonstrated by Motamedi, Sankur, Durville, Southwell, Melendes, Wang, Liu, Rediker and Khoshnevisan [1997]. The optical setup of this method is presented in fig. 3.9. The input beam - e.g., from a laser-diode array - is first collimated in both the x- and the >^-direction: «x, = «v, < 1,
(3.29)
where GX^ and Gy^ are the diffusive angles after the collimation. The coUimating optics - not shown in fig. 3.9 - can consist of two cylindrical lenses, or a cylindrical and a spherical lens. The collimated beam is then incident on a diffractive optical element (DOE) that consists of an appropriate number of linear gratings (the number of which is determined below). The beam is thus divided into a linear array of sub-beams, with each sub-beam incident on a different
Fig. 3.9. Optical setup of the method of two grating arrays of Motamedi, Sankur, Durville, Southwell, Melendes, Wang, Liu, Rediker and Khoshnevisan [1997].
1, §3]
Refractive/diffractive techniques
33
A. t
Yi
(a)
X2
•
a^
(b) Fig. 3.10. Phase-space diagrams of the method of two grating arrays: (a) at the input; (b) at the output.
grating and deflected in a different direction. A second DOE is placed at the plane where the deflected sub-beams add up to form a square cross-section. The fiinction of the second DOE - which consists of the same number of linear gratings as the first DOE - is to redirect the sub-beams parallel to the optical axis and yield an output beam that has symmetric sizes and diffiisive angles in the X- and j^-directions. (Note that every linear grating in the setup of fig. 3.9 can be replaced by an appropriately oriented prism.) The phase-space diagram of the beam immediately before the first DOE is presented in fig. 3.10a. If « denotes the ratio of input beam-quality factors in the X- and j-directions then (3.30)
n= —, Xi
where we made use of eq. (3.29). The phase-space diagram of the output beam is shown in fig. 3.10b. Neglecting the fi-ee-space expansion of sub-beams between the two DOEs, we get for the output beam: X2^N'Xu ^X2
y^^-^^ dy^
^Xi 9
ay^ ,
(3.31) (3.32)
where A^^ is the number of sub-beams. From the condition X2 s m 0^2 _ ^2 _
y2 sin ay^
y2
1,
(3.33)
34
Anamorphic beam shaping for laser and diffuse light
we get for the number of sub-beams: (3.34) and for the total number of Unear gratings on the two DOEs: (3.35) where we use the fact that the sub-beam in the middle can go through both DOEs undeflected, hence no linear grating is needed at the middle of the DOEs. 3.5. The method of stacked glass plates Izawa, Uchimura, Matsui, Arichi and Yakuoh [1998] presented and demonstrated a refractive method for anamorphic beam shaping of laser-diode arrays. This technique is based on two stacks of thin, square glass plates, as shown schematically in fig. 3.11. Both stacks consist of iV glass plates {N is determined below), rotated relative to each other by an appropriate angle. The glass plates
(a)
Fig. 3.11. Optical setup of the method of stacked glass plates of Izawa, Uchimura, Matsui, Arichi and Yakuoh [1998].
1, § 3]
Refractive/diffractive techniques
35
of the first stack are parallel with the (jc,z) plane, and those of the second stack are parallel with the {y,z) plane, where z is the direction of propagation of the input beam. The method of stacked glass plates operates as follows: the input beam, which is collimated in the x-direction, is incident on the first stack of glass plates where it is divided into N sub-beams. Each sub-beam having a square cross-section undergoes two refractions (at the air-glass and at the glass-air interface) and is hence displaced in the x-direction. The displacement is different for each sub-beam as a result of the different angles of the glass plates relative to z. The j-position of the sub-beams is unchanged, because in this direction each sub-beam simply undergoes internal reflections inside the appropriate glass plate. In the second step, each sub-beam enters its appropriate glass plate in the second stack, and undergoes two refractions in the j-direction while keeping its x-position. At the output we have a beam whose x and y sizes are exchanged - neglecting a small diffusive spread in the ^-direction - while the diffusive angles in the x- and j^-directions are still the same. This concept of splitting the input beam into sub-beams and rearranging the position of the sub-beams while keeping their diffusive angles iii the x- and j-directions unchanged is similar to the two-mirror technique of Clarkson and Hanna [1996]. Here, however, all sub-beams undergo the same number of refractions and have approximately the same path length, making the output beam cross-section more symmetrical. The phase-space diagrams of the beam at three different locations of such a beam-shaper are presented in fig. 3.12. At the input the beam is elongated in the ^-direction and collimated in the x-direction (o^j ^ 0). The effect of the first stack of glass plates is to displace each sub-beam in the x-direction, as indicated by the dashed arrows for the two extreme sub-beams in fig. 3.12a. Figure 3.12b shows the phase-space diagram of the beam between the two stacks. The size and orientations of the glass plates of the first stack are chosen such that each sub-beam is displaced by exactly its width with respect to the adjacent sub-beam. The second stack of glass plates displaces each sub-beam in the j-direction, as indicated by the two broken arrows for the two extreme sub-beams in fig. 3.12b. Finally, at the output the beam will be elongated in the x-direction and more diffusive in the j;-direction, as shown in fig. 3.12c, hence with a proper choice of N the beam-quality factors in the two lateral directions can be equalized. For the beam transformations we have the following relations:
«X3 = «X2 = «xi ~ 0,
a^3 = ay^ = ay,.
(3.37)
36
Anamorphic beam shaping for laser and diffuse light
[1,§4
1 ^ Otxl^
(a)
(b)
<^3
(c) Fig. 3.12. Phase-space diagrams of the method of stacked glass plates: (a) at the input; (b) after the first stack of glass plates; (c) at the output.
In the second equation (3.36) we neglected the diffusive spread in thej^-direction. This is a valid approximation if (Z) + L) sino^j
(3.38)
where D is the distance between the two stacks, and L is the side of one glass plate. To make the output beam-quality factors equal in the x- and >^-directions we combine eqs. (3.36) and (3.37) to obtain a condition for N, identical to that of eq. (2.4).
§ 4. Adiabatic techniques A completely different approach to anamorphic beam shaping is to slowly change the phase-space areas in the two transverse directions, by using adiabatic transformations. The slow, adiabatic change ensures that the four-dimensional phase-space volume, and hence the brightness, are conserved. One of the main advantages of the adiabatic approach to beam shaping can be illustrated with the
1,§4]
37
Adiabatic techniques
adiabatic beam-homogenizer tube, also called a kaleidoscope (Miyamoto and Maruo [1989]). This is a straight tube with square cross-section, made up of four reflecting walls. If such a device is used for homogenizing an illumination system that is composed of many independent sources then the homogeneity of the output beam is protected against fluctuations of the input sources. For example, destruction of one of the input sources results in an overall intensity reduction, instead of a dark spot, at the output. In general, it is much simpler to apply the adiabatic approach for equalizing the beam-quality factors in the two transverse directions than for the inverse operation of one-dimensional concentration, since most adiabatic techniques operate on principles that involve statistical equalization of the diffusive angles of the beam in the x- and^-directions. This is achieved through mechanisms such as multiple reflections in a tube or mode mixing in a fiber. The main advantage of adiabatic techniques over the techniques discussed above is their robustness, i.e., the fact that the uniformity and symmetry of the output beam is fairly insensitive to small changes in the input beam.
4.1. The stubbed-waveguide technique An adiabatic technique for making the beam-quality factors of a laser-diode bar symmetrical in x and y was proposed and demonstrated by Xu, Prabhu, Lu and Ueda [2001]. Their setup is shown in fig. 4.1. First the beam of a laserfy LDB
*x
LS
I
WG j;::i
ZII::::::EE-
(a)
LS d)y
WG
LDB....-
(b)
Fig. 4.1. Optical setup of the stubbed-waveguide technique of Xu, Prabhu, Lu and Ueda [2001] (LDB: laser diode bar, LS: lens system, WG: waveguide).
38
Anamorphic beam shaping for laser and diffuse light
(a)
[1? § 4
(b)
Fig. 4.2. Phase-space diagrams of the stubbed-waveguide technique of Xu, Prabhu, Lu and Ueda [2001]: (a) at the input of the WG; (b) at the output of the WG.
diode bar (LDB) - with highly asymmetric beam-quaUty factors in x and y is incident on a lens system (LS) that consists of cylindrical and spherical lenses. The lens system focuses the beam into a rectangular spot, the phasespace diagram of which is shown in fig. 4.2a. The focused beam enters a stubbed waveguide (WG) that has a circular cross-section. If the waveguide is long enough, at the output the beam will not only have a smooth circular intensity cross-section, but symmetrical diffusive angles as well, as shown in fig. 4.2b. The operation of the waveguide can be expressed with the following equations: A2 sin ax2 sin Gy^ ^ A\ sin a^^ sin ay,,
(4.1)
where Ai and A2 are the cross-sectional areas of the beam at the input and output of the waveguide, respectively. The inequality (4.1) is the consequence of the fact that the input beam does not entirely cover the input facet of the waveguide, hence some brightness is lost. Note that it is a simple waveguide that is entirely responsible for the anamorphic transformation in this technique. The formation of the smooth and symmetrical output beam shown in fig. 4.2b is the result of mode-coupling and mode-scrambling processes inside the waveguide. Xu, Prabhu, Lu and Ueda [2001] demonstrated experimentally that a complete equalization of the transverse beam-quality factors can be achieved with a waveguide having a length-to-diameter ratio of ~300. This idea of a circular beam-homogenizing waveguide can also be implemented in macroscopic systems for incoherent diffiise light, with a long, hollow tube having reflecting inner walls. Here the beam-homogenizing and coupling mechanism is the multiple reflections inside the tube. In such macroscopic systems the tube can have a diameter of several centimeters. Note that in macroscopic systems the exact geometry of the tube is of crucial importance. For
1, § 4]
Adiabatic techniques
39
example, a reflecting straight tube with a square, instead of circular, cross-section (the kaleidoscope mentioned at the beginning of § 4) is not capable of coupling the beam-quality factors between the x- and y-directions, and conserves PSAj^ and PSA^ individually; this geometry - which can be used for homogenizing the intensity cross-section for an originally symmetrical beam (Miyamoto and Maruo [1989]) - is not suitable for anamorphic transformations. On the other hand, if the same geometry of a tube with square cross-section is used as a waveguide instead of as a macroscopic reflecting kaleidoscope, it is capable of equalizing the beam-quality factors in the two transverse directions. The reason for this is that mode mixing can occur even in a waveguide of such geometry, as was demonstrated by Liithy and Weber [1995]). In their system, a completely symmetric output beam was achieved in a square fiber having a length-to-width ratio of -4000.
4.2. The fiber-bundle method Zbinden and Balmer [1990] proposed and demonstrated a technique which combined several fibers of circular cross-section to yield a system capable of manipulating the diffusive angles and the lateral dimensions of the beam independently. In this method the elongated beam of a laser-diode bar is led into a linear array of fibers so that the joint fiber cross-sections cover the entire emitter area. At the output the fibers are closely packed into a nearly circular bundle. The phase-space diagrams of the beam at the input and output of the fiber coupler are shown in fig. 4.3. At the input the beam is asymmetric both in diffusive angle and in lateral dimensions, and PSA^;^ > PSA^^j. At the output the beam cross-section is made symmetrical by arranging the fibers into a nearly
i
Otxl^^
0^x2
(a)
(b)
Fig. 4.3. Phase-space diagrams of the fiber-bundle method used by Zbinden and Balmer [1990] and by Liithy and Weber [1995]: (a) at the input; (b) at the output.
40
Anamorphic beam shaping for laser and diffuse light
[1? § 4
circular bundle. At the same time, the diffusive angles are also made symmetrical in each individual fiber by the mode-mixing process. The output beam has thus symmetrical beam-quality factors in the x- and y-directions. Note that in the fiber-bundle method, the lateral dimensions and the diffusive angles of the beam are manipulated entirely independently, by different mechanisms. Liithy and Weber [1995] proposed and demonstrated a similar technique for a line-to-bundle converter. In their setup, however, the individual fibers had square cross-sections, thus making it possible to pack them more closely, with virtually no dead space between them, and hence no loss in brightness. As we noted earlier, fibers with square cross-section are also capable of performing mode mixing, and hence the equalization of the x and y diffusive angles is possible if the fibers' length-to-width ratio is large enough (^10^).
43. The method of tapered reflecting tubes Another approach to anamorphic adiabatic beam shaping is changing the shape of the guided beam by tapering the beam guide along its longitudinal axis. If the taper is slow enough - hence the term adiabatic can be used - then the fourdimensional PSV of the beam is indeed conserved, as was shown analytically by Garwin [1952], who investigated a tapered reflective light pipe geometry for liquid scintillation cells. This concept of adiabatic beam shaping of diffuse light may be viewed as an optical analogy to adiabatic motion in mechanics and quantum mechanics (Landau and Lifshitz [1972]). It is well known that the state of an atomic ensemble can be changed adiabatically - without increase in PSV OrOp, where r is the position, p is the momentum, and a denotes the standard deviation - if the shape or size of the potential well is changed slowly enough in time. It is also known that in the equilibrium state, due to interactions (collisions) between the atoms, an equipartition occurs between different directions of the momenta, resulting in perfect coupling between orthogonal directions. Since there is no interaction between light beams in diffuse, incoherent beam shaping, the analogy with atomic physics is not complete, and the conditions for coupling between orthogonal directions require more careful analysis, as will be shown below. Bokor and Davidson [2002a] proposed and demonstrated an anamorphic beam-shaping technique for diffuse and completely polychromatic light, using a tapered reflective tube. Such flexible reflecting tubes are applicable for beams with sizes of several cm, very large divergence angles and a surprisingly small number of reflections (<10). This results in a small length-to-width ratio (~10),
41
Adiabatic techniques
1. §4]
X,
(a)
\[i
(b)
Fig. 4.4. Schematic view of the tapered reflecting tubes of the method of Bokor and Davidson [2002a]: (a) type A (anamorphic) device; (b) type B (non-anamorphic) device.
and hence a compact optical system. The reflective tube design (type A) is presented in fig. 4.4a. It is constructed fi-om four flexible planar reflective surfaces of rectangular shape (made, e.g., of aluminum) that are assembled together. As seen from fig. 4.4a, all four reflecting walls are skewed and curved, hence enabling coupling between the horizontal and vertical directions of the guided beam. For comparison, we can also consider a similar tube but with rectangular cross-section (type B, see fig. 4.4b). Here, two reflecting walls are always horizontal and two are always vertical, hence no coupling will occur between the two transverse directions of the guided beam. The shape in fig. 4.4b is similar to the lens-duct design commonly used for concentrating the beam of laser-diode arrays into symmetrical spots (Feugnet, Bussac, Larat, Schwarz and Pocholle [1995], Brignon, Feugnet, Huignard and PochoUe [1998]), but maintaining the asymmetric beam-quality factors in the two transverse directions. During the experiments, in both geometries the xi-yi end facet was rigid and had a square cross-section. The x\-y\ end facet was changed, in type A by squeezing the x\-y\ end of the tube, and in type B by bringing the x\-y\ end of the two vertical walls closer together or farther apart (while retaining their flat shape). In the experiments, the tubes were used as collimators, i.e., the smallarea facet xi-^i was used as an input and the large-area facet xi-yi as an output. The adiabatic increase in beam area implies a corresponding reduction of its divergence angles, and hence the name collimator.
42
Anamorphic beam shaping for laser and diffuse light
[1? § 4
a,2
(a)
(b)
Fig. 4.5. Phase-space diagrams of method of tapered reflecting tubes of Bokor and Davidson [2002a] (type-A, anamorphic, device): (a) at the input; (b) at the output.
For the "non-coupling" device of type B the PSAs are conserved separately in the two transverse directions: xf sin al = xf sin a^^,
yf sin a^^ = y^ sin af^,
(4.3)
where the upper index B refers to the device of type B, and a^^, Oyj, ax2 and 0^2 are the diffusive angles in the x- and ^-directions, at the input and output, respectively. Since j^f =>^f, we get a^^ = o^ (this is because all reflections in the y-z plane occur between two parallel planes, conserving the angular distribution in the^-direction). If the input beam is symmetrical in diffusive angles (a^ = ^^) then •^X2 _
sinaB
^\
yY
(4.4)
On the other hand, for the type-A device coupling occurs between the rays in the y-z and x-z planes. Moreover, if the number of reflections inside the tube is sufficiently large then the angular distribution of the light at the output will be symmetrical in the x- and j^-directions: < = <•
(4.5)
The phase-space diagrams of the input and output beams for the type-A device are presented in fig. 4.5. As seen, the output beam has symmetrical beamquality factors in the jc- and >^-directions. We note, however, that the fact that the type-A device has skewed and curved walls is not a sufficient condition to ensure the perfect coupling expressed by eq. (4.5). As was noted earlier, the analogy with an atomic ensemble in thermal equilibrium - where an equation
1, §4]
43
Adiabatic techniques 4.5 4 3.5 3 2.5 2 1.5
sinax2/ 'sin a y2
•D-D
•o-o
0.5 2
/
3
5
6
Fig. 4.6. Experimental output angular uniformity as a function of input aspect ratio: diamonds, type-A (anamorphic) device; squares, type-B (non-anamorphic) device.
similar to eq. (4.5) holds for the momenta - is not complete, because of the lack of interaction between light rays. In fact, numerical ray-tracing simulations show that the device of type A achieves perfect coupling only in the collimator configuration. If it is used as a concentrator {x2-y2 is the input and x\-y\ is the output) then {y sin ay)ovx^J{x sin a;c)output - 0 . 8 (and not 1) for joutput/^output > 1For maximum coupling the exact geometry of the tube should be optimized, but numerical and experimental results show that in the collimator configuration the geometry of fig. 4.4a is close to optimal, as well as easy to fabricate. Another geometry that can be investigated is the circle-to-ellipse configuration, which was proposed for fiber optics by Scifres and Worland [1987]. It yields much lower coupling efficiency than the type-A device, and hence requires many more reflections to provide mixing between the two transverse directions. For both type A and type B, the beam intensity is uniform at the output facet, provided that there is a sufficient number of reflections inside the tube. Thus the type-A device operates as an anamorphic beam collimator that has an output beam with uniform intensity and uniform angular distribution, while the type-B device provides a stronger collimation in the x-direction and no collimation in the j-direction. Experimental verification of these considerations is presented in fig. 4.6. Measurements of sin or^/ sin ay^ (characterizing the symmetry of the output angular distribution) were conducted for a series of x\/y\ input aspect ratios. As seen in fig. 4.6, for the type-B tube sin a^c^/ sin ay^ is approximately equal to x\/y\, as expressed in eq. (4.4). On the other hand, the type-A tube provides a nearly symmetric output angular distribution for a wide range of input aspect ratios, as expressed in eq. (4.5). As an extension to the approach of adiabatic tapering, Bokor and Davidson
44
Anamorphic beam shaping for laser and diffuse light
[U § 5
[2001b] proposed a tapered gradient-index rod, instead of a reflecting tube, for adiabatic beam shaping. Here it is not the shape, but the refractive-index profile that changes adiabatically along the direction of beam propagation. For example, gradual variation from a quadratic profile into a step profile was shown to transform a diffuse beam with a Gaussian intensity profile into a uniform one, while conserving the four-dimensional PSV [defined using a modification of eq. (1.1) for non-uniform beams, with the beam sizes and angular spreads characterized by their second moments]. With appropriate design, this gradientindex approach can be suitable also for general anamorphic beam-shaping tasks, for non-uniform and non-separable light distributions.
§ 5. Applications In this section we list a number of applications of anamorphic beam shaping. We start with the beam shaping of high-power laser-diode bars - especially for end pumping solid-state lasers - as the widest recent application area. In fact, most of the techniques discussed above were invented for laserdiode-bar applications. Next, we discuss the one-dimensional concentration of broadband (e.g., solar) radiation. Finally, we mention several application areas from spectroscopy and optical metrology. 5.7. High-power laser-diode bars applied for end pumping of solid-state lasers Solid-state lasers pumped by laser-diode bars have excellent properties such as high efficiency, reliability, high output power, and - in an end-pumping geometry - compactness. Despite the high pumping power that can be achieved with laser-diode bars, their output beam is not directly suitable for end-pumping schemes, because of the poor beam quality of laser-diode bars {M^ > 1), and especially by their highly asymmetric M^ and My values. In practice, there is usually 3 orders of magnitude difference between M'^ and My. In order to get a minimum spot size with symmetrical cross-section, M^ and My should be made equal, with the smallest possible increase in M^. In many beam-shaping schemes the asymmetric beam of the laser-diode bar is divided into sub-beams, and these sub-beams are then rearranged to form a symmetric output beam. N, the number of sub-beams, is determined by eq. (2.4). We must consider, however, the fact that a laser-diode bar is usually made up of several tens of individual emitters, with dead spaces between the emitters. It is thus justifiable
1, § 5]
Applications
45
to choose N to be the number of individual emitters, instead of the expression given in eq. (2.4). This choice of N makes it possible to eliminate dead spaces between the sub-beams and hence to increase effective brightness, by making the effective PSV smaller. In the experimental scheme reported by Clarkson and Hanna [1996], which employed the two-mirror technique discussed in §2.1, the laser-diode bar consisted of 24 emitters arranged periodically, with a fill factor of 0.5. The overall beam-quality factors were M^^ ^ 1 and M^^ ^ 1300 in the two orthogonal directions. Note that ignoring the dead spaces between emitters, My^ ^ 650. After beam shaping, the beam-quality factors were M^^ ^ 42 and My^ ~ ^^' corresponding to a 2.4-fold increase in overall PSV (1.2-fold when taking into account the dead spaces at the input) and a 12% non-uniformity between the two orthogonal directions. Among the possible reasons for such an increase in PSV are the partial overlapping of sub-beams, the different pathlengths of each sub-beam, and the so-called "smile" effect - i.e. when the individual emitters are not on a straight line, but form a curve instead. The "smile" effect comes mainly from the manufacturing process of laser-diode bars, and can be corrected partially by a tilted cylindrical lens, as was demonstrated by Wetter [2001]. Liao, Du, Falter, Zhang, Quade, Loosen and Poprawe [1997] reported an experiment in which the micro step-mirror technique (see § 2.2) was used to equalize the orthogonal beam-quality factors. In their experiment the beamquality factors for the output beam of the laser-diode bar were M^^ P::^ 1.75 and My^ ?^ 41.5. From eq. (2.4), A^ = 5 was chosen as the number of reflective facets for each micro step mirror. After beam shaping, the orthogonal beam-quality factors were M^^ ^ S.2 and M^^ ~ 9» corresponding to a 9% non-uniformity, and only a 2% increase in PSV The microprism-array technique described in § 2.3 was experimentally demonstrated by Yamaguchi, Kobayashi, Saito and Chiba [1995]. The laser-diode bar consisted of 12 emitters, each 50|im wide and 200 |im long, with center-tocenter distances of 800 fxm. The overall size of the laser-diode bar was thus 10 mm X 50 ^m, and the diverging angles in the two orthogonal directions were 30° and 10"" (FWHM). The beam shaper consisted of 14 microprisms, each having a width p = 800 |im, machined from BK7 glass. After the beam shaper, the beam was focused into a symmetric spot with a diameter of 200 |jim, corresponding to a PSV increase of 3.7 (taking into account the dead spaces at the input). Several of the refractive/diffractive anamorphic beam-shaping techniques have been employed for laser-diode applications as well. The tilted cylindrical lens method (see §3.3) was experimentally implemented by Goring, Schreiber and
46
Anamorphic beam shaping for laser and diffuse light
[1? § 5
Possner [1997]. The design was such that the number of sub-beams was equal to the number of individual emitters in the laser-diode bar, whose beam-quality factor was Mf ^ 1000. After beam shaping an output beam with M^^ ^ ^^ ^^^ My^ ^ 40 was achieved, corresponding to an 8% increase in PSV The possible reasons for decrease in brightness in this method include the diode "smile" effect - this causes "crosstalk" between different sections of the redirecting element - and alignment errors (the position and tilt of the cylindrical lens have to be set with iim accuracy (Possner, Messerschmidt, Kraeplin, Hoefer and Schreiber [1999]), thus an active alignment procedure is needed). Izawa, Uchimura, Matsui, Arichi and Yakuoh [1998] reported experimental results for the method of stacked glass plates described in § 3.5. The laser-diode bar had beam-quality factors M^^ ^ 1 and M^^ ^ 2000 in the two orthogonal directions, and lateral dimensions 10 mm x 1 |im. Equation (2.4) yields N ^ 44 for the number of sub-beams (and hence 44 glass plates for each stack). In the experimental setup of Izawa et al., 15 glass plates were used for each stack, each glass plate having the dimensions 25 mm x 25 mm x 1.5 mm. This setup thus yielded a 15-fold increase in M^ and approximately a corresponding decrease in M^. The motivation behind adiabatic beam-shaping techniques was often the desire to make the beam of a laser-diode bar symmetrical. The stubbed-waveguide technique discussed in §4.1 was experimentally implemented for a laser-diode bar by Xu, Prabhu, Lu and Ueda [2001]. Here the output beam of the laserdiode bar - after being collimated in the x-direction by a fiber lens - had lateral dimensions 8 mm x 1mm, and divergence angles of 6.5mrad and 52mrad in the X- and >^-directions, respectively. The beam was then focused into a spot of 260 \im X 23 |i,m at the input of the stubbed waveguide. The stubbed waveguide was a silica rod of 250 |im diameter and 80 mm length. At the output the beam had a diameter of 250 (im and symmetrical divergence angles of 56'' in the xand >^-directions. The PSV was thus increased by 12, an increase caused mainly by the large dark area at the input facet of the stubbed waveguide (the beam did not entirely cover the input facet of the waveguide). Liithy and Weber [1995] demonstrated experimentally the fiber-bundle method described in § 4.2. Here the beam of a laser-diode bar, with lateral dimensions 10 mm X 1 |im and diffusive angles of lO'' and 40°, was shaped adiabatically with a bundle of fibers having square cross-sections. 100 fibers, each having a lateral size of 22.5 |im, were bundled together to form an output beam having a size of 225 ^im X 225 |im. The numerical aperture of the fibers was such that they accepted 98% of the input divergence. The diffusive angle of the output beam was 22"^ in both orthogonal directions, corresponding to a PSV increase of 6.4.
1, § 5]
Applications
47
5.2. One-dimensional concentration of solar radiation Often the motivation behind an anamorphic beam-shaping technique is to concentrate solar radiation in one transverse direction. Here the beam transformation is the inverse of the cases presented in §5.1. Note that since solar radiation is polychromatic, and diffractive techniques have large chromatic aberrations, we must resort to reflective, refractive or adiabatic techniques for solar applications. Such applications include the concentration of solar radiation into narrow water pipes for heating, or side-pumping of solar-pumped lasers. For such applications several two-dimensional concentrators have been proposed, ranging from simple imaging concentrators such as cylindrical lenses and cylindrical parabolic mirrors to more efficient non-imaging concentrators (Bassett, Welford and Winston [1989]). In these two-dimensional concentrators light is manipulated in one transverse dimension only; hence these concentrators are subjected to the one-dimensional version of etendue invariance and only concentrate light by a factor ^l/sina^, where a^ is the incoming divergence angle in the concentration direction (Winston and Welford [1989]). This limit is much smaller than the overall (two-dimensional) thermodynamic limit on the concentration ratio (Winston and Welford [1989]), ~l/(sin a^ sin ay) [with a^ and Qy the incoming divergence angles in the two orthogonal directions] that can be achieved with three-dimensional concentrators. The metallic retroreflector technique presented in the second part of §2.4 was demonstrated experimentally by Davidson, Khaykovich and Hasman [2000]. The input aperture of their setup was 400 mm x 400 mm, demonstrating that anamorphic techniques may be used for "large-optics" applications, and the diffusivity of the incoming white light was 0.01 rad in both transverse directions, similar to that of solar radiation. After the anamorphic concentration, the output beam had a size of 0.3 mm in the x-direction and 400mm in the ^-direction. Figure 5.1 shows the measured x cross-section of the light intensity distribution at the output of the anamorphic concentrator (solid curve). Also shown in fig. 5.1 is the light intensity distribution at the back focal plane of the parabolic mirror that was used before the retroreflector array (dashed curve). The x size of the concentrated beam is thus 13 times better than the one-dimensional thermodynamic limit 400mm • sin(O.Ol) '^ 4 mm, and 7.5 times worse than the overall (twodimensional) thermodynamic limit 400mm • sin^(O.Ol) ^ 0.04mm. The twodimensional limit was not reached, because non-ideal imaging concentrators (a parabolic mirror and a cylindrical lens) were used before and after the retroreflector array.
Anamorphic beam shaping for laser and diffuse light
48
1
1
1
1
1
r
1
T"— '
1
'—
[1,§5
-| '
100-
^ ^
II
^
II II II II 11
80-
ii ^
60-
(0
c 0) c
4020-
0- V^—\—^—\—s-H-
10
12
^ 14
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
J J J
-^
\ i , ~i
16
18
20
22
'
24
X axis [mm] Fig. 5.1. Experimental results with the metallic retroreflector technique by Davidson, Khaykovich and Hasman [2000] for one-dimensional diffuse light concentration: the jc cross-section of the light intensity distribution after anamorphic concentration (solid curve) and at the back focal plane of the parabolic mirror, before the retroreflector array (dashed curve).
5.3. High-resolution spectrometry and optical metrology Grating-based spectrometers require a narrow (or well-collimated) input to ensure high spectral resolution. For diffuse light sources such as light bulbs, arc lamps, and the Sun, a narrow input slit is thus required. However, this can decrease the input signals by many orders of magnitude. For totally incoherent light any decrease in the size of the beam impinging on the grating without such substantial loss of power would violate the optical brightness conservation theorem mentioned in § 1.1, and is therefore forbidden. The wavelength resolution of a grating that diffracts light in the x-direction IS
k _ nq dA~M?'
(5.1)
where A is the wavelength, n is the number of grating lines illuminated by the beam, q is the diffraction order, and M^ is the beam-quality factor of the illuminating beam in the x-direction. Equation (5.1) is a direct generalization of the well-known relation for a diffraction-limited plane wave (with M^ = 1) and indicates that the spectral resolution for a diffuse beam is reduced by exactly M^. Davidson, Khaykovich and Hasman [1999] proposed the following idea to improve the spectral resolution: since M? does not affect the spectral resolution.
1,§5]
49
Applications
610
620
630
640
650
660
X [nm] Fig. 5.2. Measured spectral impulse response of the grating-based spectrometer of Davidson, Khaykovich and Hasman [1999], with an anamorphic concentrator (dashed curve) and without it (solid curve). For anamorphic concentration the retroreflector technique described in § 2.4 was used.
an anamorphic transformation should be applied to the input diffuse beam to reduce M^, while conserving M^, hence improving the spectral resolution without any loss in brightness. (The reduction of M^ must of course be accompanied by an equivalent increase in My). Davidson, Khaykovich and Hasman [1999] demonstrated an experimental arrangement that improved the resolution of a grating-based spectrometer by anamorphic concentration of the input beam in the x-direction. The effective light source had a size of 25 mm x 25 mm, and diffusive angles 0.03 rad in both transverse directions, corresponding to M^^ = My^ ?^ 1185 at A = 633 nm. The experimental results are shown in fig. 5.2. With the spectrometer used in the experiment, a FWHM wavelength resolution of dA = 36.8 nm (near A = 633 nm) was measured when the input beam directly illuminated the spectrometer, without any anamorphic transformation (solid curve in fig. 5.2). Next, the Porroprism array technique described in the first part of § 2.4 was applied to concentrate the input beam in the x-direction. After the anamorphic concentration, the FWHM wavelength resolution improved to dA = 2.8nm, which corresponds to a 13-fold improvement in spectral resolution (dashed curve in fig. 5.2). We mention here briefly that anamorphic concentration of white diffuse light is suitable also for applications in optical metrology such as color-coded optical profilometry (Hasman, Keren, Davidson and Friesem [1999], Hasman and Kleiner [2001]), where the goal is to focus the incident light on a narrow
50
Anamorphic beam shaping for laser and diffuse light
[1
stripe that is axially dispersed. Both the transverse and longitudinal resolutions of the measured profile are proportional to the beam quality in one transverse direction (Hasman, Keren, Davidson and Friesem [1999]), say x, exactly as for the wavelength resolution of a grating-based spectrometer (eq. 5.1). Hasman, Keren, Davidson and Friesem [1999] achieved a diffraction-limited beam quality of M^ = 1, by passing a white light beam of M^ ^ My ^ 100 through a narrow slit, at the price of losing ^99% of its power. Using anamorphic beam shaping, as did Davidson, Khaykovich and Hasman [1999], optical resolution may be increased by significantly reducing M^, with no loss of power. Again, just like in the case of the grating spectrometer, the necessary corresponding increase in M^ does not affect the optical resolution.
§ 6. Summary We have discussed in some detail the many methods developed for anamorphic beam shaping of diffuse light. The reflective techniques described in § 2 have a wide range of applicability for any light source, including broadband radiations. However, they are much more sensitive to alignment errors than the adiabatic methods discussed in § 4. Adiabatic techniques, in general, are extremely robust, but only suitable for simple beam-shaping problems, such as symmetric collimation of an asymmetric input beam. The refi-active/diffractive techniques of §3 are similar to the reflective techniques inasmuch as these methods are all based on the splitting of the input beam into sub-beams and then rearranging the sub-beams appropriately to form the desired output beam. Dififractive methods that use optically recorded or computer-generated holographic elements often provide elegant and simple solutions for more complicated beam-shaping problems, however, they suffer from large chromatic aberrations and in many cases fi*om reduced efficiency. Most of these researches were motivated by the need to obtain a high-quality symmetric spot fi-om a high-power laser-diode bar, and anamorphic beam shaping can be expected to be a lively area of research for the coming years.
References Bassett, I.M., W.T. Welford and R. Winston, 1989, Nonimaging optics for flux concentration, in: Progress In Optics, Vol. 27, ed. E. Wolf (North-Holland, Amsterdam), ch. 3, pp. 161-226. Bokor, N., and N. Davidson, 2001a, Appl. Opt. 40, 2132. Bokor, N., and N. Davidson, 2001b, Opt. Commun. 196, 9.
1]
References
51
Bokor, N., and N. Davidson, 2001c, Appl. Opt. 40, 5825. Bokor, N., and N. Davidson, 2002a, Opt. Commun. 201, 243. Bokor, N., and N. Davidson, 2002b, J. Opt. Soc. Am. A 19, 2479. Bokor, N., R. Shechter, A.A. Friesem and N. Davidson, 2001, Opt. Commun. 191, 141. Brignon, A., G. Feugnet, J.P. Huignard and J.P. Pocholle, 1998, IEEE J. Quantum Electron. 34, 577. Clarkson, W.A., and DC. Hanna, 1996, Opt. Lett. 21, 375. Davidson, N., and A.A. Friesem, 1993a, Appl. Phys. Lett. 62, 334. Davidson, N., and A.A. Friesem, 1993b, Opt. Commun. 99, 162. Davidson, N., and L. Khaykovich, 1999, Appl. Opt. 38, 3593. Davidson, N., L. Khaykovich and E. Hasman, 1999, Opt. Lett. 24, 1835. Davidson, N., L. Khaykovich and E. Hasman, 2000, Appl. Opt. 39, 3963. Ehlers, B., K. Du, M. Baumann, H.G. Treusch, P Loosen and R. Poprawe, 1997, Beam shaping and fibre coupling of high-power diode laser arrays, Proc. SPIE 3097, 639. Feugnet, G., C. Bussac, Ch. Larat, M. Schwarz and J.P Pocholle, 1995, Opt. Lett. 20, 157. Garwin, R.L., 1952, Rev Sci. Instrum. 23, 755. Goring, R., P. Schreiber and T. Possner, 1997, Microoptical beam transformation system for highpower laser diode bars with efficient brightness conservation, Proc. SPIE 3008, 202. Graf, Th., and J.E. Balmer, 1996, Opt. Commun. 131, 77. Hasman, E., S. Keren, N. Davidson and A.A. Friesem, 1999, Opt. Lett. 24, 439. Hasman, E., and V Kleiner, 2001, Appl. Opt. 40, 1609. Izawa, T., R. Uchimura, Sh. Matsui, T. Arichi and T. Yakuoh, 1998, Efficient diode bar-pumped intracavity-doubled Nd:YV04 laser using stacked-glass plate beam shaper, in: CLEO Technical Digest, CThAl. Kritchman, E.M., 1980, Opt. Lett. 5, 35. Kritchman, E.M., 1981, Appl. Opt. 20, 1234. Kritchman, E.M., A.A. Friesem and G. YekutieH, 1979, Solar Energy 22, 119. Landau, L.D., and E.M. Lifshitz, 1972, Mechanics and Electrodynamics (Pergamon Press, Oxford) p. 108. Leger, J.R., and W.C. Goltsos, 1992, IEEE J. Quantum Electron. 28, 1088. Liao, Y., K. Du, S. Falter, J. Zhang, M. Quade, P Loosen and R. Poprawe, 1997, Appl. Opt. 36, 5872. Liithy, W., and H.P Weber, 1995, Infi-ared Phys. Technol. 36, 267. Miyamoto, I., and H. Maruo, 1989, Shaping of CO2 laser beam by Kaleidoscope (for use in laser surface hardening), Proc. SPIE 1031, 512. Motamedi, M.E., H.O. Sankur, F Durville, W.H. Southwell, R. Melendes, X. Wang, C. Liu, R. Rediker and M. Khoshnevisan, 1997, Optical transformer and collimator for efficient fiber coupling, Proc. SPIE 3008, 180. Possner, T, B. Messerschmidt, A. Kraeplin, B. Hoefer and P. Schreiber, 1999, Microoptical components for fiber coupling of high power laser diode bars, Proc. SPIE 3778, 88. Scifi-es, DR., and DPh. Worland, 1987, US Patent 4,688,884. Villareal, E I , H.J. Baker, R.H. Abram, D.R. Jones and D.R. Hall, 1999, IEEE J. Quantum Electron. 35, 267. Wetter, N.U., 2001, Opt. Laser Technol. 33, 181. Winston, R., and WT. Welford, 1989, High Collection Nonimaging Optics (Academic Press, New York). Xu, J., M. Prabhu, J. Lu and K. Ueda, 2001, Jpn. J. Appl. Phys. 40, 1279. Yamaguchi, S., T Kobayashi, Y Saito and K. Chiba, 1995, Opt. Lett. 20, 898. Zbinden, H., and J.E. Balmer, 1990, Opt. Lett. 15, 1014.
E. Wolf, Progress in Optics 45 © 2003 Elsevier Science B. V. All rights reserved
Chapter 2
Ultra-fast all-optical switching in optical networks by
Ivan Glesk, Bing C. Wang, Lei Xu, Varghese Baby and Paul R. Prucnal Department of Electrical Engineering, E-Quad, Olden Street, Princeton University, Princeton, NJ 08554, USA
53
Contents
Page § 1. Introduction
55
§ 2.
61
Use of nonlinearities in an optical fiber for all-optical switching
§ 3. Interferometric SOA-based all-optical switches
71
§ 4. All-optical switches based on passive waveguides
93
§5.
Demonstrations of all-optical switching in networks
94
§ 6.
Conclusion
HI
References
114
54
§ 1. Introduction In the backbone of today's high-performance networks, optical fibers provide enormous point-to-point communications capacity. To accommodate the rapidly growing volume of communications traffic, each fiber's capacity can be augmented by using "Dense Wavelength-Division Multiplexing" (DWDM), whereby the bandwidth in each optical fiber is increased by multiplexing multiple channels on different wavelengths (Nielsen, Stentz, Rottwitt, Vengsarkar, Hsu, Hansen, Park, Feder, Strasser, Cabot, Stulz, Kan, Judy, Sulhoff, Park, Nelson and Gruner-Nielsen [2000], Srivastava, Radic, Wolf, Centanni, Sulhoff, Kantor and Sun [2000]). Currently, commercial DWDM equipment is being deployed which multiplexes hundreds of lOGigabit/s channels into each optical fiber, enabling an aggregate throughput on the order of a few Terabit/s per fiber. Optical Time-Division Multiplexing (OTDM) is another technique used to increase the transmission speed by carrying data on narrow, single-wavelength optical pulses which are time-interleaved. DWDM and OTDM can be used together to dramatically increase the bandwidth capacity of the optical fiber. Using a hybrid OTDM/WDM scheme, transmission of 19 wavelengths at a bit rate of 160Gb/s per wavelength for an aggregate bandwidth of 3 Tb/s has been demonstrated by Kawanishi, Takara, Uchiyama, Shake and Mori [1999]. However, as the capacity of a single fiber extends to a few Terabits per second, a new communications bottleneck is emerging at the endpoints of the optical fibers, where routing and switching are performed to direct the traffic to its destination. Currently, the majority of backbone routers rely upon electronic crossbar switches to route packets. Electronic crossbars, however, do not provide sufficient capacity to interconnect multiple fibers due to their low speed and scalability. Electronic crossbars rely upon integrated devices made of silicon, gallium arsenide, and indium phosphide. The switching speed of these devices is determined by the frequency response of the material and the architecture of the transistors in the switch. Complementary metal-oxide semiconductor technologies, which are used in the majority of high performance digital devices like microprocessors, are not likely to scale beyond tens of gigahertz (Kurisu, Fukaishi, Asazawa, Nishikawa, Nakamura and Yotsutangai [1999]). The highest performance electronics based upon III-V materials like indium phosphide 55
56
Ultra-fast all-optical switching in optical networks
[2,§1
have only reached a few hundred gigahertz (Masuda, Ouchi, Terano, Suzuki, Watanage, Oka, Matsubara and Tanoue [1999]). Given these results, it does not seem likely that electronic technology will achieve the terahertz speed needed for terabit routers required for optical networks in the near future. The following subsections describe the general architecture of DWDM and OTDM systems and present the need for all-optical switches in these systems.
1.1. Dense wavelength-division multiplexing (DWDM) Figure 1 shows a diagram of a simplified generic WDM network link. Each electronic data channel, which in itself consists of numerous slower data channels time-division multiplexed together electronically, is transmitted on a different wavelength. In most long-haul DWDM systems, the transmitter is externally modulated. An external modulator is used to modulate a narrowlinewidth continuous wave (CW) laser, commonly a distributed feedback (DFB) semiconductor diode laser. This eliminates the chirp associated with direct modulation of a semiconductor laser and reduces the effect of optical fiber dispersion on the signal, thereby increasing the transmission distance and bit rate. Each of the CW lasers emits a narrow-linewidth optical output at a different wavelength, and the different wavelength channels are combined using a power coupler before entering the optical fiber. At the receiving end of the optical (Petal) CW laser
MOD
Nxl CW laser ^2
(Data2) I
AWG Optical fiber
MOD
l"|
Rx1 H Rx2 RxN
(PataN) CW laser
1/ -•
MOD
n LJ[
n
n n
Multiple-A. transmission Fig. 1. Block diagram of point-to-point WDM network.
2, § 1]
Introduction
57
fiber, an arrayed waveguide grating (AWG) splits the different wavelength channels into different physical ports. The different wavelength channels are then individually converted to electrical signals using optical receivers. Figure 1 represents a simplified DWDM point-to-point link. Actual systems may have some variations from this simplified model at the transmitting and receiving ends. For example, in certain short-distance and low-bit-rate systems where dispersion is less of an issue, the data is used to directly modulate the laser at the transmitter. Using photonic integration techniques, electro-absorption modulators have been built on the same substrate as narrow-linewidth lasers, with operating bandwidth over 40 GHz (Kawanishi, Yamauchi, Mineo, Shibuya, Murai, Yamada and Wada [2001]). At the receiver end, other techniques for wavelength channel selection, such as fiber Bragg gratings (FBG), can be used to separate the constituent channels (Set, Dabarsyah, Goh, Katoh, Takushima, Kikuchi, Okabe and Takeda [2001]). An optical switching fabric that is currently being integrated into commercial DWDM switching systems is based upon micro-electromechanical systems (MEMS). Simple embodiments of the MEMS technology include movable mirrors that route beams of light to their destinations. Lucent Technologies has demonstrated a 1296-port optical crossconnect consisting of moving mirrors, capable of directing light from one fiber to another (Ryf, Kim, Hickey, Gnauck, Carr, Pardo, Bolle, Frahm, Basavanhally, Yoh, Ramsey, Boie, George, Kraus, Lichtenwalner, Papazian, Gates, Shea, Gasparyan, Muratov, Griffith, Prybyla, Goyal, White, Lin, Ruel, Nijander, Arney, Neilson, Bishop, Kolodner, Pau, Nuzman, Weis, Kumar, Lieuwen, Aks3aik, Greywall, Lee, Soh, Mansfield, Jin, Lai, Huggins, Barr, Cirelli, Bogart, Teflfeau, Vella, Mavoori, Ramirez, Ciampa, Klemens, Morris, Boone, Liu, Rosamilia and Giles [2001]). Some of the advantages of the MEMS architecture include scalability, low power consumption, low loss, compact size, and protocol transparency. MEMS, since they avoid the electronic conversion required in standard routers, offer a simple solution to the optical switching problem. However, since MEMS are inherently mechanical, they are limited in speed. For example, the Lucent Crossconnect can only move its mirrors on a time scale of 5 ms. While this is appropriate for optical circuit switching and optical layer restoration protection switching, it is not fast enough to support switching on a packet-by-packet basis required by IP routing. Furthermore, electronic hardware must still be used to obtain the routing information to control the switch. Due to the mechanical nature of MEMS, long-term reliability and packaging are still critical issues in these systems that will have to be improved over time. Additional advances in MEMS will most likely not be able to upgrade the speed of the switches much beyond
58
Ultra-fast all-optical switching in optical networks Top View
1
[2, § 1
O V
Input A
Output A
Input B
Output B
Side View
! W!
;
^
1
Fig. 2. Directional coupler. Top view: two waveguides pass through a region of electro-optic material. W is their separation length in the coupling region and L is their length of interaction. Side view: layered structure of the device.
a few hundred microseconds. As a result, MEMS technology will not replace current electronic crossbars in IP routers on the Internet backbone. The MEMSbased switches will most likely find applications in interconnecting service providers and large cities where continuous traffic streams are established for long periods of time between fixed locations. Another optical technology that is used to switch light from multiple physical inputs to multiple physical outputs is the LiNbOs directional coupler based on the electro-optic (EO) effect. Directional couplers, by placing two optical waveguides sufficiently close together such that the optical modes fi-om one waveguide are coupled into the other, can split the incoming power in one waveguide into the two output waveguides in a preset ratio. In an active directional coupler, the waveguides pass through an electro-optic material. Figure 2 shows the schematic of an active directional coupler. A voltage can be applied across the EO material such that the two waveguides see different indices of refraction (Anwar, Themistos, Rahman and Grattan [1999]). When no voltage is applied to the EO region, the device is in the cross state where most of the power from input A exits through output B and most of the power from input B exits through output A. However, when a voltage is applied across the EO region, the device changes into the bar state, where power from input A goes to output A and power from input B goes to output B. Thus, the applied voltage can switch the device from one state to another by changing the difference in index of
2, § 1]
Introduction
59
refraction between the two waveguides such that the light passing through them will have a relative phase difference of 7t at the output of the interferometer. The voltage required to obtain the Ji phase shift depends on the dimensions of the device, such as the interaction length L of the two waveguides and their separation W, as well as the index of refraction of the device. One advantage of the LiNbOs directional-coupler-based switch is that a large number of such devices can be connected to build larger optical switching fabrics. However, the large switching voltages typically required for the Jt phase shift prevent them from operating above a few GHz. Both MEMS and LiNbOs-based switching technology require electronic control. Significant effort is being expended to ftirther increase the number of wavelengths and the bit rate per wavelength in DWDM transmission systems. State of the art electronics can drive the optical bit rates at speeds of up to 40 Gb/s. Two separate groups have demonstrated aggregate data transmission of lOTb/s in 2001 (Fukuchi, Kasamatsu, Morie, Ohhira, Ito, Sekiya, Ogasahara and Ono [2001], Bigo, Frignac, Charlet, Idler, Borne, Gross, Dischler, Poehlmann, Tran, Simonneau, Bayart, Veith, Jourdan and Hamaide [2001]). However, even at the highest electronic bandwidth, the large number of DWDM wavelength channels required to fill the optical fiber transmission bandwidth makes network management increasingly difficult and expensive. By increasing the bit rate transmitted per wavelength, the number of wavelengths can be significantly reduced. OTDM techniques can be used to increase the bit rate per wavelength beyond the electronic limits. In the next section, OTDM concepts will be introduced for a better understanding of the advantages of optical switching in OTDM networks.
L2. Optical time-division multiplexing (OTDM) As already mentioned, OTDM increases the transmission speed by carrying the data bits on temporally narrow optical pulses closely multiplexed together. Since the optical pulses occupy only a small fraction of the entire bit period, multiple data channels can share the bit period by temporally placing their pulses in different locations within the bit period. Figure 3 illustrates the concept of timeinterleaving optical pulses from different channels. In OTDM, the electronic baseband signal is modulated on narrow pulses that occupy a small fraction of the bit period. Then the optical pulses from different baseband channels are temporally delayed into different timeslots, and the pulses are then interleaved in time to create an aggregate line rate equal to the number of channels multiplied by the rate of the baseband electronic channel. For example, sixteen 10 Gb/s
60
Ultra-fast all-optical switching in optical networks
[2, § 1
(Datal)
" MOD LCData2:)
Optical Pulse Source
MOD
n. Delay T„ /-\'
I
IXN
N x l V_K
Optical fiber
V^
(
/T
I DeMUX r H R x l \ DeMUX [~H
°X 2
I DeMUX | ~ ^
r^X N
y
Delay(N-1)To (DataN)
MOD h
N timeslot channels interleaved in time Fig. 3. Block diagram of point-to-point OTDM network.
baseband signals can be multiplexed together, using 2ps pulses, to form a 160Gb/s single wavelength channel. One of the most common methods for generating picosecond optical pulses at GigaHertz repetition rates involves actively mode-locking an erbium-doped fiber laser (EDFL). The pulse power at the output of a commercial EDFL is typically lOdBm or higher. The high output power allows one laser to power anywhere from ten to one hundred different OTDM channels, depending on the power requirement of the particular application. Other sources commonly used for OTDM pulse generation includes gain-switched DFB lasers and modelocked semiconductor lasers. At the receiver end of an OTDM connection, the individual timeslot channels are demultiplexed into the baseband rate for further signal processing or conversion into the electronic domain. Since the aggregate data rates at the receiver are much higher than the electronic rates, all-optical methods are needed for demultiplexing different OTDM channels. Demultiplexing techniques using various all-optical switches will be discussed in the next few sections. Recently, several experimental demonstrations have shown that OTDM can meet many of the demanding needs of a high-performance switching fabric, including fiill connectivity, low latency, high aggregate throughput, reliability, and scalability (Barry, Chan, Hall, Kintzer, Moores, Rauschenbach, Swanson, Adams, Doerr, Finn, Haus, Ippen, Wong and Haner [1996], Lucek, Gunning, Moodie, Smith and Pitcher [1997], Tsukada, Zhong, Matsunaga, Asobe and Oohara [1996], Deng, Runser, Toliver, Coldwell, Zhou, Glesk and Prucnal [1998], Deng, Runser, Toliver, Glesk and Prucnal [2000]).
2, § 2]
Use of nonlinearities in an optical fiber for all-optical switching
61
As described earlier, electronic technology is not expected to solve the switching bottleneck in next-generation optical networks. Growing demand for higher switching speeds has led to the development of all-optical mechanisms for ultrafast switching. All-optical switches could be employed in optical networks for different switching applications such as OTDM demultiplexing, packet routing in networks and clock extraction. All-optical devices are also finding applications in high-speed all-optical signal processing such as wavelength conversion, alloptical 3R regeneration (re-amplification, re-shaping, re-timing), and high-speed all-optical analog sampling. The next few sections explore the different switching schemes using various devices and materials that are being developed today.
§ 2. Use of nonlinearities in an optical fiber for all-optical switching All-optical switching generally takes advantage of a nonlinear interaction between optical waves. The nonlinearity in a variety of materials including fiber, semiconductors and crystals like lithium niobate has been used to demonstrate all-optical switching. A brief summary of some of the key techniques developed for ultra-fast switching and experimental results demonstrating their switching capabilities are discussed here. 2.1. All-optical switches based on four-wave mixing (FWM) Four-wave mixing (FWM) is a parametric phenomenon arising from thirdorder optical nonlinearity and has been used to develop all-optical switches. To understand the origin of FWM, let us consider the basic equation of third-order nonlinearity: P{t) = X^'^^E{tf.
(2.1)
Here, P{t) is the instantaneous polarization, ^^^^ is the third-order nonlinearity coefficient, and E{t) is the instantaneous applied electric field. Assume that the applied field consists of three firequency components: E{t) = Ei(t) X e-^^^' + j&2(0 X e-^^^' +^3(0 x e"^""^'.
(2.2)
Substituting eq. (2.2) into eq. (2.1) shows that P(t) has frequency terms at cou C02, C03, 3coi, 3(02, 3co3, (±coi ± (02 ± (O3), (2(ji)\ ± 0^2), (2(0i ± C03), (2co2 i ft^i), (2co2 ± ^3), (2co3, ± c^i), and (20^3 ib 0^2). These time-varying
62
Ultra-fast all-optical switching in optical networks
[2, § 2
polarization components act as sources of new components of electromagnetic field at these frequencies. By general convention, all the terms in parentheses are called the FWM mixing terms, while the 3 a; 1,2,3 terms are called the thirdharmonic-generation terms. Thus, the interaction of three electromagnetic waves in a medium with thirdorder nonlinearity results in the generation of electromagnetic waves at new frequencies. However, the power created at the new frequencies by the fourwave mixing process depends on the efficiency of the parametric coupling. Higher efficiency of FWM mixing requires a matching of the frequencies as well as the wavevectors, a condition known as "phase matching". In general, this phase-matching condition is more difficult to satisfy for the terms where a single photon is produced by the annihilation of three photons, that is, for the terms (ft>i + 0^2 + ^^3), {2ci)\ + 0)2), {2wx + (^3), (20^2 + ^ i ) , (2^2 + ^3), {2(02, + ^1) and (20^3 + 0)2). The phase-matching condition is easier to satisfy for the other mixing terms - {co\ + CO2 - <^3), {2o)\ - (O2), (2co\ - CO3), (20)2 - 0)iX (20^2 - ^3), (2(0^ - Wi), (20^3 - (^2) - resulting in more power at these frequencies. FWM occurs even when the number of input waves is less than three. Thus, in the presence of two beams of frequencies a)\ and CO2 in a medium having third-order nonlinearity, FWM mixing terms are produced at the frequencies (2a; 1 - a;2) and (2a;2 -co\). Although the above equations are strictly valid only for continuous-wave beams, FWM mixing terms are generated even when the beams are pulse trains, if the pulses from the two beams at co\ and a;2 overlap temporally This technique can be used to ail-optically demultiplex a single channel from an OTDM stream. A schematic representation of this technique is shown in fig. 4. The injection of a control pulse stream at the base channel rate with the multiplexed data stream into the nonlinear medium results in the generation of the FWM mixing terms when the probe and signal pulses overlap. Thus, the information present in one channel can be duplicated onto the mixing wavelengths, which can be filtered out from the rest of the data channels. An all-optical switch using FWM was used to demultiplex a 4 Gb/s channel from a 16 Gb/s data stream by Andrekson, Olsson, Simpson, Tanbun-Ek, Logan and Haner [1991]. The experimental setup is shown in fig. 5. Both the data pulses and the control pulses were obtained from actively modelocked external cavity lasers (ECL). The ECLs produced 20-25 ps long pulses which were nearly transform-limited at 1.531 [im at a repetition rate of 4 GHz. Erbium-doped fiber amplifiers were used throughout the setup to maintain sufficient power levels. The data pulses were externally modulated using LiNb03
2, §2]
Use of nonlinearities in an optical fiber for all-optical switching
63
Multiplexed Data at co^ — • Time Control at CD. - • Time Demultiplexed channel a t (2(0j -CD2) & (20)2 - c o j )
T—I
1
r-
n
1—T—I
1
r
-^'Time
Fig. 4. Schematic representation of demultiplexing using FWM mixing. 4 Gb/s^ iML-ECLl
|MOD^
Bit rate Upconverter
Data at 16Gb/s
FWM based Demux 14 km DSF
RF Transmitter
> < : > •
-I Filter I
HBERTI
16Gb/sto4Gb/s
Optical Control Signal at 4GHz |ML-ECLh Fig. 5. Experimental setup for demultiplexing 4Gbps channel from 16Gb/s data stream. ML-ECL, mode-locked external cavity lasers; MOD, modulator; BERT, bit error rate tester; DSF, dispersion-shifted fiber.
Mach-Zehnder modulators and were multiplexed to 16Gb/s using a series of 3 dB couplers. Both the multiplexed data stream and the control pulse stream were then injected into the fiber for four-wave mixing. The average data and control power entering the fiber were 4.3 dBm and 5.2 dBm. Isolators were used throughout the setup to maintain unidirectional propagation of light. The fiber used to obtain the nonlinear interaction was a 14 km long dispersion-shifted (DS) single-mode fiber with a chromatic dispersion value of 0.05 ps/nm/km. The data and control pulse lasers were both driven from the same RF source to obtain synchronization. The wavelength separation of the data and control pulses was 0.8 nm. The FWM output was filtered using a fiber Fabry-Perot interferometer set to a wavelength corresponding to the FWM mixing fi-equency The all-optical switch demonstrated an extinction ratio of 8 dB. Error-free demultiplexing of all the four 4 Gb/s channels was obtained with a power penalty of about 0.9 dB. This power penalty, however, is not a fundamental limit and can be decreased using higher control and data pulse powers.
64
Ultra-fast all-optical switching in optical networks
[2, § 2
One disadvantage of switching using FWM is the low efficiency associated with the mixing process. For example, Andrekson, Olsson, Simpson, TanbunEk, Logan and Haner [1991] obtained an average power ratio between the FWM and the control pulses of about 6%. This low efficiency necessitates the use of huge control power for switching unlike other switches that will be described later. Another disadvantage of FWM switching is the degradation in performance due to the tendency of the control and data pulses to walk off from each other because of their different group velocities. In the experiments of Andrekson, Olsson, Simpson, Tanbun-Ek, Logan and Haner [1991], this was reduced by the use of dispersion-shifted (DS) fiber, which has very low chromatic dispersion at the operating wavelength thus minimizing the walk-off to 0.6 ps. Thus, the DS fiber serves two purposes - it reduces the effect of walk-off between the control and data signals caused by group-velocity dispersion; it also increases the nonlinear interaction by providing the condition of phase matching for the four-wave mixing process. In optical networks, data signals quickly reach an arbitrary state of polarization after transmission in the fiber. Since all-optical switches usually handle signals after transmission, it is important that they should have uniform switching performance for different polarizations. For efficient four-wave mixing, it is necessary that the data and control pulses have their polarizations aligned. This essentially makes the switch mentioned above polarization dependent. Polarization-independent operation of an FWM-based switch at lOOGb/s was obtained using a polarization-maintaining (PM), polarization-rotating fiber loop mirror (PRLM) (Morioka, Kawanishi, Uchiyama, Takara and Saruwatari [1994]). The technique essentially involved the addition of two independent four-wave mixing processes for the two orthogonal polarizations with equal efficiency, ensured by providing equal pump powers in both polarization directions. Error-free 500Gb/s to lOGb/s demultiplexing was also demonstrated using FWM in a 300 m polarization-maintaining dispersion-shifted optical fiber using low-noise 1 ps supercontinuum pulses (Morioka, Takara, Kawanishi, Kitoh and Saruwatari [1996]). A 3Tbit/s OTDMAVDM transmission experiment, using 19 WDM channels each at 160Gb/s, was demonstrated where FWM switches were used for OTDM demultiplexing (Kawanishi, Takara, Uchiyama, Shake and Mori [1999]). The transmission was done over 40 km of dispersion-shifted fiber and a gain-flattened tellurite-based erbium-doped amplifier was used for amplification. All-optical switching based on FWM in a semiconductor optical amplifier (SOA) was attempted because of the compact size of the SOA. Errorfree demultiplexing of a 6.3Gb/s channel from a lOOGb/s aggregate stream
2, § 2]
Use of nonlinearities in an optical fiber for all-optical switching
65
was demonstrated in 1994 (Kawanishi, Morioka, Kamatani, Takara, Jacob and Saruwatari [1994]). The power penalty of demultiplexing was about 5dB and the efficiency of the FWM mixing process was about 4%. A polarization-independent FWM process was demonstrated for 200Gb/s demultiplexing by using two PM fibers before and after the SOA (Morioka, Takara, Kawanishi, Uchiyama and Saruwatari [1996]). In this method, the first PM fiber separates the two orthogonal polarizations temporally, thus depolarizing the signal to suppress polarization coupling in the SOA. The two orthogonal polarizations then undergo independent FWM mixing in the SOA before being combined temporally using the second PM fiber, which has its principal axis rotated 90° with respect to that of the first PM fiber. This switch demonstrated stable, error-fi-ee operation with less than 0.5 dB polarization dependency and an FWM mixing efficiency of 15%. 2.2. All-optical switches based on cross-phase modulation (XPM) XPM-induced carrier-firequency shifts in an optical fiber is another technique that has been used for high-speed all-optical demultiplexing. In this method, optically amplified control (clock) pulses are used to shift the center wavelength of every coincident signal (data) pulse by adjusting the temporal S5mclironization such that the signal pulses coincide with the clock pulses' edges. The center fi-equency shift depends on the power of the clock pulses. At the output end of the fiber, an ordinary fi*equency filter can be used to separate out the demultiplexed data channel. A 5 Gb/s data stream was demultiplexed from a 40 Gb/s aggregate stream using this technique (Patrick and Elhs [1993]). Multiple data channels can be simultaneously demultiplexed by an extension of the same method. If the control pulses used have a width that spans multiple data channels, different data channels "see" different control pulse powers and hence have different center frequency shifts. At the output end of the fiber, the different channels can be demuhiplexed by using a grating. A 4-output demultiplexing operation was demonstrated at 60 Gb/s (Uchiyama, Kawanishi and Saruwatari [1998]). 2.3. Nonlinear loop mirror (NOLM) as an all-optical switch 2.3.1. Sagnac interferometer (fiber-loop mirror) Figure 6 shows the schematic of a Sagnac interferometer. It consists of a 2x2 directional coupler with its two ports (3 and 4) connected by a piece of fiber to
66
Ultra-fast all-optical switching in optical networks
Input
[2, § 2
1 2 Output
Fig. 6. Structure of fiber loop mirror.
form a loop. Light is input into the interferometer through port 1. In general, the coupler has a coupling ratio K/(l - K), where 0 < A^ < 1. When K = 0.5, 50% of the input light will travel clockwise (CW) around the loop and fifty percent will travel counterclockwise (CCW). In addition, light coupled across the coupler (between port 1 and port 4) suffers a | j r phase lag with respect to light traveling straight through the coupler (between port 1 and port 3). Since the light components traveling in CW and CCW directions traverse the same piece of fiber, they experience the same amount of phase change. The transmitted intensity at port 2 (called also the output port) is therefore the sum of a CW field of arbitrary phase 0 and a CCW field with relative phase (0 - JC). Assuming AT = 0.5, the two components will have equal amplitudes, resulting in a zero transmitted intensity at the output port 2. In this case all the input light is reflected back along the input port 1. Hence, this architecture is called an optical loop mirror. For K ^ 0.5, the transmission coefficient T, defined as the ratio of power output to power input, is T=l-4K(l-K).
(2.3)
Again, we assumed that the light components traveling in the CW and CCW directions experience the same amount of phase change in traversing the fiber loop. A more detailed analysis of a fiber-loop mirror, which includes coupler excess loss, fiber loss, birefiingence in the loop, etc., can be found in Mortimore [1988]. The Sagnac interferometer is inherently stable due to the fact that both CW and CCW components travel through the same piece of optical fiber and therefore do not "pick up" any additional relative phase change due to slow and random environmental fluctuations.
2, § 2]
Use of nonlinearities in an optical fiber for all-optical switching
67
2.3.2. Nonlinear optical loop mirror (NOLM) Using nonlinear effects, the CW and CCW components can be made to experience different phase changes even though they pass through the same piece of fiber. This configuration of the Sagnac interferometer is known as NonHnear Optical Loop Mirror (NOLM). An additional phase difference A0 between CW and CCW signals can be obtained due to the nonlinear interactions within the fiber loop. In this case the transmission coefficient T of the NOLM can be expressed as follows: T = l-4K(l
-K) cos2(^A0).
(2.4)
If the optical power is high enough, the signal acquires a phase change that is intensity-dependent through self-phase modulation (SPM). This phase shift (0) is given by ^ = ? ^ ^ ^ , A
(2.5)
where «2 is the nonlinear Kerr coefficient, A is the signal wavelength, E is the amplitude of the electric field inside the optical fiber, and L is the distance traversed in the fiber. If the coupling coefficient K is not balanced, the CW and the CCW components of the signal can acquire different phase shifts. When K is not 0.5, the additional phase difference between CW and CCW is j^^=^^n,\E\'L^_2nm\EfL^^_^^2nn,\EtL^^^_^^^ A
A
^^.6) A
This equation shows that for any value of Z ^ 0.5, all of the input power will emerge at the output port (port 2) whenever 2nn2\E\ L ^^^ _ ^^ ^ ^^^
^^^ ^ ^^^
^^^^
A
The minimum power transmission to port 2 occurs for m even, and is given by eq. (2.3). This type of NOLM was first proposed by Doran and Wood [1988] to study interactions between counterpropagating light, including optical solitons, in the loop. For suitability in optical switching applications, the NOLM is slightly modified from the original configuration shown in fig. 6. This modified structure
68
Ultra-fast all-optical switching in optical networks
[2, § 2
cw
Input
L---J Output
Fig. 7. Typical NOLM configuration for optical switching.
is shown in fig. 7. The NOLM uses a balanced {K = 0.5) 2x2 directional coupler. A wavelength-division multiplexer (WDM) coupler is used to introduce high-power optical control pulses into the loop, as shown in fig. 7. If a highpower optical control pulse is launched clockwise into the fiber loop such that it temporally overlaps with the CW signal pulse, the CW pulse will experience an induced nonlinear phase shift, due to cross-phase modulation (XPM), which can be expressed as (Jinno and Matsumoto [1992]): (p(t)cw = 2y / P,(t - iwaik^c) dx, (2.8) Jo Jo where y = 2;r«2/A^eff, with ^eff the effective core area of the loop fiber, Pc{t) is the power waveform of the control pulse, and Twaik is the relative delay between the input signal and the control pulse in a unit length. The CCW signal also experiences a phase shift due to the control pulse through XPM. But in the case of the CCW signal, the phase shift is constant (Jinno and Matsumoto [1992]): 0(Occw = 2yP~,L,
(2.9)
where Pc is the averaged power of the control pulse. Combining expressions (2.8) and (2.9) for the relative phase-shift difference A0(O between CW and CCW signals after traveling the loop of length L yields
(r
A0(O = 0(Ocw-0(Occw = 2y ( / Pc(t-T^^i^,x)dx-PcLj
.
(2.10)
For complete switching, the relative phase-shift difference A^(^) must be equal to Jt.
2, §2]
Use of nonlinearities in an optical fiber for all-optical switching
69
A wavelength filter is used at the output port of the NOLM to suppress the control signal. For control and signal pulses of orthogonal polarization states, a polarizer can be used at the output to filter out the control pulse. 2.3.3, Switching characteristics of NOLM Assume a Gaussian profile for the control pulse (Pc(0)? Pc(0 = ^ o e x p | - 4 1 n 2 - ( —
(2.11)
^0
where PQ is the peak power and TQ is thefixUwidth at half maximum (FWHM) of the control pulse. From eqs. (2.9) and (2.11), the CCW signal phase shift is given by 0CCW -
It \/p^y^O^^o/control,
(2.12)
where y^ontroi is the repetition rate of the control pulses. Therefore, the transmission coefficient can be expressed as
T=\-AK{\-K)cosH
yPo<
exp
_4j^2.('iz^y
dx (2.13)
Jt ~ \/ ~^ ^^^Oycontrol
Figure 8 shows the relationship between the width of the switching window (FWHM) and the walk-off time TQ when K = 0.5, L = 2000 m, and/contmi = 1 GHz. 60
K 50
- • - Control pulsewidth To=10ps - • - Control pulsewidth To=30ps
E 40 P 30•1 20^ 10sz
I 00
— I — • — I —
10
20
30
40
50
60
Walk-off time XQ (ps) Fig. 8. Relationship between width of switching window and walk-off time, for control-pulse widths of To = 10psand30ps.
70
Ultra-fast all-optical switching in optical networks
r\ ^^^
1.0 ^
0.8
1
0.6
'
•
'
'
'
H
1.0-
,
0.8
,
.
•^ 0.6-
E
1 0.4.
^
/
0.2 00 160
[2, § 2
\
-
V
y 180
200 220 Time (ps)
0.4-
CD
0.2.
y
0.0240
260
180
V
200 220 240 Time (ps)
260
Fig. 9. 30 ps switching window using (a) TQ = 30 ps, TG = 10 ps, and (b) TQ = 10 ps, r^ = 30 ps.
Here TQ is the amount of pulse walk-off between the control and input signal pulses for the entire length of the loop fiber, XQ = iwaik^- As seen in fig. 8, the width of the switching window increases when the control pulse width or the walk-off time increases. Figures 9a and 9b show the 30 ps switching windows generated by a 30ps control pulse and lOps walk-off time, and a lOps control pulse and 30 ps walk-off time, respectively. For control pulse widths much larger than the walk-off time, the width of the switching window is determined by the control pulse width and has a bell shape. For control pulse widths much smaller than the walk-off time, the width of the switching window is determined by the walk-off time, and has a square-like shape. Narrower control pulses and smaller walk-off lengths can be designed to further improve the switching performance. The peak phase shift induced in the clockwise signal of the input that is travelling with the control pulse is (Uchiyama, Morioka, Kawanishi, Takara and Saruwatari [1997]) (2.14) where erf(0^
2
r'
V^ Jo
(2.15)
Qxp(-u ) du.
Therefore, the maximum phase difference is given by A0max = 0CW " 0CCW
..'""-"R-pr
(2.16)
JT
/control
For efficient switching, A^^ax = ^' For TQ = 10 ps, TQ = 30 ps andy^ontroi = 1 GHz, we have yPoL = 4.46. Typically, in optical fiber, «2 = 3 x 10~^^ m^/W, ^eff = 50 ^im^.
2, § 3]
Interferometric SOA-based all-optical switches
71
and 7 = 2;r«2/A^eff = 2.4W-ikm'^ at 1.55 (im. Therefore, Po^=1.8Wkm. Due to the low optical nonlinearity «2, a long piece of fiber (^km) is needed to achieve efficient switching. The above discussions assumed that the control pulse retains its shape as it propagates along the fiber. However, for short pulses and long fiber loops, the effect of group-velocity dispersion (GVD) can not be neglected. This is especially true when the dispersion length of the optical fiber, ZD = ^o/ft, with ft the GVD parameter, approaches the length of the fiber loop. The interaction of dispersion and nonlinear effects results in serious distortion of the switching window since it changes both the waveform of the control pulses, through SPM, and the modulation profile of the input signal, through XPM (Jinno [1992]). An analysis of optical pulse propagation including GVD and nonlinear effects requires a more involved numerical solution of the nonlinear Schrodinger equation. The NOLM is sensitive to the input polarization due to the difference in the XPM coefficient for parallel-polarized and cross-polarized pulses. To overcome this problem it has been proposed to use twisted or circularly polarized fiber to reduce the source of polarization sensitivity. In one of these experiments, the polarization sensitivity was reduced to 0.5 dB by twisting the DSF (Liang, Lou, Stocker, Boyraz, Andersen, Islam and Nolan [1999]). Due to the ultra-fast response of the nonlinearity in optical fiber, the switching window for NOLM can be made very short using very narrow control pulses and a loop that is free of walk-off. 640 Gb/s switching in a single-channel OTDM experiment has been demonstrated with a NOLM using a 450-m long zerodispersion flattened fiber (Nakazawa, Yoshida, Yamamoto, Yamada and Sahara [1998]). The control pulse width was 1.1 ps and the walk-off between the signal and the control pulses was <100 fs. The optimum peak power for the NOLM in this experiment was 3.2 W. NOLM-based all-optical switches have demonstrated very high-speed optical switching. However, as optical fibers are passive materials with low nonlinearity, high optical powers and long interaction lengths are required to achieve efficient switching. The long fiber length of these devices requires special design to reduce phase mismatch. The optical tunability range of the data wavelength is fairly low once the phase matching is optimized. Also, optical fiber cannot be integrated, which further limits its development. § 3. Interferometric SOA-based all-optical switches In the past ten years, the semiconductor optical amplifier (SOA) has become a key component for optical signal processing in optical communication and
72
Ultra-fast all-optical switching in optical networks
[2, § 3
networks. SOA-based devices are very compact and offer the possibility for monolithic integration together with other photonic devices. Also, the short length of the SOA (less than 1 mm) has reduced the problem of phase mismatch. SOAs are widely used as building blocks in all-optical demultiplexers, switches, wavelength converters, memories, dispersion compensators, etc. With the further development of SOA fabrication technologies, SOAs with very low polarization sensitivity (less than 1 dB) and high saturation output power (10 dBm) have been developed, resulting in improved performance. 3.1. Gain dynamics in SOA The operation of a SOA depends primarily on the creation of a carrier population inversion that ensures that the stimulated emission is more prevalent than absorption. The population inversion is usually achieved by electric current injection in the p-n junction where the generated electron-hole pairs recombine by means of stimulated emission. An approach similar to that published by Agrawal and Olsson [1989] can be used to describe the gain dynamics in the SOA. In a rate-equation approximation, the response of the SOA gain medium to the optical field E can be described by the carrier-density rate equation:
|^=DV^A..4,-^-^^^^|^P, at
qV
Tc
(3.1)
hcoo
where N is the carrier density, D is the diffusion coefficient, / is the injection current, q is the electron charge, V is the active volume, Tc is the spontaneous carrier lifetime, ^a;o is the photon energy, a is the gain coefficient, and iVo is the carrier density required for transparency. The propagation of the electromagnetic field inside the SOA is governed by the wave equations V ^ ^ - ( i ± 2 ^ ^ = 0,
(3.2)
where c is the velocity of light in vacuum, and the dielectric constant includes two parts: the background refi*active index nb and the contribution of the charge carriers X- / is a fiinction of the carrier density, and can be expressed by ;^(^) = _!?f (a + i)a(N-Nol (3.3) wo where n is the effective mode index. The carrier-induced index change is taken into account through a linewidth enhancement parameter (a). For a SOA, a is typically in the range of 3-8.
2, § 3]
Interferometric SOA-based all-optical switches
73
The carrier-density equation (3.1) can be simplified by noting that the width and the thickness of the active region of the SOA are generally smaller while the amplifier length is much larger than the diffusion length. Since the carrier density is nearly uniform along the transverse dimension, carrier diffusion along that direction can be neglected. By averaging over the active-region dimensions, the rate equation reduces to dt qV r^ hcoo ^ ^ ^^ where A(z, t) is the slowly varying envelope associated with the optical pulses, and the gain is defined by g{N) = ra{N-N^\
(3.5)
where F is the mode confinement factor. Combining eqs. (3.4) and (3.5) yields dl ^ go-g
_ g\Af
^^ ^^
where £'sat is the saturation energy of the amplifier and go is the small-signal gain. Separating A{z, t) into its constituent amplitude and phase parts, ^ = VPexp(i0),
(3.7)
yields a simple relation between the amplitudes and phases of the input and output pulses from the SOA for the case when the internal loss of the SOA is much less than the gain: Pouxir) =Pin(r)exp[Kr)],
(3.8)
0out(r) = 0 i n ( r ) - i a / z ( r ) .
(3.9)
Here the subscripts "in" and "out" represent the input and output pulses, respectively. The time r is measured in a reference fi-ame moving with the pulse. The fixnction h represents the integrated gain over the transverse direction of the SOA, and is defined by K'^)= / g(^,r)dz. (3.10) /o From eq. (3.6), it follows that h{r) is the solution of the ordinary differential equation &h gpL-h Pin(r) — =— 7 ^ [ e x p ( / i ) - 1]. (3.11) The calculated gain change using a short optical pulse at the input is shown in fig. 10. Esat was taken to be 1 pJ, and goL was 4.5, corresponding to a gain of 90 (~e'*.5). The pulse width used for the calculations was 10 ps.
74
[2, §3
Ultra-fast all-optical switching in optical networks 100 10
's
«°'' Q. X
60
6
c
40-
4 Input optical power Gain (T^=300ps) Gain (T'=100PS)
03
O
20-
O
•o
o" Q)_ O CD
2 0
0200
400
600
800
Time (ps) Fig. 10. Dynamics of SOA gain change upon passing of a short optical pulse, for carrier lifetimes of 100psand300ps .
When a short optical pulse is injected into the SOA, the carrier density decreases due to enhanced stimulated emission. For high-energy input optical pulses, the carrier-density decrease can be very large, resulting in significant decrease in the gain of the SOA. This process is known as gain saturation. As seen in fig. 10, the saturation time can be as short as a few picoseconds. Afi;er the short pulse has passed, the gain of the SOA recovers due to the injection of the carriers by the electrical current. The recovery time is decided by the carrier lifetime, which is typically several hundred picoseconds. Figure 10 shows the gain change of the SOA for carrier lifetimes of lOOps and 300 ps. Although the recovery time of this kind of resonant optical nonlinearity is long, the turn-on time of the saturation process can be much shorter. Different kinds of devices using SOAs have been developed to exploit these two processes to build ultra-fast all-optical gates and switches. 3.2. All-optical switches based on NOLM with built-in SOA: SLALOM and TOAD As already mentioned in §2, NOLM is a very good candidate for ultra-fast all-optical switching applications since it utilizes the ultra-fast nonlinearity in fiber. Therefore, the switching window of a NOLM can be made very short (less than 1 ps). However, the nonlinear interaction used in NOLMs is relatively weak. Hence, the operation of the NOLM usually requires high control-pulse energies and long fiber loops to generate a significant phase shift. Although there are many techniques to reduce the requirements for both the control-pulse energy and the loop length (Yamada, Suzuki and Nakazawa [1994], Asobe, Oohara, Yokohama
2, § 3 ]
75
Interferometric SOA-based all-optical switches
Output Fig. 11. Schematic diagram of TOAD.
and Kaino [1996]), practical NOLMs for commercial optical communication systems are yet to be realized. Experiments with SOAs inserted into the loop of the Sagnac interferometer demonstrated that low-energy optical pulses could change the gain of the amplifiers sufficiently to produce significant phase shifts in subsequent pulses passing through the amplifier (O'Neill and Webb [1990]). Since these integrated semiconductor amplifiers were very short (
76
Ultra-fast all-optical switching in optical networks
[2, §3
this configuration, the switching window closes earlier than the recovery time of the SOA as the SOA is moved closer to the midpoint. Figure 11 shows a schematic diagram of such a device, known as a Terahertz Optical Asymmetric Demultiplexer (TOAD) (Sokoloff, Prucnal, Glesk and Kane [1993a,b]).
3.2.1. Working principles of TOAD When a data pulse enters the loop, it is split into CW and CCW traveling components by a 50:50 coupler. The two components pass through the SOA at different times as they counterpropagate around the loop, and recombine interferometrically at the 50:50 coupler. For the TOAD geometry shown in fig. 11, the CCW component of the data pulse reaches the SOA earlier than the CW component. In the absence of a control pulse, both pulse components experience the same effective medium as they propagate around the loop, and the data pulse is reflected back toward the input port as in a loop mirror. If a high-energy control pulse is injected into the loop, it saturates the SOA and changes its index of reft-action. If the control pulse is timed such that it arrives at the SOA after the CCW data pulse and before the CW pulse, a differential phase shift can be achieved between the two counterpropagating data pulses. This differential phase shift can be used to switch the data pulses to the output port. Figure 12 shows the schematic of the timing sequence of the CCW data pulse, the control pulse, the CW data pulse component, and the SOA gain change.
Fig. 12. Schematic of timing sequence of the control, CW data and CCW data pulses at the SOA and the resulting SOA gain change for optical switching.
2, § 3]
Interferometric SOA-based all-optical switches
11
The SOA gain for CCW and CW pulses is marked by "A" and "B" in fig. 12, respectively. In the presence of a control pulse, the CCW pulse will experience unsaturated gain of the SOA and the CW pulse will experience saturated gain. Therefore, the two components also experience a large phase difference. When the difference is n, the recombined pulse is transmitted out of the TOAD. Subsequent data pulses experience a gain and a refractive index that are slowly recovering; hence CW and CCW pulses have only a small phase difference and so are reflected on recombination at the coupler. A polarization or wavelength filter can be used at the output to discriminate the switched data signal from the control pulse. The output signal of the TOAD can be described by the following interferometric equation: PouM
= \Pin{t)\
G c w ( 0 + ^CCW(0 X
-
2 A / G C W ( 0 G^CCWCO cos
(3.12)
0cw(O - 0ccw(O
The input data signal is represented by P{n{t), and it is assumed that the interferometer is balanced so that there is initially no signal at the output. In the presence of control signals, the two components of the input signal (CW and CCW) that interfere within the interferometer experience a time-dependent gain, Gcw,ccw(0> ^^^ phase shift, 0cw,ccw(O? ^s they traverse the SOA. From eqs. (3.8) and (3.9), we have Gcw(0 = exp(/zcw(0), 0cw(O = -\ahc^{t\ and Gccw(0 = exp(/?ccw(0). 0ccw(O = -\ahccwit). Due to the asymmetric position of the SOA in the TOAD loop, the CW pulse will reach the SOA later than the CCW pulse by a time delay of 2AxsoA/
78
Ultra-fast all-optical switching in optical networks
[2, §3
4.0 • Phase change for CW pulses Phase change for CCW pulses
100
200
300
400
500
600
Time (ps) u-
c o
.
L
1
1
'
1
'
.
1
1
I
T^
•
1
•
T
8-
'(fi CO
"E (fi c
+-•
64-
Q
<
O
2-
(b) 0100
•
200
300
400
500
600
Time (ps) Fig. 13. (a) Calculated phase evolution for CCW and CW pulses; (b) corresponding TOAD transmission.
3.2.2, Working characteristics of TOAD Very narrow switching windows can be achieved by precisely controlHng the offset position of the SOA. Demultiplexing of a single channel from a 250 Gb/s data stream was demonstrated by Glesk, Sokoloff and Prucnal [1994a]. Errorfree demultiplexing from a continuous 160 Gb/s data stream was demonstrated by Suzuki, Iwatsuki, Nishi and Saruwatari [1994]. Since the size of the device depends only upon the length of the SOA and its offset from the center position in the loop, compact TOADs based on discrete components have been constructed with loop lengths of less than 1 meter. The TOAD is an ideal candidate for use in optical communication networks because it requires low control-pulse energy. To achieve frill switching, the differential phase shift between the counterpropagating data pulses should
2, § 3]
Interferometric SOA-based all-optical switches
79
approach n. For a 500|JLm long InGaAsP SOA («SOA = 3.3) at 1.55 ^m, the index change is on the order of 10~^. It has been estimated that the change in carrier density associated with this phase shift is on the order of 10^^cm~^ (Cotter, Manning, Blow, Elhs, Kelly, Nesset, Phillips, Poustie and Rogers [1999]), which is approximately a 10% change in the carrier density of an appropriately biased SOA in population inversion. Both theoretical calculations (Manning, Ellis, Poustie and Blow [1997], Kang, Glesk and Prucnal [1996]) and experimental measurements (Kang, Chang, Glesk and Prucnal [1996b]) have verified that a control pulse with 10% of the SOA saturation energy is sufficient to induce a JZ phase shift. Depending upon the SOA parameters, current bias, and wavelength, control-pulse energies of <500 fl are typically needed. This value can fluctuate by nearly 20% without significantly degrading the signal-to-noise ratio (SNR) at the switched output (Zhou, Kang, Glesk and Prucnal [1999]). Amplified spontaneous emission (ASE) noise, data pulse saturation, and component tolerances can also affect the SNR of the TOAD output (Zhou, Kang, Glesk and Prucnal [1999]). As compared to other forms of optical amplification, SOAs have relatively high noise figures, approaching 6 to 8 dB. However, by employing an optical bandpass filter, error-free operation of the demultiplexer was achieved (Deng, Runser, Toliver, Coldwell, Zhou, Glesk and Prucnal [1998]). Although sufficient data-pulse energy is needed for detection above the background noise, the maximum allowable data energy is constrained by the saturation characteristics of the SOA. Theoretical and experimental studies of the data signal saturation effect constrain the data energy to approximately not more than 5-10% of the control signal (Kang, Glesk and Prucnal [1996]). This rule, however, depends heavily upon the OTDM aggregate bit rate as well as the TOAD operating characteristics. The TOAD is robust to temperature variations and can be operated reliably without stabilization as data signals propagating in both directions around the loop experience the same effective medium. The architecture can also be adapted to accommodate other short, highly nonlinear materials in place of the SOA to operate the device with different wavelengths or to achieve performance enhancements. The TOAD device and its variations may prove to be the most practical approach to all-optical switching as they can be integrated using a variety of techniques that are discussed later in this section.
3.3. SOA-based Mach-Zehnder all-optical switch geometries Although the TOAD is based upon the Sagnac interferometer, other interfer-
80
Ultra-fast all-optical switching in optical networks
[2, § 3
•;
A
Input
Control
^ — — ( S O A K ^ — •
Output
Control 1 Input - ^ - ^ ^
j Control 2 - i - >
ym^SOk)--^
^
Filter |-> Output
^>^C-CS0A)---^^^^ ^ (b)
Fig. 14. All-optical switches with Mach-Zehnder geometries: (a) coUiding-pulse MachZehnder (CPMZ) switch; (b) symmetric Mach-Zehnder (SMZ) switch.
ometric configurations are possible using a similar operating principle. These architectures improve the integratability and performance of the device, although they may require active stabilization if constructed fi-om discrete components. Two variations of the switch in a Mach-Zehnder interferometer configuration are shown in fig. 14. The input signal is split 50:50 into two arms of the MachZehnder interferometer structure. Afi;er passing through the SOA the two signals recombine at the output port. In the absence of control signals, the two arms can be balanced so that the two components experience destructive interference and are rejected fi-om the output port. When control pulses are injected into the interferometer, a differential phase shift is briefly introduced between the two arms of the interferometer, causing an input data pulse to be switched to the output port. Similar to the TOAD, subsequent data pulses that pass through the switch see the slow recovery of both SOAs and are rejected. The difference between the two Mach-Zehnder geometries shown is in the propagation direction of control and data signals. In the colliding-pulse Mach-Zehnder (CPMZ) shown in fig. 14a, the data and control signals counterpropagate through the interferometer (Glesk, Chang, Kang, Prucnal and Boncek [1995]). As a result, a filter is not needed at the output to reject the control, and the control pulses can be coupled into the interferometer without introducing additional coupling losses. The nominal width of the switching window for the CPMZ is determined by the distance between the midpoints of the SOAs such that Twin = 2AxsoA/<^fiber-
2, §3]
Interferometric SOA-based all-optical switches
The other architecture, known as the symmetric Mach-Zehnder (SMZ), shown in fig. 14b, requires a filter at the output port to reject the control from the switched data signal since data and control signals co-propagate. Assuming the SOAs are positioned in the same relative location within the interferometer, the nominal switching window for the SMZ is determined by the temporal control pulse separation, A^cs? of Control 1 and Control 2 prior to entering the interferometer such that Twin = A^cs-
Although the nominal width of the switching window provides an estimate of the temporal width of the switching window, it does not account for the finite length of the SOAs. While the SOA length has little effect on the SMZ geometry, the minimum achievable switching windows for both the TOAD and CPMZ are constrained by the length of the SOAs (Kang, Chang, Glesk and Prucnal [1996a], Toliver, Runser, Glesk and Prucnal [2000]).
3.4. Experimental demonstrations of the TOAD and the Mach-Zehnder-based all-optical switches Here we describe the experimental demonstrations of the switching performance of the TOAD and the Mach-Zehnder-based all-optical switches (CPMZ and SMZ). All three switches were constructed from off-the-shelf discrete components. The SOA-based nonlinear element was an Alcatel 1901 SOA biased at approximately 100 mA. The switching windows were measured by using the scanning pump-probe apparatus shown in fig. 15. Polarization controllers (PC) were used internally within each switch in order to align the interferometer for the proper condition for interference. A 1.55 |U,m mode-locked erbiumdoped fiber laser (EDFL) was used to generate a continuous stream of 2-3 ps EDFA
Control input
All-optical
•At-
PC
EDFLW>
A^jtc!:^..
-^^^^^^-^ControU Data input
Mechanical vibrator
ODLi PC 000
Output
-q Data
u-] 4 — OSC
PC -£OCL^Control 2\
Fig. 15. Schematic of experimental setup for characterization of ultra-fast all-optical switches; OSC, oscilloscope.
82
Ultra-fast all-optical switching in optical networks
[2, § 3
optical pulses at a 10 GHz repetition rate. The pulse stream was amplified by an erbium-doped fiber amplifier (EDFA) and optically split into control signals and data signals for injection into the optical switch under test. For the SMZ switch configuration (which requires two control pulses) an optical delay line (ODL) was used to set the relative offset between the pulses to control the desired width of the switching window. In the TOAD and CPMZ configurations, an optical delay line inside the interferometer was used to change the SOA offset and set the switching window. For the SMZ and TOAD configurations, polarization controllers were used to set orthogonal polarization states for the input control and data pulses. A polarization filter was used at the output of the switch to separate the switched data pulses from the control pulses. A mechanical vibrator was used in the setup to periodically scan the data pulses over a 40 ps range with respect to the control pulses. The output of the system is a convolution of the data pulses with the transfer function of the switching window induced by the control pulse. This technique provides a means of rapidly characterizing the switching window. While the TOAD is based upon the inherently stable Sagnac interferometer, thermal variations in the optical fiber cause the output of the Mach-Zehnder switches to fluctuate slowly in time. By performing the scan at a rate faster than the thermal variations, switching windows of the fiber-based Mach-Zehnder geometries can be obtained without resorting to complex stabilization techniques. (Note that thermal variations do not significantly affect the stability of any of these interferometers if integrated devices with short optical pathlengths are used.) The switching window provides information regarding the shape, amplitude, and temporal width of the optical transfer function. This characterization is instrumental in determining the optical demultiplexing and sampling bandwidth of the switch. For the results described here, various pulse energies, widths, and repetition rates were used to demonstrate the effect these parameters have on the switching performance. For the TOAD windows shown in fig. 16a, the data and control pulses were set to energies of 5 and 20 fJ, respectively. The long rising edge of the switching window is due to the finite length of the SOA whereas the sharp falling edge is limited only by the widths of the data and control pulses. The switching window was found to decrease nearly linearly with decreasing SOA offset position within the loop. As the switching window is reduced further until the offset becomes less than the length of the SOA, a portion of the SOA straddles the midpoint of the loop. This decreases the available nonlinearity, resulting in a decreased contrast ratio and a decreased amplitude of the switching window. Additionally, at extremely small offsets, the width of the switching window does not decrease further since the finite widths of the data
3]
Interferometric SOA-based all-optical switches
10
83
20 30 Time (ps)
Fig. 16. Experimental measurements of temporal switching window for (a) TOAD, (b) CPMZ and (c) SMZ, with successively reduced window sizes.
and control pulses become the dominant limiting factors. This trend continued until the effective switching offset was 0 ps, when the switching window nearly vanished. The shortest switching window achieved with this experiment was about 3.8 ps at FWHM. The CPMZ switch was investigated under similar circumstances. In this case the data and control pulses were approximately 1.6ps in width with similar pulse energies as before. The results of these scans are shown in fig. 16b. In contrast to the TOAD, the counterpropagating geometry significantly increases both the rise times and the fall times of the window edges. This is primarily a fimction of the finite SOA lengths, which were approximately 500 (im. The architecture of the CPMZ limits its performance to minimum switching window widths of about 8 ps for this SOA length. The SMZ architecture is best suited for high-bandwidth applications. Due to its co-propagating nature, the switching window of the SMZ has the unique characteristic of both sharp rising edges and sharp falling edges, and therefore exhibits no dependence upon the length of the SOAs. Ultra-short pulses of approximately 500 fs at a 2.5-GHz repetition rate were used to evaluate the device. The data and control pulse energies were set to 1 6 0 and 200 fJ, respectively, at the input ports of the switch. The results of the experiment are
84
Ultra-fast all-optical switching in optical networks
[2, § 3
shown in fig. 16c. For A^cs > 6ps, the peak of the switching window flattens out, indicating that the data pulse is completely switched for the duration of the window. For smaller control-pulse separations, the amplitude gradually decreases due to the finite temporal widths of both the control and data pulses. The smallest detectable window was 1.5 ps. This performance indicates that the SMZ architecture may be suitable for demultiplexing fi-om a 660-Gb/s data stream. Reducing the optical switching window fiirther to sub-picosecond regimes may be challenging due to SOA gain compression (Tajima, Nakamura, Ueno, Sasaki, Sugimoto, Kato, Shimoda, Hatakeyama, Tamanuki and Sasaki [2000]). Further investigation and measurements of these dynamics are an important area of future research. The efficiency of all of these optical switches can vary depending upon the device operating characteristics and the geometry used. For the switching windows described in this section, the flat portion of the window amplitude can correspond to a switching efficiency of greater than 100% due to the gain imparted on the data signal by the actively biased optical amplifier. The SOA must be biased above the transparency point to achieve gain in the switched output. Experimental demonstrations have reported 6dB gain in the switched output for a TOAD window width of lOps (Deng, Runser, Toliver, Coldwell, Zhou, Glesk and Prucnal [1998]). As the switching window is reduced, the contrast and efficiency both decrease due to the finite pulse widths and SOA lengths used in the device. For the TOAD, a 4ps switching window used to demultiplex a single channel fi-om a 250Gb/s data stream achieved 14% efficiency at 1.3 fxm (Glesk, Sokoloff and Prucnal [1994a]). As SOA fabrication and design techniques improve, greater switching efficiencies for narrower switching windows are expected.
3.5. Cascaded TOAD all-optical switch As explained previously, the minimum width of the TOAD switching window is limited by the finite length of the SOA. If the offset of the SOA fi-om the center is decreased such that the SOA starts to straddle the center of the loop, the effective SOA length seen by the two counterpropagating pulses is reduced. The decrease in effective SOA length leads to a reduction in the contrast ratio of the TOAD switching and thus, an excess power penalty. The effective length of the SOA required for producing the relative n phase shift places a practical limitation on the width of the switching window of the TOAD to be greater than the propagation time of the pulse through the SOA. To overcome this
2, §3]
85
Interferometric SOA-based all-optical switches 10
0
5
10
time (ps)
0
5
time (ps)
Fig. 17. TOAD switching windows with SOA and control port at (a) same side, (b) different sides.
limitation imposed on the minimal achievable switching window a new method was developed and demonstrated. A narrower switching window was obtained by cascading two TOADs with SOAs on opposite sides of the fiber loop (Wang, Baby, Tong, Xu, Friedman, Runser, Glesk and Prucnal [2002]). This approach overcomes the limitation on the minimal switching window imposed by the finite length of the SOA. One characteristic of the TOAD switching window is that the rising and falling edges have different slopes (Sokoloff, Glesk, Prucnal and Boncek [1994], Toliver, Runser, Glesk and Prucnal [2000]). The slope of each edge is determined by the position of the SOA within the fiber loop with respect to the control port (see fig. 17). If the SOA is placed on the same side of the fiber loop as the control port then the rising edge of the switching window is very steep, limited only by the clock pulse width. The slope of the falling edge of the switching window is a result of the clock and the counterpropagating data pulses meeting inside the SOA and is thus related to the propagation time of the pulse through the SOA. If the SOA is placed on the side of the loop opposite to the control port, the two edges are interchanged and the falling edge of the switching window is much steeper than the rising edge. Figure 17 shows the shape of the switching window for SOAs on different sides of the loop. By cascading two TOADs with a time shift of 5 between their switching windows, the resulting transfer fiinction, called Cascade(r, (5), becomes the product of the two constituent ones with the time shift 8 taken into account: Cascade(^, S) = S W A ( 0 X SWB(^ - (5),
(3.13)
where S W A ( 0 and S W B ( 0 are the individual switching windows of the two cascaded TOADs. In such a configuration, one TOAD has the SOA and the control port on the same side of the loop, the other has them at opposite sides. Their switching
86
[2, §3
Ultra-fast all-optical switching in optical networks
Cascade (t, 6)
n
•nr\
n
r\
SOA
SOA
Filter f—•
Fig. 18. Principle of new optical switch (cascaded TOAD) based on overiap of two TOAD switching windows. 60
T
45
^
30
^
15
7
r
1
1
\ \
l
-3
1—
\
/
\ V'
/.-.-80 ,
-2
-6.51
1
- 1 0
'
•.
..
• V
1 2 time (ps)
, ^
'-' \ \ •.
\
3
\
\
4
5
6
•
Fig. 19. Simulated transfer function Cascade(^(5) resulting from cascading two TOADs with 8ps switching windows, for different delay offsets b.
windows are then placed such that the sharp edges overlap, as shown in fig. 18. This results in a switching window size limited only by the optical pulse width of the clock and data. To study this effect, simulations were done by Wang, Baby, Tong, Xu, Friedman, Runser, Glesk and Prucnal [2002] using a model for the gain and the phase changes developed by Kang, Chang, Glesk and Prucnal [1996a]. The simulations assumed Gaussian pulse shape for optical pulses, with pulse widths of 1 ps and 1.5 ps for the input clock and input data pulses, respectively. The SOA was assumed to be 500 ^im long with a recovery time of 200ps. Figure 19 shows the simulated resulting transfer function Cascade(^, &) (switching window) of two
2, §3]
87
Interferometric SOA-based all-optical switches
cascaded TOADs for different delay offsets d between two 8-ps wide TOAD switching windows. The switching window ampHtude remains fairly constant until the width is decreased to 1.4ps.
3.6. The Ultra-fast Nonlinear Interferometer (UNI) all-optical switch The UNI (Ultra-fast Nonlinear Interferometer), like the other interferometric switches mentioned above, is based on the gain saturation of SOA by high-energy control pulses. The UNI uses one SOA in a single-arm interferometer (SAI) structure as shown in fig. 20. SAI structures were originally developed for femtosecond pump-probe measurements of nonlinear processes in semiconductor waveguides. In the switches mentioned above, the temporal width of the switching window was determined by the relative time delay with which the two interfering data components pass through the saturated SOA. This relative time delay is determined either by the temporal offset between the control pulses, as in the SMZ geometry, or by a relative displacement between the SOAs in the two interferometric arms equivalent to a temporal offset between the data pulses, as in the TOAD and CPMZ geometries. In the UNI, a polarization-sensitive delay element is used to provide a temporal offset between the two orthogonal data polarizations. The principle of operation of the UNI is as follows. An input signal (data pulse) enters the switch through a polarization-sensitive optical isolator (PSI). Pulses are split by a polarization-sensitive delay [length of polarizationmaintaining (PM) birefringent fiber] into two orthogonal components, which will also separate them temporally. Due to the large difference in refractive indices for the orthogonally polarized modes in the birefi-ingent fiber, these two data components will be delayed fi-om each other by an offset determined by the length of the birefi-ingent fiber and by the degree of birefringence. PM Fiber PSI
Output Signal
Control
Fig. 20. Block diagram of UNI.
88
Ultra-fast all-optical switching in optical networks
[2, § 3
If fix and Hy are the mode indices for the orthogonal fiber modes, then the temporal offset (At) is given by At= \nx-ny\ - ,
(3.14)
where L is the length of the PM fiber and c is the velocity of light in vacuum. A typical birefiingent fiber has |«jc - «^| ^ 10~^. Thus, for an offset of a few picoseconds, a few meters of birefiingent fiber are required. Now, the two data signal components travel through the SOA into which high-energy control pulses are injected using the 50:50 coupler. If the control pulses are injected into the SOA in such a way that they arrive at the SOA in between the arrival of the two data pulse components, the two components acquire different phase shifts due to the saturation of the SOA by the highenergy control pulse. The two signal components are then re-timed to overlap in a second polarization-sensitive delay (another length of birefiingent fiber) and are subsequently combined interferometrically by use of a polarizer set at 45*" relative to the orthogonal signal polarizations. The control pulse is filtered out at the output of the device. A counterpropagating structure can be used to avoid the wavelength- or polarization-discriminating element at the output. Since the UNI is a single-arm interferometer, all the signals travel along the same path and are exposed to identical fluctuations in optical pathlength. Therefore, the device is stable and no active interferometric bias stabilization is necessary (Patel, Rauschenbach and Hall [1996]). The temporal width of the switching window is determined by the temporal offset produced by the polarization-sensitive delay element. Thus, changing the length of the birefiingent fiber can vary the width of the switching window of the UNI. The UNI has been used to demultiplex a lOGb/s data channel fi-om a 20Gb/s and a 40Gb/s stream (Patel, Rauschenbach and Hall [1996]). A simple extension of the UNI structure was attempted to fiirther improve the switching performance. This folded UNI geometry is shown in fig. 21 (Schubert, Diez, Berger, Ludwig, Feiste, Weber, Toptchiyski, Petermann and Krajinovic [2001]). Data pulses entering the switch pass through a polarization controller (PCI) which rotates the polarization such that the pulses pass through a polarization beam splitter to the PM fiber. The second polarization controller (PC2) aligns the polarization such that the orthogonal polarization components of pulses separated after passing through the PM fiber have equal intensity. Both data components pass through the SOA in the same direction. The third polarization controller (PC3) is used to rotate the polarization to ensure that the delay between the two pulses is reversed and that they are combined
2, § 3]
89
Interferometric SOA-based all-optical switches Control
PM Fiber Signal
^
PBS
, PC1
•
^
Demultiplexed Output ^
^
£1
WDM
cfe
Fig. 21. Folded UNI design proposed by Schubert, Diez, Berger, Ludwig, Feiste, Weber, Toptchiyski, Petermann and Krajinovic [2001].
interferometrically at the polarization beam splitter. The data pulse gets switched to the input port in the absence of the control pulse and to the output port in the presence of the control pulse. In comparison with the UNI geometry originally proposed by Patel, Rauschenbach and Hall [1996], the folded UNI structure is more stable. This is because the data pulses are split and recombined, including all random phase shifts caused by temperature fluctuations, through the same piece of fiber. In the original linear scheme, random phase shifts are not fiilly compensated as they might differ in the two PM fibers. Also, replacing a single piece of PM fiber can vary the duration of the switching window
3.7. Gain-transparent SOA-based all-optical switch In all of the interferometric switches using semiconductor optical amplifiers (SOA) that have been described in the previous section, optical control pulses are used to deplete the carriers in the SOA. The carrier depletion causes a gain change in the SOA and thus a change in the refractive index determined by the Kramers-Kronig relations. The phase change experienced by the data pulses due to the change in the refi*active index is used for interferometric switching. The temporal width of the switching window itself is determined by the relative time delay between the carrier depletions introduced in the interferometric arms. However, this "conventional" operation of the SOA-based interferometric switches has some disadvantages. The gain change that is created simultaneously with the phase change results in reduced extinction ratio for the demultiplexed data channel and an amplitude modulation of the data channels that are not
90
[2, §3
Ultra-fast all-optical switching in optical networks control control
\
phase change only
both gain and phase change
\ >
A (^im)
Negligible gain and phase dispersion data
11
/
->-
A (^im)
Fig. 22. "Conventional" and gain-transparent approach in operating SOA-based interferometric switches. Ax ,
GT SOA-Switch
1.3^mSOA Control Out
Control 1.3|im
Optical circulator
Data In IIII M
DeMuxed Data
^^
1.55^m
Fig. 23. Schematic diagram of GT-SOA switch.
demultiplexed. In addition, the data signal quality is degraded by the addition of amplified spontaneous emission (ASE) noise. To a large extent this can be eliminated by a dual-wavelength operation technique proposed by Diez, Ludwig and Weber [1999]. Here, the data signal is chosen at a wavelength that is far off the gain and ASE peaks of the SOA, while the control pulses are chosen at a wavelength close to the peak of the gain spectrum of the SOA (see fig. 22b). Experimental demonstration of this idea was demonstrated using TOAD/SLALOM switch architecture with the SOA gain peak and the controlsignal wavelength at 1.3 [xm and the data signal at 1.55 jJim (fig. 22b). This type of all-optical demultiplexer was named Gain-Transparent SOA switch (GT-SOA Switch) (Diez, Ludwig and Weber [1998, 1999]). A schematic diagram of the GT-SOA switch is presented in fig. 23. The wavelength combination used was determined solely by the availability of components. As the data wavelength is far away firom the gain peak of the SOA, the data signal experiences negligible amplitude change. However, the data signal at 1.55 fim still experiences a strong phase change due to the effect of the
2, § 3]
Interferometric SOA-based all-optical switches
91
control pulses at 1.3 |im. This strong phase change has also been used in all-optical wavelength conversion from 1.3 |im to 1.55 [xm (Lacey, Pendock and Tucker [1996]). Also, since the data wavelength is far away from the ASE maximum of the SOA, a very low amount of noise is added, resulting in a lower noise figure. Since in this dual-wavelength technique the data wavelength is far from the gain and ASE peaks of the SOA, the SOA is essentially transparent to the data signal (i.e., the energy of the data pulses is less than the band-gap energy of the material). Hence, this effect is called the gain-transparent (GT) effect. The gain-transparent operation of these SOA-based interferometric switches has several advantages over conventional operation. Due to the addition of less ASE noise at the data wavelength, the switch has a low noise figure. This, coupled with the enhanced switching contrast due to negligible gain change, results in better signal-to-noise ratio. Also, since the data and control pulses are widely separated in wavelength, low-loss WDM couplers can be used instead of fiber couplers to combine data and control signals. This further increases the signal-to-noise ratio of the switch. The negligible gain change also results in reduced intensity modulation of the transmitted data channels. This latter aspect is very significant in optical add/drop multiplexers where, irrespective of how the channels are added, it is necessary that the "drop" process does not affect the transmitted channels (Diez, Ludwig and Weber [1999]). In the "conventional" mode, since the SOA carrier population recovers slowly, the transmitted data channels "see" a transmittance function determined by the gain recovery of the SOA, resulting in intensity modulation. In the gain-transparent mode, there is no intensity modulation since there is no gain change at the data wavelength. Ideally, in the conventional operation of SOA-based interferometric switches, the population inversion of the SOA is not affected by the data pulses, and only the control pulses are able to provide carrier depletion sufficient for switching. However, this is true only for low-energy data pulses. At high energies, the data pulses deplete the carrier population in the SOA, resulting in switching even in the absence of a control pulse. This process is called "self-switching". In the GT operation of SOA-based interferometric switches the SOA is transparent to the data wavelength, hence the data pulses do not deplete the carrier population, and self-switching is not possible. Therefore, the switching performance of a switch in the gain-transparent mode is independent of data input power. The performance of a Mach-Zehnder all-optical switch operating in the gain-transparent mode was studied by Diez, Schubert, Ludwig, Ehrke, Feiste, Schmidt and Weber [2000]. The switch exhibited high linearity
92
Ultra-fast all-optical switching in optical networks
[2, § 3
of output data signal energy with input data signal energy over a range of 50 dB, limited only by the noise of the detection system. The transparency of the SOA to the data signal wavelength also results in low crosstalk since the effect of each data channel on the gain dynamics of the SOA, which in turn can affect the other data channels resulting in crosstalk, is negligible.
3.8. Performance enhancement of SOA-based all-optical interferometric switches and their integration on a single chip 3.8.1. Reduction of SOA recovery time SOA-based all-optical switches have attracted large research interest for applications in the areas of optical communication, ultra-fast all-optical switching and processing. They have demonstrated the capability of high-speed operation including all-optical demultiplexing of data streams approaching 1 Tb/s even though they rely on an active resonant nonlinearity with a long recovery time, typically between 50 ps and 1 ns. However, since the optical switching function is based on gain saturation in a SOA, the repetition rate of the demultiplexing operation is limited by the recovery time of the SOA. To increase the clocking speed of these devices, it is necessary to reduce the recovery time of the SOA. It has been demonstrated that SOA recovery time can be significantly reduced if an additional CW beam is injected directly into the SOA (Manning and Davies [1994]) during switch operation. It has been estimated that this method may enable the optical switch to function at clocking rates approaching 100 GHz (Manning, Davies, Cotter and Lucek [1994]). 3.8.2. Towards an integrated all-optical switch Photonic integration of SOA-based switches is essential for practical, highperformance all-optical switches for commercial systems. Various research groups have reported integration of all-optical switches based on the Sagnac, Mach-Zehnder, and Michelson interferometers, using both monolithic and hybrid technologies. A monolithically integrated Sagnac configuration of the all-optical interferometric switch was used for demultiplexing from 20Gb/s to lOGb/s and 20 Gb/s to 5 Gb/s by Jahn, Agrawal, Pieper, Ehrke, Franke, Furst and Weinert [1996]. Also, both the CPMZ and SMZ geometries have been integrated
2, § 4]
All-optical switches based on passive waveguides
93
and subsequently demonstrated as high-speed all-optical demultiplexers (Hess, Caraccia-Gross, Vogt, Gamper, Besse, Duelk, Gini, Melchior, Mikkelsen, Vaa, Jepsen, Stubkjaer and Bouchoule [1998], Wolfson, Kloch, Fjelde, Janz, Dagens and Renaud [2000], Studenkov, Gokhale, Wei, Lin, Glesk, Prucnal and Forrest [2001]). An all-optical switch based on the Michelson interferometer was integrated and used to demonstrate demultiplexing from 20 Gb/s to 5 Gb/s by Mikkelsen, Vaa, Storkfelt, Durhuus, Joergensen, Pedersen, Danielsen, Stubkjaer, Gustavsson and van Berlo [1995]. The best reported performance for integrated all-optical demultiplexers has been achieved using the Mach-Zehnder configuration. Both the CPMZ fabricated by the Heinrich Hertz Institute (HHI) in Germany and the SMZ fabricated by Alcatel in France have been used to demultiplex optical data from 40 Gb/s down to 10 Gb/s (Jahn, Agrawal, Arbert, Ehrke and Franke [1995], Wolfson, Kloch, Fjelde, Janz, Dagens and Renaud [2000]). An integrated CPMZ demultiplexer with a switching window of approximately 20 ps was also demonstrated (Studenkov, Gokhale, Wei, Lin, Glesk, Prucnal and Forrest [2001]).
§ 4. All-optical switches based on passive waveguides As mentioned earlier, the viability of NOLM-based switches was limited by the large lengths of fiber required due to the weak nonlinearity in fiber. Other passive waveguides, made of semiconductor material or lithium niobate, have shown relatively large nonlinearity. Optical switches based on such waveguides have been actively researched due to their integration compatibility, potentially low switching energies, and their low amounts of noise (as compared to active SOAbased switches). Tajima, Nakamura, Hamao and Sugimoto [1994] demonstrated optical switching using the resonant nonlinear band-filling effect in a GaAs waveguide. The control signal wavelength was set slightly below or at the band edge to efficiently generate photocarriers in the waveguide. Data signal wavelength was set a few tens of nanometers below the band edge to achieve both a high signal transmittance and a relatively large change in nonlinear refractive index. Like the gain saturation and recovery in the SOA, the band-filling effect is highly efficient but has a slow relaxation process. Different interferometric configurations have also been proposed and demonstrated for ultra-fast switching. A symmetric Mach-Zehnder structure using nonlinear waveguides is shown in fig. 24. The nonlinear element used typically is a strip-loaded GaAs waveguide
94
Ultra-fast all-optical switching in optical networks
[2, § 5
Control 1
Signal Input
Nonlinear portion 1
Signal output ^^/^ oab
^^
iMM»mi#
Control 2
^^
%^^
—•
Nonlinear portion 2
Fig. 24. Symmetric Mach-Zehnder all-optical switch using nonlinear passive waveguide.
consisting of a 0.5|j-m thick GaAs core and Alj^Gai-jcAs (x = 0.1) cladding (Nakamura, Tajima and Sugimoto [1994]). Other symmetric Mach-Zehnder configurations based on the band-filling effect have also been demonstrated: a polarization-discriminating symmetric Mach-Zehnder (PD-SMZ) all-optical switch (Tajima, Nakamura and Sugimoto [1995]), and a delayed interference signal wavelength converter (DISC) (Ueno, Nakamura, Tajima and Kitamura [1998]). A 200 fs switching window was demonstrated with a PD-SMZ optical switch using a passive InGaAsP bulk waveguide. The application potential for these switches is limited since the data and control pulse wavelengths have to be correlated for the generation of efficient band-filling effects. Lithium niobate is another possible candidate as a nonlinear element for building all-optical switches. Some initial results have shown potentially efficient switching with very low control energy in an annealed proton-exchanged waveguide formed in periodically poled lithium niobate (PPLN) (Parameswaran, Fujimura, Chou and Fejer [2000]). The switching operation in these waveguides is based on sum-frequency mixing using second-order nonlinear interactions. The nonlinear interactions in lithium niobate offer the advantages of ultra-fast response, potentially low switching energy, wide wavelength range of operation and integration compatibility.
§ 5. Demonstrations of all-optical switching in networks In the previous sections we described several switching devices and nonlinear materials which could be suitable for fast all-optical switching in future alloptical networks. Although there is presently no consensus on the exact approach towards the "next generation" of all-optical networks, a variety of promising proposals exist. One approach includes the utilization of OTDM technology, introduced in § 1, to increase the single-wavelength transmission
2, § 5]
Demonstrations of all-optical switching in networks
95
rate of WDM systems as well as to provide ultra-fast all-optical switching platforms. WDM transmission typically uses the non-retum-to-zero (NRZ) data format. OTDM, on the other hand, uses narrow pulses in the RZ format. Therefore, if OTDM systems are used to switch WDM channels, high-speed all-optical data format converters are needed to translate between the different types of optical data formats. Fourteen parallel wavelength channels were simultaneously switched by Mathason, Shi, Nitta, Alphonse, Abeles, Connolly and Delfyett [1999] using a switch based on the same principle as the TOAD, thus providing an interface for OTDM switching. Sub-nanosecond switching between wavelengths was demonstrated using a TOAD and an OTDM channel selector by converting multiple-wavelength optical pulses into time-interleaved OTDM channels (Wang, Baby, Xu, Glesk and Prucnal [2002]). Devices other than SOAs have also been used to demonstrate fast switching and WDM/OTDM conversion. A 80Gb/s data stream was demultiplexed into lOGb/s channels using XPM in optical fiber (Olsson and Blumenthal [2001]). The same all-optical switch was also used to demonstrate wavelength-selective multicasting (Rau, Olsson and Blumenthal [2001]), thus enabling the transmitter to send the same data into multiple selective WDM channels. In addition to the OTDM/WDM hybrid approach, another type of optical network which employs ultra-fast all-optical switching is optical packet switching. While the Internet traffic that occupies a large portion of today's network is packet based, the optical networks that form the physical layer are circuit based. Optical packet-switching networks are needed to reduce the inefficiency associated with the transmission of packet-based data over a circuit-based optical network. However, packet switching in the optical domain continues to be a challenging problem since removing and reading the packet headers require switching and synchronization on a nanosecond or sub-nanosecond time scale. One common approach to ease the transition into optical packet networks involves attaching a label to the packet that can be used for simple forwarding and switching instructions. These optical labels are then stripped and replaced when the packet leaves the router to be forwarded to its next destination in the network. Typically, a small portion of the optical packet's power is tapped so that the appropriate information can be read electronically for processing information (Blumenthal [2001a,b]). Using an integrated SOA-based Michelson interferometric all-optical switch, optical label switching at lOGb/s was demonstrated by Fjelde, Kloch, Wolfson, Dagens, Coquelin, Guillemot, Gaborit, Poingt and Renaud [2001]. The next several sections describe in detail the use of ultra-high-speed alloptical switches in several applications.
96
Ultra-fast all-optical switching in optical networks
[2, §5
5.1. Demonstrations of all-optical clock extraction in self-clocking OTDM network As seen earlier, all-optical demultiplexers in OTDM networks require a frame clock for their operation. Conventional electronic clock-recovery schemes may not have sufficient speed for ultra-fast OTDM networks with throughputs of hundreds of gigabits per second. All-optical timing-recovery schemes suitable for ultra-fast systems have been demonstrated either by injection locking of a mode-locked laser (Smith and Lucek [1992]) or by using an optical phaselocked loop (Kamatani and Kawanishi [1996]). In the next section we describe an alternative technique for clock extraction known as self-clocking (Perrier and Prucnal [1989], Deng, Glesk, Kang and Prucnal [1997]). 5.1.1. Clock/data separation using all-optical thresholder in self-clocked OTDM system In self-clocking OTDM networks, several different methods have been proposed to distinguish between the clock and data pulses. The simple scheme in fig. 25a assigns different wavelengths to the clock and data pulses. The clock and data pulses can be separated at the receiver end by simple wavelength filters or wavelength routers. However, this method is affected by clock skew caused due
I Hi i
Wavelength filter / router
(a)
i
•^
±LL • •
PBS
^1
Clock Data
II Clock
-rrrr <")
* i^il
Data
.
Thresholding Device
1 * "
* ^
Clock Data
(c) Fig. 25. Self-clocking schemes using (a) polarization, (b) wavelength, (c) intensity discrimination.
2, § 5]
Demonstrations of all-optical switching in networks
97
to fiber dispersion, which degrades its performance especially at high bit rates. A second scheme, shown in fig. 25b, uses orthogonal polarization states for clock and data pulses to distinguish the two (Glesk and Prucnal [1995]). However, proper operation of this scheme requires carefiil control of the polarization using either expensive PM fibers or other methods throughout the network to maintain the orthogonal polarizations for clock and data pulses. Another promising method, represented in fig. 25c, distinguishes between the clock and data pulses by their difference in intensity. The high-intensity clock pulse occupies a designated time slot in the OTDM data frame or packet. At the receiver end, an all-optical thresholder is required to extract the clock pulses. The next subsection describes the operation of an all-optical SOA-based thresholder.
5.1.2. Ultra-fast all-optical thresholder Figure 26 shows the schematic of an ultra-fast all-optical thresholder demonstrated by Deng, Glesk, Kang and Prucnal [1997]. The switch is based on an unbalanced {K ^ 0.5) Sagnac interferometer with a SOA placed asymmetrically within the loop. The input coupler splits the incoming pulses into two components, CW and CCW, of different intensities. Both components of the lower-energy data pulses do not have sufficient energy to saturate the SOA. Hence, they both acquire the same phase change as they traverse the SOA. Due to this zero differential phase shift, they are reflected back at the coupler. The weaker component of the high-energy clock pulses, CW in the case of fig. 26, passes through the SOA without saturating the SOA. Due to the asymmetric position of the SOA in the loop, the stronger, CCW, component enters the SOA a short time later, and causes gain saturation. Therefore, the
Variable 2x2 coupler Input
Transmitted port (Clock) Reflected port (Data)
Fig. 26. Ultra-fast all-optical thresholder for clock and data separation.
98
Ultra-fast all-optical switching in optical networks
[2, § 5
CCW clock pulse component experiences a phase change different from the CW clock pulse component. This differential phase shift causes most of the clock pulse to be directed to the transmitted port at the coupler. By properly choosing the splitting ratio K of the coupler, different gains for the two counterpropagating pulses can be balanced by the intensity difference caused by the uneven splitting at the coupler, thus enhancing the switched output of the device. It was calculated and experimentally confirmed that for the SOA used in the experiment, the optimal splitting ratio is AT = 0.2 (Deng, Glesk, Kang and Prucnal [1997]). To ensure proper operation of the thresholder, the SOA is positioned in the loop in such a way that the weaker pulse arrives at the SOA before the stronger pulse. As shown in fig. 26, the SOA is asymmetrically placed toward the "weaker" branch (K < 0.5) of the coupler. To avoid the data pulses, which follow the clock pulses, to be switched out to the transmitted port, they must arrive at the SOA after the switching window created by the clock pulse is closed. To ensure this, the offset AJC of the SOA must satisfy the following requirement: 2AxsoA/
2, §5]
99
Demonstrations of all-optical switching in networks Clock DeMux; TOAD ] - • Out
Clock & DATA Separation SOA
Clock Data
II
Fig. 27. Experimental setup for clock/data separation using the ultra-fast all-optical thresholder. MOD, modulator, PBS, polarization beam splitter, PC, polarization controller. self-clocked cell 10 ns
H
^
1 2 1
ClockJ
Data
w^ JX
4-^
2 ^ 1 10 ps 100 ps
Fig. 28. Timing diagram for clock and data in the self-clocked cell. The cell contains a clock pulse and 2 aggregated TDM channels (CH1,CH2) at lOOGb/s in a 10ns time frame.
separation is determined by the driving current of the SOA and the spHtting ratio of the unbalanced loop mirror. After clock and data separation, the transmitted signal, comprised of the clock signal, was sent to the control port of the TOAD. The reflected signal from the thresholding device was used as the data input to the TOAD as shown in the bottom part of fig. 27. The extracted clock signal was temporally aligned to ensure the demultiplexing of time slot 2 fi-om the OTDM frame. The demultiplexed data channel 2 with a '1010- • •' pattern at 100MHz, is shown
100
Ultra-fast all-optical switching in optical networks
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in fig. 29. The pattern is the same as the modulation pattern encoded initially in time slot 2. The described results have demonstrated good performance and cascadability of the TOAD and the thresholding device. 5.2. All-optical demultiplexing and packet routing using TOAD switches The use of all-optical switching elements could provide novel solutions to alleviate the electronic bottleneck imposed on electronic/optoelectronic routing switches. Although the existing architectures largely address the rapid switching capability of OTDM systems, the interconnects still rely on some combination of electronic/optoelectronic approach. Maximum throughput with minimum delay can only be achieved in transparent all-optical networks, where all-optical address recognition will directly trigger and control an all-optical packet-routing switch. This approach will eliminate the need for optoelectronic conversion and electronic processing and will achieve the minimum latency. By combining all-optical packet-header recognition with an all-optical packetrouting switch, a 250Gb/s transparent, self-routing all-optical node was constructed and demonstrated by Glesk, Sokoloffand Prucnal [1994b,c] and Glesk, Kang and Prucnal [1997a,b]. The schematic design of a 1x2 all-optical selfrouting optical node is shown in fig. 30. OTDM packets enter the input port where a polarization splitter (PS) is used to separate the optical clock fi*om the data. The first TOAD (TOAD 1), having a narrow window (<4ps) is set to ail-optically demultiplex a single address bit fi-om the incoming optical packet header. This demultiplexed address bit is then used to control a second TOAD (TOAD 2), configured as an all-optical routing switch by having its switching window set to match the length of incoming packets. If bit ' 1 ' is
2, § 5 ]
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detected in the packet address header by TOAD 1 the packet is routed to the first output port (OUT 1) of this packet-routing switch. When the address bit is '0', TOAD 2 is not triggered and the packet is reflected to the second output port (OUT 2) of TOAD 2. This method allows an all-optical node to transparently route packets to one of the two output ports simply by reading a single bit from the optical address header of incoming packets. Synchronization of the switch is achieved by using a self-clocking scheme where the optical clock propagates with the packet in an orthogonal polarization. To account for the delay caused by the all-optical address processing, a short fiber length is used to buffer the packet prior to entering the input port of TOAD 2. A SOA is used at the output of TOAD 1 to amplify the demultiplexed address pulse to a level sufficient to trigger the switching operation of TOAD 2. An optical isolator (01) is inserted between TOAD 1 and the in-line SOA to reduce optical feedback within the system. Cascading multiple 1x2 optical nodes enables the construction of large-scale, Banyan-type transparent routing networks. The offset position for the SOA in TOAD 1, which performs the address recognition, is given by AXTOADI = ^^ * <^fiber, where 1/r = 250Gb/s is the bit rate of the incoming data. The switching window of TOAD 2 is much wider than that of TOAD 1 so that it can accommodate an entire optical packet. The offset position for the SOA in TOAD 2 is set to AXTOAD2 = \T - Cfiber, where T = Nx, with A^ the number of bits in the packet. For the experimental demonstration, 1/r = 250Gb/s packets were generated from 1-ps pulses emitted from a NdrYLF laser followed by a compression stage. Figure 31 shows the timing diagram and experimental detection of the fourbit test packets used in the experiment. The first bit is a clock pulse polarized orthogonally to the subsequent data pulses in the packet. A fixed delay afi;er the polarization splitter at each node was used to align the clock with the appropriate address bit in the header. In this fashion, a packet will flow through the network
102
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and each node will read a unique bit from the header to route the packet to its destination. In the experiment, a delay of 2r was used so that address bit 2 routes the packet through the node. This bit was modulated with a T or a '0' to test the operation of the 1x2 all-optical routing switch. Since these bits are spaced at an interval of 4ps, a bandwidth-limited detector cannot distinguish the individual bits of the header. As a result, the two input packets appear as "triple height" and "double height" in fig. 32b for header patterns '111' and
2, § 5]
Demonstrations of all-optical switching in networks
103
' l O r respectively. The timing diagram and results of the routing experiment are shown in fig. 32a. Packets with a ' 1' at header bit position 2 were switched by TOAD 2 to OUT 1, whereas packets with a '0' in bit position 2 were reflected by TOAD 2 and coupled to OUT 2. The "triple height" packets were correctly switched to OUT 1, and the "double height" packets were reflected to OUT 2. The two TOADs in the serial configuration provide the key fiinctionality for the all-optical self-routing switch. While TOAD 1 uses a switching window that is much shorter in duration than the SOA recovery time, TOAD 2, which performs the all-optical packet-routing operation, has a window size long enough to pass the entire optical packet. For a 1000-bit packet at a bit rate of 250Gb/s, the duration of the packet is 4 ns, which is approximately ten times the typical recovery time of the SOA. As the SOA recovers, the amplitude of the switching window decays exponentially for such long windows (Toliver, Runser, Glesk and Prucnal [2000]). This may adversely affect the individual bits of the packet payload and may cause an amplitude reduction fi-om the beginning to the end of the packet. To accommodate long packets, other techniques may be applied to maintain a flat, uniform switching window for the duration of the packet. One scheme involves clocking TOAD 2 with several optical control pulses per packet duration. Clocking the optical switch every lOOps, for instance, will maintain an open and relatively flat switching window for the duration of the packet. Another approach is to modulate the electrical bias of the SOA. The current bias of the SOA can be decreased just after the switching event has been initiated to prevent a rapid recovery. This technique will permit the switching window of TOAD 2 to remain open for a longer time and keep its amplitude more uniform.
5.3. Demonstration of GT-SOA switch in hybrid WDM/OTDM system Present-day optical networks use DWDM techniques to increase transmission bandwidth in optical fiber. However, the large number of wavelengths involved in DWDM systems makes network management very difficult and costly. Hence, it is expected that the next generation of optical networks will use both WDM and OTDM as complementary techniques for low-cost efficient use of the huge fiber bandwidth. The development of such WDM/OTDM-based networks, however, also necessitates the development of ultra-fast all-optical switches, which can demultiplex different time channelsfi-ommany wavelengths simultaneously. This implies that the switching performance of the optical switches has to be uniform for a broad wavelength range of the data signal.
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As mentioned above, the GT-SOA switch involves the choice of data wavelengths far away from the gain and ASE peak of the SOA. This also implies that the signal wavelength falls in a band with very low gain and phase dispersion as shown in fig. 22b. This results in uniform switching window performance over a wide range of signal wavelengths as compared to that of switches in conventional operation. This was demonstrated over a wide signal wavelength range of 160mn using the GT-SOA switch by Diez, Ludwig and Weber [1999]. The GT-SOA switch was also used as an all-optical demultiplexer in a 640 Gb/s WDM/OTDM experiment involving eight wavelengths, each operating at data rates of 80 Gb/s. The block diagram of the experimental setup is shown in fig. 33. Eight WDM channels were generated by four tunable mode-locked semiconductor lasers, producing pulses of width 1.3 ps, along with a spectral slicing technique using an arrayed waveguide grating (AWG). The eight WDM channels obtained were 2nm apart, each with a pulse width of about 4ps. Each of the eight WDM pulse trains was synchronized using delay lines and modulated using the data stream at 10 Gb/s. Each of these 10 Gb/s channels was then time-domain multiplexed using a fiber delay line multiplexer. Thus, each WDM channel had a bit rate of 80 Gb/s giving an aggregate bit rate of 640 Gb/s. This data stream was then transmitted over 50 m of fiber before the receiver. The receiver consisted of two stages. In the first stage, all eight WDM channels were demultiplexed in the time domain using the GT-SOA switch. A polarization-insensitive 1.3 |im multiple-quantum-well SOA with an unsaturated fiber-to-fiber gain of 30 dB, at 1310nm and 400 mA injection current, was used in the switch. The second stage of the receiver contained a tunable wavelength filter with a bandwidth of 1 nm for separation of the eight wavelength channels. Due to the high optical bandwidth and high linearity of the GT switch, all eight WDM channels could be demultiplexed simultaneously in the time domain. Measurements of bit error rates were made for all the time slots of one wavelength and for all the wavelengths for one time slot. The measurements demonstrated a variation in receiver sensitivity of 2dB between different time slots and wavelengths (Diez, Ludwig and Weber [1998]).
5.4. High-speed OTDM-based interconnect With the growing capacity of the Internet, there is a great demand for replacing the standard electronically switched backplane routers by optical interconnects.
106
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Although DWDM has been used to support large aggregate traffic bandwidths, routing functions using DWDM will involve complicated dense wavelength conversions. Also, current DWDM laser and filter tuning techniques rely on slow technologies, which increase the hardware latency, thus decreasing the effective network bandwidth. Recent experiments have proved OTDM to be capable of fulfilling the switching and routing needs of future networks. Channel access in OTDM-based routers can be achieved using time-slot tuners and all-optical demultiplexers. This section describes the demonstration of a 100-Gb/s OTDMbased switched interconnect using a broadcast star architecture (Deng, Runser, Toliver, Coldwell, Zhou, Glesk and Prucnal [1998]). Figure 34 shows the schematic of the OTDM-based interconnect. The core structure is based on a broadcasting star architecture embedded in two layers of network interface unit (NIU): the optical and electronic layers. At the outside layer, each port of SLU NxN system contains an electronic NIU that interfaces with the incoming high-speed signal through a network interface card (NIC). The optical NIU consists of a tunable transmitter and a TOAD-based receiver, as shown in fig. 35. The Data OUT port connects the node to the central star coupler which broadcasts the data to the desired time-slot channels from each node. Meanwhile, the Data IN port receives the aggregate information that contains the bit-interleaved ultra-fast OTDM frame. To achieve global synchronization, picosecond pulses from a single mode-locked fiber laser source were amplified and distributed to the individual nodes via 1 x A^ splitters. The optical NIU consists of two basic modules, the optical transmitter and receiver. The optical transmitter is composed of a Gated Time-Slot Tuner (GTST)
Demonstrations of all-optical switching in networks
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that consists of cascaded feed-forward fiber delay lattices sandwiched between two EO modulators (Deng, Kang, Glesk and Prucnal [1997]). Each node can transmit data to a desired time slot by configuring the time delay of the data timeslot tuner in the transmitter. The optical receiver module (Rx) utilizes the TOAD to demultiplex the desired channel from the aggregated OTDM. By placing a GTST on the optical clock input to the TOAD in the receiver, each node can receive any channel by aligning the optical clock pulse to demultiplex the desired bitfi*omthe time frame. The overall node receiver consists of a TOAD-based all-optical demultiplexer and GTST in the optical NIU, and control boards and receiver electronics in the electronic NIU. The receiver demultiplexes a single data channel fi'om the lOOGb/s aggregated OTDM fi-ame. The switching window size of the TOAD was set to be 10 ps. Tuning between the OTDM channels at the receiver was achieved using a GTST on the optical clock port of the TOAD. To demonstrate the lOOGb/s demultiplexing fiinctionality of the network, an OTDM frame was created by multiplexing data from four unique nodes: Chan 1 and Chan 3 had a fixed pattern, Chan 2 had a pseudo-random pattern, and Chan 4 had an all-'O' pattern. By tuning the GTST on the clock port of the TOAD, each channel was distinctly resolved on a bandwidth-limited oscilloscope, as shown in fig. 36 (the fourth time slot is not shown as it contains '0' only). A bit error rate of 10~^^ was achieved in the experimental measurement. In this network demonstration, the GTST and driver board electronics were capable of tuning to any of the 16 times slots in the network with an average latency of 1.6 ns. This architecture thus provides very high scalability, rapid reconfigurability with high throughput of lOOGb/s, and full connectivity
108
Ultra-fast all-optical switching in optical networks
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Time [30ps/div] Fig. 36. Demuliplexed TOAD output eye diagram at the node receivers as seen on a bandwidthlimited oscilloscope.
between the different nodes. Hence, this architecture has the potential to be used as a switching fabric in the backplane of an enterprise switch or backbone optical router.
5.5. Demonstration of photonic packet switching in shufflenet network The proliferation of data networks has raised the importance of advancing beyond circuit-switched optical networks to support packet switching. To effectively utilize the fiber bandwidth, it is necessary to build transparent optical networks (TON) which eliminate the need for electro-optic conversion of packets for their routing. This type of data communication, where packets are routed in an optically transparent manner, is referred to as photonic packet switching. This section discusses the demonstration of an ultra-highcapacity packet switch, capable of processing lOOGb/s packets, exploiting all-optical switching technology developed at Princeton University. Use of OTDM techniques avoids the complicated wavelength-conversion technology required for photonic packet switching with WDM technology. The demonstrated optical node is capable of extracting the address bits fi-om a packet header and performing packet routing at lOOGb/s (Toliver, Glesk, Runser, Deng, Yu and Prucnal [1998]). In order to minimize the complexity of the routing protocols and to simplify the routing hardware, the testbed was based on a regular distributed architecture known as ShufileNet (Stone [1971]), which is shown at left in fig. 37. A schematic of the network node architecture is shown at right in fig. 37. The network node consists of five primary subsystems: (1) an optical packet generation and compression subsystem to interface the node data rate to the higher OTDM-network data rate (Deng, Kang, Glesk, Prucnal and Shin [1997]); (2) an all-optical header processor to extract routing information from
2, § 5 ]
109
Demonstrations of all-optical switching in networks
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the incoming packets; (3) an electronic routing controller to set the state of the routing switch; (4) an optical space switch to route the packets to the appropriate output port, and finally (5) a packet-detection subsystem for decompressing received packets (Toliver, Deng, Glesk and Prucnal [1999]). A short fiber delay buffer is used to store the payload while the routing controller sets the state of the routing switch. Although electronic circuitry is used for routing, the payload remains in the optical domain throughout the routing process. The latency of photonic packet routing can be significantly reduced by keeping the number of address bits processed at each node to a minimum. In this case, the routing protocol developed for the POND network only requires two binary control bits in order to determine whether the packet should be directed to port Out 0 or port Out 1, or received at the Rx port (Seo, Yu and Prucnal [1997]). The routing header that is appended to the front of a packet consists of a sequence of four such routing instructions, for a total of eight routing bits in the case of the 8-node ShuffleNet. Since this header is encoded at the same 100-Gb/s data rate as the packet payload, an array of two TOADs is used in the all-optical header processor to extract the routing control bits as the packet enters the node. For example, to route a packet from Node 0 to Node 7 in the POND network, a possible path is through intermediate Nodes 4 and 1. The packet output routing sequence for the path 0, 4, 1,7 should be Out 0, Out 1, Out 1, and Rx, respectively. The eight-bit header address sequence corresponding
110
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Ultra-fast all-optical switching in optical networks
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to this path is {00}, {10}, {10} and {11} (Toliver, Glesk, Runser, Deng, Yu and Prucnal [1998]). A single node of the POND network was constructed to serve as a testbed for evaluating the key optical technologies required for implementing an entire network. To demonstrate the functional operation of the testbed, a sequence of 100-Gb/s packets was injected into the routing node to simulate the path through Nodes 0, 4, 1, and 7. Since only one node was constructed, the physical network node had a dynamic virtual address that was updated for each packet detected to reflect the simulated path through the network. To test the entire routing path from Node 0 to Node 7, a sequence of four 100-Gb/s packets, separated by 160 ns, was injected into the node. The demultiplexed header bits are shown in fig. 38a. The waveforms, TOAD 0 and TOAD 1, are the TOAD output signals as measured on a bandwidth-limited photodetector. These outputs, after digital thresholding, are displayed in the lower traces, Thr 0 and Thr 1. As each packet enters the node, a pair of header bits fi:om a specific routing group is demultiplexed in parallel by TOAD 0 and TOAD 1. In fig. 38a, the address sequence is correctly demultiplexed as {00}, {10}, {10}, and {11}. For example, both TOAD 0 and TOAD 1 demultiplex a ' 1 ' from the header of Packet 4 and read the address sequence {11}. These control bits are sent to the routing controller to set the state of a 4x4 lithium niobate space switch. In this demonstration, the routing latency was approximately 20 ns, limited by the clock speed of the programmable EPLD used as the routing controller (Yu, Toliver, Runser, Deng, Zhou, Glesk and Prucnal [1998]). By replacing this chip with a high-performance application-specific integrated circuit (ASIC), the routing controller decision time may be reduced further. The signals detected at the three outputs of the routing switch are shown
2, § 6]
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in fig. 38b. Note that each 100-Gb/s packet appears as a single pulse due to bandwidth-limited detection. However, it can be seen that the packets are routed correctly according to the routing instructions in the header. To implement this OTDM-based packet switching system on a larger scale, optical packet synchronization will be required to synchronize packet routing throughout the network. Self-clocking schemes, as described in §5.1, can achieve local bit-level synchronization despite the asynchronous arrival of highspeed optical TDM packets. In addition to the self-clocking schemes based on wavelength, amplitude, and polarization discrimination, already discussed in § 5.1, time-slot discrimination was demonstrated by Toliver, Glesk and Prucnal [2000].
§ 6. Conclusion As discussed in this chapter, rapid advances in DWDM technology, such as the development of erbium-doped fiber amplifiers, have resulted in the deployment of commercial multi-wavelength optical networks with aggregate throughputs exceeding terabits per second. As a result of these advances, there is a growing discrepancy between the electronic data rate generated by individual customers and the aggregate rate of data transported in optical networks. This discrepancy, known as "electronic bottleneck", is most apparent at points in the network where it is necessary to add/drop, switch or process the optical data. Using all-optical technology to process the data can provide performance benefits in terms of speed, and in some cases, parallelism. This is because in some cases the data can be processed efficiently in optical form by handling only a few bits of information, while the bulk of the data remains untouched. In contrast, if it is not necessary to process all of the data, converting it all to electronic form before processing can be quite costly, inefficient and unnecessary. All-optical switching is an essential technology for building fixture ultra-fast optical networks. Any advancement in this area is expected to have a major impact on the entire network performance. With continued efforts by academia and industry, significant progress is being made toward building ultra-fast alloptical integrated switches for different applications in current optical networks. In this chapter, we have reviewed and discussed several approaches to building ultra-fast all-optical switches for different applications, in which the data, which remains in optical form while passing through a switch, is controlled via an optical nonlinearity, using an optical control signal. A brief summary of some
112
Ultra-fast all-optical switching in optical networks
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of the approaches discussed is given in table 1, including switches based on nonlinearities in optical fiber, SOAs, and passive waveguides. Although speed is one of the most important attributes of these all-optical switches, with demonstrated speeds as fast as 640Gb/s with a nonlinear optical loop mirror, there are several other equally important characteristics that will determine whether any of these technologies will be practical to use in deployed networks. Aside from the generally high cost of these technologies (which is likely to decrease as these technologies mature), other considerations that will determine their feasibility include their size, the optical switching energy required, the sensitivity to polarization and environmental fluctuations, cascadability and scalability. For example, it is unlikely that a single gate requiring a long length of optical fiber will be preferred over a semiconductor device that can be integrated in arrays and manufactured in large volumes. It is also less practical to implement a switch requiring a large optical control energy than a switch requiring an optical control energy that is comparable to the sensitivity of a typical optoelectronic receiver. Fully fimctional optical switches (Glesk, Runser and Prucnal [2001]) in networks will require switching in both space, time and wavelength. While wavelength-domain techniques are now the dominant approach to multiplexing data, time-domain switching, the subject of this chapter, continues to outperform wavelength switching in terms of speed, switching energy, latency, and scalability. Thus, in applications that require scalable, rapidly reconfigurable switching, optical time-domain switching can provide superior performance compared to space and wavelength switching. As discussed in this chapter and illustrated in table 1, the class of SOA-based all-optical switches provide superior performance, feasibility and polarization independence. If they operate based on the differential onset of a fast optical nonlinearity (for example, the TOAD), they can operate at ultra-high speeds because their switching window is much shorter than the recovery time of the SOA. The gain of the SOA, together with the phase sensitivity of the optical interferometer, allows these switches to be controlled with extremely low switching energy. They are integratable and cascadable. Several systems demonstrations were discussed in this chapter that illustrate the use of various ultra-fast all-optical switches, in a laboratory testbed setting, for optical switching at hundreds of gigabits per second. Looking toward the future of optical networks, it is likely that signal processing and switching will continue to utilize electronics wherever possible, particularly in network areas where low latency and high switching speed are not required. However,
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[2
in network areas where switching must be performed at aggregate data rates, all-optical switching is likely to be required.
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E. Wolf, Progress in Optics 45 © 2003 Elsevier Science B.V All rights reserved
Chapter 3
Generation of dark hollow beams and their applications by
Jianping Yin^'^, Weijian Gao^ and Yifu Zhu^ Key Laboratory for Optical and Magnetic Resonance Spectroscopy, Department of Physics, East China Normal University, Shanghai 200062, China; •^Department of Physics, Suzhou University, Suzhou 215006, China; ^Department of Physics, Florida International University, Miami, FL 33199, USA
119
Contents
Page § 1. Introduction
121
§ 2.
Definition of a dark hollow beam and its parameters
122
§3.
Generation of dark hollow beams
123
§4.
Classification of dark hollow beams
158
§5.
Applications of DHBs in modem optics
165
§6.
Applications of DHBs in atom optics
173
§ 7. Applications of DHBs in coherent matter-wave optics
188
§ 8.
196
Summary and outlook
Acknowledgments
199
References
199
120
§ 1. Introduction In recent years, various methods have been developed to generate laser beams with zero central intensity such as Laguerre-Gaussian beams, high-order Bessel-Gaussian beams, high-order Mathieu beams, doughnut hollow beams, LPoi-mode output hollow beam, localized hollow beams, and so on. This family of special light beams is called dark hollow beams (DHBs) (Yin, Noh, Lee, Kim, Wang and Jhe [1997]). DHBs have some unique physical properties, such as a barrel-shaped intensity distribution, a helical wavefront, and center phase singularity, and may carry spin and orbital angular momentum and exhibit spatial propagation invariance. DHBs may be used as optical pipes, optical tweezers and spanners, and have become a powerful tool in the manipulation and control of microscopic particles (such as micrometer-sized particles, nanometer-sized particles, biological cells, and so on). Therefore, they have many important and extensive applications in laser optics, binary optics, computer-generated holography (CGH), optical trapping of particles, materials science, biological and medical sciences, and so on. There are certain advantages in using DHBs to trap, guide and transport neutral atoms. For example, atoms confined near the dark center of the DHBs suffer the minimal light-shift effect of atomic internal levels, and experience a low photon-scattering rate, a lower photon-assisted-collision rate, and no attractive potential from van der Waals interaction of material walls and so on. Consequently, a higher atomic density and lower temperature may be obtained for the confined atomic sample. Therefore, DHBs, as atomic pipes, tweezers, refi*igerators and motors, have practical and important applications in cooling and trapping of neutral atoms, and in manipulation and control of cold atoms (Yin, Noh, Lee, Kim, Wang and Jhe [1997]). Since the 1990s, a variety of methods have been used to generate DHBs, and various techniques using DHBs in optical traps for microscopic particles, in guiding, cooling and trapping for neutral atoms, as well as in manipulation and control of coherent matter waves [i.e., Bose-Einstein condensates (BEC)] have been developed. In § 2 we will characterize DHBs and their physical parameters. In § 3, various experimental schemes for DHB generation are described. In § 4, classifications of DHBs and their areas of applicability are briefly discussed. 121
122
Generation of dark hollow beams and their applications
[3,
In §§ 5, 6 and 7 we review applications of DHBs in modem optics, atom optics and coherent matter-wave optics. The final section summarizes the current stage of DHB research and presents a perspective outlook for future studies and applications of DHBs.
§ 2. Definition of a dark hollow beam and its parameters A dark hollow beam (DHB), as its name implies, is an annular (ring-shaped) light beam with a null intensity center on the beam axis in the propagation direction. Figure 1 shows the radial intensity distribution of a dark hollow beam. In addition to some general light-beam parameters, such as frequency VL, power PQ and divergent angle a, several special parameters can be defined to characterize the spatial properties of a DHB as follows (see fig. 1): (1) The dark spot size (DSS) is defined as the fiiU width at half maximum (FWHM) of the radial intensity distribution inside the notch of the DHB.
Fig. 1. Radial intensity distribution of a dark hollow beam.
3, § 3]
Generation of dark hollow beams
123
(2) The beam width (WDHB) is defined as the fiill width at 1/e^ of the maximal value of the radial intensity distribution outside the core of the DHB. (3) The beam radius (ro) is defined as the distance between the position of the maximal radial intensity and the center of the light beam. (4) The ring-beam width (Wr) is defined as the fiill width at 1/e^ of the maximal value of the radial intensity distribution, i.e., Wr = ^DHB-2ro. (5) The width-radius ratio (WRR) is defined as the ratio of the ring-beam width Wr to its beam radius ro, i.e., WRR=^^^^^-2.
(2.1)
It will be shown in subsection 6.2 that the WRR is a direct measure of the efficiency of DHB-induced Sisyphus cooling for neutral atoms. Two other parameters are defined (Spektor, Piestun and Shamir [1996]) as (1) The normalized curvature at the point of minimum intensity,
hd^r
(2.2)
Here ro is the location of the minimum in the radial direction r where (dI/dr)r = ro = 0, and IQ is the reference intensity of the DHB. (2) The normalized slope at the inflection point of the r-dependent fimction describing the intensity distribution inside the notch,
lodr
(2.3)
where ri is the location of the maximum slope where (d^I/dr^)r = ro = 0. DHBs also possess orbital angular momentum, as reviewed by Allen, Padgett and Babiker [1999]. Different DHBs possess different optical angular momenta, which may be used to rotate microscopic particles (see § 5.2) or cold atoms (see § 6.4) as an optical or atomic motor.
§ 3. Generation of dark hollow beams In this section, the methods and basic principles for generating DHBs are introduced and reviewed in detail.
124
Generation of dark hollow beams and their applications
[3, § 3
3.1. Transverse-mode selection With a judiciously designed cavity, the TEMQJ mode may be predominantly excited and the laser output beam is therefore a DHB. By this method, Rioux, Belanger and Cormier [1977] used a conical cavity to generate a DHB of a CO2 laser (10.6 (im) from an output ring-shaped transverse mode. Tamm and Weiss [1990] generated a stationary TEMQJ doughnut beam in a singlelongitudinal-mode He-Ne laser cavity by transverse-mode selection, and used this beam to study the laser bistability and optical switching of the spatial patterns. White, Smith, Heckenberg, Rubinsztein-Dunlop, McDufif and Weiss [1991] used a Na2 molecule vapor laser in a ring cavity to produce a TEMQI doughnut beam at a wavelength A = 0.525 ^im by altering the length of the laser cavity. Wang and Littman [1993] reported an experiment that typifies the basic configuration and requirements of DHB generation with transverse-mode selection. They produced a hollow beam with a variable-radius ring by selectively exciting a TEMQJ transverse mode in a side-pumped, grazing-incidence dye laser pumped by a N2 laser (Wang and Littman [1993]). The ring radius of the TEMQJ mode can be changed by tuning the laser frequency. The experimental setup is shown in fig. 2. Laser oscillation was established between the reflection surfaces of a Fabry-Perot (FP) etalon and a frequency selector composed of a diffraction grating and a tuning mirror. A negative lens is used to diffuse the beam in the cavity. If the laser is operated at a fixed wavelength A, and the FP etalon has a thickness d and an index of refraction «, the maximum beam transmission through the FP etalon occurs at an incident angle 6 given by 2ndcos6 = mX,
(3.1)
where m is an integer. In reciprocal length units (e.g., cm~^), the quantity l/(2nd) is the free spectral range of the FP etalon. Because of the axial symmetry of the Pump Light
a
Tuning Mirror
»Cylindrical Lens
Fabry-Perot etalon
Negative Lens
Diffraction Grating Dye Cell
Fig. 2. Laser cavity scheme for generation of TEMQJ modes with variable-radius rings.
3, § 3]
Generation of dark hollow beams
125
cavity configuration, the laser output forms a ring-shaped beam (i.e., a doughnut beam) with the intensity distribution specified by
2jrw"^(z) w^{z)
2r^ 'w\z)
(3.2)
where z2 w2(z) = w 2 ( l + ^ )
(3.3)
is the beam waist on the z plane, and WQ is the beam waist on the plane z = 0. is the output power of the hollow laser beam, ZR = JTWQ/A is the Rayleigh length, and z is the propagation distancefiromthe beam waist WQ at z = 0. When the laser is tuned to a wavelength A and the maximum transmission is assumed to occur at 0 = 0, then,
PQ
2nd = WQAQ,
(3.4)
where mo is an integer. The next transmission maximum will occur at an angle 0o = a r c c o s ( ' ^ ^ ^ ^ y
(3.5)
The angles of other transmission maxima, corresponding to orders m ^ mo - 2, can also be obtained. By tuning the laser to a shorter wavelength, the ring of m = mo order is scanned to a larger angle. In order to produce the laser output beam with a single ring at all times and still maintain the maximum scanning range of the ring size, the negative lens should be chosen so that the angular divergence of the beam is just below ^oIn the experimental setup of fig. 2, a N2 laser (337 nm) is focused by a cylindrical lens with a 10 cm focal length to pump a stirred dye cell with a Rhodamine 6G concentration of 2.5xlO~^M in a methanol solution. A solid FP etalon with a fi-ee spectral range (FSR) of 1.6 cm"^ was placed ~7cm away fi-om the opposing side of the dye cell, which leads to 60 ^ 0.0136 rad. A negative lens with a focal distance / = -25 mm was used in the cavity, and the half angular divergence of the intracavity beam was 0.012 rad. The negative lens is roughly halfway between the cell and the FP etalon, and is slightly tilted to prevent lasing from its surface reflection. The active gain region is 10 mm long and '^2w = 0.1 mm in diameter, resulting in a half-diffraction angle of X/(jTw) = 0.004 rad for light emerging from the dye cell, which is further
126
Generation of dark hollow beams and their applications
o
[3, § 3
o
Fig. 3. Output mode patterns at four diiferent oscillation frequencies viewed in the focal plane of a lens outside the cavity. (From Wang and Littman [1993], Opt. Lett. 18, 767, reprinted with permission.)
expanded to ~0.012rad by the intracavity negative lens. In the experiment, a holographic grating with a groove density of 1800 lines/mm and a length of 5 cm was used. The center of the grating was ~7 cm away from the dye cell. The experiment showed that when the laser frequency was continuously tuned by rotating the tuning mirror, the laser output beam was a DHB with a continuously varying dark spot size (DSS) as shown in fig. 3, where the corresponding laser frequency was (clockwise from top left) ca = o^o ~ 17 240.0 cm~^ 0) = COQ-0.4cm~^ CO = OJQ-0.8cm~^ and co = COQ- 1.2cm~^ respectively. 3.2. Geometric optical method 3.2.1. Generation of ring-shaped hollow beams With a double-cone prism, Ito, Sakaki, Jhe and Ohtsu [1997] produced a DHB with a double-Gaussian intensity profile; their experimental apparatus is shown in fig. 4. The principle of the geometric optical method is rather simple: a Gaussian laser beam is divided symmetrically into two beam paths through the first refraction at the apex of a double-cone prism, and after the second refraction a nondivergent doughnut-shaped hollow beam emerges from the other side of the prism. The intensity of a TEMQO Gaussian beam with power PQ, beam waist WQ and wavelength A can be described by ^G(r,z)=
exp KW^{z)
Ir^
w\z)\
(3.6)
where z is the axial propagating distance, r is the radial position, and w(z) is the beam waist at position z, related to the characteristic length ZQ by w(z) = woWl + ^ ,
-V
(3.7)
3, § 3 ]
Generation of dark hollow beams (a)
127
-•
Gaussian
Double-cone Prism Doughnut
0) 3 r
-1
0 1 Position (mnn)
Fig. 4. (a) Conversion of a Gaussian beam into a ring-shaped hollow beam with a double-cone prism. (b) Relative intensity profile of the Gaussian beam (fi-om a He-Ne laser) with waist 2wo = 0.7 mm. (c) Relative intensity profile of the ring-shaped hollow beam converted by a double-cone prism of length 2L = 4.25 mm and fiill apex angle Id = 90°; the inset shows a cross-sectional image of the ring-shaped hollow beam. (From Ito, Sakaki, Jhe and Ohtsu [1997], Phys. Rev A 56, 712, reprinted with permission.)
where Ji
^0 =
(3.8)
JWQ.
Assuming the length and the full apex angle of the double-cone prism are 2L and 20, respectively, and its refractive index is «, a simple ray-optics analysis shows that the intensity of the DHB can be expressed approximately as the sum of the intensities /±(r,z) of two modified Gaussian beams /(r,z) = / 4 r , z ) + /_(r,z)
(3.9)
where I±{r.z) =
jtw±(r,z)
exp
2(rTRof
(3.10)
128
Generation of dark hollow beams and their applications
[3, § 3
The waists w±{r,z) of the two modified Gaussian beams at a coordinate (r,z) are given by
w±(r,z) = woWl +
^
.
(3.11)
Moreover, the ring diameter 2RQ, defined in fig. 4a, is determined by the length L and the angle 6 of the prism as follows: R^=Lsm2e(\---J^
).
(3.12)
The DHB produced with a double-cone prism (2Z = 4.3 mm; 26 = 90°) is shown in fig. 4c, while fig. 4b shows the intensity distribution of the input Gaussian beam firom a He-Ne laser, with beam waist WQ = 0.35 mm. The inner and outer diameters of the DHB produced are 0.6 mm and 1.4 mm (measured at 1/e of the maximum intensity), respectively. The measured ring diameter 2Ro = 1.0 mm is in good agreement with that derived firom eq. (3.12). An optical axicon (a single cone-shaped prism) has been known to be useful in converting a Gaussian beam into a DHB since the 1950s (Mcleod [1954, 1960], Belanger and Rioux [1978]). Recently, several groups have used an axicon setup to generate ring-shaped DHBs suitable for manipulating neutral atoms. Manek, Ovchinnikov and Grimm [1998] used a single axicon combined with a spherical lens to produce a DHB with a beam diameter of 2ro = 800 jim and a ring width of Wr = 35 \im (the corresponding width-radius ratio (WRR) is ~ ^ ) . Song, Milam and Hill [1999] used three axicons (each with 3"" base angle) together with a simple lens to generate a DHB at A = 852 nm fi-om a diode laser. In their setup, a hollow beam was generated by the first axicon, and the lens had different focal points for the inner and outer rims of the beam ring. Two additional axicons were used to control the core diameter and the ring thickness. A unique feature of this setup is that most of the diffraction resulting from the top of the first axicon is located outside the core, so the generated DHB has steep walls and a darker core. Yan, Yin and Zhu [2000] used a pair of antireflection-coated axicons, one convex and the other concave, to generate a well-collimated DHB with a dark-center diameter adjustable firom ~1 mm to 1 cm. Arlt, Kuhn and Dholakia [2001] used a lens-axicon system to realize the spatial transformation of an arbitrary highorder Laguerre-Gaussian beam (LGB) into an ultra-narrow annular beam with a FWHM of ~50 |jim, and a width-radius ratio (WRR) of ~ ^ . Recently, using two asymmetric conical prisms of I'' and 20° base angles, Yin, Gao, Wang, Long
3, § 3 ]
129
Generation of dark hollow beams
and Wang [2002] generated a DHB with a double-Gaussian intensity profile and obtained an ultra-narrow annular beam with a width-radius ratio (WRR) of ~ ^ . 3.2.2. Generation of high-order Bessel beams Nondivergent light beams, such as high-order Bessel beams, are useful for guiding and collimating of neutral atoms. Arlt and Dholakia [2000] generated high-order Bessel beams with orders / = 1 to 4 by illuminating an axicon with a LGB. The principle of converting a LGB to a high-order Bessel beam is simple. When a single-ringed LGB with azimuthal mode index / is used to illuminate an axicon placed at its beam waist, the field distribution behind the axicon can be expressed by using the stationary phase method to evaluate the Fresnel diffraction integral as £(r,0,z)=i^exp[i^(z+^
/ d r V \A ( V 2 — j e x p ( - 4 j exp(i^,rO exp(ifc0 L X / d^' exp(i/^) exp \-ikr'r cos(^ - (l)')/z\ , (3.13) where ^ is a normalization constant, and the term exp(-i^^r^O represents the phase retardation caused by the axicon. The field amplitude of the LGB (set /> = 0) has been factorized into its radial and azimuthal components. Integrating over the azimuthal angle (j)' and the radial position r', the output intensity in the transverse plane behind the axicon is proportional to 2z2
/(r,z) ex z'^^' exp - ^
J/(^,r),
(3.14)
where Zmax is the "propagation distance" defined by ^max = W o M : ^ ,
kr = k(n -
1)7,
(3.15)
where « = 1.5 and y = r are the refractive index and internal angle of the axicon, respectively. Equation (3.14) shows that, close to the optical axis, the beam generated by an axicon illuminated with a LGB with the azimuthal mode index / approximates to a Bessel beam of order /.
130
Generation of dark hollow beams and their applications LaguerreGaussian beam
Axicon
[3, §3
High-order Bessel beam
Aperture Fig. 5. Illuminating an axicon with a Laguerre-Gaussian beam generates an approximate higherorder Bessel beam within the shaded region.
Fig. 6. Experimental beam cross-section (355 fxm x 355 |im) of Bessel beams of orders / = 1,2,3,4 (from left to right) at a distance z = 14 cm behind the axicon. The radial profiles shown are averages of 40 azimuthal sections. The radius of the inner ring increases with order from r\ =21.2 \im to r4 = 6\.2 jjim. (From Arh and Dholakia [2000], Opt. Commun. 177, 297, reprinted with permission.)
The experimental setup of Arlt and Dholakia [2000] is shown in fig. 5. The incident LGB was produced by illuminating a computer-generated hologram (He, Heckenberg and Rubinsztein-Dunlop [1995]) with a Gaussian beam (a linearly polarized He-Ne laser beam). The diffraction efficiency of the hologram was about 30% and the LGB had a waist size of wo = 2.5 mm. The axicon was placed at the beam waist of the LGB. A hard aperture of radius R = 5 mm was used to filter out the other diffraction orders of the hologram. The spatial intensity distribution of the Bessel beams so produced was observed by magnifying the beam with a X10 microscope objective and then recording it with a CCD camera. Figure 6 shows the observed profiles for four Bessel beams with / = 1,2,3 and 4 at the same distance behind the axicon. The averaged radial profiles (dotted lines) of the Bessel beams agree with the theoretical intensity profiles (solid lines). Recently, King, Hogervorst, Kazak, Khilo and Ryzhevich [2001] used a biaxial crystal to
131
Generation of dark hollow beams
3, § 3 ]
realize the transformation of a Bessel beam from zeroth to first order or from first order to second order, and produced two Bessel beams of/i(r,z) and J2(r,z). 3,2.3. Generation of localized hollow beams In pursuit of a single hollow-beam atom trap with a large and controllable trapping volume, Cacciapuoti, de Angelis, Pierattini and Tino [2001] used the lens-axicon setup shown in fig. 7 to generate an optical bottle beam with a null intensity region surrounded in all directions by light walls. A collimated Lens Axicon
Trapping Region
0.10 i 0.00
-O.lOi -0.10 0.00 0.10 (b)
0.40
(c)
0.60 0.80 1.00 1.20 z position (mm)
1.80
Fig. 7. (a) Schematic of the experimental arrangement for the generation of a localized hollow beam. The first lens produces a virtual ring of radius r. The second lens images this ring and generates a dark region of radius R. (b) Cross section of the hollow-beam profile in the image plane of the second lens, (c) Section of the beam profile in a plane containing the optical axis. (From Cacciapuoti, de Angelis, Pierattini and Tino [2001], Eur. Phys. J. D 14, 373, reprinted with permission.)
Gaussian laser beam passes through the first converging lens with focal length/i and then enters the axicon. If the distance d between the first lens and the axicon is greater than the focal length/i of the lens, the optical rays produce a "virtual" ring-shaped intensity profile (i.e., a divergent hollow beam) in the back focal plane of the first lens. After this, the conical hollow beam is imaged by the second lens (/2), and a ring-shaped bottle beam with a dark region surrounded by light is formed near the focal point of the second lens. If the magnification of the second imaging lens is G, the radius of the ring-shaped beam in the image plane is given by
R=
Ga{n-\){d-f,\
(3.16)
132
Generation of dark hollow beams and their applications
[3, § 3
where a and n are the base angle and refractive index, respectively. The threedimensional (3D) closed dark region can be adjusted easily by changing the parameters d.d'Ji, etc. With a = lOOmrad, n = 1.51,/i = 12.5mm,^ = 10cm, d = 14 cm and d^ = 93 cm, they obtained a 3D closed bottle beam with a core diameter of 50 |im and an axial length of 1.2 mm; the corresponding aspect ratio is 24. With a = lOOmrad, n = 1.51,/i = 12.5mm,/2 = 15cm, d = 15cm and d^ = 106 cm, the 3D closed dark region had a diameter of 280|Lim and an axial length of 3.92 mm, with a corresponding aspect ratio of 14. Similar lens-axicon systems have been used by other researchers to produce a localized hollow beam near the imaging plane of a lens. Kulin, Aubin, Christe, Peker, Rolston and Orozco [2001] used a lens-axicon system to generate a 3D closed hollow beam with an elongated-diamond-like dark region with diameter 1.5 mm and axial length 15 cm. The volume of the dark region is about 80 mm^, which may be used to trap more than 10^ cold atoms. Herman and Wiggins [2002] proposed a new method to produce and magnify a hollow beam by using a simple spherically aberrating lens and a projection lens. They performed a theoretical analysis and showed that the efficient loading of cold atoms from a standard magneto-optical trap (MOT) into a single hollow-beam trap may be realized. Recently, Kaplan, Friedman and Davidson [2002] generated a hollow beam with an optimized dark region by using a hybrid system composed of a telescope, two refractive axicons and a binary phase element (BPE). They obtained a dark region with a diameter of ~1 mm and an axial length of ^3 mm (aspect ratio ^ 3), which matches the spherical size (2r = 1 mm) of a standard MOT.
3.3. Mode conversion For most laser cavities with rectangular symmetry the transverse field distribution of the laser beam can be described by a superposition of Hermite-Gaussian (HG) modes that form a complete basis set, with which an arbitrary field distribution can be described by an appropriate superposition of modes with indices n and m. The field amplitude of HG modes is given by
»V(Z)-''V
•••2(z2+z2)
(3.17) X exp[-(« + m+\)\l)]H„
'3]
133
Generation of dark hollow beams
where w{z) =
fe)-
/2(z2+zi) kz^
(3.18)
\l){z) = arctan
Hn{x) and Hm{y) are Hermite polynomials of orders n and m, respectively, C ^ is a normalization constant, k is the wavenumber, ZR is the Rayleigh length, (« + m + 1) t/; is the Gouy phase shift, and w(z) is the radius at which the Gaussian term falls to 1/e of its on-axis value at the propagating distance z from the beam waist (z = 0). Similarly, if a LGB travels along the z-direction and is polarized principally in the x-direction, the electric field of the LGB can be expressed as E{R) = iw
^
i dupj
Akz
(3.19)
(I * 0),
where the function Upj may be written in cylindrical coordinates as u^%r,,p,z) =
(1 +zVz2 )l/2
w{z)
Ll
Ir" w\z)
exp
w^(z)
i^^z ^'^^PI o/ 2 ^ 2A I exp(-i/0)exp i(2/? + / + 1) arctan [2(z2+z|)J L V^R (3.20) where L ^p is the associated Laguerre polynomial, and C^l^ is a normalization factor; the integer variables p and / are the radial and azimuthal mode indices, respectively. The existence of phase factor &^^ shows that the high-order LGBs possess an orbital angular momentum Ih per photon (Padgett and Allen [1999], Babiker, Power and Allen [1994]). Abramochkin and Volostnikov [1991] considered the mathematical transformation of laser beams undergoing the influence of astigmatism, and found an integral transformation of Hermite-Gaussian beams (HGBs) to LGBs. They recognized that passing the HGBs through a cylindrical lens can perform the beam conversion desired. This transformation of HGBs to LGBs with an astigmatic laser-mode converter was studied by Danakis and Aravind [1992], Allen, Beijersbergen, Spreeuw and Woerdman [1992] and Beijersbergen, Allen, van der Veen and Woerdman [1993]. Their work shows that any LG mode can be expressed in terms of HG modes as follows u^^i(x,y,z) = J2 k=0
i^b(n,m,k)u^1„_f^,^(x,y,z),
(3.21)
134
[3, §3
Generation of dark hollow beams and their applications d, — f \ » — d,—H-« I
^ M, W
dj
*\*-c/,
»| « t/j-H
IK\ U K
I HeNe
M,
A^'
/.
J 3
J 4
J5
Fig. 8. Experimental arrangement for jTi mode conversion. Two mutually perpendicular wires (W) are placed perpendicular to the mode axis inside a He-Ne laser consisting of two mirrors and a He-Ne gas tube. 00
01
02
03
11
12
13
22
Fig. 9. Experimental results obtained with the ^n mode converter. The top row shows the input ^Gnm modes, the bottom row shows the output LG„^ modes, for the n, m listed above the modes. (From Beijersbergen, Allen, van der Veen and Woerdman [1993], Opt. Commun. 96, 123, reprinted with permission.)
with the transformation coefficient b(n, w, k) =
(n + m)\k\
Ji^d^
[{l-tni^tri
=
(3.22)
Beijersbergen, Allen, van der Veen and Woerdman [1993] performed the first experimental study with an astigmatic laser-mode converter. Their experimental setup and results are shown in figs. 8 and 9, respectively. In fig. 8, the two mirrors M\ (R = 600 mm) and M2 (R = 431 mm) form the cavity of a He-Ne laser. BS is a beam splitter;/] and/5 are two lenses with short focal length, used to observe the input HG beam before the mode converter and the output LG beam after the mode converter. Two cylindrical lenses / and^i form the mode converter. /2 is a mode-matched lens. Other parameters are: di = 525 mm, ^2 = 306 mm, c/3 = 225 mm, t/4 = 176 mm, and ds = 27 mm. The top row in fig. 9 shows the input HG modes and the bottom row shows the corresponding output LG modes from the converter. Later, Kuga, Torii, Shiokawa, Hirano, Shimizu and Sasada [1997] used an astigmatic mode converter composed of a pair of cylindrical lenses (focal length / = 25 mm) separated a distance d = Vlf, to generate a LG03 hollow beam fi-om a HG03 laser beam. Courtial and Padgett [1999] analyzed the performance of such a mode converter and studied the transformation and propagation properties
3, § 3]
Generation of dark hollow beams
135
of the converted LGB. Also, Allen, Courtial and Padgett [1999] and O'Neil and Courtial [2000] studied the mode transformations in terms of the constituent HG or LG modes, and developed the matrix formulations to describe the mode transformations, which can be used to construct a novel variable-phase-shift mode converter that offers better performance for certain applications. 3.4. Optical holographic method In the early 1990s, Lee, Stewart, Choi and Fenichel [1994] used an amplified holographic reconstruction technique to convert a Gaussian beam into a hollow beam similar to the first-order Bessel beam {J\). It is well known that high-order Bessel beams are DHBs with zero central intensity except the zeroth-order Bessel beam, JQ- The /Q beam has the form /o(^) ^ '^o((^p), with p the radial distance, and can be written as -^oW == X— 2jt J
Qxp(bc sin 6) dO,
(3.23)
0
while the «th-order Bessel function J„(x) is given by 2JT
Jn(x) =^
fexp[i(xsin6-nd)]dd
(3.24)
0
or
0
It can be seen from eqs. (3.23)-(3.25) that higher-order Bessel beams can be obtained firom the zeroth-order Bessel beam by adding a phase factor exp(-i«0) or Qxp(ind). For example, the first-order Bessel beam, / i , can be expressed as 2jt
In
•^1W ^ w~ I exp[i(x sin 6 - 0)], dd = -— exp(i0) exp(ix sin 6) dd, 2JT J lit J 0
0
(3.26) or from eq. (3.25), lit
lit
-^1W = ir-^ / exp[i(x cos 6 + 0)] dO = —^ / exp(+i0) exp(ix cos 6) dd. zjti J Im J 0
0
(3.27) Experimentally, the exp(±id) phase factor in the integrands of eq. (3.26) or eq. (3.27) can be obtained by adding a circular wedged phase plate to the
136
Generation of dark hollow beams and their applications
[3, § 3
configuration of the JQ Bessel beam (Dumun, Miceli and Eberly [1987]). This phase plate could be a disk whose optical thickness increases linearly with polar angle. Alternatively, Dumun's annular slit can be replaced with a wedged annular slit. However, such circular phase plates are not commercially available, which made it difficult to generate the J\ beam until Lee, Stewart, Choi and Fenichel [1994] developed a new technique to produce a Ji-like hollow beam called the J\' beam. Their technique uses an annular slit with a phase plate in firont of it. This will generate a beam pattern Ji/ that can be represented by eq. (3.23) plus an additional phase factor exp[i0(0)]fi*omthe phase plate in the integrand, i.e.,
-^( W =J-f exp(i00) exp(ijc sin 6) d0, 2Jt J
(3.28)
0
with the phase factor exp[i0(0)] given by f exp(iO) = +1 Qxp[i(l)(d)]=\ "^^ [ exp[i(;r±Inn)] = - 1
0
(3.29)
Since exp(ijc sin 6) = Jo(x) + 2 [J2(x) cos(20) + Mx) cos(40) + • • •]
(3.30)
+ 2i[yi(x)sin(10) + y3sin(30)+ • • ] , then fi-om eqs. (3.28)-(3.30) one obtains Ji(x) = ^ [Ji(x)+ i J3(x)+ ly5(^)+ • • •] •
(3.31)
Thus, Ji'ix) is a superposition of Ji(jc) and higher-order odd-numbered J„(x) weighted by the order number n. In many experiments, one is interested mainly in the region within the first maximum (jc = 1.8412) of the J\ beam, in which Ji(x) > Jn(x) and J„{x) > Jn+2{x). Thus, J\{x) is close to Jy(x). Figure 10 shows the experimental setup used to generate J( beams (Lee, Stewart, Choi and Fenichel [1994]), which can be divided into three components: (a) production of the JQ beam; (b) holographic amplification; and (c) reconstruction of the / / beam. A half-wavelength phase difference was achieved by a slightly tilted glass microscope slide (thickness 0.25 mm). Holographic addition was accomplished by the use of the auxiliary beam splitters BS2 and BS3 and the holographic plate H2. Holographic plates Hi and H2 were initially exposed
Generation of dark hollow beams
3, § 3 ]
137
Annular slit -r
A Lens A
Laser
^-_^d
-L M (a)
[
^BS
(c)
/-
^"T"'"'
%.M
sNr^«^
RiO.lmW
Fig. 10. Generation and amplification of / ( beam: (a) production of JQ beam; (b) holographic amplification; (c) combination of the two processes, resulting in holographic addition.
138
Generation of dark hollow beams and their applications
[3, § 3
at the same time. First H2 was developed and repositioned, while Hi was kept shielded from light. Two beams, one the original signal beam, the other the image beam generated by the reference beam R2 out of hologram H2, are projected and superimposed on screen S. The rotating phase plate PP will only rotate the signal beam pattern, leaving the image beam unchanged. The rotated signal beam, re-exposed on Hi, averages out the circular fluctuations. The dimensions of the experimental setup were as follows: annular slit diameter d = 2.5 mm, annular slit width Ad = 10 |jim, lens focal length/ = 140mm, and lens diameter 2R = 7.5 mm. The intensity pattern of the / / beam was observed, and the measured first zero-ring diameters for the / [ beam were in the range of 70.3-78.0 |im, while the calculated value is 75.6 [im. The efficiency of Gaussian to Bessel beam conversion was ~50%, and the propagation distance over which intensity remained uniform was about 60 cm. 3.5. Computer-generated-hologram method 3.5.1. Generation of Laguerre-Gaussian or vortex hollow beams The wavefronts of a hollow beam contain dislocations of the periodic structure in which the surface of constant phase has a helicoidal form. Such a screw dislocation in a wavefront requires the amplitude of the electric field to become zero on the dislocation axis. The phase of the wave changes by ImJt for one revolution around the dislocation axis, where the integer m (positive or negative) is called the topological charge, or the order of the dislocation (Bazhenov, Soskin and Vasnetsov [1992]). On the optical axis the phase is undefined (singular), and the radiation intensity must be zero (Heckenberg, McDuff, Smith and White [1992]). When a plane wave interferes with such a helicoidal wave with dislocated wavefronts, the interference pattern will have a dislocation structure. This suggests an efficient method for generating a doughnut hollow beam (i.e., a LGB) by using a computer-generated hologram. Taking a screw-dislocation wavefront as a simple example, the amplitude £* of a helicoidal wave in cylindrical coordinates can be expressed as E{r, 6,z) exp(-ife) = E{)r exp(i9) exp (Fiiz) - _ ^ . 1 exp(-i^z), V ^i(^)/
(3.32)
where E^ is the real amplitude, and k = 2JI/X is the modulus of the wavevector. The fiinctions F\{z) and F2{z) describe the divergence of the beam in space and slow phase drift. The presence of the multiplier exp(i0) shows that the phase of
3, § 3 ]
Generation of dark hollow beams
-1.5 -1.0 -0.5
0
0.5
1.0 1.5
139
-1.5 -1.0 -0.5
(b)
0
0.5
1.0 1.5
(C)
Fig. 11. Dislocation grating structures: (a) screw dislocation grating; (b) first-order dislocation grating; (c) second-order dislocation grating.
the wave changes by In with a round trip on any circle in the (r, d) plane around the z axis. In the paraxial approximation, the complex amplitude E is given by E(r, 6, z) = Eor exp(i0) exp 2 In
kA/2 \ ikA/2.
. A + 2z/ik
(3.33)
When this screw-dislocation wave interferes with a plane wave E(z) = EQQ"^^ propagating in the same direction, i.e., the angle between the two wavevectors is 0, the total intensity distribution has the following form: /(r, d, z) = El + [Eor Qxp(P)f + lE^ exp(P) cos(0),
(3.34)
where P = \n{ (l) = d-
auf
z^ + CjkAf^ 9
A^ + (2z/kf
(3.35)
2z/k
^^ + (2z/yl)2-^^^^^^(f l^^-
The position of the interfering maximum is defined by the condition cos 0 = 1, or 0 = 2nJt(n = 1,2,3,...), this describes the spiral structure of the interference fi"inge shown in fig. 11a. The direction of circulation of the spiral line depends on the sign of 6 in the term exp(i0) of eq. (3.33).
140
Generation of dark hollow beams and their applications
[3, § 3
If the plane wave is incident on a screen at an angle 2q), the equation for determining the interference maxima in Cartesian coordinates is a r c t a n ( - ) - ( x ^>-)^2 ^(2z/)^)2-Cretan|^-j
(3.36)
- kx sin(2g}) - 2kz sin((p) + JT = Inn. The calculated interference pattern is shown in fig. lib. If rexp(i0) in eq. (3.33) is replaced by r'"exp(i/w0), with m = 2,3,4,..., a higherorder, synthesized grating with higher-order screw dislocation is obtained. Figure He shows the pattern of the interference fiinge between a second-order screw dislocation and a plane wave. Thus one can generate doughnut hollow beams (LGB) of various orders with the computer-generated-hologram method. The experimental procedure is as follows. First, setting a proper constant A, one produces a two-dimensional (2D) grating on the computer screen according to eq. (3.34). Second, one photographs the 2D grating fi-om this computer screen. Third, after reducing the gratings (the reduction factor being determined by the size of the interference patterns on the computer screen and the dimensions of the desired grating), a suitable hologram is prepared. When a reference plane wave is diffracted from the hologram, a doughnut hollow beam (LGB) will be generated behind the hologram. Heckenberg, McDufF, Smith and White [1992] were the first to use computergenerated holograms (CGH) to produce a TEMQJ doughnut hollow beam. A series of experiments followed. (Bazhenov, Soskin and Vasnetsov [1992], Basistiy, Bazhenov, Soskin and Vasnetsov [1993], and others). Later, some higher-order LGBs were produced by He, Heckenberg and Rubinsztein-Dunlop [1995], Clifford, Arlt, Courtial and Dholakia [1998] and Arlt, Dholakia, Allen and Padgett [1998]. Furthermore, optical vortex hollow beams were generated by computer-generated holography (Sacks, Rozas and Swartzlander [1998]). More recently, we used three different dislocation gratings (a screw dislocation grating, a first-order dislocation grating and a second-order dislocation grating, see fig. 11) to produce three doughnut-like hollow beams, with the experimental results shown in fig. 12. The doughnut beam in fig. 12a is generated by the screw dislocation grating (see fig. 11a), the doughnut beams in figs. 12b and c are produced by the first- and second-order straight-stripe dislocation gratings (see figs. 1 lb,c) respectively, and the corresponding intensity distribution can be described approximately by a TEMQ^ doughnut hollow beam (/ = 1,2,...). In our experiment, the grating size is 2 mm x 2 mm, the grating density is 13 lines/mm, and the observation distance fi*om the grating plane is 1 m (with a He-Ne laser at 0.6328 ^im). Our results show that a screw dislocation grating
3, § 3]
Generation of dark hollow beams
141
Fig. 12. Doughnut hollow beam generated by (a) screw dislocation grating, (b) first-order dislocation grating, and (c) second-order dislocation grating, respectively (z = 1 m, A = 0.6328 |im).
will generate a doughnut hollow beam in zeroth-order diffraction (dark spot size (DSS) ^ 0.6 mm at z = Im). A straight-stripe dislocation grating will produce two doughnut beams in ± Ist-order diffraction, and the DSS {^ 1.0 mm at z = 1 m) of the doughnut beam generated by the second-order dislocation grating is about twice the DSS {^ 0.5 mm at z = 1 m) of that produced by the first-order dislocation grating.
3.5.2. Generation of higher-order Bessel beams The «th-order Bessel beam J„ is related to the zeroth-order Bessel beam Jo by a phase factor exp(ibi«0) in the integrand (see eq. (3.24). A Bessel beam is free of refraction, so its intensity profile will be invariant during the wave propagating, which is usefril in applications such as optical alignment, ray-path tracing and laser guiding of atoms. It is straightforward to produce higher-order Bessel beams with the computergenerated-hologram (CGH) method. Vasara, Turunen and Friberg [1989] developed a scalar-wave theory to express the intensity distribution of a general nondiffracting beam behind the CGH, using two binary amplitude-coded CGHs to generate two Bessel beams of 1st and 6th order. Paterson and Smith [1996] designed an axicon-type CGH to produce two approximate higher-order Bessel beams (Ji-like and Jio-like) and developed a new theoretical method to explain
142
[3, §3
Generation of dark hollow beams and their applications
the experimental results. They showed that the amplitude transmission function of a finite axicon-type CGH can be approximated as U{r,e,0) =
Wo exp(i«0) exp(-ipo^) 0
if r < ro,
(3.37)
if r > ro,
where ro is the radius of the hologram and po is the radial spatial frequency of the Bessel wave. With the Fresnel approximation in the Kirchhoff integral, the amplitude of the Bessel beam generated at the position z is given by U{r,d,z) = —- exp ik \ z-\lAz
Iz
ro ''0
Xy
r
/ expl'iA:^')
r''=0
2jr
y V(r\ Q\ 0) exp|
.ib-r'cos(0'-0)
-^de'l dr'.
.
.«' = 0
(3.38) Substituting U(r', q', 0) from eq. (3.37) into eq. (3.38) and evaluating the angular integral yields ^2 ^
U(r,d,z)=
^-—exp ik I z + iAz 2z
exp[i«(0- i:/r)] (3.39)
^rr'
d/.
exp /-' = 0
Following the method of Vasara, Turunen and Friberg [1989] and using the principle of the stationary phase (Bom and Wolf [1999]), the above radial integral can be evaluated and the intensity distribution of the «th-order Bessel beam (y„-like) is given by IJTzpi I(r,d,z) oc WQ
i-i
Jn(Por)+\
j'iPor) (3.40)
for r < rmax(= zpo/k), r nk/pl- Here J„{por) = 2 |P^(por)| sin{po^ + arg[P^(por)]}
(3.41)
is a function with the same envelope as the wth-order Bessel function, but with the phase of the cosinusoidal wave shifted by ^Jt, and r^ax = zpo/k. The Bessel function in eq. (3.40) is given by Jn(por) = P^ipQr) exp(ipor) + P (por) exp(-ipor).
(3.42)
3, § 3]
Generation of dark hollow beams
143
Fig. 13. Ji Bessel beam produced by an axicon-type CGH at two propagation distances: (a) z = 430 mm (for which Tj^ax = 0.87 mm) and (b) z = 1300 mm (r^ax = 2.62 mm) recorded with a CCD camera. Reconstruction wavelength A = 633 nm. (From Paterson and Smith [1996], Opt. Commun. 124, 121, reprinted with permission.)
and we have P^(por) = [P-(por)r,
(3.43)
where P^(por) and P~(por) are slowly varying complex functions. Paterson and Smith [1996] showed theoretically that the output intensity distribution (7„-like) from the axicon-type CGH is very close to that of an ideal Bessel beam (7„), but the deviation will increase with rising order of the Bessel beam. Moreover, the finite aperture of the holographic element (CGH) results in a finite propagation-invariant distance while the ideal Bessel beam is invariant in propagation. The maximum propagation-invariant distance derived from geometric optics is given by ^max " ( ~ ) ^0.
(3.44)
As an example, the maximum propagating distance is z^ax ~ 4.9 m for the generated Ji-like beam with A = 633 ^m, ro = 9.9mm and po = 2jr/(0Ajt)mm~\ which is in good agreement with the experimental result (~4.8m). Paterson and Smith [1996] designed and fabricated several axicon-type computer-generated holographic elements that were written as first-order binary amplitude holograms by the fringe-following method with a carrier frequency to separate the diffraction orders. The holograms were used to produce Bessel beams (/„-like) with various values of n, po and ro; their experimental results are shown in figs. 13 and 14. The cross-section of the output Ji-like beam at two different propagating distances for the parameters A = 633 ^im, ro = 9.9 mm and po = 2jt/(0. Ijt) mm~^
144
Generation of dark hollow beams and their applications
[3, §3
100 nm
Fig. 14. (a) Central region of an axicon-type binary hologram designed to produce an approximate JiO Bessel beam, (b) Corresponding output beam at z = 900 mm recorded with a CCD camera. (From Paterson and Smith [1996], Opt. Commun. 124, 121, reprinted with permission.)
is shown in fig. 13. Note that the intensity profile of the Ji-like beam at the center is approximately that of the J\ beam: propagation invariance was observed for the generated Ji-like beam at distances z fi-om 0.2 m up to ~5m, which is about equal to z^ax ^ 4.9 m predicted by eq. (3.44); also, the diameter of the central hole (peak to peak) agrees with the theoretical prediction of 2ro ?^3.5/po ^ 0.175 mm. Figure 14 shows the axicon-type binary hologram designed to generate the yio-like beam and its experimental result. The behavior of the Jio-like beam is similar to that of the J\ -like beam, with similar propagating invariance over a finite range, but its central dark spot (ro ^ n/po ^ 0.6 mm) is larger than that of the Ji-like beam by a factor of ~7. Their study shows that the «th-order Bessel beam has a dislocation with polarity n at its center, and the intensity near the center of such beams for poro < n varies as (por)^". Higher-order Bessel beams consist of a dark disk of radius ro < n/po surrounded by the characteristic ring structure of the Bessel beams. The Bessel beams with different orders have different hole profiles, ranging from parabolic intensity holes (/ oc r^) to flatbottomed holes with well-defined diameters. 3.5.3, Generation of localized hollow beams Vinas, Jaroszewicz, Kolodziejczyk and Sypek [1992] proposed and demonstrated a novel method to generate a localized hollow beam by using a computergenerated circular holographic zone plate. They designed a circular zone plate with a Jt phase jump and manufactured it using computer-generated holography. When a focused Gaussian beam passes through the holographic zone plate, the destructive interference in the middle of the focal spot generates a localized hollow beam near the focal point. The study shows that the dark spot size (DSS)
Generation of dark hollow beams
3, § 3 ]
145
Fig. 15. (a) Circular Fresnel zone plate with a phase jump, (b) Intensity profile across the focal spot. (From Vinas, Jaroszewicz, Kolodziejczyk and Sypek [1992], Appl. Opt. 31, 192, reprinted with permission.)
resulting from the zone plate in the focal plane is smaller than the general focal spot of the Gaussian beam. The radial and axial amplitude distributions of the light field generated by the zone plate in the focal plane can be calculated as particular cases of the diffraction integral in the Fresnel approximation. A monochromatic plane wave with wavelength A is assumed to illuminate perpendicularly a circular zone plate. The amplitude distribution in the focal plane is given by
UipJ)
iJzCp
oo
(3.45)
ir)Jo(2jtrp)rdr,
where p = 1/A/(XQ + ylY^^ is the spatial frequency, r is the radial coordinate in the aperture plane, Pt(r) is the transmission frinction, JQ is the zeroth-order Bessel frinction, and Cp is a constant phase factor. The axial distribution is given by oo
U(0,z) =
IJTCZ
(r)exp - i
)b-2
2(1//-1/z)
rdr,
(3.46)
where Q is another phase factor. Figure 15a shows the computer-generated circular holographic zone plate (Vinas, Jaroszewicz, Kolodziejczyk and Sypek [1992]). The zone plate was manufactured as a binary amplitude hologram with the diffraction efficiency of ~65%, divided into symmetrical parts with equal areas, and a ^A step along the circle at r = \^flA is introduced, which results in the function Pxir) = circ(2rZ4) - 2circ(2A/2Z4), where A is the diameter of the
146
Generation of dark hollow beams and their applications
[3, § 3
aperture. The intensity distribution in the focal plane for the zone plate with a circular phase jump is derived from eq. (3.45) as
,
(3.47)
where t = Ap/Xf, and J\ is the first-order Bessel fiinction. Equation (3.47) shows that the radial intensity distribution has a J\ -like profile. The first maximum is for t = 1.136, and the halfwidth (HW) of the first maximum is equal to 0.727, which is 141.4% of the HW value for a general focal spot. From eq. (3.46), the axial intensity distribution is given by I{0,z) = IoU^
sinc'l^.
M^).,.(...M_-).
(3.S,
Theoretical study shows that a ftirther reduction of the focal spot size can be obtained when the central region of the zone plate is obscured. The localized hollow beam generated with the computer-generated circular holographic zone plate is shown infig.15b (other parameters: ^ = 6 m m ; / = 75 cm). Jaroszewicz and Kolodziejczyk [1993] designed a computer-generated spiral holographic zone plate and obtained a localized hollow beam; they discussed the dependence of the radial intensity profile on the obstruction ratio G = 1 - A^Z4, where /SA = A- A\, with Ai the diameter of the obscured part near the central region of the zone plate. Recently, Arlt and Padgett [2000] produced a novel computer-generated hologram and produced a localized hollow beam (they called it an optical bottle beam) near the focal point. Their CGH is based on the interference fringe between a Gaussian beam and a LG beam (I = 0, p = 2). The destructive interference of these two beams results in a focused beam with localized intensity null at the focus. 3.5.4. Generation of higher-order Mathieu beams More recently, a new type of hollow beams - higher-order Mathieu beams was proposed and studied by Gutierrez-Vega, Iturbe-Castillo and Chavez-Cerda [2000], Chavez-Cerda, Gutierrez-Vega and New [2001] and Gutierrez-Vega, Iturbe-Castillo, Tepichin, Rodriguez-Dagnino, Chavez-Cerda and New [2001]. Mathieu beams are dififraction-fi*ee beams with elliptical symmetry defined by an elliptic parameter q. A detailed description of Mathieu beams will be presented in §4.10. Two zeroth-order Mathieu beams with elliptic parameters of ^ = 25 and q = 595
3, § 3]
Generation of dark hollow beams
147
were generated by the geometric optical method (Gutierrez-Vega, Iturbe-Castillo, Tepichin, Rodriguez-Dagnino, Chavez-Cerda and New [2001]). Similar to the zeroth-order Bessel beam, the zeroth-order Mathieu beam has a maximum central intensity distribution, and its electric field amplitude in the elliptical coordinates (§, rj,z) is given by E(^, rj,z; q) = CMBJCO(^, q) Ccoiv, q) exp(iA:^z),
(3.49)
where CMB is a normalization constant, JQo(^,q) and Ceo(V,q) are the zeroorder radial and angular Mathieu fianctions of the first kind, respectively, and the elliptical coordinates are defined by the relations ( X = h cosh § cos T], y = h sinh ^ sin r],
(3.50)
^ z = z.
^ e [0, oo) and rj e [0,2jt) are the radial and angular variables, respectively, and 2h is the interfocal separation. The elliptical parameter q = \h^k} carries information about the transverse spatial firequency ky and the ellipticity of the coordinate system through h. Three high-order Mathieu hollow beams (m = 2, q= 1.5, kr = 9.6x10'^ m~^; m = 5, q = l, kr = 7.2xl0'^m-i; m = 2, q = 20, kr = 9.6xl0'^m-^) were produced with the computer-generated hologram (Chavez-Cerda, Padgett, Allison, New, Gutierrez-Vega, O'Neil, Mac Vicar and Courtial [2002]). The corresponding experimental setup and results are shown in figs. 16 and 17, respectively. In the experiment, the computer-generated phase holograms are designed to represent the phase distribution of the desired higher-order Mathieu beams (see fig. 16a) propagating at a small angle relative to the optical axis. Using a slide writer, these holograms were manufactured by first developing a photographic negative of a grey-scale representation of this phase function, then making a contact print onto holographic film, and then bleaching the film, thereby transforming grey levels into optical thickness. When the phase hologram is illuminated by a Gaussian beam truncated by a circular aperture, a higher-order Mathieu beam is generated in the first-order diffractive light (see fig. 17). Higherorder Mathieu beams carry orbital angular momentum. Calculations showed that for m = 6 and q = 0, IS, 21 and 72, the orbital angular momentum per photon is 8Lz = 6h, 4.92h, 4A0h and 6.32^, respectively. Due to the complex phase structure, the orbital angular momentum per photon bL^ of high-order Mathieu beams can be fi-actional in h.
148
Generation of dark hollow beams and their applications
t?i = 2, <7 = 1.5, Av = 4 1 0 ^ m - '
m = 5, q = 1,
[3, §3
m = 5,q = 20,
(a)
(b) Fig. 16. (a) Greyscale representation of the optical thickness of some Mathieu-beam holograms, (b) Schematic of the experiment for holographic generation of higher-order Mathieu beams. The phase hologram //j is illuminated by a beam truncated by the circular aperture ^ i. In the focal plane of lens L\ (fi = 600 mm), the circular aperture A2 picks out the bright ring of the first diffraction order, which is then turned into a diffraction-free beam by the lens L2 (fz = 250 mm). The CCD array C allows the intensity cross-sections of the resulting beam to be recorded at various distances z behind the image of the hologram, Hj. (From Chavez-Cerda, Padgett, Allison, New, Gutierrez-Vega, O'Neil, Mac Vicar and Courtial [2002], J. Opt. B 4, S52, reprinted with permission.)
5.6. Method of micron-sized hollow fiber Yin, Noh, Lee, Kim, Wang and Jhe [1997] proposed a new method to generate a doughnut-like hollow beam by using a micron-sized hollow optical fiber (HOF). A hollow-fiber waveguide consists of a hollow core and a cylindrical cladding. Let the inner and outer radius of the hollow core be a and b; the core thickness, d, is thus d = b - a. The cladding can be taken to be infinite. In the cylindrical coordinate system (r,d,z) and under the weakly-guiding approximation, the transverse component ^tC^) of the electric field propagating along the z-axis in a hollow fiber is given by (Wang, Yin, Gao, Zhu and Wang [1998]) AIm{vr) sm{md) r
(3.51)
where the relative transverse propagation constant u is defined as 1/2
u= (k^ni-p^)
(3.52)
3, § 3 ]
149
Generation of dark hollow beams z/mm
m = 2,q=l.5 yt,«9.610^m-i
m = 5,q = 7
m = 5,q = 20
40
80
120
160
Fig. 17. Intensity cross-section of different experimentally generated higher-order Mathieu beams at different distances z behind the plane H2 (see fig. 16b). The three different beams correspond to the three holograms shown in fig. 16a. The area represented by each image is about 0.5 mm by 0.4 mm. (From Chavez-Cerda, Padgett, Allison, New, Gutierrez-Vega, O'Neil, Mac Vicar and Courtial [2002], J. Opt. B 4, S52, reprinted with permission.)
and the relative transverse attenuation constants v and w are defined as vl/2
(3.53) (3.54)
with 13 and k the propagation constant and the wavenumber (k =
2jt/X),
150
Generation of dark hollow beams and their applications
[3,
respectively. The constants A, B, C and D are determined by the boundary conditions of the electric field atr = a and r = b = a + d. In eq. (3.51), Jm and Nm are the Bessel fianctions of the first and second kind of order m, and /^ and K^ are the modified Bessel ftinctions of the first and second kind of order m. The HOF parameters used in the experiment of Yin, Noh, Lee, Kim, Wang and Jhe [1997] are: «2 = 145, a = 3.5 |im, d = 3.8 |im; the relative difference in refractive index between the core material (n\) and the cladding («2), An = (n] - nlyin^, is 0.18%, and the laser wavelength A = 0.78 iim. Since An(= 0.18%)
r
^core(^) = BJo(ur) + CYo(ur) I ^ciaciding(^) = DKo(wr)
a ^r
(3.55)
^b,
b
From Fraunhofer diffraction theory and in the scalar approximation, the electric field distribution of the LPoi-mode output beam infi-eespace is given by
I
AIo(ur)rJi(2jTrp)dr h
+ / [BMur) + CYQ{ur)\ rJx{2jtrp)dr
(3.56)
0
+
DKo(wr)rJ](2JTrp)dr,
I
where exp i ^ ^ ^ ^ )
c' - IJT-
uz
^Mm-^2^)],
P = ^.
Numerical integration of eq. (3.56) yields the electric field distribution in free space as a function of the radial position; it shows that the output intensity of the LPoi mode at the axial center is zero, i.e., the LPoi-mode output beam is a dark hollow beam with a large divergent angle (2.7x 10"^ rad) and the maximum
3]
Generation of dark hollow
151
beams
CCDArray L^ . L5
MO^*'
^
[ID4<»s|' VAi
t
Screen
Fig. 18. Experimental setup for generating a LPoi-mode output hollow beam by using a micron-sized hollow fiber.
intensity of the output LPoi-mode beam occurs at the radial distance TQ. A DHB with a smaller divergent angle can be obtained if a microscope objective (MO) with suitable magnification M and short focal length/ is used to collimate the LPoi-mode output beam. Experimentally, a micron-sized HOP was used to produce a doughnut-like hollow beam by Yin, Noh, Lee, Kim, Wang and Jhe [1997] (see fig. 18). Two lenses Li and L2 form a telescope system with a magnification factor of about 20 X, which is used to perform both expansion and collimation of the incident beam. VAi, VA2 and L3 are two variable apertures and a focusing lens, respectively. The inner (outer) diameter of the hollow core is 7 (14.6) |im, and the outer diameter of the fiber cladding is 123.4 |im. The numerical aperture, NA = wosin^A = n\ sin^t = W\ - ^2]^^^ i^ about 0.124, which restricts the relative optimum incident angle d^ to a value ^ 7 . Because of this small angle, they used a lens with 100 mm focal length as the final focusing lens in order to get a higher input-output coupling efficiency of the hollow fiber. Figure 19 shows the radial intensity distribution of the LPoi-mode DHB at z = 800mm. The intensity distribution of the LPoi-mode output DHB, similar to that of the TEMQ^ doughnut beam, is approximately given by (Yin and Zhu [1999]) I{r.Z) = Ah
Jtw\z)
exp
2r^
w\z)
(3.57)
where Po is the laser power, w{zf- = WQ[1 + A:2(Z/ZR)^] is the beam waist on the z-plane, ZR = (Jtwl/X) is the Rayleigh length, and WQ is the beam waist on the plane z = 0. The constants ki and ^2 are modified parameters that are determined by fitting the numerical results from the exact Fresnel diffraction integral to the beam model given by eq. (3.57) or by fitting the experimental
152
Generation of dark hollow beams and their applications ^-ntr.agg .^^uo^^
[3, §3
^ mm.
Fig. 19. Radial intensity profile of LPoi-mode output hollow beam using a M-20x objective lens at z = 800 mm; the corresponding DSS is -200 |im.
data to the model. The results show that the DSS of the DHB collimated by the M-20x objective is about 50 ^im and 200 |im at z = 100 mm and z = 800 mm, and the relative divergent angle at the near field is about 6.5x 10~^, whereas the divergent angle at the far field is about 4.0x10"^. Under identical conditions, the divergent angle of a collimated DHB in this experiment is smaller than that of a standard doughnut beam by ~6 times. 3.7. Method of radial-distributed Jl-phase plate Chaloupka, Fisher, Kessler and Meyerhofer [1997] proposed a segmented waveplate method to generate a localized hollow beam (LHB) near the focal point and studied the characteristics of a tunable LHB both theoretically and experimentally (Chaloupka and Meyerhofer [2000]). Recently, Ozeri, Khaykovich and Davidson [1999] used a circular jr-phase plate with two zones to produce a LHB and demonstrated a single-beam blue-detuned optical dipole trap for cold atoms. The experimental procedure is as follows: first, a circular ;r-phase plate, which imposes a phase difference of exactly Ji radians between the central and outer part of the plate, is fabricated; second, a collimated beam is passed through the jT-phase plate and then focused by a lens. The destructive interference between the inner and outer parts of the beam results in a dark region around the focus. In the experiment, a collimated Gaussian beam with 1/e^ radius wo and power PQ was passed through a circular ;r-phase plate with radius b and clipped by a circular aperture with radius a. The radii a and b are chosen such that the integrated amplitude of light is zero at the focal point, yielding the following condition: b = woy^-ln[i(l+exp(-aVw2))].
(3.58)
3, § 3]
Generation of dark hollow beams
153
To prepare a circular phase plate, they evaporated a thin dielectric layer (e.g., MgF2, n = 1.38) through a 4-mm diameter radial mask on a glass substrate using a commercial optical coating machine. W i t h / = 250mm, A = 779nm, Wo = 6 mm and a = 5 mm, they obtained a dark region with axial length 5.8 mm and radial diameter 46|Jim, and found that around the dark focal point the intensity is well approximated by a harmonic potential in the axial direction and a fourth-power potential in the radial direction. Soon afterwards, Ozeri, Khaykovich, Friedman and Davidson [2000] used a binary phase element to generate a LHB with a larger dark region and realized a large-volume dark optical trap for cold atoms. Recently, Yin, Gao, Wang, Long and Wang [2002] proposed using two new ;r-phase plates (figs. 20a,b) to generate the LHBs. Figure 21 shows the 2D intensity contour distribution of the focused LHBs generated in this way; the wavelength, A = 0.5012 jim, is the averaged value of two Ar+ laser lines (A = 0.488 [im and 0.5145 |im). The experimental setup is shown in fig. 20c. For a JTT-phase plate with three phase zones (0, jr, 0), the amplitude distribution of the LHB near the focal point of the l e n s / is given by 1
E(p,u)=
-—^^
1-e-i
/ rexp(V)Jo(pOexp (-2^^^^)
^^
0
-2 j — ^
(3.59) / rexp(V)Jo(pOexp ("2^^^ ; ^^'
where In f D\
Ijt f D^^
are the simplified radial and axial coordinates in the imaging space, respectively. D is the effective diameter of the incident beam, r is the radial coordinate on the lens plane, / is the lens focal length, and /, a and b are the normalized radii of the three zones of the phase plate. When Z? = 0, eq. (3.59) gives the amplitude distribution of a ;r-phase plate with two phase zones (>7r, 0). The corresponding intensity distribution is I{p,u)=\E{p,u)\\
(3.61)
Figure 21 shows that a closed LHB can be generated around the focal point of the lens and the 3D hollow space can be compressed or expanded by changing
154
[3, §3
Generation of dark hollow beams and their applications
(a)
(b) LHB
GLB
rQ\ ^ ^
7i-Phase Plate
Lens
Fig. 20. Structure of :n"-phase plates with radial jr-phase distribution: (a) two-zone structure {jt, 0); (b) three-zone structure (0, IT, 0); (c) principle of experimental setup for generating a localized hollow laser beam. ^=0^5012^ RA=0.02 a=1 vb=0.62
-3000 -2000 -1000 (a)
0
1000
axial distance (urn)
2000
-3000 -2000 -1000
3000 (b)
0
1000
2000
3000
axial distance ([.im)
Fig. 21. (a) Contours of 2D intensity distribution of a localized-hollow beam near the focal point of the lens produced by a jr-phase plate with two-zone structure (JT, 0). (b) Contours of 2D intensity distribution of a localized-hollow beam near the focal point of the lens produced by a jr-phase plate with three-zone structure (0, Ji, 0).
the focal length / of the lens (or by changing the effective diameter D of the incident beam). When the focal length/ is reduced from/ = 250mm to / = 50 mm, the radial dark spot size (DSS) of the localized hollow beam is
3, § 3]
Generation of dark hollow beams
155
decreased by a factor of 5, whereas its axial DSS is reduced by a factor of 25. The relationship between the relative aperture RA (= D/2f) and the DSS in the radial and axial directions of the LHB is derived from eqs. (3.59)-(3.61); the following relationship for the J^r-phase plate with two phase zones is deduced: DSSLl, « 0.784 {'-]
m,
(3.62)
Jim,
(3.63)
,2
DSSfi,
« 2.948(^1
whereas for the jr-phase plate with three phase zones, it becomes DSS<,1, « 1.634 (^l^
iim,
(3.64)
DSSfi,
jxm.
(3.65)
«^ 5.864 (^^)
Equations (3.62)-(3.65) show that the radial DSSradiai is proportional to the focal length / of the lens and inversely proportional to the diameter D of the incident beam, while the axial DSSaxiai is proportional to / ^ and inversely proportional to D^. This shows that the 3D dark trapping volume can be easily changed by altering the relative aperture RA (= D/2f). Moreover, the DSS^^^ of the three-zone j^-phase plate is larger than the DSS^^^ of the two-zone JT-phase plate. This tells us that a ;r-phase plate with more zones should be used in order to obtain a LHB with a larger dark volume. 3.8. Method of azimuthal-distributed Jt-phase plate When a DHB is tightly focused by a lens, the intensity distribution of the focused DHB near the focal point will usually be a Gaussian, not a doughnut, due to the lens diffraction and constructive interference effect. Wang, Fang, Wang, Feng and Wang [1992] proposed a method using azimuthal-distributed :^-phase plates to generate a TEMQ^ doughnut beam near the focal point of a lens. Beijersbergen, Coerwinkel, Kristensen and Woerdman [1994] used a spiral phase plate to convert a TEMoo Gaussian beam at A = 633 nm into a helicalwavefront beam. Turnbull, Robertson, Smith, Allen and Padgett [1996] also used a spiral phase plate to realize the transformation of fundamental HG modes into LG modes at millimeter-wave frequencies. Recently, Yin, Gao, Wang, Long and Wang [2002] analyzed both radial and axial intensity distributions of a focused
156
Generation of dark hollow beams and their applications
[3, § 3
beam that passed through two azimuthal-distributed jr-phase plates, and found that irrespective of whether the incident beam is Gaussian or doughnut (or other DHB), the radial intensity profile is always doughnut-like at any z-plane, and the axial intensity profile is always zero on the z-axis, i.e., I(r=0,z)
= 0.
(3.66)
Such a beam may be called a "focused hollow beam" (FHB). The principle of generating a focused hollow beam using an azimuthaldistributed JT-phase plate can be understood as follows: when a TEMoo-mode Gaussian beam passes through an azimuthal-distributed JT-phase plate (figs. 22a,b) and then is focused by a lens with a focal length/, a focused hollow beam will be formed behind the lens due to completely destructive interference at the beam center from all of the 0-phase and JT-phase elements on the phase plate. The electric field distribution of the FHB generated with the experimental setup shown in fig. 22c can be described by
E{x,y,z)=
-—-y-exp
X
.i{\u
2.T
1
/
/ exp{~i [upcosiO- 9o) +
0
{up^+(!>]}pdddp,
Ro/R
(3.67) where 2JT
fR\
IJTR
iJiR
Here A is the diffraction constant,/ is the focal length of the lens, R is the radius of the lens, RQ is the radius of the small hole at the center of the lens, A is the laser wavelength, ^o is the initial phase, and 0 is the phase distribution of the JT-phase plate. Figure 23 shows the 2D intensity distribution of the FHB in the focal plane of the lens. The DSS of the FHB can be controlled by changing the focal length of the lens or the diameter D of the incident beam. DHBs may also be generated by other optical methods; examples are a hybrid method using two axicons and a JT-phase plate (Machavariani, Davidson, Hasman, Blit, Ishaaya and Friesem [2002]), a space dark-soliton method (Tikhonenko and Akhmediev [1996], Anastassiou, Pigier, Segev, Kip, Eugenieve and Christodoulides [2001]), an optical vortex-state method (Mamaev,
157
Generation of dark hollow beams
3, § 3 ]
T GLB -D-D
i ;r-Phase Plate (c)
/
Lens
Fig. 22. Structure of jr-phase plates with angular phase distribution: (a) phase increasing from 0 to In continuously; (b) phase evenly divided into 32 parts on circle; (c) principle of experimental setup for producing a focused dark hollow beam.
1.0n
12 -9 -6 -3 0
(a)
3
6
9
-3 0 3 y(|im)
12
^(^^^
(b)
Fig. 23. 2D intensity distribution of the focused hollow beam on the focal plane [i.e., the {x,y) plane] of the lens for the jr-phase plates shown in (a) fig. 22a and (b) fig. 22b. Here RQ/R = 0.05, and
KA = R/F-0.2.
Saffinan and Zozulya [1996, 1997]), a focused rotating Gaussian-beam method (Friedman, Khaykovich, Ozeri and Davidson [2000]), and others (Yu, Lu and Harrison [1998], Chen, Lan and Wang [2001]).
158
Generation of dark hollow beams and their applications
[3, § 4
§ 4. Classification of dark hollow beams Since Laguerre-Gaussian beams form a complete orthonormal basis set for paraxial light beams, an arbitrary DHB may be written as a superposition of Laguerre-Gaussian modes if the electric field amplitude and phase of the DHB are known. However, for the convenience of later discussions on DHB applications, DHBs can be roughly classified into ten different types according to their radial intensity distribution. These ten types of DHBs, their basic characteristics and their areas of applicability are described briefly in the following. 4.1. Hollow Gaussian beam (HGB) The simplest hollow laser beam can be generated by blocking the central part of a Gaussian beam with an opaque disc or by using a hollow mirror with a small hole in its center to reflect a Gaussian beam. Such a hollow beam may be called hollow Gaussian beam (HGB). Because of the power loss, the conversion efficiency is low. The radial intensity distribution of the central part of the HGB is determined by Fresnel or Fraunhofer diffraction of the disc (or the small hole), while the distribution of the remaining part of the HGB is still Gaussian. HGBs have been applied, for instance, in studies of third-harmonic generation (THG) in a ring pump beam (Glushko, Kryzhanovsky and Sarkisyan [1993]), of multiphoton ionization of xenon atoms (Peet and Tsubin [1997]), of dark magneto-optical traps (dark MOT) (Ketterle, Davis, Joffe, Martin and Pritchard [1993]). 4.2. Laguerre-Gaussian beam (LGB) The LGB is a divergent hollow beam, which can be produced fi-om a HermiteGaussian beam by the mode-converter method (§3.3), or from a Gaussian beam by the computer-generated-hologram method (§3.5.2). The electric field amplitude of a linearly polarized LGB propagating in the z-direction is given in cylindrical coordinates by 2p\
Po
^
r2 •-
1-^
wHz) xx;i
2r2
w^(z)
exp(i/0),
- - -1"^^ 2r'
wHz)
(4.1)
3, § 4]
Classification of dark hollow beams
159
where PQ is the beam power, / {p) is the azimuthal (radial) mode index, w{z) = Wo [l-\-(Xz/jrcol)^] is the beam waist at z (WQ being the beam waist at z = 0), and Lj, is the generalized Laguerre polynomial. Note that the radial mode index p is related to the number of concentric rings (p + I) in the intensity cross-section, and the azimuthal mode index / describes the charge of the phase singularity. Simpson, Dholakia, Allen and Padgett [1997] pointed out that a linearly polarized LGB possesses an orbital angular momentum (0AM) of Ih per photon. Including spin angular momentum, the total angular momentum (TAM) per photon for a LGB is then TAM = {Ua,)h,
(4.2)
where Oz = 0 (±1) for linearly (circularly) polarized light. However, for a tightly focused beam the polarization state is no longer well defined, and the total angular momentum of the beam is given by -11
h, (4.3) 7—r + 1 2p + / + 1 Recently, the intrinsic nature of the 0AM of a LGB and and the extrinsic nature of its rotational firequency shift were explained and measured by O'Neil, Mac Vicar, Allen and Padgett [2002], Courtial, Dholakia, Robertson, Allen and Padgett [1998] and Courtial, Robertson, Dholakia, Allen and Padgett [1998]. Their experimental results agree with eq. (4.3), and the rotational frequency shift is given by TAM = I+Oz + OA
Ao) = QJ = Q{l + Oz\
(4.4)
where Q is the angular velocity between source and observer. The LGBs form a complete set of optical modes. An arbitrary light beam may be represented as a superposition of LGBs. Recent extensive studies of LGBs have not only revealed their ftindamentally important properties but have also led to many useful applications such as optical tweezers and optical spanners for manipulating and controlling microscopic particles (including nanoparticles, biological cells and neutral atoms). 4.3. Higher-order Bess el beam or Bessel-Gaussian beam (BGB) A high-order Bessel beam is a diffraction-free, hollow laser beam (except for the zeroth-order Bessel beam), and its field amplitude is given by Ei(r, 0, z) = —~ Qxip{ikzz)Ji(krr) exp(i/0), w(z)
(4.5)
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Generation of dark hollow beams and their applications
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where ^ is a normalization constant, k^ and k^ are the transverse and longitudinal components of the wavevector k (k = In/X = y/kfTlc^), and Ji is the /th-order Bessel beam with azimuthal index /. It is impossible to produce an ideal Bessel beam because of the infinite amount of energy carried by the beam. Practically, a Gaussian profile should be used to modulate the Bessel beam, and thus form a real Bessel-Gaussian beam (BOB) (Bagini, Frezza, Santarsiero, Schettini and Spagnola [1996]). BGBs do not carry an infinite amount of energy and can be realized experimentally by the geometric optical method (§3.2.2), the optical holographic method (§3.4), computer-generated holographic method (§3.5.3), and others. The electric field amplitude of a linearly polarized /th-order BGB is given by A f Ei{r,(t),z) = —— Qxp(ik,z)Ji(krr)Qxp(il(l))exp( W{Z)
\
P j W'^^Z)^
The zero central intensity of high-order BGBs originates from the phase singularity of the charge / associated with the azimuthal phase term exp(i/^). The radius of the inner ring, r/ = pi/kr, is determined by the position pi of the first maximum of the /th-order Bessel fiinction and increases with the order /. On the other hand, the transverse intensity distribution of a BGB is always a ring of radius kr, independent of the order /. It is only the azimuthal phase term exp(i/0) that distinguishes BGBs of different orders. Similar to the high-order LGB, a high-order BGB with mode index / carries orbital angular momentum Ih per photon. A detailed discussion of the optical angular momentum of the BGB and related applications can be found in a recent paper by Volke-Sepulveda, Garces-Chavez, Chavez-Cerda, Allen and Dholakia [2002]. Owing to their propagation-invariant feature, BGBs have been explored in light guiding of microscopic particles (including neutral atoms, molecules, biological cells), laser collimation of cold atoms, and atom-optical lithography. 4.4. TEMli doughnut hollow beams (DHB) The TEMQ^ (/ = 1,2,...) doughnut beam is a divergent hollow beam and can be generated by transverse mode selection from the laser cavity (§ 3.2.1) or by the computer-generated-hologram method (§3.5.1). The electric field amplitude of the TEMQ^ doughnut beam is given by 2/+ip
r
^(^^'-)=\/^^;7Tr(^exp
^2
w\z)\
1
exp(i/0),
where P and w{z) are the power and radius of the TEMJj doughnut beam.
(4.7)
3, § 4]
Classification of dark hollow beams
161
In fact, TEMQ/ doughnut beams are LGBs with mode index /? = 0; only because of their special generation schemes and extensive applications do we regard them as a different kind of hollow beam. So the TEMQ^ D H B S also possess orbital angular momentum Ih per photon. Similar to LGBs, the TEMQ; D H B S have been used in optical tweezers and optical spanners to confine and manipulate microscopic particles. 4.5. LPoi-mode hollow beam (LPHB) The LPo/-mode (/ = 1,2,3,...) output hollow beams are produced by the hollowfiber method and are divergent. By using a microscopic objective to collimate the LPoi-mode output beam, a well-collimated doughnut-like hollow beam with a divergent angle about six times smaller than that of the TEMQ^ doughnut beam can be produced (assume the other beam parameters are the same). The electric field amplitude of the LPoi-mode output beam can be written approximately as E(r, 0, z) = \\ —
J— exp
'w\z)
exp(i0),
(4.8)
and w(z) = woJl+k2[^Y
(4.9)
where PQ is the laser power, and ki and k2 are two modifying parameters determined by fitting the numerical results from the exact Fresnel diffraction integral to the beam model given by eq. (4.8) or by fitting the experimental data to the model. The LPoi-mode output hollow beam possesses an orbital angular momentum of h per photon. If a LPo/-mode (/ = 2,3, •) in a micron-sized HOP is selectively excited and output, a higher-order LPo/-mode output hollow beam can be generated with orbital angular momentum Ih per photon. LPo/-mode output hollow beams have been used in laser guiding, collimating, funneling, trapping and cooling of neutral atoms, and may also be used in applications involving optical tweezers and spanners for microscopic particles. 4.6. Double-Gaussian-profile hollow beam (DGHB) This type of hollow laser beams has a double Gaussian radial-intensity distribution and can be generated by the geometric optical method with a single
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Generation of dark hollow beams and their applications
[3, § 4
or a pair of conical prisms (§ 3.2.1). The intensity profile of the DGHB is given by I{r,z)= — I —-—-exp Jt [w^(r,z)
^7—T f' wf(r,z) J J (4.10) where w±(r,z) are the waists of two modified Gaussian beams at coordinate (r,z). The ring width and width-radius ratio (WRR) of a DGHB can vary in a large range. Such ring-shaped hollow beams may also possess optical angular momentum, but a rigorous expression for the orbital angular momentum is yet to be found. The ring width of the DGHB can be made very small (e.g., w = 50 |im; WRR = ^ (Arlt, Kuhn and Dholakia [2001]), and the width-radius ratio (WRR) of the DGHB can be smaller than j ^ . Such a hollow beam will have an enhanced intensity gradient, which is particularly usefiil in laser cooling and trapping of neutral atom based on intensity-gradient forces. [
J-—+-2"—-exp w;(r,z) J wf(r,z) [
4.7. Focused hollow beam (FHB) When a hollow laser beam is focused by a lens, the beam profile near the focal point will usually approach a Gaussian due to the lens diffraction and the constructive interference effect. However, when a hollow beam, even a Gaussian beam, passes through a special ;r-phase plate and is then focused by a lens, the focused beam will have a hollow beam profile near the focal point and also elsewhere as a result of the destructive interference at the beam axis. The electric field amplitude of the FHB can be described by eqs. (4.8) and (4.9). Because the FHB has a very small DSS in its focal plane, it may be used to focus an atomic beam and form an atomic lens, which should be usefiil in atom lithography It may be possible to study adiabatic compression (heating) and expansion (cooling) of atoms in the FHB as well as prepare an ail-optically cooled and trapped Bose-Einstein condensate (BEC) in a pair of crossed FHBs (see fig. 39b, below). 4.8. Localized hollow beam (LHB) A localized hollow beam (LHB) can be produced by the geometric optical method (§ 3.2.3), the computer-generated-hologram method (§ 3.5.4), or by the method of a radial-distributed JT-phase plate (§3.8). The intensity distribution of the LHB is usually complicated. The LHB produced by the radial-distributed
3, § 4]
Classification of dark hollow beams
163
:/r-phase plate can be calculated numerically from eqs. (3.59)-(3.61). Calculations by Ozeri, Khaykovich and Davidson [1999] as well as by us show that around the dark focal point, the axial and radial intensity can be approximated by I{z)^kxz\
I{r)^k2r\
(4.11)
where k{ and k2 are two fitting coefficients. Since the LHB has a 3D closed dark hollow region and a large intensity gradient, it can be used to trap microscopic particles and neutral atoms. In particular, the intensity gradient near the focal point of a LHB may be used to efficiently cool neutral atoms by LHB-induced Sisyphus cooling (see § 6.2), which may be useful for realizing an optically trapped and cooled BEC. 4.9. Higher-order Mathieu hollow beams (MHB) There are three kinds of diffraction-free light beams: first, the well-known uniform plane waves with an infinite plane wavefront; second, Bessel beams with circular symmetry; and third, the recently defined Mathieu beams with elliptical symmetry, being the elliptical generalization of Bessel beams. Just as Bessel beams are related to the TE and TM modes of circular waveguides, the Mathieu beams are related to the TE and TM modes of elliptical waveguides. The holographic generation of high-order Mathieu beams and its basic characteristics were studied both theoretically (Chavez-Cerda, Gutierrez-Vega and New [2001]) and experimentally (Chavez-Cerda, Padgett, Allison, New, GutierrezVega, O'Neil, Mac Vicar and Courtial [2002]). From these works, the definition, parameters, propagation properties and orbital angular momentum (0AM) of high-order Mathieu beams are summarized as follows: (1) Definition: the electric field amplitude of high-order Mathieu beams in elliptical coordinates (^, r],z) is given by E(^,r],z;q)'^^
= [^m-^e,m(?;^)Ce,m(^;^) + L5mAm(§;^)'^e,m(^;^)]
X exp(i^^z), (4.12) where A^ and 5 ^ are two coefficients; Je,m and Jem are mth-order radial Mathieu functions of the first kind, which are the even and odd solutions of the eccentric radial (modified) Mathieu equation, respectively; and CQ^m and SQ^m are two wth-order angular Mathieu functions of the first kind, which are the even and odd solutions of the eccentric angular (ordinary) Mathieu equation, respectively.
164
Generation of dark hollow beams and their applications
[3, § 4
(2) Parameters: Mathieu beams are characterized by the "elhpticity" parameter q, and the integer m, which is termed the mode order (index) or charge. Bessel beams, which are circularly symmetric, are the limiting case of Mathieu beams at ^ = 0. The center of a Mathieu beam then contains a so-called optical vortex of charge m. A vortex is a screw dislocation in the phase front of the beam, the pitch of which is m wavelengths, corresponding to a phase change of 2mK around the vortex center. For a moderate increase in q, the mth-order vortex separates into m vortices of order 1, all of which are positioned close to the beam axis. As q increases beyond a critical value given by ^c ^ w^ - 1, numerous additional vortices appear in various positions across the beam. It can seen from fig. 17 (above) that for ellipticity parameters ^ < 10, high-order Mathieu beams are multiple-ring hollow beams. (3) Propagation properties'. Ideal Mathieu beams are propagation-invariant. But generated Mathieu beams (see fig. 17) are diffraction-free only over a finite distance Zmax, the so-called propagation-invariant region, behind which the center of the beam rapidly becomes fainter. This is a well-known effect due to the finite size of the beam. The propagation-invariant distance is given by ^max = - ^ , (4.13) tant7 where r is the radius of the beam in the plane of the aperture and 6 is the inclination angle of the beam's plane-wave components with respect to the optical axis. (4) Orbital angular momentum (0AM): Due to the complex phase structure in high-order Mathieu beams, it is not possible to find a closed-form expression for the angular momentum of Mathieu beams. However, the OAM per photon in high-order Mathieu beams, bL^, can be calculated numerically by _n^lAx,y)\E(x,y)\^AxAy
--
n\Eix,y)\^dxAy
'
^^''^^
where lz = \r xpi\\s the OAM flux, and^i is the momentum density. This shows that the OAM per photon in the beam, bL^, can be written as the integral over the local OAM per photon, 64, weighted by the normalized energy density (or intensity). Therefore, the OAM per photon in highorder Mathieu beams can be calculated quite easily from the values of E{x,y) at a finite number of points on a beam cross-section.
3, § 5]
Applications ofDHBs in modern optics
165
4.10. Double-rectangular-profile hollow beams (DRHB) A double-rectangular-profile hollow beam (DRHB) would be an ideal DHB with an intensity distribution (see fig. 24) r0 I(r) = ^ / o 0
r
Ar = b - a <€, a,
(4.15)
r>b,
but such a beam has not been generated experimentally yet. This kind of DHB is potentially usefiil in atom-optics studies; for example, the DRHB will lead to better Sisyphus cooling for neutral atoms. Further research may lead to a practical way of generating DRHBs.
-b-
r —•
Fig. 24. Ideal dark hollow beam with double-rectangular intensity profile.
§ 5. Applications of DHBs in modern optics 5.1. Optical trap for microscopic particles (optical tweezers) Ashkin and Dziedzic [1971] developed an optical technique for trapping microscopic dielectric particles. The technique uses a tightly focused laser beam that creates a high electric field gradient near the focal point and exerts a force on dielectric particles inside the laser beam towards the beam focus. If the numerical aperture of the focusing optics and the laser power are sufficiently high, the gradient force can overcome the forces due to Brownian motion, gravity, and forward-scattering light force, and result in a three-dimensional optical trap for the particles. They trapped a micron-sized glass sphere in three dimensions and referred to this trap as optical tweezers (Ashkin, Dziedzic, Bjorkholm and Chu [1986]).
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Generation of dark hollow beams and their applications
[3, § 5
Optical tweezers offer a non-contact, intrinsically sterile method for manipulating microscopic objects and can be incorporated into a conventional microscope arrangement. The same objective lens can be used to focus the laser beam and view the trapped objects. Such single-beam traps are widely used in biology for manipulating biological and dielectric particles (Svoboda and Block [1994]). Other applications can be found in aerosol science (Ashkin and Dziedzic [1975]) and chemistry (Misawa, Kitamura and Masuhara [1991]). The total force exerted on the dielectric particles by the focused laser beam has two components: the gradient force component due to the electric field gradient and the light-scattering force component due the absorption of photons by the particles. The gradient component is directed towards the beam focus and constitutes the trapping force. The scattering force is directed along the beam propagation direction and is detrimental to the optical trap if the laser beam is propagating downward, as is usually the case in a conventional microscope configuration. In geometric optics, the gradient force can be understood in terms of the light refi-action upon transmission through the dielectric particle. When the particle is on the beam axis, the axial trapping force is produced from the refraction of the off-axis rays within the beam. On-axis rays are detrimental to axial trapping as their back-scattered component results in a force on the particle in the beam propagation direction. Consequently, if a light beam with a doughnut-type mode instead of a Gaussian mode is used, the trapping efficiency is expected to be higher since the on-axis light intensity is zero in a doughnut beam (Ashkin [1999], Simpson, McGloin, Dholakia, Allen and Padgett [1998], Chang and Lee [1988]). Furthermore, particles are trapped near the high-intensity focal region of the beam and thus are susceptible to optical damage through heating from light absorption. These considerations lead to the adaptation of optical tweezers by using hollow laser beams (Sato, Harada and Waseda [1994], Gahagan and Swartzlander [1996], Simpson, Allen and Padgett [1996]). There are three useful features of using hollow beams to form optical tweezers. First, as mentioned above, the null intensity of the on-axis region reduces the light-scattering force, and under the same operating conditions, the optical damage of trapped particles due to heating, which is particularly important for trapping biological samples, is minimized (Simpson, McGloin, Dholakia, Allen and Padgett [1998]). Second, optical tweezers constructed with a Gaussian beam can only trap microscopic particles having a relative refractive index greater than unity, while optical tweezers made from a hollow beam permit trapping of the particles in the dark center near the focal point, therefore the relative refractive index of the particles can be smaller than unity (Gahagan and Swartzlander
3, § 5]
Applications ofDHBs in modern optics
167
[1999]). Third, the special phase structure of hollow laser beams is shown to be associated with the orbital angular momentum of photons and can induce rotation in absorbing particles through angular momentum transfer from the hollow beam (Allen, Beijersbergen, Spreeuw and Woerdman [1992]). In an electric field E, the gradient force exerted on a particle with polarizability a is given by (Ashkin and Dziedzic [1971]) Fg = - ^ « o « V ^ ^
(5.1)
where no is the refractive index of the medium in which the particle is suspended. The electric field of a hollow laser beam propagating along the z-direction can be written as (Chang and Lee [1988])
|W(Z)|2
expji^z + i ^^
1R{Z)
-
|>V(Z)|2
- i0(z)},
(5.2)
where a is a constant, r is the radial distance, k is the wavevector, and >v(z), R{z) and 0(z) are the beam width, the beam curvature, and the phase term, respectively. Several approaches have been used to model optical tweezers, such as electromagnetic-wave representations (Barton and Alexander [1989], Wright, Sonek and Berns [1994], Ren, Grehan and Gouesbet [1994]) and ray-optics representations (Ashkin [1992], Roosen and Imbert [1976]), as well as hybrid models with elements from both methods (Gussgard, Kindmo and Brevik [1992]). The minimum gradient force F^ required to hold a particle of density Pp and diameter d against gravity and thermal motion is given by (Feigner, Muller and Schliwa [1995]) F^=F,^\jt
2kT (pp - p o ) d'g+—,
(5.3)
where F^ is the light-scattering force, po is the density of the surrounding medium and T is the ambient temperature. The gradient force exerted on a dielectric particle depends on the index of refraction of the particle, rip and that of the surrounding medium, no. It is instructive to consider the picture of optical ray refi*action and understand direction of the gradient force from the conservation of linear momentum. As shown in figs. 25 and 26, depending on the incident angle and location of the particle, the gradient force behaves differently for low-index particles (n^ < no) and high-index particles (n^ > no). A low-index particle is trapped above the
168
Generation of dark hollow beams and their applications
[3, § 5
6
(b)
(a)
Fig. 25. (a) Gradient and scattering forces exerted on a low-index («p < n^) particle by an arbitrary ray. Fg, gradient force; Fg, scattering force, (b) A low-index particle at the equilibrium position in a focused hollow beam. Zf, beam focus; Zp, trap center.
(b) Fig. 26. (a) Gradient and scattering forces exerted on a high-index particle by an arbitrary ray. (b) A high-index particle at the equilibrium position in a focused hollow beam. Zf, beam focus; Zp, trap center.
beam focal point (fig. 25b), while a high-index particle is trapped right below the beam focus (fig. 26b). A typical experimental setup for hollow-beam optical tweezers is shown in fig. 27a; the equilibrium positions for trapped particles are shown schematically in fig. 27b. Sato, Harada and Waseda [1994] succeeded in trapping metal particles of radius 7|im in two dimensions with a hollow TEMQ^ beam. Gahagan and Swartzlander [1998] used a hollow beam to trap low-index microparticles: water droplets («p = 1.33, Pp = 1.00 g/m^, radius 2-40 jJim) in acetophenone
169
Applications ofDHBs in modern optics
5]
M
M>
Particle with low refractive index
Objective lens Sample cell
Particle with high refractive index White light
Hollow beam (a)
(b)
Fig. 27. (a) Experimental setup for hollow-beam tweezers, (b) Schematic diagram of the trap center showing that a high-index particle is trapped near the beam focus while a low-index particle is trapped above the beam focal point.
(no = 1.53, po = 1.02 g/m£) and hollow glass spheres (radius 5-15 \im) in acetophenone. In a later experiment, they trapped simultaneously both low-index and high-index microparticles (Sato, Harada and Waseda [1994]). Simpson, Allen and Padgett [1996] used a hollow beam with a pure Laguerre-Gaussian mode to trap silica spheres (1-5 |jim). As the off-axis light rays contribute more effectively to the gradient force, they observed higher axial trapping efficiency in comparison with a Gaussian beam trap with same operating parameters. More recently, Reicherter, Haist, Wagemann and Tiziani [1999] have used doughnut-beam tweezers to simultaneously trap three polystyrene particles in two dimensions. 5.2. Optical motor for microscopic particles (optical spanners) Chang and Lee [1988] studied the radiation force and torque exerted on a stratified sphere in a circularly polarized TEMQJ laser beam; they found that there is a radiation torque on a transparent hollow sphere in the doughnut-beam trap, which can be used to rotate a microsphere. Allen, Beijersbergen, Spreeuw and Woerdman [1992] showed that it is possible to produce an optical beam with orbital angular momentum. In particular, a Laguerre-Gaussian beam (LGB) with an azimuthal-mode index / has a phase dependence of exp(i/0) and each photon in the beam carries an orbital angular momentum Ih. Although it is well known that photons carry intrinsic spin angular momentum ±h, this was the first time that properties of the orbital angular momentum of photons were revealed and clarified. In general, a hollow beam can be described as a superposition of LGB modes and its orbital angular momentum will be the weighted average of the orbital angular momentumfi*omthe constituent LGB modes. Including the spin angular momentum, which is ±h for circularly polarized
170
Generation of dark hollow beams and their applications
[3, § 5
Fig. 28. Pattern aberrations resulting from misalignment of a Gaussian beam (GB) relative to a LG beam (LGB) of order / = 3, for three different misalignments: (a) w^^ = JCOQ^^ which makes it difficult to resolve the pattern at the focus; (b) focus displaced longitudinally by 5ZR (Rayleigh range of the LG beam); (c) focus displaced transversely by a one LG beam radius 0)^^. (From MacDonald, Paterson, Volke-Sepulveda, Arlt, Sibbett and Dholakia [2002], Science 296, 1101, reprinted with permission.)
light and 0 for linearly polarized light, gives for the total angular momentum of a circularly polarized LGB (Padgett and Allen [2000])
J,=L, +
S,^{l±\)h.
(5.4)
When absorbing microparticles are trapped in optical tweezers made of a LGB, the photon angular momentum will be transferred to the particles and causes the particles to rotate. Such optical tweezers can therefore be called an optical spanner. By changing the sense of the circular polarization, the properties of the orbital angular momentum of the LGB can be characterized. Transfer of orbital angular momentum was demonstrated by He, Friese, Heckenberg and Rubinsztein-Dunlop [1995] and Friese, Enger, Rubinsztein-Dunlop and Heckenberg [1996]. They used micron-sized particles of highly absorbing ceramic powder. Due to the large scattering force, the particle could only be confined in two dimensions near the intensity null on the beam axis; the restraint in the axial direction was provided by the microscope cover-slip. Quantitative measurements of the torque, the amount of light absorbed, and the angular momentum per photon were not possible. Nevertheless, when the particles were trapped with a LGB of azimuthal mode index / = 3, rotation of the particles was observed and attributed to the transfer of orbital angular momentum. In a later refinement, Simpson, Dholakia, Allen and Padgett [1997] used partially
3, §5]
111
Applications ofDHBs in modern optics
absorbing Teflon particles suspended in alcohol, enabling them to compare the transfer of both spin and orbital angular momentum in the same beam. As these particles were largely transparent, the gradient force was sufficient to trap them in three dimensions and hence isolate them from the sample cell walls. They found that a power of a few tens of mW resulted in a rotation frequency of a few Hz and furthermore, by changing the sense of circular polarization of the beam they could change the total angular momentum transferred from 0 to 2h per photon. When a Laguerre-Gaussian beam interferes with a Gaussian beam, the azimuthal phase variation of the LG beam is transformed into an azimuthal intensity variation of the spiral pattern with / nodes. If the relative phase of the two beams changes periodically in time, the interference pattern under different conditions (see fig. 28) will rotate around the beam axis. Thus if microparticles are trapped in the spiral arms of the pattern, the particles will rotate when the interference pattern is rotated by changing the relative optical path length. Recently (fig. 29), several microscopic particles such as silica microspheres, 0.15 s ^
0.00 s
y
:-^-y
. xJ
:1^ ,
^S
0.0 s
0.7 s .
0.0 s
5,0.
% (a)
. -^ (b)
Ifl
^x ^ /
Fig. 29. (a) Rotation of 2D trapped objects in a LGB (/ = 2) interference pattern with a plane wave. (A) Rotation of two trapped 1- ^im silica spheres with a rate of 7 Hz. (B) Rotation of a 5- |im-long glass rod. (C) Rotation of a Chinese hamster chromosome. The elapsed time t (in seconds) is indicated by the scale at the top of each sequence of images, (b) Rotation of three trapped silica spheres with a diameter of 5 |im in a LGB (/ = 3) interference pattern with a plane wave. The slight deformity (indicated by arrow) on one of the spheres allows us to view the degree of rotation of the structure. (From MacDonald, Paterson, Volke-Sepulveda, Arlt, Sibbett and Dholakia [2002], Science 296, 1101, reprinted with permission.)
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Generation of dark hollow beams and their applications
[3, § 5
microscopic glass rods and propellers, and chromosomes inside the spiral pattern were trapped in this way, and particle rotation frequencies in excess of 5 Hz were observed (Paterson, MacDonald, Arlt, Sibbett, Bryant and Dholakia [2001], MacDonald, Paterson, Sibbett and Dholakia [2001], MacDonald, Paterson, Volke-Sepulveda, Arlt, Sibbett and Dholakia [2002], Galajda and Ormos [2002], MacDonald, Volke-Sepulveda, Paterson, Arh, Sibbett and Dholakia [2002]).
5.3. Light guiding light (optical hosepipes) In recent years, there has been considerable progress in manipulating the dispersive properties of an absorbing medium in which one optical field can control the spatial propagation of another optical field (Moseley, Shepherd, Fulton, Sinclair and Dunn [1995]). This effect can be understood with reference to a model of the refi-active index for a V system. Moseley, Shepherd, Fulton, Sinclair and Dunn [1995] represented the rubidium D lines as a V-type threelevel atomic system, for the case of a strong pump beam centered on the 1 -^ 2 transition and a weak probe beam near the 1 ^> 3 transition; the resulting refractive index is given by
^^l-^^l'l 4^fi0 V
'/h^\
^
(55)
2(l+///sa,)y 4 2 + [ l ( r , 3 + r , 2 ) ] ' '
where /in is the electric dipole moment of the 1—^3 transition, A^ is the density of Rb atoms interacting with the pump beam, A^i =N {\- (///sat)/ [2(1 + ///sat)]} is the population density of level 1, zl = cOp - con is the probe detuning, and / and /sat are the pump beam intensity and the saturation intensity, respectively. When a probe beam is tuned to the red (negative A) of the longest-wavelength Rb D2 line (the probe beam experiences a refractive index greater than unity), and the doughnut beam is tuned to pump the longest wavelength of the Di transition, the probe beam is guided into the dark center of the doughnut beam. Therefore, in a three-level atomic medium coupled by a pump field and a probe field, the absorptive and dispersive properties of the probe laser beam depend on the pump laser intensity and the pump frequency detuning from the atomic transition frequency. For a pump laser with spatial intensity variation, the probe absorption and dispersion become spatially dependent. With a DHB acting as the pump laser, the probe beam may be confined and guided inside the DHB under appropriate conditions, thus realizing an "optical waveguide" written by a laser beam in an atomic medium.
3, § 6]
Applications ofDHBs in atom optics
173
Truscott, Friese, Heckenberg and Rubinsztein-Dunlop [1999] demonstrated that a LGB can guide a Gaussian light beam (light guiding light) via the cross-dispersion effect in rubidium atoms. Their experimental results showed that the LGB acting as a resonant coupling field creates a spatially dependent dispersion profile for a weak Gaussian beam that can confine and guide the reddetuned Gaussian beam along the hollow center of the LG beam. Qualitative understanding of their experimental results is provided by theoretical calculations based on the spatial dependence of the probe dispersion in a three-level V-type system. Later on, Kapoor and Agarwal [2000] and Andersen, Friese, Truscott, Ficek, Drummond, Heckenberg and Rubinsztein-Dunlop [2001] refined the model system to include hyperfine structure and Doppler broadening, and made more detailed calculations of the spatially dependent absorption and dispersion profiles. These calculations fiirther clarify the nature of the spatially dependent cross dispersion and agree qualitatively with the experimental measurements. However, quantitative comparison with experiment needs to include propagation effects, i.e., it requires calculations that include equations for the atomic polarization and Maxwell equations for wave propagation. In addition to the interesting applications of DHBs mentioned above, DHBs may also be used to explore second- or third-harmonic generation (Glushko, Kryzhanovsky and Sarkisyan [1993], Courtial, Dholakia, Allen and Padgett [1997]), to study parametric downconversion (Arlt, Dholakia, Allen and Padgett [1999]), to realize a DHB electron trap (Chaloupka and Meyerhofer [1999, 2000], and to manipulate the multiphoton ionization of atoms (Peet and Tsubin [1997]).
§ 6. Applications of DHBs in atom optics 6.1. Dipole force trap for cold atoms (atomic tweezers) When a two-level atom moves in an inhomogeneous light field, it will experience an optical dipole force; this results in an optical trapping potential U{r)=\hd\n\\+ ' '
^^^^^' \+A{6/r)
(6.1)
where d = (D(-0)^- kvz is the detuning of the laser frequency coi from the atomic resonance frequency c^a, including the Doppler shift Jwz. I{f) is the intensity distribution of the laser beam, and /§ and F are the saturation intensity and natural linewidth of the atomic transition, respectively. When the light field is
174
Generation of dark hollow beams and their applications
Laguerre-Gaussian beam
[3, § 6
Plugging beams /-C7 y^ V / /MOT
Fig. 30. Schematic illustration of the doughnut-beam trap for cold atoms.
red-detuned (d < 0), the potential is attractive, and the atoms will be attracted to the maximum of the light field. Therefore, atoms may be trapped or guided (i.e., a 2D trap) in a red-detuned Gaussian beam. When the light field is blue-detuned (6 > 0), the potential is repulsive, and the atoms will be repelled to the minimum of the light field. Therefore, atoms may be trapped or guided in a blue-detuned dark hollow beam. Blue-detuned DHBs may be used to realize various optical dipole traps for cold atoms, such as a doughnut-beam trap, a surface dipole trap, a gravito-optical trap, a single-beam dark optical trap, and so on. As early as 1986, a doughnut-beam trap (i.e., a DHB corner cube trap) for neutral ^^K atoms was proposed and studied theoretically by Yang, Stwalley, Heneghan, Bahns, Wang and Hess [1986]. In their scheme, the 2D confinement of cold K atoms is provided by a blue-detuned doughnut beam fi*om a highpower alexandrite laser. The ends of the cylindrical trap are closed by reflecting the doughnut beam back on itself with two mirrors, so forming a 3D bottle atom trap between two mirrors. Afterwards, three DHB dipole traps for cold atoms - a doughnut-beam trap (Kuga, Torii, Shiokawa, Hirano, Shimizu and Sasada [1997]), a gravitooptical surface trap (Ovchinnikov, Manek and Grimm [1997]), and a single-beam gravito-optical trap (Yin, Zhu and Wang [1998a], Ovchinnikov, Manek, Sidorov, Wasik and Grimm [1998]) - were proposed and demonstrated experimentally. In the doughnut-beam trapping scheme (fig. 30), a 3D dipole trap for cold atoms was constructed by a far blue-detuned LGB propagating along the z-direction (2D trap) and two far blue-detuned plug beams overlapping the LGB in the transverse direction. The cold ^^Rb atoms were confined in the 3D dark core of the LGB. This doughnut-beam trap confined about 10^ atoms at a temperature of -18 |iK.
175
Applications ofDHBs in atom optics
3, §6]
repumping beam
f i t
atoms in MOT w
4 ' atoms in GOST
3
W
Prism w
5..
ew laser beam
hollow beam Fig. 31. Experimental setup of the gravito-optical surface trap (GOST) for atoms.
In the gravito-optical surface trap (GOST, fig. 31), the 3D confinement of cold atoms was achieved by a far blue-detuned, nearly coUimated DHB (for horizontal confinement) and blue-detuned evanescent light with the help of the gravity field (vertical confinement). After cooling down to about 3 jiK by ID evanescentwave cooling, an ultracold atomic sample (^10^ ^^^Cs atoms) with a mean height of ^20 [im was trapped above the surface of the dielectric prism used for the evanescent-wave generation. In the proposed single-beam gravito-optical trap (GOT) scheme (fig. 32), a 3D dipole trap for cold atoms can be formed by a blue-detuned conical DHB and the gravity field, or by a blue-detuned pyramidal DHB and the gravity field (Yin, Gao, Wang, Zhu and Wang [2000]). Calculations indicate that an ultracold atomic sample with a temperature of a few |iK may be obtained with DHB-induced intensity-gradient cooling (see § 6.2) or by polarization-gradient cooling. Recently, some other DHB traps, such as a localized-hollow-beam trap with a holographic phase plate (Ozeri, Khaykovich and Davidson [1999]), a largevolume single-beam dark optical trap using binary phase elements (Ozeri, Khaykovich, Friedman and Davidson [2000]), a focused doughnut-beam trap (Webster, Hechenblaikner, Hopkings, Arlt and Foot [2000]), and a singlehollow-beam optical trap formed by an axicon (Kulin, Aubin, Christe, Peker, Rolston and Orozco [2001]), were proposed and demonstrated. Moreover, two all-optical traps with a blue-detuned DHB for preparing quantum-degenerate gases were studied by Engler, Manek, Moslener, Nill, Ovchinnikov, Schuenmann, Zielonkowski, Weidemuller and Grimm [1998]; they discussed the evaporative
176
Generation of dark hollow beams and their applications
[3, § 6
Weak repumping beam
Doughnut hollow beam Fig. 32. Scheme of a gravito-optical atom trap with a pyramidal hollow beam. Abbreviations: BSP, black square plate; PHB, pyramidal hollow beam; MOT, magneto-optical trap; GOT, gravitooptical trap. ZGOT (^MOT) is the position of the GOT (MOT). / is the focal length of the lens, which can be adjusted.
cooling and sympathetic cooling of ^^^Cs atoms in two dark optical traps and the possibility to realize an all-optical-type '^^Cs atomic BEC.
6.2. Cooling of neutral atoms (atomic refrigerator) Because of the high intensity gradient in standing-wave light or in evanescentwave light, atoms moving in these light fields will be cooled down to nearly the recoil temperature by intensity-gradient-induced Sisyphus cooling. Obviously, the intensity gradient inside a DHB can also be used to cool the atoms guided or trapped in the DHB. The idea of DHB-induced Sisyphus cooling (i.e., intensitygradient cooling, IGC) was proposed in 1997 by Yin, Noh, Lee, Kim, Wang and Jhe [1997], and analyzed for a DHB atomic fiinnel (Yin, Zhu and Wang [1998b]), DHB atomic guiding (Yin, Zhu, Jhe and Wang [1998]), a DHB-ftinnel atom trap (Morsch and Meacher [1998]), a DHB gravito-optical atom trap (Yin and Zhu [1998a]), and others (Yin and Zhu [1998b]). However, the intensity gradient inside a doughnut beam or other DHBs is far smaller than that of standingwave or evanescent-wave light, and DHB-induced Sisyphus cooling is not very efficient, as verified experimentally by Ovchinnikov, Manek, Sidorov, Wasik and Grimm [1998].
3, § 6]
Applications ofDHBs in atom optics
111
Ovchinnikov, Soding and Grimm [1995] pointed out that the 3D equiUbrium temperature of the trapped atoms in a conical GOT will be higher than that in a pyramidal GOT by a factor of 20 because of the conservation of the angular momentum L^ along the symmetry axis. This implies that efficient intensity-gradient cooling may be possible with a pyramidal-type DHB. Yin, Gao, Wang and Wang [2001] thus proposed an improved pyramidal-hollowbeam GOT (PHB GOT) to trap alkali atoms (see fig. 32). When the central part of a collimated DHB propagating upwards is blocked by a black square plate, a dark rectangular-hollow beam is produced after the plate, and the rectangular-hollow beam is focused by a lens with a variable focal length / . A divergent PHB with an improved intensity gradient is then generated. A bluedetuned plug beam is propagating transversely and overlaps with the PHB above the focal point of the lens, forming a PHB GOT right above the plug beam. In order to introduce Sisyphus cooling (i.e., intensity gradient cooling) and geometrical cooling (Soding, Grimm and Ovchinnikov [1995]), a weak, nearresonant repumping beam is propagating down and overlapping with the PHB. As cold atoms are loaded into the PHB GOT from a standard MOT and bounce inside the PHB, they experience the PHB-induced Sisyphus cooling and the repumping-beam-induced geometrical cooling. The cooling rate, the 3D-equilibrium temperature, the trapping volume and the atomic density in the PHB GOT can be controlled easily by changing the divergent angle of the PHB (i.e., by changing the focal length/ of the lens), or adjusting the position of the GOT in the PHB (i.e., by moving the plug beam up or down). The fact that the central part of the DHB in the PHB GOT is blocked by a black square plate implies that the intensity distribution inside the rectangular hollow beam will be determined by Fresnel diffraction of the square plate. Calculation reveals that the intensity distribution inside the rectangular hollow beam still has a Gaussian profile, but the intensity waist is about 200 |im when the beam waist of the collimated DHB is 10 mm (the width of the square plate is 10 mm). So the spatial intensity distribution inside the rectangular-hollow beam is compressed 50 times and the intensity gradient is greatly increased. When the rectangular-hollow beam is focused by the lens, the spatial width of the intensity distribution inside the PHB near the focal point will be compressed to a few micrometers, further increasing the intensity gradient. Therefore, more efficient Sisyphus cooling for cold atoms is obtained in this improved PHB dipole trap. Inclusion of the PHB-induced Sisyphus cooling, the repumping-beam-induced geometrical cooling, the spontaneous-emission heating as well as the recoilinduced heating from the absorption of the PHB, the plug beam and the repumping-beam photons, yields a relation for estimating the final equilibrium
178
Generation of dark hollow beams and their applications
[3, §6
35
25 -ic 20
iot
• • • •• ••**•.
0.0^^5 T o
For ^^Rb atoms
1.5
2.0 2.5 3.0
3.5 4.0 4.5
5.0
f(s) 30t^ 25 ;• 20 ^
15: 10
For ' " t s atoms •
•• •
5
(b) 0 0.0
0.5
1.0
1.5
2.0
2.5 3.0
3.5 4.0
4.5 f ).0
t(s) Fig. 33. Monte-Carlo results for the dynamic process of improved PHB cooling for (a) ^^Rb atoms and (b) '^^Cs atoms.
rms momentum prms of the cold atoms in the trap near the focal point of the PHB: 1
^1hfs
3 5 + zlhfs \hk
\2 )
sin e qr \hk
1 )
^RPB
0,
(6.2)
1 - ^PHB
where ^RPB (^PHB) is the branching ratio for the transitions of an atom after absorbing a photon from the repumping beam (the PHB); for alkali atoms, ^RPB ~ 0.60 and ^PHB ~ 0.75 (Soding, Grimm and Ovchinnikov [1995]). z\hfs is the hyperfine splitting between the two hyperfine ground states, and 6 is the trapping angle of the PHB GOT, which can be varied from a few degrees to ^60^" by adjusting the focal length/ of the lens. The results of Monte-Carlo simulations for the dynamic process of improved PHB cooling are shown in fig. 33. For the ^'^Rb atoms in fig. 33a, the simulation parameters were A^ = 200 atoms, A = 0.78 |im, TMOT = 120 |iK, ZMOT = 1 nmi, DMOT = 1mm, laser power PQ = 1000 mW, d = 1.0 GHz, 6 = 30^ yielding
3, § 6]
Applications ofDHBs in atom optics
179
~ 190|im, prms ^ 4.5 hk (T ^ 2.5 fiK), which is consistent with the result derived from eq. (6.2) (prms = 3.9hk). For ^^^Cs atoms (fig. 33b), with (^GOT)
A^ = 200 atoms, A = 0.852 |im, TMOT = 60|JiK, ZMOT = 1mm, DMOT = 1mm,
Po = 500 mW, (5 - 3.0 GHz, 6 = 30^ we obtained (ZGOT) ^ nOjixm, Prms ~ 5.3 hk (T ^ 1.8 |iK), again consistent with the result firom eq. (6.2) (Prms = 3.%hk). It can also be seen firom fig. 33 that the PHB cooling time is '-2.5 s for ^'^Rb atoms and ^1.5 s for ^^^Cs atoms, consistent with predicted resuhs (r ?^ 2.1 s and 1.2 s). This shows that in the PHB GOT, the trapped cold atoms can be cooled directly to ~2 (JiK from the MOT's temperature (60-100 \\K) by the improved intensity-gradient cooling and the repumping-beam-induced geometrical cooling. An ultracold atomic sample with a temperature of ^2 |iK and a density of 10^^-10^^ atoms/cm^ may be obtained in the PHB GOT. More recently, we also studied the dynamical process of intensity-gradientinduced Sisyphus cooling of ^^Rb atoms in a blue-detuned localized hollow beam (LHB) by using Monte-Carlo simulations. We set the following parameters: LHB power and detuning P^HB ^ 5mW and d = 10 GHz; initial momentum of the ellipsoidal MOT />MOT ^ ^^^^ (~30|iK); short and long axes of the MOT auoi = 5^m and buoi = 100 |im; radius and power of the repumping beam T^RPB = 2.5 |im and PRPB = 0.3 |iW. The resulting final temperature of cooled atoms is ^6 |iK, in good agreement with the theoretically predicted value (5.9|LiK), and the corresponding cooling time is ~0.2s. This shows that the intensity-gradient cooling effect from the LHB itself is very efficient because the LHB power used is only 5 mW and the cooling time is only 0.2 s. The Doppler shift for a two-level atom with a translation velocity v moving in a counter-propagating laser beam can be written as 4 ' G ^ - ^LG + ^LG =-kv
= -krVr " hv,,
(6.3)
where d^Q and dl^ are the radial and axial Doppler shifts, kr and k^ are the radial and axial wavevector components of the light wave, and Vr and v^ are the radial and axial velocity components of the atom. The velocity-dependent damping force exerted on the atoms from the red-detuned laser beam is F = -pu,
(6.4)
where (3 is the damping coefficient, which is related to the detuning 6, the saturation parameter s and the atomic natural linewidth F as well as the spatial intensity profile of the beam. For atoms in a LGB, the azimuthal Doppler shift
180
Generation of dark hollow beams and their applications
[3, § 6
in cylindrical coordinates, R = (r, z, 0), is given by (Allen, Babiker and Power [1994])
k.^^(x-^^\.^'P^'^'^^^ 2{z^+zl)\
{z^+zD)
{z^+zD
I (6.5) where ZR = JTVVQ/A and z = {z\^- z^)/z, t;^ is the azimuthal velocity component of the moving atom, and p and / are the mode indices of the LGB. Clearly, the first and second terms in eq. (6.5) are the usual radial and axial Doppler shifts (^LQ and (5{^Q), whereas the third term is a new, azimuthal, Doppler shift, which can be defined as ^LG = --/0
= -l^r-
(6.6)
The azimuthal Doppler shift is proportional to both the orbital angular momentum quantum number / of the LGB and the angular velocity Wr of the atoms at the radial position r in the LGB. Thus, when atoms move in two counter-propagating LGBs with orbital angular momenta / of the same sign (in a constant axial magnetic field), they will be cooled by azimuthal Doppler cooling in addition to the usual Doppler cooling (Babiker, Lembessis, Lai and Allen [1996]). Recently, polarization-gradient cooling of metastable Ne atoms and pulsed polarization-gradient cooling of ^^Rb atoms in a blue-detuned doughnut beam were studied experimentally by Kuppens, Rauner, Schiffer, Sengstock and Ertmer [1998] and Torii, Shiokawa, Hirano, Kuga, Shimizu and Sasada [1998], respectively. The optical-potential evaporative cooling of ^^^Cs atoms in a gravito-optical surface trap made of a far blue-detuned DHB was demonstrated by Hammes, Rychtarik, Druzhunina, Moslener, Manek and Grimm [2000], who obtained an ultracold and dense atomic sample with temperature 300 nK and phase-space density ^3x10""^, an increase by almost two orders of magnitude. 6.3. Manipulation and control of cold atoms Atoms moving in a blue-detuned DHB experience a transverse (2D) repulsive dipole force from the DHB. If the repulsive potential is greater than the transverse kinetic energy of the atoms, the atoms can be confined near the dark center and guided along the propagating direction of the DHB. Therefore, blue-detuned DHBs can be applied in studies of atomic wave guides, atomic funnels, atomic lenses, atomic motors, atomic-beam collimators and atomic-beam splitters.
3, §6]
Applications ofDHBs in atom optics
181
6.3.1. Atomic wave guide Since there is no van der Waals attractive potential, a DHB atomic guide has advantages over a HOF atomic guide (Marksteiner, Savage, Zoller and Rolston [1994], Renn, Donley, Cornell, Wieman and Anderson [1996], Ito, Nakaki, Sakaki, Ohtsu, Lee and Jhe [1996]). The development of efficient all-light guiding of cold atoms is of interest for applications related to transport of large numbers of cold atoms (or BEC matter waves) from one location to another, such as atom lithography, atom microscopy, atom interferometers, atomic lenses, atomic funnels, and atom fountains. Guiding of cold atoms in a blue-detuned DHB was first proposed by Yin, Noh, Lee, Kim, Wang and Jhe [1997] and Ito, Sakaki, Jhe and Ohtsu [1997], and analyzed both theoretically (Yin, Zhu, Jhe and Wang [1998], Yin, Zhu, Wang, Wang and Jhe [1998]) and experimentally (Kuppens, Rauner, Schiffer, Sengstock and Ertmer [1998] and Yin, Lin, Lee, Nha, Noh, Wang, Oh, Pack and Jhe [1998]). A DHB atomic guiding scheme is shown in fig. 34. A bluedetuned DHB with a small divergent angle (a ^ 10"^rad) is generated by the micro-collimating technique (using a M-20x objective lens L) from the output beam of the LPQI mode selectively excited by a Gaussian laser beam in a micronsized HOF (Yin, Noh, Lee, Kim, Wang and Jhe [1997], Yin, Kim, Zhu and Jhe
A/4 Plate c ^ Mirror ^ - ^ PBS WRPB
Fig. 34. Schematic diagram of atomic guiding and cooling in a dark hollow laser beam. Abbreviations: DHB, dark hollow beam; MOT, magneto-optical trap; PBS, polarized beam splitter; L, lens (M-40x microscope objective); WRPB, weak repxmiping beam; HOF, hollow optical fiber; GB, Gaussian beam.
182
Generation of dark hollow beams and their applications Copropagating Counterpropagating
50 NO
I 20 ^ O
10 0- • * • -4 -2
0
2
4
6
8
10 12 14 16
Detuning 62 (GHz) Fig. 35. Guiding efficiency versus frequency detuning in (a) the copropagating and (b) the counterpropagating scheme, with the same hollow-beam power and initial temperature. The solid curves represent numerical simulation results. (From Xu, Kim, Jhe and Kwon [2001], Phys. Rev A 64, 063401-1, reprinted with permission.)
[1998]). The efficiency of conversion from the Gaussian beam to the DHB is typically ~50%. The DHB propagates upwards in the z-direction and overlaps with a MOT. When the MOT is turned off, the cold atoms are loaded into the DHB and guided downwards inside the DHB under the action of gravity. To obtain a higher guiding efficiency, a near-resonant, weak repumping beam is propagating along the z-direction, overlapping with the DHB. Further theoretical and experimental studies on the atomic guiding in a bluedetuned DHB were done by Xu, Minogin, Lee, Wang and Jhe [1999], Xu, Wang and Jhe [2000] and Xu, Kim, Jhe and Kwon [2001]. Figure 35 shows typical theoretical and experimental results on the guiding efficiency of cold atoms in a DHB propagating upwards (the so-called co-propagating scheme) or downwards (counter-propagating scheme). The results show that at small detuning, atoms are most efficiently guided in the co-propagating scheme (for example, the maximum guiding efficiency is ~50% at a detuning of 2 GHz). For the counter-propagating scheme, the guiding efficiency is quite low at small detuning. But at large detuning, the two schemes give similar guiding efficiencies and the maximum efficiency of 23% is obtained at a detuning of 10 GHz in the counter-propagating scheme. Monte-Carlo simulation shows that the efficiency can be increased to 80% with an improved hollow beam mode. A long-distance, all-light atomic guide using a blue-detuned DHB was also demonstrated experimentally by Song, Milam and Hill [1999]: a guiding efficiency of ~10%) was obtained. DHB guiding of a continuous slow atomic beam was demonstrated by Yan, Yin and Zhu [2000] with a guiding efficiency as high as 80%.
3, §6]
Applications of DHBs in atom optics
183
6.3.2. Atomic funnels Ultracold, intense and continuous coherent atomic beams are useful in the fields of atom optics, such as atom interferometry, atom lithography and atom holography. A dense ensemble of cold atoms can be obtained from a MOT. However, when the MOT is turned off, the cold atoms diffuse in all directions. It is desirable to have a suitable atomic funnel that extracts the cold atoms from a MOT and generates a bright, compressed cold atom beam for various applications, such as atom interferometers and fiber atom waveguides. Ito, Sakaki, Jhe and Ohtsu [1997] proposed a scheme using an atomic funnel with evanescent light and a DHB in tandem to efficiently collect and guide cold atoms loaded from a MOT. This hybrid atomic funnel collects atoms released from the MOT and feeds them into a blue-detuned DHB, thus forming a well-collimated cold atomic beam. Attachment of the funnel to a HOP greatly increases the efficiency of loading cold atoms into the fiber. Monte-Carlo simulation shows that the maximum collecting efficiency of the proposed funnel is ^50% and the final temperature of ^^Rb atoms is ~80 |JiK, which is higher than the initial temperature of 10 |JIK in the MOT. Thus, the guided atoms in the funnel are heated and the atomic coherence is degraded. Yin, Zhu and Wang [1998b] proposed an atomic funnel using a hollow fiber and hollow beam in tandem, as shown in fig. 36. The proposed atomic funnel consists of a short micron-sized hollow fiber and a blue-detuned Gaussian laser beam. The Gaussian beam is coupled into the hollow fiber and selectively excites a LPoi-mode in the fiber. Both evanescent-wave light inside the hollow region of the fiber and a divergent DHB in free space are generated. The DHB propagates along the z-direction and the cold atoms from a standard MOT are loaded into the DHB. A repumping laser beam propagating opposite to the DHB overlaps
Fig. 36. Schematic diagram of an atom-fiber fiimiel. Abbreviations: HOF, hollow optical fiber; DHB, dark hollow beam; MOT, magneto-optical trap; WRPB, weak repumping beam, Lj, lens; M, mirror; O)^,(J\^,(DY^, fi-equencies of atomic resonance of the ^^Rb-D2 line, of the guiding laser, and of the repumping laser, respectively.
184
Generation of dark hollow beams and their applications
[3, § 6
the blue-detuned DHB, and a closed Sisyphus-cooling cycle can be introduced for three-level atoms. As the cold atoms are guided through the DHB, their temperatures will be reduced, and at the same time the atoms will be focused by the convergent DHB before entering the hollow optical fiber. Theoretical study shows that an ultracold, dense atomic sample with a temperature of ^2 |iK from the output facet of the hollow fiber can be obtained, with a guiding efficiency of about 95%. 6.3.3. Atomic lens To realize atomic microscopy or atom lithography with a spatial resolution of a few A, an atomic beam must be tightly focused by an atomic lens that is capable of focusing it into a region of several A. With an atomic beam treated as de Broglie waves, Balykin and Letokhov [1987] found that the optical dipole potential from a blue-detuned doughnut beam is similar to that of an object lens in photon optics or electron optics, which can be used to form an all-light atomic lens. The atomic lens scheme proposed by Balykin and Letokhov is shown in fig. 37. Theoretical study shows that for power PQ and wavelength A of the doughnut beam, the focusing length/ of atomic lens is given by wl
-^
XdhMv,
ijthPo
n
'
^
where u^ is the longitudinal atomic velocity and M is the atomic mass. The atomic lens has several de-Broglie-wave aberrations: ^dif ^ \ ^\ j^J y
JAIPQ
2
f^v,\
d^chr == T 3 V ^z / «sph = [ - ( ! - i « ) {\p^) + (1 - 4a) (^p"^)]
(diffusive aberration),
(chromatic aberration), (spherical aberration),
(6.8)
(6.9) (6.10)
where a = (PoP^) / (jchd^wl). For WQ = X,f = 15.6A, a = \wo, PQ = IW, Uz = 2.2xl0'*cni/s and Au^/uz = 10~^, an atomic lens with a resolution of a few A can be formed; the corresponding thin-lens condition is ^4/3
/ > ^ ,
(6.11)
with a the diameter and AB the de-Broglie wavelength of the atomic beam. At AB = 1 A and a = \wo4, the focal length of the lens is as follows: ( i ) / > 1.4A for Wo = A, ( i i ) / ^ 28A for wo = lOA, and (iii)/ ^ 616A for WQ = lOOA.
3, §6]
Applications ofDHBs in atom optics
185
.r
Fig. 37. Laser field configuration for focusing an atomic beam to a spot size equal to the de Broglie wavelength: (a) set-up of the laser and atomic beam; (b) cross-sectional intensity profile of the TEMQpmode laser.
DHB focusing of an atomic beam was also studied by using the pathintegral method (Gallatin and Gould [1991]), and by a particle-optics approach (McClelland and Scheinfein [1991]). DHB focusing of a cold atomic beam was demonstrated experimentally by Schiffer, Rauner, Kuppens, Zinner, Sengstock and Ertmer [1998] and Yan, Yin and Zhu [2000]. Some other schemes for laser focusing of atomic beams that have been proposed and analyzed are an atomic lens using a blue-detuned Gaussian beam and a ;r-phase plate (Wang, Fang, Wang, Feng and Wang [1992]) and a conical atomic lens with a conical mirror and a blue-detuned DHB (Dubetsky and Berman [1998]). An analytical solution for the laser focusing of an atomic beam, giving the focal position and the focal width, was found by Klimov and Letokhov [1999].
6.4. Atomic motors As discussed in § 5.2, a hollow beam (such as a LGB) possessing both spin and orbital angular momentum can be used to rotate microparticles. Similarly, atoms moving in such a DHB should exhibit new rotational effects, in addition to the normal translational effects. These effects induce changes in both the internal and the external motions of the atoms. For the internal motion, the Doppler shift for a moving atom has an additional contribution called the azimuthal Doppler shift (Allen, Babiker and Power [1994]), which is directly proportional to the orbital angular momentum Ih of the LGB (see eq. 6.6). For a two-level atom moving in linearly polarized light, the dissipative force exerted on the atom is given by ^diss)
=hkr-
i + / + (A/r)^
(6.12)
186
Generation of dark hollow beams and their applications
[3, § 6
where / = InG/F^ (with 2nG representing the power broadening), and hk is the Hnear momentum of the plane-polarized light. In the saturation limit / —> oo, we have the well-known result Fd,ss)^Mr.
(6.13)
The dissipative force is independent of the atomic position and has zero torque (^diss) = ('^ X Fdiss) = 0 relative to the beam direction. However, for a two-level atom moving in a linearly polarized LGB, the dissipative force in cylindrical coordinates R = (r,(l),z) is given by Babiker, Power and Allen [1994] as follows:
'-lo^'^^^ifT^^k^)'
^'''^
where VOLG = -
kr' / 2(z2+z2)V
—r-
z
2z' \ iz'+zi)J
(2;7 + /+l)zR (z^+zi)
r
(6.15) The force depends on the atom position, and it results in a nonzero torque given by (7diss)LG '=(fx Fdiss) \
= z {rF^)^Q ,
(6.16)
/ LG
where F^ = hk,r,
k, = l/r,
(I> = X/K
(6.17)
The magnitude of the torque is given by
where Ifi is the orbital angular momentum of the LGB. In the saturation limit, eq. (6.18) becomes
\{n,ss)^^\^ihr,
(6.19)
which is analogous to eq. (6.13). So, a two-level atom moving in a LGB is subject to a light-induced torque, T, around the beam axis, which is proportional to the
3, § 6]
Applications ofDHBs in atom optics
187
orbital angular momentum / of the LGB. Therefore, a linearly polarized LGB (or other DHB), can be used as an atomic motor (rotator) to control the rotational motion of atoms in a DHB trap. Lately, atomic motion in various DHB configurations has been studied theoretically by many groups; some interesting results have been obtained and novel applications have been proposed. For example, Power, Allen, Babiker and Lembessis [1995] studied the internal and gross motions of an atom in light beams possessing orbital angular momentum, and presented a general theoretical method. Wright, lessen and Lapeyere [1996] investigated 2D motion of cold atoms in a near-resonant annular laser beam and showed that the atomic motion can be divided into vibrational and rotational normal motions, which is analogous to a 2D molecule. Allen, Babiker, Lai and Lembessis [1996] studied the steady-state dynamics of atoms in multiple LGBs. Lai, Babiker and Allen [1997] discussed radiation forces on a two-level atom in a (T+-cr_ configuration of LGBs and showed that the atom experiences either a static torque or a purely velocity-dependent torque, depending on the relative signs of the orbital angular momenta of the two LGBs. In particular, the purely velocitydependent torque can be used under certain conditions to cool the trapped atoms. Masalov [1997] studied the transfer of a high angular momentum from a photon to an atom in a DHB. Lembessis [1999] investigated the interaction of a mobile atom with a linearly polarized LGB. Babiker and Al-Awfi [1999] studied light-induced rotational effects of atomic guiding in a hollow cylindrical waveguide. Recently, Tabosa and Petrov [1999] used an optical pumping method to realize the transfer of orbital angular momentum of a DHB to cold Cs atoms in a MOT, and observed this momentum transfer with a nondegenerate four-wave mixing technique. The experiment shows that the orbital angular momentum of light can be transferred, via optical pumping in the cold atomic sample, from one beam with a frequency (0\ to another one with a different frequency a>2- Furthermore, the experiment also implies that DHBs carrying orbital angular momentum can be used to excite vortex states in a BEC, which was proposed by Marzlin, Zhang and Wright [1997]. In addition to the applications of DHBs in atom optics mentioned above, an atomic-beam collimator and atom lithography based on DHB guiding and DHB-induced Sisyphus cooling of cold atoms were proposed and studied theoretically by Yin and Zhu [1998b]. Recently, guiding, collimating (atomicbeam collimator), focusing (atomic lens) and splitting (atomic-beam splitter) of a continuous, low-velocity ^^Rb atomic beam with a blue-detuned DHB was demonstrated experimentally by Yan, Yin and Zhu [2000]. Moreover, DHBs
188
Generation of dark hollow beams and their applications
[3, § 7
can also be used to prepare a high-density, dark MOT (Ketterle, Davis, Joffe, Martin and Pritchard [1993]) and a doughnut-mode MOT (Snadden, Bell, Clarke, Riis and Mclntyre [1997]), to study ultrasensitive two-photon spectroscopy (Khaykovich, Friedman, Baluschev, Fathi and Davidson [2000]), and to make a high-precision, atomic-fountain-based clock (Hu and Yin [2001]). § 7. Applications of DHBs in coherent matter-wave optics 7.1. All-optical route to BEC in DHB traps An ail-optically cooled and trapped BEC idea using a DHB was proposed by Yin and Zhu [1998a] and Yin, Zhu and Wang [1999]. Recently, an all-optical-type ^^Rb atomic BEC in a red-detuned CO2 laser trap using optical-potential evaporative cooling was obtained by Barrett, Sauer and Chapman [2001]. An ail-optically cooled and trapped BEC using a bluedetuned DHB has not been realized yet. Nevertheless, it is interesting to discuss the potential applications of a far-blue-detuned DHB in an all-optical-type BEC. A DHB with blue detuning greater than 10^-10^ GHz and laser power higher than 500 mW may be called a "far-blue-detuned DHB". Obviously, many such far-blue-detuned DHBs can be obtained, e.g., from an Ar^ laser (A = 0.488 |xm and 0.5145 |Jim; output power ^20-30 W), a frequency-doubled YAG laser (A = 0.53 |im; output power > 5W), a tunable dye laser (A = 0.4-0.7 ^im; output power ~1 W), a tunable Ti:sapphire laser (A = 0.6-0.8 |im; output power ~1 W), and even from some tunable diode lasers with output power >500mW. It is reasonable to expect that high-power far-blue-detuned DHBs should have important applications in atom optics and an optically trapped BEC. As one example, we consider the optical potential of a DHB generated from an Ar^ laser for ^^^Cs atoms confined in a PHB GOT, which is given by
U^ra)-^'-^'^.
(7.)
12 6 Is where I(r, Z) is the intensity of the trapping beam and Is is the saturation intensity of the atoms. When 6 = 2.468 x 10^ GHz, PQ = 20W (30 W) and 0 = 30^ the trapping potential Umax > 70 |jiK (105 jiK) is far greater than TGOT ^ 1.8 [xK and also much greater than the gravitational potential mgz/kB = 20.4 |i,K. The calculated trapping potential of the Ar+ hollow beam for ^^Rb atoms also far exceeds JGOT ~ 2.5 (IK and the gravitational potential (13.5 [iK). Therefore, the trapping potential of the Ar^ hollow beam for alkali atoms is high enough to collect and trap most of the cold atoms from the PHB GOT.
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Weak repumping beam
Ar hollow beam
Doughnut hollow beam
Fig. 38. Scheme of pyramidal-hollow-beam GOT and Ar+ conical hollow-beam trap. Abbreviations: MOT, magneto-optical trap; GOT, gravito-optical trap; PHB, pyramidal hollow beam; PBS, polarized beam splitter; BSP, black square plate. 6 is the trapping angle.
Since the blue detuning (-^10^ GHz) of the Ar+ laser relative to the resonant frequency of alkali atoms is very large, the heating from spontaneous emission of atoms in the blue-detuned Ar+ hollow-beam trap can be neglected, and the photon scattering rate (i.e., Rayleigh scattering) is only about 10~^-10"^s"\ which is similar to that in the red-detuned CO2 laser trap (A = 10.6 |Jim) (Takekoshi, Yeh and Knize [1995], Barrett, Sauer and Chapman [2001]). So the blue-detuned Ar+ DHB trap should be useful for the exploration of an all-optical-type EEC. All-optical schemes for realizing a ^^^Cs atomic EEC have been proposed and discussed by Engler, Manek, Moslener, Nill, Ovchinnikov, Schuenmann, Zielonkowski, Weidemuller and Grimm [1998], among others. As an example, we consider an all-optical-type EEC in a PHE GOT (see § 6.2). When the PHE GOT has reached its 3D equilibrium temperature (~2 ^K), one can block the PHE to load the cold atoms from the PHE GOT into an Ar+ hollowbeam trap (see fig. 38), into a crossed Ar+ hollow-beam trap (see fig. 39), or into an Ar+ localized-hoUow-beam trap (see fig. 40), and then use a bluedetuned plug beam for the Ar+ hollow-beam trap (fig. 38) or change the focal length/ of the lens (figs. 39 and 40) to compress the atomic sample in the trap. Subsequently, optical-potential evaporative cooling (or Raman cooling) is applied to further cool the trapped atoms, yielding an ultracold and dense atomic sample in the Ar+ hollow-beam trap. Using the experimental result of Chu's group -
190
[3, §7
Generation of dark hollow beams and their applications DHB,
Tl-PP Lens 4 GOT * PB - Cfc^
Lens
^
HI
\
71-PP
(a)
1/
Cold atoms
(b)
DHB,
Fig. 39. Two schemes of crossed Ar^ hollow-beam traps for alkali atoms. In (a), DHBj, DHB2 and PB are two counter-propagating dark hollow beams and a plug beam respectively. GOT indicates the cold atoms in the gravito-optical trap. f\ Ul)^^ the focal length of lens Li (L2), which can be changed continuously. In (b), GB and :7r-PP stand for Gaussian beam and azimuthal-distributed ;r-phase plate, respectively.
QIM
CD""" 7c-Phase plate
Lens
Fig. 40. Schematic of Raman cooling of alkali atoms in an Ar^ localized-hollow-beam trap. Abbreviations: GLB, Gaussian laser beam; LHB, localized hollow beam; RCP, Raman cooling pulse, RPB, repumping beam; MOT/GOT/OM, magneto-optical trap/gravito-optical trap/optical molasses.
T ^ 1.7 Tree for optical-potential evaporative cooling (Adams, Lee, Davidson, Kasevich and Chu [1995]), T ^ OATrec for Raman cooling (Lee, Adams, Kasevich and Chu [1996]) - one finds for ^^Rb atoms an ultracold and dense alkali atomic sample with an atomic density above the BEC critical density and a temperature of ^0.35 |iK (evaporative) or '--0.08 |iK (Raman). These temperatures are about equal to or lower than the BEC temperatures for ^^Rb atoms in the magnetic trap. Therefore the realization of an ail-optically cooled and trapped BEC in the Ar+ hollow-beam trap, like the preparation of an all-optical-type
3, § 7]
Applications ofDHBs in coherent matter-wave optics
191
^^Rb atomic BEC in a red-detuned CO2 laser trap (Barrett, Sauer and Chapman [2001]), may be attained by using optical-potential evaporative cooling or Raman cooling (Yin and Zhu [1998a], Yin, Zhu and Wang [1999], Yin, Gao, Wang and Wang [2001]). The singlet and triplet scattering lengths of ground-state ^^^Cs atoms in a magnetic trap are negative, which results in a giant spin-relaxation rate and threebody recombination rate. So far, all attempts to prepare a magnetically trapped ^^^Cs BEC by rf-induced evaporative cooling have failed. Also, no all-optical Cs BEC has yet been observed in the red-detuned YAG Gaussian-beam trap because of the large photon-scattering rate for trapped ^^^Cs atoms (Adams, Lee, Davidson, Kasevich and Chu [1995]). However, the photon scattering rate of trapped atoms in an Ar^ hollow beam is extremely low, about lO^^-lO'^^s^^ If one uses the Feshbach resonance technique (Inouye, Andrews, Stenger, Miesner, Stamper-Kum and Ketterle [1998]) in an Ar^ hollow-beam trap, one can change the scattering length of ^^^Cs atoms from negative to positive, and then may realize an optically trapped ^^^Cs BEC (Yin, Gao and Wang [2000], Yin, Gao, Wang, Long and Wang [2002]), similarly to the preparation of a ^^Rb atomic BEC with a negative scattering length (Cornish, Claussen, Roberts, Cornell and Wieman [2000]). As an Ar+ hollow beam (or other far-detuned DHB) may be used to trap two (or more) alkali atom samples at the same time, it can be used to perform sympathetic cooling between two isotope samples (or two different alkali atoms), which may result in a two-sample GOT or two-sample BEC (Pu and Bigelow [1998a,b]). In particular, the Ar+ hollow-beam GOT may be used to perform sympathetic cooling between an ^^Rb atomic BEC and an ultracold ^^^Cs atomic sample at ~1 fxK, which can then be used to explore the possibility of an optically trapped ^^^Cs BEC (Yin, Gao, Wang, Long and Wang [2002]). This is similar to the realization of a "^^K BEC in a magnetic trap by sympathetic cooling between an ^''Rb atomic BEC and an ultracold "^^K atomic sample (Modugno, Ferrari, Roati, Brecha, Simoni and Inguscio [2001]).
7.2. Manipulation and control of BEC 7.2.1. Blue-detuned waveguide for BEC Of particular interest are new designs capable of unprecedented sensitivity for atom holography, atom lithography and atom interferometers, in which single-mode waveguides will play essential roles, especially for lower-dimension applications such as ID configuration for coherent matter waves, controlled
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Generation of dark hollow beams and their applications
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collision in strongly confined ID systems, and studies of fundamental excitations like dark solitons in ID or quasi-ID regimes. An advantage of optical dipole traps, in comparison with magnetic traps, is the possibility of trapping atoms with arbitrary magnetic sublevels, or trapping several different atomic species simultaneously. A blue-detuned LGB de-Broglie waveguide was used to realize a BEC waveguide and to observe the dynamic evolution of BEC matter waves in the loading and waveguiding process by Bongs, Burger, Dettmer, Sengstock and Ertmer [2000]. They used a far bluedetuned LGB of the first order with a power of 1 W at 532 nm (obtained from a frequency-doubled YAG laser) and a beam waist of WQ ~ lOjJim, and demonstrated a loading efficiency of ~100% from the BEC into the LGB. The waveguide was aligned with the long axis of the BEC with a slight tilt allowing for the gravitational acceleration. The experiment shows that due to the nonadiabatic transfer, a few transverse modes of the waveguide were occupied, leading to axial heating. The evolution of the BEC inside the LGB is affected by gravity, and the atomic ensemble diffuses due to the initial thermal expansion and the further heating effects. Recently, a hollow-beam BEC waveguide in a specially designed hybrid optical-dipole/magnetic trap was proposed and demonstrated experimentally by Bongs, Burger, Dettmer, Hellweg, Arlt, Ertmer and Sengstock [2001]. Such DHB BEC waveguides open new possibilities for studying different regimes of the ID quantum gas, spinor BEC, and a guided matter-wave interferometer. 7.2.2. Red-detuned ring trap for BECs Recently, the generation and exploration of quantized vortices on atomic mesoscopic rings have attracted much interest in BEC studies. Such studies require a toroidal-shaped BEC. Since a red-detuned DHB attracts the atoms to its intensity maximum, it can be used to realize a toroidal optical dipole trap for cold atoms or BECs. Wright, Arlt and Dholakia [2000] proposed a toroidal optical trap with a red-detuned, focused LGB for a 2D BEC. They discussed how to load a BEC into such a ring-shaped trap from various initial conditions and studied the performance characteristics of this dipole trap for BEC as a function of the azimuthal mode index /. Theoretical studies show that LGBs with orbital angular momentum Ih per photon offer a flexible way to forming toroidal dipole traps in 2D atomic BECs, and cold atoms can be loaded into the ring-shaped optical trap under conditions representative of normal magnetic traps. A variety of basic and applied studies may be performed by using such traps, such as generating ring vortices and
3, § 7]
Applications ofDHBs in coherent matter-wave optics
193
persistent currents on a torus, studies of multi-component, ring-shaped BECs, realization of Tonk's gas and dark solitons, creation of coaxial toroidal traps for BECs (multiple concentric rings), studies of the radial tunneling between condensates, and preparation of a circular atomic grating (Wright, Arlt and Dholakia [2000], Tempere, Devreese and Abraham [2001]). 7.2.3. High-field-seeking trap fi)r BEC So far, most BECs have been prepared by rf-induced evaporative cooling in various static magnetic traps. The trapped atoms are in an excited hyperfine state (a weak-field-seeking state) and can relax to the ground state via spin-flip collisions. A high-field-seeking trap (a hybrid magnetic/optical trap), which is composed of a current-carrying wire with a diameter of 5 (im (40 mA current) and a focused, blue-detuned LGB with a diameter of 20 |Jim, was proposed by Close and Zhang [1999] to form a Cs BEC in the true hyperfine ground state at a magnetic field maximum. In this scheme, the current-carrying wire generates an attractive magnetic potential for cold Cs atoms in the strong-field-seeking state, and the blue-detuned LGB produces a repulsive potential such that the total potential for the trapped atoms is ^ W = ^ e x p — o \ w"^J \w/
-—, r
(7.2)
where a = -gYmYliB^- The current-carrying wire is enclosed by the trapped atomic cloud and the attractive potentialfi*omthe wire-generated magnetic field has a maximum at the surface of the wire. The atoms are repelled fi*om the surface of the wire by the blue-detuned LGB. So the trap is toroidal and the cold Cs atoms are trapped in a thin annulus. 7.3. Output coupling of coherent matter waves Since the realization of the first ^^Rb BEC in 1995, the preparation of the output of atom lasers has become a hot research subject in coherent matter-wave optics. With a crossed Ar+ hollow-beam trap and a pair of stimulated-Raman-transition beams (see fig. 41), an ail-optically controlled output of an atom laser can be realized, its output coupling principle being similar to that proposed by Moy, Hope and Savage [1997] and demonstrated experimentally by Hagley, Deng, Kozuma, Wen, Helmerson, Rolston and Phillips [1999]. In fig. 41, the model of output coupling of atom laser is composed of atoms with three energy levels.
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Generation of dark hollow beams and their applications
[3, § 7
DHB
Raman Beam 2
Raman Beam 1 Atom Laser
Fig. 41. Output coupling scheme for an atom laser based on stimulated Raman transitions. DHB, dark hollow beam; BEC, Bose-Einstein condensation.
in which level 11) is the ground state of condensed atoms, which are pre-prepared by BEC techniques, level |3) is the output level of the atom laser, and level |2) mediates the output coupling Raman transition from level |1) to level |3), which will be accomplished by a pair of Raman pulses {(0\ and o>i) and couples condensed ground-state atoms out of the DHB trap system. From scattering theory, the Raman transition rate (i.e., the single-atom rate constant) is given by (Moy, Hope and Savage [1997])
where Q\ and Q2 are the Rabi frequencies of the two Raman laser beams (ft^i and CO2), respectively. A = (Oxi - o)x = 0^3 - ^ is the detuning, and l/fo is the single-atom loss rate from the system due to the photon recoil (i.e., momentum kick, 2 hk) and can be estimated from mL where m is the atomic mass and L is the length of the interaction region for the atom in the DHB trap cavity. Then, the total output rate of the atom laser beam is given by Ro=N,Rn,
(7.5)
where A^i is the population of BEC atoms in level |1), that is, the number of condensed atoms in the BEC trap. For typical parameters, A = 2jt x 1.6 GHz,
3, § 7]
Applications ofDHBs in coherent matter-wave optics
195
Mirror
BECO> DHB
Atom laser
Plug beam (elliptic light sheet) Fig. 42. Output coupling scheme for an atom laser based on the quantum tunneling effect. DHB, dark hollow beam; EEC, Bose-Einstein condensation.
Qx = Ijt X 50kHz, Q2 ^ In X 1.6MHz, ^ = 1.5x10-^ s and Nx = 5x10^ atoms, we may obtain an atom laser with an output rate of 6.25 x 10"^ s~^ and a corresponding output time of ~8 s for a single preparation of BEC atoms (i.e., a single pump). Similarly, by use of a doughnut-beam trapping scheme (Kuga, Torii, Shiokawa, Hirano, Shimizu and Sasada [1997]) and changing the intensity of the outgoing plug beam (that is an elliptic light sheet, a dipole-potential mirror), an intensitycontrollable atom laser can be realized (see fig. 42) owing to quantum tunneling effects and the repulsive potential between the Bose-condensed atoms; this is the so-called output coupling mechanism of quantum-mechanical tunneling (Wiseman and Collett [1995]). In fig. 42, the intensity modulator is used to control the output flux of the atom laser, that is, to control the height of the potential barrier for condensed-atom tunneling. A similar scheme for a continuous output of coherent matter waves (an atom laser) from a BEC in a blue-detuned LGB dipole trap, which acts simultaneously as a matter waveguide, was proposed by Bongs, Burger, Dettmer, Sengstock and Ertmer [2000]. To form a cavity for the atom laser, two additional dipole repulsive-potential mirrors are used to close the LGB BEC waveguide. The reflectivity of the dipole mirrors is velocity dependent so that, by applying Bragg pulses to transfer appropriate amounts of momentum to the condensate, it should be possible to extract parts of the condensate from the atom laser cavity. A main advantage of this scheme would be the direct coupling of the coherent matter waves into the doughnut waveguide. In addition, a far blue-detuned DHB can be used to study spinor BEC (Ho [1998]), vortices in BECs (Tempere, Devreese and Abraham [2001] and dark
196
Generation of dark hollow beams and their applications
[3, § 8
solitons in BEC even to form a guided BEC matter-wave funnel, a matter-wave rotator, a matter-wave circular grating, a matter-wave splitter, interferometers and so on. § 8. Summary and outlook 8.1. Summary A light beam with a ring-shaped intensity distribution may be defined as a "dark hollow beam (DHB)". Such light beams usually possess both spin and orbital angular momentum and are best described in terms of superposition of LGB modes. DHBs are characterized by some practical parameters such as the dark spot size (DSS), the beam width (^DHB)? the beam radius (ro), the ringbeam width {Wr), and the width-radius ratio (WRR). According to the radial intensity distribution, DHBs can be roughly classified into ten different types, each having some special characteristics and certain possible areas of applicability. For example, Laguerre-Gaussian beams, doughnut hollow beams, LPoi-mode output hollow beams and double-Gaussianprofile hollow beams may be used to trap and manipulate microparticles as well as to guide, fiinnel and trap cold atoms. Due to their propagation-invariant property, higher-order Bessel-Gaussian beams and higher-order Mathieu beams are particularly applicable in guiding and collimating cold atoms, which may be used for atom lithography. The focused hollow beam and the localized hollow beam are well suited for focusing and trapping atoms. The double-rectangularprofile hollow beam has an extremely high intensity gradient and may be used to provide DHB-induced Sisyphus cooling for cold atoms. In recent years, many methods have been used to produce various types of the DHBs. Transverse-mode selection has been used to generate higher-order LGBs (including doughnut hollow beams). The geometric optical method has been used to generate ring-shaped hollow beams, higher-order Bessel beams and localized hollow beams. Mode conversion has been used to transform HGBs into higher-order LGBs. The computer-generated-hologram method has been used to produce higher-order LGBs (including doughnut beams and vortex hollow beams), higher-order Bessel beams, localized hollow beams and higher-order Mathieu beams. The optical holographic method and micron-sized-HOF method have been used to generate higher-order Bessel beams and doughnut-like hollow beams. The jr-phase-plate method has been used to produce focused hollow beams and localized hollow beams. DHBs have three useful features for trapping microparticles:
3, § 8]
Summary and outlook
197
(1) The null intensity of the on-axis region reduces the light-scattering force, which minimizes the optical damage to trapped particles caused by absorptive heating, which is particularly important for trapping biological samples. (2) A focused hollow beam can trap high-index and/or low-index microparticles, thus widening the types of applicable microparticles such as nanometer- or micrometer-sized particles, atomic clusters and biological samples or living cells. (3) A DHB can be used to trap partially absorbing particles and transfer orbital angular momentum from photons to the particles, thus realizing optical spanners. DHBs have been used in cooling and trapping of neutral atoms. Atom-optics devices, such as atomic waveguides, atomic tweezers and atomic motors, may be realized with DHBs. Far-blue-detuned DHBs may be used to build an optical trap and implement optical-potential evaporative cooling, which may lead to ailoptically cooled and trapped BEC and other applications in the manipulation of coherent matter waves. Far-blue-detuned DHBs may also be used to trap two or more alkali atomic species at the same time and perform sympathetic cooling between the species, which may lead to the two-species BEC, or spinor BECs.
8.2. Outlook If efficient intensity-gradient-induced Sisyphus cooling can be realized in a DHB configuration, it may open new possibilities for the application of DHBs in atom optics. So far, the double-rectangular-profile hollow beam seems to be promising in this regard as discussed in § 4.10. Recently, Arieli [2000] proposed a continuous phase plate to realize beam shaping and pattern generation. The proposed phase plate was designed using the inverse phase-contrast method, which may be used to design a new phase plate for producing a doublerectangular-profile beam. Due to the complexity of the phase structure of some hollow beams, such as the ring-shaped hollow beam, the double-Gaussian-profile hollow beam, and the double-rectangular-profile beam as well as the higher-order Mathieu beams, the properties of the orbital angular momentum for those DHBs have not been well understood, and no closed-form expression for the orbital angular momentum in those beams is available. It will be interesting and worthwhile to fully understand the orbital angular momentum in those beams. The optical damage to trapped particles caused by absorptive heating is detrimental for trapping biological samples, particularly for living cells. The null
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Generation of dark hollow beams and their applications
[3, § 8
central intensity of a DHB reduces the light-scattering force and minimizes the optical damage for trapped biological samples in DHB tweezers. It is expected that DHBs tweezers will play an important role in the manipulation of biological samples and living cells in biology and biochemistry, and may be useful for nonlinear spectroscopic studies of trapped microparticles. Recently, focused Gaussian-beam tweezers were used to study the Raman spectrum of highly refractive and nontransparent particles or single trapped biological cells (Xie and Li [2002], Xie, Dinno and Li [2002]). The experiment demonstrates a high sensitivity and permits real-time spectroscopic measurements of the biological samples. The experimental Raman technique may provide a valuable tool for studying fundamental cellular processes and to obtain diagnoses of cellular disorders. It may be advantageous to adapt the Raman technique to DHB tweezers, which may have better spatial performance than Gaussian-beam tweezers. Recently, focused YAG Gaussian-beam laser tweezers (A = 1.064 |Jim) were used to transport a ^^Na atomic BEC sample over a distance up to 44 cm from a loffe-Pritchard-type magnetic trap into a Z-wire magnetic microtrap (Gustavson, Chikkatur, Leanhardt, Gorlitz, Gupta, Pritchard and Ketterle [2002]). This transport technique avoids the optical and mechanical access constraints of conventional condensate experiments and creates many new scientific opportunities. On the other hand, a CO2 laser Gaussian-beam trap (A = 10.6 [jim) was proposed to realize all-optical production of a degenerate ^Li Fermi gas (Granade, Gehm, O'Hara and Thomas [2002]). This degenerate two-component mixture is ideal for exploring mechanisms of superconductivity ranging from Cooper pairing to the BEC of strongly bound pairs. Such possibilities should also exist for suitable blue-detuned DHB traps. Laguerre-Gaussian beams possess orbital angular momentum and form a complete basis set. They can be used to represent an arbitrary light beam within the paraxial ray regime of light propagation. If one can manipulate and control the superposition of the LG modes and realize quantum entanglement for different LG modes, then multi-dimensional entangled states are created, which will be of considerable importance in quantum information processing. Amaut and Barbosa [2000] were the first to study theoretically the orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by spontaneous parametric down-conversion. Recently, entanglement of the orbital angular momentum states of photons in a parametric down-conversion experiment has been demonstrated by Mair, Vaziri, Weihs and Zeilinger [2001]. Molina-Terriza, Torres and Tomer [2002] proposed a scheme to prepare photons in multidimensional LG states. Their approach is based on the phase-coherent superposition of LG beams with dislocated centers, which produces a multipearl.
3]
References
199
necklace-shaped light field with nested vortices in each pearl. They showed that arbitrary superpositions of the orbital angular momentum states may be obtained, which allows the addition and removal of specific projections of the orbital angular momentum states. Other studies on superposition and entanglement of the angular momentum states of photons or atoms in DHBs have been done by Franke-Arnold, Bamett, Padgett and Allen [2002], Vaziri, Weihs and Zeilinger [2002] and Muthukrishana and Stroud [2002]. It is conceivable that the photons with entangled orbital angular momentum can be extended to multidimensional, multiparticle regimes and capacity-increased quantum information processing such as quantum cryptography with higher alphabets and quantum teleportation. Some molecules in light may acquire a large dipole moment due to the ac Stark effect. Similarly as with applications of DHBs in ultracold atomic physic and atom optics, the DHB technique may prove to be usefiil in ultracold molecular physics and molecule optics. Blue-detuned DHBs may some day be used to guide, focus, collimate, trap and rotate cold molecules, and to form various molecular optics elements such as molecular tweezers, traps, hosepipes, fimnels, lenses, motors, and so on.
Acknowledgments We are gratefiil to Prof. Emil Wolf for reading and commenting on a preliminary version of this review. We would like to thank Drs. J. Courtial, S. ChavezCerda, and M. MacDonald for helpfiil discussions. We also thank all authors who provided original figures, and the publishers of the Journals in which illustrations originally appeared for their permissions to reproduce them. J. Yin would like to thank Professors Yuzhu Wang, Yiqiu Wang and W Jhe for their friendly co-work during his visit and acknowledges support from the National Natural Science Foundation of China (Grants No. 69878019 and 10174050), the Natural Science Foundation of Jiangsu Province (Grants No. DK97139 and OOKJB140001), and the Fostering Foundation of New-century Academic Leader from The Educational Department of Jiangsu Province. Y. Zhu acknowledges support fi-om the Office of Naval Research (N00014-01-1-0754).
References Abramochkin, E., and V Volostnikov, 1991, Opt. Commun. 83, 123. Adams, C.S., H.J. Lee, N. Davidson, M. Kasevich and S. Chu, 1995, Phys. Rev. Lett. 74, 3577. Allen, L., M. Babiker, W.K. Lai and VE. Lembessis, 1996, Phys. Rev A 54, 4259.
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E. Wolf, Progress in Optics 45 © 2003 Elsevier Science B. V. All rights reserved
Chapter 4
Two-photon lasers by
Daniel J. Gauthier Department of Physics, Duke University, Durham, NC 27708, USA
205
Contents
Page § 1. Introduction
207
§2.
Two-photon processes
210
§3.
Simple models of amplification and lasing
218
§ 4.
Two-photon amplification and lasing
231
§ 5. The two-photon maser
235
§ 6.
The dressed-state two-photon laser
239
§ 7.
The Raman two-photon laser
251
§ 8.
Quantum-statistical and nonlinear dynamical properties . . . .
260
§ 9. Future prospects
267
Acknowledgments
267
References
268
206
§ 1. Introduction An amazing property of laser light that makes it so useful for a variety of applications is its high directionality, spectral purity, and high power. This high degree of spatial and temporal coherence arises from a complex interplay between several physical processes taking place within a laser, such as the fundamental lightmatter interactions of absorption, spontaneous emission and stimulated emission, and the effects of feedback due to the reflecting surfaces of the optical resonator (Mandel and Wolf [1995]). The degree of coherence of light generated by lasers can be altered significantly and often in a surprising manner by modifying the properties of the resonator or the light-matter interactions. Of great current interest is the generation of specific quantum states of light that possess desired correlations among the emitted photons (related to the «th-order spatial and temporal coherence functions characterizing the light) for applications in quantum information (Bouwmeester, Ekert and Zeilinger [2000]), for example. One fascinating technique for modifying the coherence properties of laser light is to operate the device in the 'cavity quantum electrodynamics regime' where only a small number of atoms and photons interact with and are strongly coupled to the optical resonator. For example, the threshold, stability, and quantumstatistical properties of a single-atom maser (Meschede, Walther and Miiller [1985]) or laser (An, Childs, Dasari and Feld [1994]) operating in this regime are very different from their multi-atom counterparts. The coherence properties of laser light can also be modified by exploiting different types of light-matter interactions. For example, it has been known since the early days of quantum mechanics that there exist two-photon analogies to the standard one-photon processes of absorption, emission, and stimulated emission. To understand how two-photon effects can change the properties of laser light, I first review the characteristics of the one-photon processes shown in figs, la-c depicting the interaction of light with an atom possessing an upper state \b') and lower state \a'). Spontaneous emission (fig. la) occurs when an atom in the upper state decays at a rate ^^^ to the lower state and emits a photon whose frequency co is essentially equal to the transition frequency cOb'a' • For a typical allowed electric dipole transition, ^^^ ?^ 10^ s~^ Absorption (fig. lb) occurs when an atom in the lower state is promoted to the upper state and the photon is annihilated, where 207
208
Two-photon lasers
[4, § 1
one-photon processes (a)
(b)
(c)
—•:—lb'> \
iir"'b'>
CO
CO
5^la'>
^—lb'>
T
CO
—•—la'>
\
«
'^la'>
two-photon processes (d)
(e)
(f)
lb> \
CO'
(g)
1—lb> ^
'
^
—i^—lb> ^'
^
rj^
—^^—lb> ®'
)
co'
Fig. 1. Light-matter interactions. One-photon (a) spontaneous emission, (b) absorption, and (c) stimulated emission. Two-photon (d) spontaneous emission, (e) absoqjtion, (f) singly stimulated emission, and (g) stimulated emission.
the absorption rate W^^, is proportional to the incident photon flux. Stimulated emission (fig. Ic) occurs when an incident photon forces an atom to jump from the upper state to the lower state and two photons are scattered by the atom. The scattered photons have the same fi-equency, phase and direction as the incident photon, which gives the laser its unique coherence properties. The stimulated emission rate ^^/^, is also proportional to the incident photon flux, and proceeds most efficiently when co = cob'a' and when the states have opposite parity (electric-dipole-allowed transition). Based on thermodynamic considerations, it was shown by Einstein [1917] that ^ j , ^ , = WlJ^,. By analogy, consider the two-photon processes that occur when light interacts with an atom possessing an upper state \b) and lower state |fl), which typically have the same parity so that one-photon transitions between the states are electric-dipole forbidden. Spontaneous emission (fig. Id) occurs when an atom in the upper state decays to the lower state at a rate A^^ and emits two photons of frequency co' and w". For this process to be allowed, the atom must possess additional auxiliary states |/) with opposite parity (real or continuum states). Note that the fi-equencies of the scattered photons can take on any value so long as co' + co" ^ cota, where co^a is the two-photon transition fi-equency. The dashed line in the figure represents a 'virtual state' that has opposite parity (associated with the real states |/)) whose lifetime is determined by the Heisenberg uncertainty principle, crudely given by A/IAE"!, where AJE" is the energy difference between the virtual state and the closest opposite-parity
4, § 1]
Introduction
209
auxiliary state. For typical metastable states where two-photon spontaneous emission is the dominant decay mechanism, A^^ ^ls~^. Two-photon absorption (fig. le) occurs when an atom in the lower state is promoted to the upper state and two incident photons are annihilated, where the absorption rate Wj is proportional to the flux of photons at frequency CD' multiplied by the flux of photons at fi*equency (0^\ A new decay mechanism for the upper state that has no correspondence to any of the one-photon process is singly-stimulated spontaneous emission (fig. If). In this process, an incident photon of frequency co' ^ ^Wba induces the emission of two photons at (O^ and a single photon at the complementary fi-equency co'^ = Wba - co\ In the stimulated emission process (fig. Ig), two incident photons force the atom to the lower state and four photons are scattered by the atom. The stimulated emission rate W^^ = wll is also proportional to the flux of photons at frequency co' multiplied by the flux of photons at fi-equency O)", and proceeds most efficiently when (o' + w" c::: (Oba- The scattered photons have the same frequency, phase and direction as the incident photons, which give the laser coherence properties different from those of normal one-photon lasers. For the case when there is only a single monochromatic beam of light incident on the atom so that CD' = co" ^ \(Oba, the degenerate two-photon stimulated emission rate is proportional to the square of the incident photon flux. Working independently, both Sorokin and Braslau [1964] and Prokhorov [1965] suggested that it would be of great interest to develop a 'two-photon laser' based on the two-photon stimulated emission process. While replacing the standard one-photon stimulated emission process in a laser by a high-order one might be expected to give rise to subtle differences observable only at the quantum level, it has been predicted that there will be dramatic changes in both the microscopic and macroscopic laser behaviors even when many atoms participate in the lasing process. One reason for these differences is that the stimulated emission rate depends quadratically on the incident photon flux (for the degenerate two-photon laser where co' = o)"), resulting in an inherently nonlinear light-matter interaction. As an example, consider the effects of such a nonlinearity on the threshold behavior of the degenerate two-photon laser. Briefly, the threshold condition for all lasers is that the round-trip gain must equal the round-trip loss. For onephoton lasers this yields the well-known result that lasing will commence when a uniquely defined minimum inversion density (proportional to the gain) is attained via sufficient pumping. The situation is more complicated for the two-photon laser because the unsaturated gain increases with increasing inversion density and with increasing cavity photon number. This results in a threshold condition
210
Two-photon lasers
[4, § 2
specified by a uniquely defined minimum inversion density and cavity photon number so that it cannot turn on unless quantum fluctuations or an injected field bring the intracavity light above the critical value. In addition, once the minimum photon number exists in the cavity, the photon number undergoes a run-away process, growing rapidly until the two-photon transition is saturated (Sorokin and Braslau [1964]). Therefore, the two-photon laser operates in the saturated regime (a source of optical nonlinearity) even at the laser threshold, giving rise to the possibility that the laser will display dynamical instabilities. The primary purpose of this chapter is to review the research on two-photon quantum processes (§§2,4), leading up to the development and characterization of continuous-wave two-photon masers (§5) and lasers (§§6,7). An experimental perspective will be emphasized, with a brief description of simple theoretical models of two-photon lasers to develop a conceptual foundation for their behavior (§ 3). The vast body of theoretical research on two-photon lasers will not be covered in detail, highlighting only the early work and some of the more recent resuks (§ 8).
§ 2. Two-photon processes The concept of stimulated emission of photons was first postulated by Einstein [1917] in the development of an alternate derivation of Planck's black-body radiation law, but only encompassed the idea of the one-photon process shown in figs. la-c. The first discussion of two-quantum processes in the interaction between electromagnetic radiation and atoms appeared in an initial report by Goppert-Mayer [1929] and in a later article (Goppert-Mayer [1931]) that summarized the results of her doctoral dissertation. In these studies, the interactions of the atom and field were treated fiilly quantum-mechanically using second-order perturbation theory (one of the earlier uses of this approach). The concept of two-photon quantum processes was used soon thereafter to explain the decay rate of metastable atomic states. Breit and Teller [1940] predicted that the dominant mechanism for the decay of the hydrogen 2Si/2 state is via two-photon spontaneous emission (fig. Id) to the IS 1/2 ground state, giving rise to a predicted lifetime of the order of 0.1s. The probability of such spontaneous two-photon transitions is low, and it is smaller than those of allowed one-photon transitions by a factor of the order of a{aZf {^ 10~^ for hydrogen), where a is the fine-structure constant and Z is the atomic number (Bethe and Salpeter [1977]). The lifetime predictions were later refined by Shapiro and Breit [1959] and Tung, Ye, Salamo and Chan [1984]; confirmation
4, § 2]
Two-photon processes
211
of the predictions took many years and required the use of sophisticated experimental techniques because perturbations from external static electric fields and collisions significantly shorten the lifetime, thereby obscuring the twophoton decay process. Observation of two-photon absorption and stimulated emission processes (figs, le and Ig, respectively) required the advent of the laser, which could produce intensities high enough to increase the transition probability to detectable levels. In this section, I briefly review the early research investigating two-quantum processes. 2.1. Spontaneous emission The study of two-photon spontaneous emission has been of interest for a number of years because of its possible role in dictating the decay of metastable states (Breit and Teller [1940]), for a possible explanation for the continuous spectrum observed in planetary nebulae (Spitzer and Greenstein [1951]), and for the search of a possible electric dipole moment of the electron (Salpeter [1958]). The first direct detection of two-photon spontaneous emission was by Lipeles, Novick and Tolk [1965] using the 2Si/2 state of singly ionized helium. They used ionized helium rather than atomic hydrogen because the two-photon lifetime and Stark quenching rate are both more than an order of magnitude smaller. They verified the prediction by Spitzer and Greenstein [1951] that the photons are emitted in a continuous broad spectrum peaked at \(j)ba and followed a (1 + cos^ 6) angular distribution for polarization-insensitive detectors, where 6 is the angle between the wavevectors of the scattered photons. A review of the status of the experiments as of the late 1960s was given by Novick [1969]. In contrast to the decay of the hydrogenic 2Si/2 state, the emission spectrum and angular distribution are expected to be quite different for the two-photon decay of the helium-like 2^S triplet state due to quantum interferences arising in the twophoton matrix element (Dalgarno [1969]), although the predicted lifetime of 10 years for atomic helium imposes severe experimental challenges. The decay rate of metastable states due to two-photon processes can be enhanced significantly by singly stimulated spontaneous emission using an intense laser beam (see fig. If) as discussed by Lipeles, Gampel and Novick [1962] and later by Abella, Lipeles and Tolk [1963]. The first experiment to observe such emission was by Yatsiv, Rokni and Barak [1968] ^ who studied
^ Note the typographical error in the title of the paper by Yatsiv, Rokni and Barak [1968]: proton should be replaced by photon.
212
Two-photon lasers
[4, § 2
the interaction of intense laser beams with a dense potassium vapor. To say the least, their experiment is rather complicated because multiple intense fields were used to populate the upper state as well as to singly stimulate the spontaneous emission. Specifically, a beam generated by a ruby laser and a second beam generated by Raman-shifting some of the ruby light in nitrobenzene were overlapped and passed through a 1-m-long heated potassium vapor cell. By coincidence, the sum of the ruby laser fi*equency and the Raman-shifted frequency is essentially equal to the 4Si/2 to 6S1/2 two-photon transition frequency; the 6S1/2 and 4Si/2 states served as the upper and lower states shown in fig. If for studying the singly stimulated spontaneous emission process. The field at frequency oj' needed to induce this process was simultaneously selfgenerated in the potassium vapor by stimulated atomic Raman scattering of the ruby laser light from the potassium 4P3/2 state (Rokni and Yatsiv [1967]), which was populated by exciting the 4Si/2 -^4P3/2 transition whose resonance frequency overlaps, by coincidence, the frequency of the nitrobenzene-shifted ruby laser light. Evidence for singly stimulated two-photon emission was the observation of the complementary frequency w'^ = coba - o)', which was 10cm~^ above the 5P3/2-4S1/2 resonance line. In hindsight, there is a chance that this emission was due to a different process known as phase-matched four-wave mixing (Shen [1984]) where the incident laser beams and the selfgenerated atomic-Raman-shifted light interacted via a third-order nonlinear susceptibility to generate the complementary frequency, ft is known that twophoton resonant four-wave mixing occurs readily in strongly driven atomic vapors (see, for example, Malcuit, Gauthier and Boyd [1985]), including the specific transitions studied by Yatsiv, Rokni and Barak [1968] (Efthimiopoulos, Movsessian, Katharakis, Merlemis and Chrissopoulou [1996]). To circumvent the complexity in the possible interpretation of the experiment by Yatsiv, Rokni and Barak [1968], Braunlich and Lambropoulos [1970] investigated singly stimulated two-photon emission using a beam of metastable deuterium atoms and a single intense, 55-joule laser beam generated by a neodymium-doped-glass laser at a wavelength of 1.054 |im. The atomic beam of metastable deuterium atoms was produced by a charge-exchange reaction between a beam of deuterons and a Cs vapor. Evidence for the enhanced emission was provided by the observation of light scattered by the atoms at a wavelength of 1,373 A, which is equal to the wavelength of the complementary photon. They observed on average ~30 photons per 100 laser shots at this wavelength, which agrees well with the theoretically predicted rate of 15 photons per 100 laser shots considering the uncertainty in the efficiency of the collection optics. Related experiments aimed at generating intense vacuum-ultraviolet light were conducted
4, § 2]
Two-photon processes
213
by Zych, Lukasik, Young and Harris [1978] using a high-density hehum glowdischarge. In this experiment, the 2s ^S ^^-^ Is^ ^S transition of metastable helium underwent singly stimulated two-photon emission by irradiating the atoms with pulses from a picosecond NdiYAG laser at a wavelength of 1.064 ^im, leading to the spontaneous generation of photons at a wavelength of 637 A. Research on spontaneous two-photon emission continues to date, mainly for understanding the spectral distribution of light emitted by astrophysical objects and for fundamental tests of physics. For example, Stancil and Copeland [1993] have investigated theoretically the dependence of the lifetime of the hydrogenic 2Si/2 state in an ultra-strong magnetic field that might be produced in the vicinity of white dwarfs or pulsars. They found that the rate can be enhanced enormously, to a value approaching 10^ s~^ at fields of the order of 10^ T, because it is resonantly enhanced by the sublevels of the 2P states that are Zeeman-shifted to energies between the IS and 2S states by the magnetic field. Also, knowledge of the lifetimes of metastable helium-like ions is important for fundamental experiments using highly charged heavyion accelerators, including parity violation in helium-like uranium (Munger and Gould [1986]) and measurement of the nuclear magnetic moments of Coulombexcited nuclear states (Labzowsky, Nefiodov, Plunien, Soff and Liesen [2000]). A general relativistic theory for the two-photon spontaneous emission rates in hehum-like ions in support of these experiments was recently presented by Sanots, Patte, Parente and Indelicato [2001].
2.2. Absorption Of all the two-photon processes, absorption has found the widest range of applications, from precision measurement of physical constants (Niering, Holzwarth, Reichert, Pokasov, Udem, Weitz, Hansch, Lemonde, Santarelli, Abgrall, Laurent, Salomon and Clairon [2000]) to a new type of high-resolution microscopy that has revolutionized the study of living organisms in three spatial dimensions (Denk, Strickler and Webb [1990]). It was first observed experimentally by Kaiser and Garrett [1961] soon after the development of the laser. They passed the red light of a ruby laser beam (wavelength 694 nm) through a CaF2:Er^+ crystal and observed blue fluorescent light (wavelength 425 nm) emanating from the crystal. Their explanation of the fluorescence generation is as follows: the Er^^ ion is promoted from the ground 4f state to the broad 5d excited state via two-photon absorption, then relaxes to the bottom of the 5d band, then decays to the ground state via one-photon spontaneous
214
Two-photon lasers
[4, § 2
emission of a blue fluorescent photon. Support for their conjecture was the observation that the intensity of the fluorescent Hght scales quadratically with the intensity of the ruby laser beam. In addition, an estimate for the twophoton absorption cross-section based on the experimental parameters was in surprisingly good agreement with the theoretical estimate of Kleinman [1962], who showed how to make simplifying assumptions in evaluating the complete theory of Goppert-Mayer [1931] so that it could be compared to experiments. They also pointed out that the blue light could not be due to the phase-matched process of second-harmonic generation, which was reported only four weeks before their experiment (Franken, Hill, Peters and Weinreich [1961]), because the CaF2:Er^^ crystal possesses a center of inversion and hence second-harmonic generation should be forbidden. Numerous observations of two-photon absorption were reported by several groups with access to a ruby laser soon after the experiments of Kaiser and Garrett [1961]. One notable experiment by Abella [1962] used a ruby laser beam to excite the 6S1/2 —> 9D3/2 two-photon transition in a dilute rubidium vapor, which subsequently decayed back to the ground state via two step-wise onephoton spontaneous emission events. Because the atomic structure was well known for rubidium, a quantitative comparison to theoretical predictions could be make. It was observed that the fluorescent intensity was a factor of 100 smaller than expected, even correcting for the multi-mode nature of the ruby laser beam, suggesting that non-radiative quenching of the 9D3/2 was taking place. Similarly, Hall, Robinson and Branscomb [1965] studied photodetachment of T via twophoton excitation. Since I" only has a single bound level, the two-photonexcited detachment rate is sensitive to the bound-to-continuum transition matrix elements. They found that the experimentally measured rate was a factor of 36 larger than the predicted value (Geltman [1962]), suggesting that treating the free-electron states as plane waves is not appropriate. There have been many subsequent observations of two-photon absorption in every imaginable material and for a wide range of applications. Several discussions are available in textbooks such as those by Levenson [1982] and Stenholm [1984].
2.3. Stimulated emission and lasing Soon after the realization of the first laser in 1960, there was an explosion of research on every new and imaginable effect that might occur when light from a ruby laser was focused to high intensity and passed through matter in
4, § 2]
Two-photon processes
215
various forms and states. In addition, researchers were scrambling to develop new laser sources to extend the accessible wavelength range and to tailor the properties of laser light for various possible applications. It is therefore not too surprising that the concept for the two-photon laser was developed independently and essentially at the same time by scientists in the USA at the IBM Research Laboratories and in the former USSR at the Lebedev Institute in Moscow. The first public discussion of the work on two-photon lasers coming out of the Lebedev Institute appears to have been in December 1964 during the Nobel prize acceptance speech of Prokhorov [1965], who was being recognized for his contribution to the development of the one-photon laser. In his speech, he summarized the research leading up to the discovery of the laser, then went on to describe enthusiastically the possibility of developing a two-photon laser. He was especially interested in two properties of such quantum oscillators: They should display a faster growth of the field density in comparison to usual lasers, and they should produce simultaneously two different frequencies w' and o)" (where oj' + 0)'' = Wba)' Since the specific lasing frequencies are set by the boundary conditions imposed on the electromagnetic field by the surfaces comprising the optical resonator, such a laser could be a source of broadly and continuously tunable radiation, which would have been of great use in the area of molecular spectroscopy and controlling chemical reactions. It was also mentioned that an auxiliary laser would be needed to provide a sufficient number of photons to initiate lasing, as described briefly in § 1. He closed with a statement that the development of a tunable two-photon laser would be difficult, but that it was extremely interesting and might revolutionize the chemical industry. In hindsight, his comment on the difficulty of achieving two-photon lasing was quite appropriate; it took over fifteen years for researchers to overcome the several technical challenges that delayed the development of the first two-photon quantum oscillators. It appears that work on the two-photon laser at the Lebedev Institute pre-dates the Nobel lecture considerably. Specifically, the published lecture (Prokhorov [1965]) cites a patent disclosure with a filing date of December 1963. The patent was granted by the Soviet government sometime between the time of a brief paper describing the possibility of a two-photon laser by Selivanenko [1966] (which was submitted for publication in May 1964, one month after the work in the United States was first published), and a longer paper by Kirsanov and Selivanenko [1967] in which the patent was referenced. The longer paper gives more details of the operating characteristics of the two-photon laser and contrasts its behavior to a parametric oscillator. Note that the term two-photon oscillator was often used in the Soviet literature to describe both a two-photon laser and
216
Two-photon lasers
(a)
Jf
[4, §2 l»
P^.o
18
p^.1
c^^^^^
17
°
^f \f
f
16
o
k C
15
^a
yr f 3t
J J
^ _ ^ ^ q ( 0 ) = 1.0q,J
0), 14
^^"^^-^O-S
\f
0
5
10 time (ns)
15
Fig. 2. (a) Scheme for realization of a two-photon laser using A-ions as a primer for two-photon stimulated emission from B-ions. (b) Theoretically predicted giant pulse generation in a two-photon laser. Figure based on Smith and Sorokin [1966].
a parametric oscillator even though their behaviors are very different (see, for example, Gurevich and Kheifets [1967]). I have not yet been able to obtain a copy of the two-photon laser patent, with my search hindered by the recent passing of Prof Prokhorov, but it seems to make claims about the ability of the twophoton laser to generate intense continuously tunability pulses of light, based on the later published reports. Additional information summarizing the work on two-photon lasers from the Lebedev Institute around this time period can be found in the publication by Butylkin, Kaplan, Khronopulo and Yakubovich [1977]. The IBM group (Sorokin and Braslau [1964]) also appreciated the potential usefulness of the two-photon laser for generating intense pulses of light. The first paper focused on the operating characteristics of a degenerate two-photon laser, including the generation of a giant pulse. They also devised a different technique for achieving a critical photon density (called 'priming photons') for initiating lasing. Rather than providing priming photons from an auxiliary laser beam that was injected into the resonator, as suggested by Prokhorov, they envisioned a laser medium containing two different species of ions doped in a solid host, one possessing a one-photon transition (transition frequency w^) and the other possessing a two-photon transition (transition frequency a^). A population inversion is obtained in both species by exciting the ions to broad absorption bands via flash-lamp pumping, which subsequently decay to the upper laser transitions, as shown in fig. 2a for the case of a non-degenerate twophoton laser. One-photon lasing on the A-species ions gives rise to emission at
4, § 2]
Two-photon processes
217
frequency O^A, which triggers two-photon emission at both C^A and (DQ such that (o^A + ct>c ~ ft^e). They proposed that lasing could be obtained with technology available at the time using CaF2:Yb^^ as the two-photon medium and one of many rare-earth ions doped in one of several crystals as the one-photon medium for supplying the priming photons. Note that Sorokin and Braslau [1964] only treated the case of a degenerate two-photon laser, which was later generalized by Garwin [1964] to account for non-degenerate operation of the laser. Both reports are summarized in greater detail by Smith and Sorokin [1966]. They used a simple rate-equation model to describe the operating characteristics of the combined effects of both atomic species and the coupling of the atoms to the optical resonator, as described in greater detail in § 3. For degenerate operation of the two-photon laser, they found that two-photon amplification gives rise to an extremely fast increase in the number of laser photons once a critical number of photons ^min in the resonator is attained due to the one-photon lasing species. They described this as an avalanche effect, leading to a giant pulse that continues until most of the energy is extracted from the B-species atoms. Figure 2b shows the predictions of Sorokin and Braslau [1964] for the temporal evolution of the photon number ^ as a fiinction of the initial number of priming photons ^(0) provided by the A-species atoms. Once ^(0) > ^min, it is seen that the photon number increases by three orders of magnitude on sub-nanosecond time scales, then decreases exponentially at the cavity decay rate. Gordon and Moskvin [1976] also predicted giant-pulse operation in a two-photon laser for essentially the same configuration, although they appear to be unaware of the previous work of Sorokin and Braslau [1964]. In a related theoretical study on the short-pulse generation capabilities of a two-photon laser, Letokhov [1968] suggested that passing a short laser pulse through a two-photon amplifier will give rise to a dramatic pulse shortening whose ultimate duration would only be limited by saturation of the two-photon transition (see also Selivanenko [1966]). Lethokov also points out that significant pulse shortening can be expected for intensities much lower than that needed to operate a two-photon laser, and hence an amplifier based on two-quantum processes might be more usefiil than a laser. More recently, Heatley, fronside and Firth [1993] investigated theoretically ultrashort-pulse generation in a twophoton laser in which the bandwidth of the gain overlapped many longitudinal modes of the optical resonator. Taking into account linear dispersion and twophoton gain saturation, they found that the laser is capable of producing subpicosecond pulses via phase locking of the longitudinal modes. The two-photon laser appeared to be an experimentalist's dream because it should be continuously tunable over a broad range (recall that there were only
218
Two-photon lasers
[4, § 3
fixed-frequency lasers in the early to mid 1960s), operate at high power and store large energies. The high energy extraction capabilities were so promising that two-photon lasers and amplifiers were thoroughly scrutinized for use in laser-induced thermonuclear fusion experiments (Carman [1975]). On a more fundamental side, the two-photon laser challenges our understanding of the interaction of light with matter because it is a highly nonlinear, far from equilibrium system that cannot be analyzed easily using standard perturbation techniques. Unfortunately, achieving two-photon lasing was stymied by a lack of suitable gain media. In § 4, the early experimental work on measuring twophoton amplification and lasing will be summarized afi;er a brief introduction to simple models of the two-photon processes in the next section.
§ 3. Simple models of amplification and lasing Before reviewing the experimental work on two-photon amplification and lasing, it is usefial to consider simple models of these systems to gain an understanding of how they are different from one-photon systems and why it has been difficult to realize them in experiments. I first consider a rate-equation model of the interaction, then go on to consider how coherent effects enter in two-photon processes.
3.1. Rate-equation model In a situation where there is large dephasing of the coherences of the atomic energy levels, a simple rate-equation model of lasers is known to give results that are not too different from that obtained by more complete approaches (Allen and Eberly [1987]). Sorokin and Braslau [1964] used this approach in the first description of the operating characteristics of the two-photon laser. For simplicity, consider degenerate operation of a two-photon laser (mode volume V) filled homogeneously with atoms possessing a two-photon transition as shown in figs. Id-g. The number of photons in the cavity is denoted by q and the total number of atoms in the upper (lower) energy level is denoted by Nb (No) so that the inversion is given by AA'^ = Ni,-Na. The primary difference between one- and two-photon lasers is that the stimulated emission rate
4, § 3]
Simple models of amplification and lasing
219
is proportional to the square of the photon number, where B^^"^ is the two-photon rate coefficient. Following Concannon and Gauthier [1994], the photon number and inversion evolve according to ^"^ -B^^\^^N-K[q-q,^{t)l d^
(3.2)
-j^
(3.3)
and = -2B^'^q'AN-y\\{AN-ANo),
where ANo is the inversion density in the absence of the field due to the pump process, yy is the atomic inversion decay rate, y\\ANo is the pump rate, K is the cavity decay rate, and qmjit) is the photon number injected into the cavity by an external source. It is seen from eq. (3.2) that the photon number increases due to the two-photon stimulated emission process and by injection from the external source, and decreases due to linear loss through the cavity mirrors. The possibility of two-photon spontaneous emission processes at the laser frequency is ignored because the emission rates are extremely small in the optical regime (Holm and Sargent [1986]). This approximation is not valid for two-photon masers where the stimulated and spontaneous rates are comparable (Davidovich, Raimond, Brune and Haroche [1987]). From eq. (3.3), it is seen that the inversion decreases in response to the stimulated emission process and due to other radiative (at firequencies distinct from the laser firequency) and nonradiative decay mechanisms, and increases due to the pump process. An crude understanding of the laser turn-on behavior can be obtained by investigating the initial transient behavior of the laser using eq. (3.2) under the assumption that the inversion is not depleted during the turn-on, as discussed by Sorokin and Braslau [1964] and Schubert and Wiederhold [1979], and summarized in Schubert and Wilhelmi [1986]. With an initial inversion AN{0) and qinj = 0, the photon number increases when the initial number of photons in the cavity is greater than (3.4) For an initial photon number ^(0) exceeding the minimum value, the temporal evolution of the laser is given by q(t) = ^I^ , ^^^ l - [ l - ^ ( 0 ) / ^ ™ „ ] e « '
(3.5) ^ ^
220
Two-photon lasers
[4, § 3
which diverges in a time
tdiv
In
^(Oy^min ?(0)/
(3.6)
Of course, saturation of the inversion prevents any divergence in the photon number. Nevertheless, eq. (3.6) serves as a reasonable estimate of the time at which the photon number undergoes the explosive growth shown in fig. 2b. For the laser materials considered by Sorokin and Braslau [1964], they estimated that K = 2.5x 10^ s-\ B^^^ = 3.6x IQ-^^ s-^ and AN(0) = 2x10^^ atoms, so that ^min = 3.5x10^"^ photons, which they believed could have been obtained with existing technology. A later report (Smith and Sorokin [1966]) mentions that was probably overestimated by at least an order of magnitude. Even though the photon number can experience growth when ^(0) > qmin, sustained two-photon laser oscillation can occur only when there also exists a sufficient number of inverted atoms in the resonator. The conditions for sustained laser operation must take into account the mutual interaction of the photons and atoms, which can be determined by investigating the steady-state solutions to eqs. (3.2) and (3.3) and their stability, following the work of Ning [1991] and Concannon and Gauthier [1994]. It is found that the steady-state inversion is given by
where
^-=Vj5i
(3-8)
is the two-photon saturation photon number. Equation (3.7) is dramatically different from the steady-state inversion for a one-photon laser (Svelto [1989]) where the inversion is clamped above the laser threshold. The saturation photon number typical for the rare-earth elements can be found using yy = 1.6x10^ s~^ and the rate coefficient given above, yielding ^sat = 4.7x10^"* photons. The saturation intensity is given in terms of the saturation photon number through the relation ^at = chcjq^JV, For co = 2.7xl0^^s-^ V = 10"^ m^ and ^sat = 4.7x10^"^ photons, one has /gat = 3.8x10^^ W/m^, an intensity that was easily attainable from lasers in the mid 1960s.
221
Simple models of amplification and lasing
3]
5o2.0 ;zi
1
1
1 •
< 1.5 iz; <
/
\
.
/
/ / / /
1.0
c o •p-l w
> d 2.0
X
(b)
/ 0.5 0.0
/ L.
0.0
/
/ / /
\
A
/'
/
•
0.5
1 1.0
/
••
••'
-j
V^ .^^_^____^ ''
1 1.5
J 2.0
p u m p r a t e , AN^/AN^*' Fig. 3. Predicted steady-state response of (a) photon number and (b) inversion as a function of pump rate. From Concannon and Gauthier [1994].
The steady-state solution for the photon number is very different from that of one-photon lasers. Concannon and Gauthier [1994] found three solutions given by (3.9)
9 i = 0, and 4K-
ANo±jANi-
(3.10)
for the case when ^inj(0 = 0. The physically meaningful (real) steady-state solutions represented by these equations are shown in fig. 3 where they are plotted as a fiinction of the pump rate. The threshold conditions for sustained laser action can be determined from eq. (3.10) by finding the minimum value of AA/^ that admits a positive photon number. Upon inspection, the minimum inversion is given by t h _ 4^sat^
AA^r =
(3.11)
y\\
This yields q^ = ^sat and ANss = 2^AN^^ at threshold. For the estimated laser parameters given above, AN^ = 2.9x 10^^ inverted atoms. To turn the laser on at the minimum threshold inversion (AA^^ = AA/^^), an initial photon number approximately equal to K ^min —
B(^)AN'^
= 1/7 2^sat
(3.12)
must be present in the cavity. The photon number and inversion will evolve toward their steady-state values given above, as governed by eqs. (3.3) and (3.2).
222
Two-photon lasers
[4, § 3
Equations (3.11) and (3.12) represent the dual threshold conditions for achieving two-photon lasing, in agreement with the heuristic discussion of the threshold behavior presented in § 1. A similar condition has been found for the case of a non-degenerate two-photon laser by Hoshimiya, Yamagishi, Tanno and Inaba [1978]. For the case when photons are injected into the resonator over many cavity lifetimes, the minimum number of injected photons can be somewhat less than that given by eq. (3.12), as discussed below (see eq. 3.13). The discontinuous threshold behavior shown in fig. 3 is indicative of a firstorder phase transition (sometimes referred to as a hard mode of excitation; see Butylkin, Kaplan, Khronopulo and Yakubovich [1977]), which is very different from the smooth turn-on behavior of normal one-photon lasers. Note from fig. 3a that the two-photon gain is saturated at threshold, again in sharp contrast to the typical one-photon laser which operates very far below saturation. Figure 3b also shows that the inversion is never constant, unlike the behavior of one-photon lasers where the inversion clamps above threshold (Svelto [1989]). Based on past experience with one-photon lasers (see, for instance, Weiss and Vilaseca [1991]), it is expect that the steady-state solutions may be unstable because the laser operates in the saturated regime. Concannon and Gauthier [1994] performed a linear stability analysis of the steady-state solutions and found that: (1) the zero-photon solution (q^^.AN^^) is always stable (twophoton spontaneous emission, neglected here, can destabilize this solution); (2) the (q~^, ATV^") solution, where the photon number decreases with increasing pump rate, is always unstable; (3) the (q'^^,AN^J solution is always stable for a 'good' cavity (y\\/K > 1); and (4) the (q'^^,AN^J solution is unstable for a 'bad' cavity (y\\/K: < 1) for pumping just above threshold but stabilizes for higher pump rates. Ovadia and Sargent [1984], Ning and Haken [1989a] and Ning [1991] have found that the degenerate laser without ac Stark shifts is stable for a good cavity, may display instabilities for a moderately good cavity, and is always unstable for a bad cavity when coherent effects are taken into account, as will be described in greater detail in § 8. To address how an injected field that remains on for several cavity lifetimes can turn on the laser, Concannon [1996] determined the steady-state solutions of eqs. (3.2) and (3.3) and their stability properties when a continuous-wave beam of photons is injected into the cavity (^inj ^ 0). The stability of the solutions can be quite complex, especially in the bad-cavity limit. However, the analysis is straightforward if the laser is initially 'off', with ANo > AAA* and ^inj is increased slowly. This behavior is illustrated in fig. 4a where the three solutions for the steady-state photon number are shown when ^inj ^ 0 and ANo = 1.2A/V*. The low-power solution (dashed curve) increases as qinj increases and is stable
O
0.00
ft
111
Simple models of amplification and lasing
3]
0.04
0.08
0.12
injected photon n u m b e r , q^^^j/q sat
pl
p u m p r a t e , ANQ/ANQ
Fig. 4. Predicted behavior of the two-photon laser with an injected signal, (a) Photon number as a function of the injected number of photons, (b) Minimum number of photons that have to be injected into the two-photon laser to switch it from the off to the on state. From Concannon and Gauthier [1994].
until it reaches a critical value ^-Jj. '^ O.ll^sat- This point defines the minimum injected photon number necessary to initiate lasing. Increasing ^inj beyond ^jjjj forces the laser to switch from the low-power solution (now unstable) to the high-power solution (stable); the laser turns on. The behavior of the laser as ^inj decreases depends on the quality of the cavity. For a good cavity, the laser continues to operate at high power as ^inj decreases. For a bad cavity, the laser only continues to operate at high power if the solution is stable when q\^^ = 0. Concannon and Gauthier [1994] studied the stability behavior for other values of the pump rate and found that the injection threshold ^jjjj decreases for higher values of the pump rate as shown in fig. 4b and is given approximately by
(3.13) ^sat
when AA^o ^ AA^*. At the minimum pump rate (AA^o = AA^^^) one has 0.134^sat- In practice, a pulse (peak photon number q\^^ is injected ^inj into the laser rather than a continuous-wave beam. When the laser can (cannot) adiabatically follow the temporal variation of the pulse, ^j^ = ^Jjjj {q\^^ > ^Jjjj). The transient behavior of the laser described above was explored by Concannon and Gauthier [1994] through numerical integration of eqs. (3.2) and (3.3). Figure 5 shows how the laser responds to injected trigger pulses for a good cavity {y\\/K = 2) when the pump rate is greater than the threshold pump rate (ANo = I.IAN^^) and when there are no photons in the cavity initially. For a weak trigger pulse (peak amplitude q^^- = 0.1 qsaufig-5a) the laser is not driven above threshold, while for a slightly stronger pulse (peak amplitude ^J^ = 0,12^sat,
224
[4, §3
Two-photon lasers 0.2
120
120
time, t / t Fig. 5. Predicted temporal evolution of the photon number for (a) weak and (b) slightly stronger injected signal. The two-photon laser turns on (b) only when a sufficient number of photons are injected into the cavity. From Concannon and Gauthier [1994].
fig. 5b) the laser is driven above threshold and attains a constant amplitude after the injected pulse switches off. This is in good agreement with the injection threshold ^|Jjj ^ 0.11 ^sat for this pump rate using eq. (3.13). The most important conclusion that can be drawn fi-om the rate-equation model is that the intensity circulating in the laser resonator is always greater than or equal to the two-photon saturation intensity. Hence, the gain medium must have a low two-photon saturation intensity to avoid the high intensities that tend to magnify competing nonlinear optical processes. Fortunately, low saturation intensities go hand-in-hand with large two-photon gain, as discussed below. 3.2. Coherent effects in two-photon amplification The rate-equation model described above does a reasonable job of capturing some of the essential characteristics of the two-photon laser. Additional important features arise from the coherent driving of the two-photon dipole moment (Takatsuji [1975]), which becomes relevant when the dipole dephasing rate is comparable to the population decay rate. To determine the influence of these coherent effects, a semi-classical theory of the interaction of the laser field with the atoms is quite revealing as described by Allen and Stroud [1982] and Meystre and Sargent [1991], for example. Following Gauthier and Concannon [1994], consider the interaction of a monochromatic field E{rJ) = [e^* exp(-ia>0 + c.c] with a three-level atomic system possessing an inverted two-photon transition \b) ^-^ |a) as shown in fig. 6a. Since these states have the same parity, they are not connected by a normal one-photon transition and hence the electric-dipole matrix element connecting them is zero. A real intermediate level |/) with opposite parity is located near the virtual
4, §3]
Simple models of amplification and lasing
a)
225
b) ii'>; (0
•
-•
lb>
lb>
li> COAS
•la> Fig. 6. Simple three-level energy-level diagram for investigating two-photon amplification. Population is pumped fi-om state \a) to state IZ?) at a rate R. (b) The competing process of anti-Stokes Raman scattering where the two-photon laser field offirequencyo) generates a new field at fi-equency
intermediate level of the two-photon transition (detuning Ata = co - cOfa, Bohr frequency cota for the |/) <^ \a) transition) to enhance the two-photon transition rate. Since Aia is assumed to be small, other intermediate levels can be ignored. The electric-dipole matrix element of the \b) ^ \i) (\i) ^^ \a)) transition is denoted by ^bt (fita) and the corresponding Rabi frequency is given by Qbi = l^bi • ^E/h {Qia = Ipiia ' eE/h). Population in the upper level can decay spontaneously to the intermediate level (rate YM) which subsequently decays to the lower level (rate yta). For simplicity, assume that %« > Ybt so that essentially no population builds up in the intermediate level; Roldan, de Valcarcel and Vilaseca [1993] has explored the consequences of this approximation. Note that this model can be generalized to account for many intermediate states (Meystre and Sargent [1991]), but the physics of the problem is primarily embodied in this simple analysis. The only nonzero elements of the density matrix are: the populations paa and pbb\ the one-photon coherences pM and pia\ and the two-photon coherence pbaThe equations of motion for the density matrix are simplified using the oneand two-photon rotating-wave approximations (Narducci, Eidson, Furcinitti and Eteson [1977], Allen and Stroud [1982]) and by adiabatic elimination of the one-photon coherences (performed formally by setting pbi = pia = 0), which is valid as long as Aia > Ybt, ^bi, Qa- It is found that the equations of motion for the two-photon population inversion w = (pbb - Paa) and the slowly varying two-photon coherence Oba = Pba Qxp(i2(jot) are given by dw
~di
-711 (w - Weq) - i[0* a^^ - cr^Z)^],
(3.14)
226
Two-photon lasers
[4, § 3
and ^
= [i{A2-d,)-Y^]o,a-'-wQ,
(3.15)
In these equations, the inversion decay rate is denoted by yy ^ {YM + R), Weq is the equilibrium inversion in the absence of the field, A2 = 2co- cota is the two-photon detuning, yj_ is the two-photon coherence dephasing rate,
is the two-photon Rabi frequency, and _ \Qbi\'-\Qta\' 4Aia
_
(||i6/-e|^-||i,,-6|^),^„ h^Aia
is the ac Stark shift of the two-photon transition (Liao and Bjorkholm [1975]), where e is the polarization unit vector. Note that eqs. (3.14) and (3.15) are nearly identical to the corresponding equations for a driven two-level atom and are often collectively referred to as the two-photon two-level model (Meystre and Sargent [1991]). Important new features of this model are the proportionality of the twophoton Rabi fi-equency to the intensity rather than the field amplitude, and the presence of the ac Stark shift 4 of the transition frequency, which is also proportional to the intensity of the field. As can be seenfi-omeq. (3.15), maximal atom-field coupling occurs when the detuning of the field is adjusted to the ac-Stark-shifted resonance (A2 = ds) rather than to the bare-atom resonance (A2 = 0). This shift plays an important role in the nonlinear and quantumdynamical properties of two-photon lasers, as will be discussed in § 8, and gives rise to optical bistability when a two-photon absorber is placed in an optical resonator (Giacobino, Devaud, Biraben and Grynberg [1980]), and to cavity-less optical bistability for beams counterpropagating through a two-photon amplifier (Gingras and Denariez-Roberge [1985]). The optical field induces a polarization in the atoms as determined by eqs. (3.15) and (3.14), which acts back on the field in a self-consistent manner through Maxwell's equations. The macroscopic polarization of the medium is given by P(r,t) = rj{fi) = r]Tr(pfi), where rj is the atomic number density and the angle brackets denote a quantum-mechanical expectation value. Concannon [1996] has investigated the case of a continuous-wave beam of light passing through a low-
Simple models of amplification and lasing
4, § 3 ]
111
O
lin/«sat
Fig. 7. Predicted two-photon amplification as a function of the input intensity.
gain two-photon amplifier and finds that the output (/out) and input (/in) intensities are related through /out /in
^_
/in^^'^L
(3.18)
l+(/in//sat)2'
where the two-photon gain coefficient is given by G(2) =
(3.19)
and the saturation intensity is defined through the relation \Hi
n
e|^ IA-. ^|'7±
^v/^4^2^yii[(z^2-4)2 + r i ] '
(3.20)
^sat
Figure 7 shows the predicted laser beam amplification as a fiinction of the input intensity. It is seen that there is no gain for zero input intensity, in contrast to a one-photon amplifier where the gain is independent of the input intensity until saturation sets in. From the figure, it is also seen that the gain takes on its maximum value I^^xG^^^L/1 at /in = /sat- In light of the scaling of G^^^ (see eq. 3.19), the maximum gain is inversely proportional to the saturation intensity, which can be minimized by taking z\2 = ^s, using transitions with large dipole matrix elements, and using an atom with nearly degenerate transitions frequencies so that Aia is small (or by operating the laser in the non-degenerate mode with thefi*equenciesadjusted so that Aia is small). While the situation of a continuous-wave beam leads to a simplified analysis, it is also of interest to consider how pulses of light propagate through a twophoton amplifier since it is easier to attain the high peak intensities needed to
228
Two-photon lasers
[4, § 3
reach the saturation intensity with pulsed lasers. Narducci, Eidson, Furcinitti and Eteson [1977] considered the case of an ultrashort pulse propagating through a two-photon amplifier, where the pulse width is much smaller than the transition dephasing rate. They found that there are no asymptotically stable solutions; the pulse grows in amplitude, compresses in duration, and eventually breaks up into multiple pulses. The results obtained fi-om the density-matrix formalism are connected to the rate equations used in the previous section under conditions when there is large dephasing of the two-photon coherence {F > Z\2,4,7)- In this case, the coherence can be adiabatically eliminated fi-om eq. (3.14) using eq. (3.15) with Oba = 0. By comparison with eq. (3.3), it is found that the two-photon rate coefficient is given by
From eq. (3.21), it is straightforward to determine the saturation photon number using eq. (3.8) and the threshold inversion using eq. (3.11). It was an understanding of the scaling of G^^^ on the dipole matrix elements and the intermediate-state detuning that led to the development of the first twophoton quantum oscillators. 3.3. Competing processes Based on the preliminary estimates by Sorokin and Braslau [1964] as described above, it would appear to be straightforward to build a two-photon laser. Unfortunately, experimental realization of this new quantum oscillator was hindered by competing nonlinear optical effects that prevented the attainment of an inversion on the two-photon transition or obscured the lasing process. To lowest order in perturbation theory, the two-photon stimulated emission process is a third-order nonlinear optical effect and hence any other third-order effect can compete with it. Competing processes include self-focusing of the transverse profile of a laser beam (Carman [1975]), which degrades the spatial coherence of the beam and limits its intensity, and photo-ionization of the atoms in the gain medium by excitation of the atomfi*omthe state \b) to the continuum by absorbing one or more photons (Bay, Elk and Lambropoulos [1995]). It was found in studies of candidate two-photon amplifying media that the competing effects usually dominate. Another primary competing process is normal one-photon lasing that can occur on the \b) -^ |/) transition for the case when the state \i) is lower
4, § 3]
Simple models of amplification and lasing
229
in energy than the state \b). For large Aia, the two-photon lasing frequency is spectrally isolated from the possible one-photon lasing frequency and the latter can be avoided by using an optical resonator constructed with mirrors that are highly transparent at the one-photon resonance and highly reflective at the twophoton lasing frequency when the one-photon gain is sufficiently low For small Aia, which enhances the two-photon stimulated emission rate, the possible oneand two-photon lasing frequencies will be close. The one-photon lasing can be avoided by using a short optical resonator constructed with ultra-low-loss, high-reflectivity mirrors, such as that used in cavity quantum electrodynamics experiments (see, for example. An, Sones, Fang-Yen, Dasari and Feld [1997]), that selectively enhances the two-photon laser frequency and not the one-photon laser frequency. For this approach to succeed, no mode of the cavity (longitudinal or transverse) can overlap with the one-photon amplification resonance, which is difficult to achieve in practice because the mode frequencies tend to be dense except in mode-degenerate cavities. Even without an optical resonator, the one-photon amplification process can deplete the two-photon inversion when the gain is high, giving rise to the quantum-initiated processes of superfluorescence or amplified spontaneous emission (superfluorescence occurs when the inversion is created on a time scale that is fast in comparison to the dipole dephasing time of the transition whereas amplified spontaneous emission dominates when the excitation is slow, as discussed by Malcuit, Maki, Simkin and Boyd [1987]). These competing processes were first pointed out by Nakatsuka, Okada and Masahiro [1974] and discussed in detail by Grischkowsky, Loy and Liao [1975] in their proposal for creating a two-photon inversion by the adiabatic rapid-passage technique. The threshold for superfluorescence or amplified spontaneous emission occurs when the one-photon intensity gain coefficient G^^^ '^ 30/L so that the singlepass increase in the intensity of a beam propagating through the medium of length L is '^ exp(30). The only way to avoid this competing process is to reduce the atomic number density (which also reduces the two-photon gain!). To estimate the limitation imposed by this competing process, consider a two-photon transition where \^ibi' ^\ ^ \Mia ' ^\ and the two-photon coherence dephasing rate is similar to the population decay rate (y± ^ yy). The ratio of the one-photon gain coefficient to the maximum two-photon gain is then given approximately by
^'"^
^ 1 ^
(322)
where y^ is the coherence dephasing rate for the upper transition. For one
230
Two-photon lasers
[4, § 3
proposed scheme involving the 6P3/2 ^ 4P3/2 two-photon transition of atomic potassium (Bay and Lambropoulos [1994]), this ratio is of the order of 10^ for degenerate two-photon lasing so that the maximum two-photon gain is ~3 x 10"^ when the one-photon gain is set to the superfluorescent limit (G^^^L ?^ 30). To achieve lasing using such a small gain requires the use of an ultra-low-loss optical resonator. Petrosyan and Lambropoulos [1999] have shown that twophoton lasing can coexist with one-photon lasing that occurs on an adjacent mode of the optical resonator, suggesting that this competing effect need not be devastating. Another important competing process, identified by Carman [1975], is stimulated anti-Stokes Raman scattering of the two-photon laser field fi'om the inverted two-photon transition, as shown schematically in fig. 6b. In this process, a photon is annihilated and an anti-Stokes photon is generated as the atom undergoes a transition fi-om state \b) to state |a), thereby depleting the atomic inversion and removing photons fi*om the two-photon laser field. For most materials, it was found that the rate of anti-Stokes Raman scattering far exceeds that for two-photon stimulated emission because the intermediate virtual state of the former process is closer to a real atomic transitions |/') in comparison to the latter so that it enjoys a larger resonance enhancement. For example, Glownia, Arjavalingam and Sorokin [1985] found that stimulated antiStokes Raman scattering dominates two-photon stimulated emission in laserpumped vapor-phase triethylenediamine. Carman [1975] suggested that twophoton amplification might dominate when an intense pulse produced by a CO2 laser (wavelength 10.6 |im, used in laser-ftision experiments at Los Alamos in the 1970s and 1980s) passes through a gas of excited iodine, acting as an "afterburner" by increasing the energy by a factor of ten. Yet another competing process that may have played a role in the first experiments on singly stimulated two-photon emission and two-photon lasing is parametric wave mixing. It can arise in experiments when the laser beams that are used to create the two-photon inversion interact simultaneously with the photons involved in the stimulated-emission process, and is especially of concern when the pumping and stimulating fields travel along the same axis. In wave-mixing processes, the atom begins and ends in the same quantum state, there is a well-defined phase relation among the interacting fields, and the momentum of the fields must be conserved for efficient generation (the phasematching condition). Ascertaining the presence of parametric wave mixing can be complicated by the presence of superfluorescence or amplified spontaneous emission that produce intense beams of light self-generated in the medium whose phase fluctuates because the process is initiated or sustained by quantum noise.
4, § 4]
Two-photon amplification and lasing
231
The self-generated fields can then interact with the incident pump laser beams via parametric wave mixing, generating new fields that might obscure or prevent two-photon lasing. In most situations, parametric wave mixing can be suppressed using an experimental geometry where the pump laser beam(s) propagate in a direction orthogonal to the beam(s) undergoing two-photon amplification or lasing because the interaction cannot be phase-matched. This technique was used by Gauthier, Wu, Morin and Mossberg [1992] in the realization of the dressedstate two-photon laser and by Pfister, Brown, Stenner and Gauthier [2001] in the realization of the Raman two-photon laser. Competing nonlinear optical process limited the usefiilness of most candidate two-photon gain media, hampering research on two-photon lasers, although not for lack of effort (see, for example, Bethune, Lankard and Sorokin [1978] and Glownia, Gnass and Sorokin [1996]). In the intervening years, researchers developed one-photon lasers that had high energy, high power, short pulse durations, and tunability. Thus, many of the original applications envisioned for two-photon lasers were filled by one-photon lasers. Only with the understanding that two-photon lasers should produce beams of light with novel and potentially usefiil quantum-statistical properties was interest in them rekindled, as will be described in § 8. § 4. Two-photon amplification and lasing Overcoming the competing effects described in the previous section required a thorough understanding of two-photon processes and novel experimental techniques. The first reported observation of laser beam amplification due to the stimulated emission process was by Loy [1978] at IBM who created a transient inversion on the ammonia Vi = 0~ <^ V2 = 2' (J = 5, K = 4, M = ±5) two-photon transition and succeeded in amplifying a CO2 laser beam (10|im wavelength) by -0.2% over a brief 25 ns time interval. The inversion was created using the method of two-photon adiabatic inversion in a manner similar to that put forth by Grischkowsky and Loy [1975] and Grischkowsky, Loy and Liao [1975], also of IBM. In the experiment, two laser pulses counterpropagated through a 40-cm-long low-pressure ammonia cell to both create and probe the inversion. As the pulses entered the cell, the two-photon transition frequency was linearly swept through resonance with the sum firequency of the two beams via the dc Stark effect over an interval of 200-300 ns, and then was quickly swept back to the its original firequency over an interval of ~ 50 ns. Complete inversion of the twophoton transition occurred when the sweep interval was short in comparison to
232
Two-photon lasers
[4, § 4
the lifetime of the upper state (~4 [is at an ammonia pressure of 9 mTorr) and the rate of the frequency change was adiabatic, that is, less than the square of the two-photon Rabi frequency (see eq. 3.16). The first condition was satisfied by keeping the ammonia pressure low (9 mTorr) and the second condition was satisfied by adjusting the time over which the transition frequency was swept and the intensities of the laser pulses. During the 200-300 ns linear sweep of the transition frequency, both conditions were satisfied; most of the population of the lower energy state was transferred to the excited state and large two-photon absorption of the laser pulses was observed. During the resetting of the transition frequency, most of the population was returned to the lower state, resulting in a brief interval over which the laser pulse was amplified. At higher ammonia vapor pressures, the magnitude of the two-photon gain decreased because the inversion decay rate increased from collisions so that the inversion decayed before the resetting of the atomic frequency. Because the ammonia system displayed low gain and low energy-storage capabilities, Loy concluded that it would not be a useful candidate for building a two-photon laser. Taking a different approach to the problem, Schlemmer, Frolich and Welling [1980] investigated continuous-wave non-degenerate two-photon amplification that was resonantly enhanced by an intermediate atomic level. In their experiment, they measured simultaneous lasing on the neon 3s2 ^ 3 p 4 (3.39 (im transition wavelength) and 3p4 ^ 2s2 (2.40 (jim transition wavelength) cascade transition using two optical resonators, each of which supported normal onephoton lasing on one of the transitions. They observed that the power tuning curve of one of the lasers contained an off-resonance feature when the frequency of the other laser was tuned to the edge of its tuning range so that the sum of the two laser frequencies was equal to the 3s2 <-> 2s2 two-photon transition frequency. For this detuned operation and a ring-laser geometry, they also found that the emission direction for the two lasers was unidirectional and in opposite directions, which they attributed to the reduction of the Doppler effect for counterpropagating beams. While it is clear that two-photon amplification was playing a role in their experiment, they did not attempt to ascertain the relative contributions of two-photon and off-resonant one-photon amplification. I am not aware of any follow-up experimental research to determine whether two-photon lasing could be obtained in this system for larger intermediate-state detunings, which is unfortunate because the neon system appears to be very promising for making a non-degenerate two-photon laser in the infrared part of the spectrum. Soon after these experiments in ammonia, Nikolaus, Zhang and Toschek [1981] reported the observation of two-photon lasing in a laser-pumped.
4, § 4]
Two-photon amplification and lasing
233
high-density Hthium vapor. In the experiment, 4 ns pulses generated by two tunable dye lasers counterpropagated through a lithium heat pipe operated at a temperature of 720°C, corresponding to an atomic number density of ~10^^ cm"^. The frequency (JOI of dye laser 1 was tuned to the high-frequency side of the one-photon transition, and the frequency (O2 of dye laser 2 was adjusted so that 0)1 + 0)2 was equal to the 2s^S^->4f^F^ transition. The laser pulses transferred most of the population to the 4f ^F^, giving rise to prompt superfluorescent emission that quickly transferred population to the 3d^D state, resulting in a transient two-photon inversion between the 3d^D upper state and the 2s ^S ground state. Note that two-photon excitation of the 4f^F^ should be forbidden because the 2p ^P^ ^ 4f ^F^ transition is electric-dipole forbidden; they presumed that it occurred due to the small electric quadrupole moment connecting the states. Subsequent detailed analysis by Sparbier, Boiler and Toschek [1996] indicated that the lasers actually excited a nearby Li2 molecular state, which rapidly dissociated, leaving atomic lithium in the excited 4f ^F^ as well as other nearby states. Once a two-photon inversion was established, photons from dye laser 1 at frequency o)\ singly stimulated emission at frequency 0)^ that was observed to propagate along the same axis, but opposite to the propagation direction of the stimulating beam. In a single pass through the vapor, the light atfrequencyo)jc attained a maximum peak power of 30 W. Nikolaus, Zhang and Toschek [1981] referred to this light as 'laser emission', although no optical resonator surrounded the medium^; it is more reminiscent of the super-radiant light generated by nitrogen or excimer lasers (Svelto [1989]). Nikolaus, Zhang and Toschek [1981] stated that the interaction can not be due to a parametric wave-mixing process because such an interaction could not be phase-matched with a geometry of counterpropagating pump beams, and the emission showed no phase correlation with the beam produced by dye laser 2, although they did not describe how this measurement was performed. In a second experiment, they injected light from a third dye laser whose frequency a>3 was set to precisely one-half the 3d ^D <-^ 2s ^ S two-photon transition frequency. They observed that the beam was amplified by up to 20%, which they attribute to two-photon amplification. These experiments using laser-pumped lithium vapor were met by some skepticism from the IBM group. Jackson and Wynne [1982] conducted experiments in
^ This experiment is similar to the eariier observation of singly stimulated two-photon emission by Yatsiv, Rokni and Barak [1968], with the exception that there is greater flexibility in choosing the frequency W\ because it is generated by a tunable laser.
234
Two-photon lasers
[4, § 4
laser-pumped sodium vapor and observed light generated at co^ whose frequency characteristics were analogous to that observed by Nikolaus, Zhang and Toschek [1981]. Rather than attributing the generated light to two-photon emission, they showed that it could be due to parametric wave mixing. The interaction can be phase-matched even with the counterpropagating pump-beam geometry, but only for the case when the generated light is emitted counterpropagating with the pump beam at co\, consistent with their observation. Jackson and Wynne [1982] noted that two-photon emission should be generated in both directions, inconsistent with their observation. In a separate experiment, they were not able to observe two-photon amplification when a third laser was tuned to the analogous two-photon transition. Since Nikolaus, Zhang and Toschek [1981] only reported on measurements of generated light that counterpropagated with (0\, the claim of Jackson and Wynne [1982] could not be tested. Following up on the experiments of Jackson and Wynne [1982], Gao, Eidson, Squicciarini and Narducci [1984] observed singly stimulated two-photon emission on the sodium 8P <-^ 3P two-photon transition, which was inverted by single-photon excitation of the 3S ^-^ 8P transition. Their experiment was unique in that they only attempted to obtain two-photon emission after the pump laser pulse had exited the vapor but before the inversion decayed via cascade onephoton spontaneous emission, thereby suppressing all possible phase-matched interactions involving the pump beam. They observed singly stimulated emission at 1.14 fi,m that was correlated and counterpropagated with the stimulating beam at 0.74 [O-m. They did not observe any singly stimulated emission that copropagated along the direction of the stimulating beam. They attributed this asymmetry of the emission to the near cancelation of the Doppler broadening of the two-photon transition, thereby increasing the gain, which occurs only for the counterpropagating geometry. Thus, they demonstrated that an asymmetry in the emission direction does not necessarily indicate a phase-matched interaction, as put forth by Jackson and Wynne [1982]. Many years later, Sparbier, Boiler and Toschek [1996] performed a series of detailed experiments in laser-pumped lithium to shed additional light on the emission observed by Nikolaus, Zhang and Toschek [1981]. One of the main findings is that the pump lasers actually excited a lithium dimer state that dissociated, leaving behind atomic lithium in the 4f ^F^ state. They suggest that this excitation mechanisms suppressed parametric wave-mixing processes involving this state since dissociation is believed to be an incoherent process that does not depend on the relative phase of the exciting fields under their experimental conditions. In addition, analysis of the spectrum and state of polarization of the generated light and its dependence on the tuning of the
4, § 5]
The two-photon maser
235
pump laser beam frequencies revealed that two-photon emission and parametric wave mixing coexisted with nearly equal strength. They observed that the emission direction was asymmetric, consistent with the experiments of both Jackson and Wynne [1982] and Gao, Eidson, Squicciarini and Narducci [1984]. They did not attempt to observe the two-photon emission process after the pump laser beams had exited the lithium vapor cell to test whether the presence of the pump beams was required for the observed emission, but it seems clear that several scattering processes, including two-photon stimulated emission, occurred simultaneously. These early experiments bring to light the great difficulties in constructing two-photon lasers in the presence of competing effects and point to the need for a drastically different experimental approach^.
§ 5. The two-photon maser The crucial advance that revolutionized research on two-photon quantum oscillators was the use of experimental techniques developed for research in cavity quantum electrodynamics (CQED); see, for example, Berman [1994]. In such experiments, only a few atoms at a time, often produced by an atomic beam source or a cloud of cooled and trapped atoms, pass through a small-volume ultrahigh-g resonator. No windows can be placed in the resonator to contain the atoms because the small reflection or scattering losses from the window surfaces would seriously degrade the cavity Q, and hence the resonator must be placed in the same vacuum chamber as the atom source. In the mid 1980s, experiments were underway at I'Ecole Normale Superieure in Paris and at the Max-Planck-Institut fur Quantenoptik in Garching that combined the ultralarge electric-dipole moments characteristic of alkalimetal Rydberg atomic transitions in the microwave part of the spectrum with ultrahigh-g superconducting cavities for studying the strong-coupling regime of CQED. In the CQED research most related to the present discussion, Brune, Raimond and Haroche [1987] investigated the possibility of building a two-photon maser based on nSyi'^in - l)Si/2 two-photon transitions that are enhanced
^ I note that some authors (see, e.g., Ovadia and Sargent [1984], Swain [1988] and Ning and Haken [1989a]) classify an experiment by Grynberg, Giacobino and Biraben [1981] as an observation of two-photon lasing. However, this experiment used two-photon excitation of an atomic state that served as the upper level of a normal one-photon laser.
236
Two-photon lasers
[4, § 5
by the {n ~ \)Vy2 intermediate state, where n is the principal quantum number. They discovered that the Rydberg states of the alkali-metal atoms are especially favorable for enhancing the degenerate-frequency two-photon stimulated emission rate because the intermediate state detuning Aia is nearly zero for one value of the principal quantum number, denoted by HQ, due to the slow variation of the quantum defects of the energy levels as a function of n. Thus, by taking n = «o ± 1, it is possible to spectrally separate the degenerate two-photon resonance from the competing one-photon resonance while keeping the rate large. For rubidium, (rio - 1) = 38 and Aja/lJt = 39 MHz for (« - 1) = 39, which should be compared to the two-photon maser frequency co/2jt = 68.42 GHz, and cavity linewidths of the order of 1 kHz for a cavity Q of 10^ that were available at the time. Using typical parameters for the rubidium system and available superconducting microwave resonators, Brune, Raimond and Haroche [1987] estimated an amazingly low threshold for two-photon masing of only one atom and a few tens of photons on average in the resonator! Thus, the two-photon maser would operate in what is known at the 'micromaser' regime. They envisioned that a dilute stream of atoms would pass through the resonator in a time ^int on average with a mean time ^at between atoms, where t^t > ^int so that there is at most one atom in the cavity on average. The cavity lifetime ^cav = S / ^ was assumed to be much longer than both ^t and ^int so that the atoms, one at a time, would interact with the cavity photons that had been left behind from previous atoms. The threshold conditions for masing, expressed in terms of these quantities, are given by ""'-' kt
AVehA,. ^ ^ ,
''-^ " ^-
^'-'^ (5.2)
where q is the mean cavity photon number and Q is the two-photon Rabi frequency, given approximately by eq. (3.16). Threshold conditions (5.1) and (5.2) are equivalent to the conditions (3.4) (with AN(0) = AA^th) ^^^ (3.11), respectively, discussed in § 3. For the rubidium parameters given above, tint = 25 (xs, (fibi • e) = 1,443^^0, (fiia • e) = 1,479^^^, Q = 10^ (^cav = 0.23 ms), and V = 70 mm^, the threshold conditions predict t^t = 14.3 |bis and q = 32. Brune, Raimond and Haroche [1987] also pointed out that the two-photon spontaneous emission rate for this atomic transition is relatively large so that the threshold condition for the minimum number of photons can be satisfied by spontaneously generated photons emitted into the cavity and does not require
5]
The two-photon maser
237
injection of an external field. They developed a full quantum theory for the two-photon maser that takes into account the ac Stark shifts, and predicted that the time it takes for the mean photon number to go above threshold will show a significant lethargy because the probability for a large enough quantum fluctuation was somewhat low In follow-up studies, Davidovich, Raimond, Brune and Haroche [1987] and Davidovich, Raimond, Brune and Haroche [1988] fiirther developed and analyzed a quantum theory of the two-photon micromaser. Using a master equation approach for describing the complete state of the field in the cavity, they found that the photon-number statistics are sub-Poissonian for a wide range of atomic injection times and hence the device could be used to generate squeezed states of the electromagnetic field (see § 8).
5.1. Realization of the two-photon maser The first demonstration of a two-photon maser based on this proposal was reported by Brune, Raimond, Goy, Davidovich and Haroche [1987] who used the rubidium atomic transitions discussed in the example above. The energylevel structure for the two-photon transition and the experimental setup is shown in fig. 8. A collimated thermal atomic beam of rubidium atoms passed through three laser beams that propagated in a direction orthogonal to the motion of the atoms to cancel Doppler broadening. The lasers sequentially excited the
ionization signal Fig. 8. Two-photon maser. Top panel: energy-level diagram of Rb showing the relevant states and excitation pathway. Bottom panel: experimental setup showing the atomic and optical pumping beams, superconducting microwave cavity, and field-ionization region. From Brune, Raimond, Goy, Davidovich and Haroche [1987].
238
Two-photon lasers 40P
'^ =» — • b)
.i
40S
39S
1
1 cr
r-1
2
i
LU CT TO O
3
1 ..^/^~V^_^
Z
o
LU UCO
^ o TIME
• // / / //
Q:
»—
•^ n
1
\\ \ \ \\, \
\
>0.2
»— UJ
x2
1
c;
Z 0.4
_ j
d)
[4, §5
. ^__^^„^
1 v'»
• 1 .85 FREQUENCY
.V
^"-.^._^_^ 1 68 415.90 (MHz )
.95
0.5 ^ts
Fig. 9. Two-photon maser operating characteristics. Population of the atomic states when the maser operates (a) below threshold and (b) above threshold, (c) Tuning the two-photon maser through the gain resonance. From Brune, Raimond, Goy, Davidovich and Haroche [1987].
atoms from the ground 5Si/2 state to the 4OP3/2 Rydberg state just before they entered a liquid-He-cooled superconducting niobium microwave resonator with a ig of 10^ at a temperature of 1.7 K. Just as the atoms entered the resonator structure, but before they entered the resonator itself, a coherent microwave field transferred the atoms from the 4OP3/2 state to the 40Si/2 state, which served as the upper state of the two-photon transition. The excited atoms entered the resonator and interacted with the cavity field. The frequency of the resonator was adjusted precisely to the two-photon transition frequency (measured to be 68.41587±0.00001 GHz in a separate experiment) by mechanical deformation of the resonator. Note that the lifetime of the upper state is long in comparison to the transit time of an atom through the resonator so that the number of atoms in this state should remain constant unless the interaction between the atom and cavity de-excites the atom due to lasing processes, for example. For such a high-ig resonator, very few photons are emitted by the cavity so the researchers instead measured the number of atoms in each atomic state. After the atoms exited the resonator, they passed through field-ionization plates upon which a linearly increased voltage was applied. In this manner, the ionization current is directly proportional to the applied voltage, which in turn is directly proportional to time during the voltage sweep. Figure 9a shows the ionization current as a ftinction of time for a low flux of atoms [1/fat = (8±4)x 10"^ atoms s~^], where it is seen that rubidium atoms were only found in the 40P and 40S states, indicating that the maser was below threshold. For larger atomic fluxes [1/fat = (2ibl)xl0^ atoms s~^ fig. 9b], more than half the atoms that entered in the 40S state exited in the 39S state, indicating that the laser was above threshold. Furthermore, the measured threshold atomic flux
4, § 6]
The dressed-state two-photon laser
239
agreed well with the predicted value of 1/^at = 1/14.3 ^s = 7x10"^ atoms s"^. The maser also displayed a very strong dependence on the frequency of the microwave resonator as shown in fig. 9c when the atomic flux was high. The relative transfer of population from the upper to the lower state was monitored and it was found that high transfer (indicating masing) occurred over a tuning range of only ~40 kHz. This tuning range is much smaller than the detuning Aia from the competing detuned one-photon \b) <^ \i) resonance, providing further evidence that they were observing pure two-photon masing. Finally, they observed very long turn-on times of the maser (up to a few seconds) when the atomic flux was set close to the threshold value and the cavity initially started in the vacuum state, consistent with the predictions of Brune, Raimond and Haroche [1987]. This one experimental result has led to numerous theoretical investigations of the two-photon micromaser because the physical parameters of the experimental system are well known and only a small number of atoms and photons are present in the resonator. It has become a universal system for investigating the nonlinear interaction between light and matter from first principles. Recent theoretical research, for example, includes a study of cooperativity when two atoms are present in the resonator simultaneously (Ashraf and Toor [2000]), the influence of atomic decay on the linewidth of a two-photon micromaser (Hamza and Qamar [2000]), the effects of two-photon atomic-motion-induced amplification of radiation (Zhang and He [1999]), and the existence of trapping states and the generation of a photon-number state (Alexanian, Bose and Chow [1998]). A complete review of the body of research on theoretical aspects of two-photon micromasers is beyond the scope of this chapter.
§ 6. The dressed-state two-photon laser The major breakthrough in the development of optical two-photon quantum oscillators came with the realization that it is possible to 'engineer' a nearideal two-photon gain medium that possesses a small and adjustable detuning, as discussed by Lewenstein, Zhu and Mossberg [1990]. It consists of a two-level atom with an electric-dipole-allowed transition driven by an intense, continuouswave, near-resonant laser field. The coupling is so strong between the atom and the intense field that it make more sense to think of it as a composite atom-field system (a 'dressed' atom) rather than separate entities. By combining dressed atoms with optical CQED experimental techniques (Kimble [1994]), they showed that it is possible to realize a two-photon laser that operates with only
240
[4, § 6
Two-photon lasers 0.010
-
'
•
• (b)i
two—photon gain
0.005 0.000
_j\i\»
\i
-0.005 -0.010 -50
-25
25
50
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p r o b e - p u m p detuning,
1
-25
i/N
^
W-
1
1
25
~
50
{cj-co^)/y^g
Fig. 10. Predicted laser beam amplification fi-om strongly driven two-level atoms for (a) a weak and (b) an intense probing field.
a few thousand atoms and photons in the cavity, and that it is possible to avoid competing nonlinear effects by using an appropriate experimental geometry. It has been known for quite some time that the absorption spectrum of a weak probe beam interacting with a collection of dressed atoms has a spectrally narrow gain feature for the case when the dressing laser frequency is detuned from the bare atomic resonance frequency (Boyd [1992]). Consider a two-level atom with a ground (excited) level denoted by \g) (\e)) driven by an intense 'dressing' laser field E(^(t) = [Efi exp(-ia;dO + c.c.]. The atomic states have opposite parity with transition matrix element fi^g. The population decay rate (coherence dephasing rate) is denoted by y!^ (y^f) and it is assumed for simplicity that the atom experiences only natural broadening so that y^f = ^/n^. The interaction strength of the driven-atom system is characterized by the generalized Rabi frequency Q^ = J^l +^d' where Q^ = l^^g • E^/h is the resonant Rabi frequency and ^d == ^d - (^eg is the detuning of the driving field from the atomic resonance frequency (Oeg. The Rabi frequency can be of the order of 6 x 10^ s~^ for a strong optical transition and a 1 W laser beam collimated to a diameter of ~1 mm. Figure 10 shows an example of a weak-probe gain spectrum calculated with the density-matrix formalism (Boyd [1992]). The gain feature occurs at a probebeam frequency 0)^= co^-Q'^ for A^ < 0, its width is equal to that of the undriven transition y j , and the maximum gain is equal to ~0.05a^g, where aeg is the linecenter absorption coefficient of the undriven system. Agarwal [1990] showed that a laser based on this one-photon gain feature is identical in every respect to a normal laser, even including the quantum-statistical properties of the generated light, although later studies show that Bloch-Siegert-like shifts can significantly
4, § 6]
The dressed-state two-photon laser
241
modify the squeezing spectrum of the laser (Zakrzewski, Lewenstein and Mossberg [1991]). Several research groups have successfully constructed onephoton lasers using dressed atoms, as summarized in the more recent work by Lezama, Zhu, Kanskar and Mossberg [1990]. Dressed atoms display phaseinsensitive optical amplification {i.e., not involving a phase-matching condition) when the angle between the propagation direction of the dressing and the propagation direction of the probe field or cavity axis is large and as close as possible to 90"" (orthogonal geometry). For the case where the pump and probe beams are nearly colinear, it is known that dressed atoms can give rise to large probe-beam amplification due to phase-matched four-wave mixing, as discussed by Boyd, Raymer, Narum and Harter [1981]. The absorption spectrum of a dressed atom becomes complex when the probebeam intensity increases as shown in fig. 10b, calculated using the procedures described by Agarwal and Nayak [1986]. Several new, intensity-dependent gain features appear that occur at co = Wd - Q'^/m (m= 1,2,3,...). The existence of these features has been known for some time; they were originally referred to as "subharmonic resonances" (see, for example, the more recent work of Papademetriou, Chakmakjian and Stroud [1992]). The interpretation that these features correspond to multi-photon stimulated emission resonances was first pointed out by Lewenstein, Zhu and Mossberg [1990]. In particular, they showed that the feature occurring at co = co^ - ^Q'^ arises from two-photon stimulated emission and could be used to realize a two-photon laser. As in the one-photon gain feature, it is possible to enhance multi-wave mixing processes near these multi-photon gain features for nearly colinear laser beam geometries. Thus, an orthogonal geometry is needed to suppress wave-mixing processes. A basic understanding of the origin of the gain features can be obtained from a perturbation-theory analysis of the interaction of the probe beam and the driven atom. It is found that the gain features arise from hyper-Raman scattering processes, as shown in fig. 11. In both processes, probe-beam photons (solid arrows) are created at the expense of pump-beam photons (dashed arrows) when the fields induce a transition from the ground level \g) to the excited level \e). Hence, the 'upper' ('lower') laser level is actually the ground (excited) level of the atom for the driven-atom gain medium. The 'inversion' between the upper and lower laser levels is maintained by spontaneous emission from \e) to \g). Note that there is never an inversion between the excited and ground levels of the atom. In the one-photon amplification process (fig. 1 la), one probe-beam photon is created and two dressing-field photons are annihilated, while in the two-photon amplification process (fig. 1 lb), two probe-beam photons are created and three dressing-field photons are annihilated. This analysis does not accurately predict
Fig. 11. Perturbative scattering diagrams for (a) one- and (b) two-photon laser beam amplification in laser-driven two-level atoms.
the location of the gain resonances shown in fig. 10b because the perturbative analysis does not converge for such a strong atom-field interaction. A quantitative (and intuitive) understanding of the origin of the gain features can be obtained using the dressed-state interpretation of the driving-field-atom interaction. The analysis properly accounts for the interaction of the intense pump field with the atom and treats the probe field and spontaneous emission as perturbations. As described by Cohen-Tannoudji, Dupont-Roc and Grynberg [1992] and Compagno, Passante and Persico [1995], the dressed states are the energy eigenstates of the Hamiltonian describing the two-level atom, the single quantized mode of the dressing field, and their interaction. They are linear superpositions of the atom-field product states and are denoted by |±,«), where n is the photon occupation number (volume) of the dressing mode. The dressed states consist of an infinite ladder of levels separated by the energy of a dressing photon ho)^, and each rung of the ladder is a doublet. For a driving field produced by an intense continuous-wave laser beam, described as a coherent state, only a narrow distribution of dressed levels high up the ladder are occupied. They are 'locally periodic' since their relative populations and the splitting of the doublets are independent of «. In this case, the doublet splitting is given by the generalized Rabi fi-equency O^. The interaction of the dressed states with the other field modes is treated using perturbation theory. It is found that all matrix elements are zero except for those that connect one rung of the ladder with its nearest neighbor. Considering the dressed-state selection rules, it is found that each level on a rung of the ladder decays spontaneously to both levels one step down the ladder. After several spontaneous emission events, the relative populations of a dressed-state doublet reach equilibrium in response to spontaneous decays into and out of the doublet, and the relative population distribution is independent of n due to the local periodicity. When the dressing field is on resonance (A^ = 0), both levels are
4, §6]
243
The dressed-state two-photon laser
a)
b)
a"i 0)d
l+,n+1> l-,n+1>
l+,n+1> l-,n+1> d
d
l+,n> l-,n>
d
-V-
d
l+,n> l-,n>
co = a ) , - Q ;
l+,n+1> l-,n+1>
l+,n+1> l-,n+1>
Fig. 12. Dressed-state energy-level diagrams showing (a) one- and (b) two-photon amplification in laser-driven two-level atoms.
equally populated; for off-resonance driving the populations become unbalanced. For example, the states \-,n) are more heavily populated that the states |+,«) when A^ < 0, as shown in fig. 12. A probe beam (frequency co) interacting with the dressed atom experiences gain or absorption when the populations of the dressed levels are unbalanced just as a bare atom will absorb or amplify a beam of light. It can be seen from fig. 12a that a probe beam interacting with the |-,« + 1) ^^ |+,w) transition will be amplified due to one-photon stimulated emission when o) = 0)^- Q'^ for Z\d < 0. This is precisely the spectral location of the gain feature shown in fig. 10a. Analogously, one-photon absorption will occur on the \-,n) ^-» |+,/2+1) transition. An intense probe beam can induce multi-photon transitions among the dressed levels. In particular, a probe beam interacting with the |-,« + 1) ^ |+,« - 1) will be amplified due to two-photon stimulated emission when co = co^- \Q'^ as shown in fig. 12b. Most importantly, the \+,n) and \-,n) intermediate levels are nearly degenerate with the virtual level of the two-photon transition, thereby enhancing significantly the two-photon rate coefficient. In addition, the twophoton gain is spectrally distinct from the regions of large one-photon gain; hence, a high-finesse optical resonator can selectively enhance the two-photon gain. It is precisely for these reasons that dressed atoms make a near ideal twophoton gain medium. Note that the driven-atom system can support m-photon gain due to m-photon stimulated emission when it undergoes a transition from one rung on the ladder to m rungs below.
244
Two-photon lasers
[4, § 6
Extending the proposal of Lewenstein, Zhu and Mossberg [1990], a thorough theoretical investigation of the dressed-state two-photon laser was undertaken as described in a series of theoretical papers by Zakrzewski, Lewenstein and Mossberg [1991]. Their effective Hamiltonian approach to derive semiclassical two-photon-laser equations includes the effect of the ac Stark shifts arising from the two-photon cavity field, but assumes that the cavity linewidth is narrow so that the two-photon process can be distinguished from the competing onephoton process. Interestingly, they find that the one- and two-photon processes are strongly anti-correlated so that once the two-photon laser turns on, it will suppress to some degree the coherence needed for one-photon lasing. In addition, they find that the ac Stark shift narrows considerably the domain of parameters over which stable laser operation exists, and causes significant intensity-dependent 'pulling' of the laser frequency. Based on an analysis of the steady-state solutions of the laser equations, they determined the threshold for sustained continuous-wave two-photon lasing. The threshold depends on the detuning of the dressing field from the atomic resonance and takes on its minimum value when A^ = -OAQ^. At this optimum value, the threshold number of dressed atoms in the cavity is Mh^25.4^,
(6.1)
and the number of photons in the cavity at the laser threshold is ^th^2.36^, g^
(6.2)
where K is the cavity decay rate, g = y^/6jtc^yf>^/a)^gVe is the atom-cavity coupling constant"^ corresponding to the Rabi frequency for a single photon in the cavity, V^ = JTWILC is the effective mode volume, Wo is the 1/e radius of the field in the cavity, and Lc is the cavity length. To estimate the threshold requirements, consider creating dressed atoms using the barium ^SQ <-^ ^Po transition {cOeg = 3.4xl0'^rads~^ yn^ = 1.19xl0^rads~^) driven with an intense laser beam (^d = 2.6x 10^ rad s~^) and the use of a high-finesse optical cavity with Wo = 66(i,m, Lc = 5 cm, and K = 5.3xl0^rads~^ yielding g = 2.8 X lO^rads"^ For this experimental arrangement, A^th :^ 4.5x10"^ atoms
^ Note that Lewenstein, Zhu and Mossberg [1990] define g as the Rabifi*equencyfor a single photon in the cavity. Many other researchers define it to be one-half the single-photon Rabi firequency
4, § 6]
The dressed-state two-photon laser
245
and ^th — 9.3x10"^ photons, which was attainable with available technology. Note that this theory of dressed-state lasers assumes that the two-level atom only experiences spontaneous-emission broadening and that the resonator is in a ring configuration. A stability analysis of the steady-state solutions revealed that the laser should be stable, at least in the vicinity of the threshold. Further above threshold, multiple effects destabilize the dressed-state two-photon laser operation. Zakrzewski, Lewenstein and Mossberg [1991] predicted that the laser will display instabilities far above threshold due to the presence of ac Stark shifts induced by the cavity field. In addition, off-resonant one-photon amplification and lasing is expected to disrupt two-photon laser operation at an atom number approximately equal to 15 times the threshold value, which still gives a considerable range for investigating its behavior. This contrasts the behavior of two-photon lasers based on inverted three-level atoms; those are unstable just above threshold and become stable far above threshold, as will be described in § 8. A later study by Zakrzewski and Lewenstein [1992] investigated the operating characteristics of the two-photon laser in the so-called 'bad-cavity' limit, where the cavity decay rate exceeds all other decay rates of the system. Under these conditions, the effective-Hamiltonian approach used by the researchers is no longer valid, which required formulating the problem in terms of generalized effective theories that are based on a hierarchy of macroscopic variables. They find that competition between two- and one-photon lasing can give rise to self-pulsing and even chaos in the intensity of the beam generated by the laser. For the case of dressing the atom with a short pulse so that the energies of the dressed states are now a fiinction of time, Zakrzewski, Segal and Lewenstein [1992] found that the light emitted by the cavity has different frequencies arising from the various dressed energy levels, which interfere and give rise to a very complex field generated by the laser. Connecting the results for the dressed-state laser to the earlier discussion in § 3, it is found that the two-photon rate coefficient is given by
at the optimum dressing field detuning. It is seen that the dressed-state and threelevel-atom rate coefficients are similar with Aia -^ \^'d^y comparing eqs. (6.3) and (3.21). Thus, the rate coefficient for a dressed atom can be considerably larger than that for a typical three-level atom with fixed energy levels because Q'^ can be small. Note that there is an optimum value of Q'^ for a particular
246
Two-photon lasers
[4, § 6
experimental situation since the one- and two-photon gain features will start to overlap as Q'^ becomes small.
6.1. Measurement of continuous-wave two-photon amplification To test the proposal of Lewenstein, Zhu and Mossberg [1990], Zhu, Wu, Morin and Mossberg [1990] used dressed ^^^Ba as the two-photon amplification medium. Under appropriate conditions, ^^^Ba is a near ideal two-level atom because the ^ So <-^ ' Po transition has zero nuclear spin and hence is not complicated by hyperfine interactions, and it possesses a large electric-dipole moment with a transition wavelength accessible by high-power tunable dye lasers. One minor complication is the presence of other barium isotopes (22% abundance in natural barium), but their resonances are all to the highfrequency side of the ^^^Ba ^SQ ^ ^Po transition frequency. In the experiments, an atomic beam source of barium was used to reduce the Doppler broadening of the transition. The Doppler effect broadens and reduces the two-photon gain feature, and requires the use of a higher-intensity dressing field because the Rabi frequency needs to exceed the Doppler width to see appreciable laser beam amplification. On the other hand, the use of an atomic beam limits the available atomic number density and the length of the gain medium. To compensate for loss in gain, they surrounded the dressed atoms with a 1-cm-long optical enhancement cavity (a confocal FabryPerot optical resonator) to increase the effective path length of the medium by approximately the cavity finesse (200 in their experiment). The dressed-state gain was determined by measuring the transmission of a probe beam that passed through the cavity. The axis of the atomic beam and enhancement cavity, and the propagation direction of the dressing laser beam, were mutually orthogonal to suppress wave-mixing-type gain processes, as shown in the top panel of fig. 12. The atomic beam was rapidly turned on and off using a mechanical chopper so that lock-in signal-averaging techniques could be employed. During the interval when the atomic beam was blocked, a standard modulation technique was used to actively lock the frequency of the probe laser to the empty-cavity resonance to compensate for laser frequency drift. The gain measurements were complicated by their choice of a locking scheme because the dispersion of the dressed atoms shifts the frequency of cavity. Therefore, the probe-beam transmission was always below what it would have been in the absence of dispersive effects, mimicking absorption. Locking the probe-beam frequency to the cavity resonance in the presence of the dressed atoms was not easily
247
The dressed-state two-photon laser
4, §6]
Pump Laser
NIN
(y)-''
Cavity-Probe Lock
I
PZT-^
PMT A1
Computer
A 2 LOOK-IN Amplifier
100
o 100 500
Frequency v^~
v^{yiRz)
Fig. 13. Experimental measurement of dressed-state two-photon laser beam amplification. Top panel: experimental setup. Bottom panel: laser beam amplification for (a) a weak and (b) an intense probing field. From Zhu, Wu, Morin and Mossberg [1990].
implemented due to the intensity-dependent gain and dispersive effects arising from the multi-photon stimulated emission and saturation processes, causing the locking loop to be unstable. Figure 13a shows the probe-beam gain spectrum for the case of a weak probe beam, Q^/ln c:^ 340 MHz, and A^ c± -100 MHz so that Q'^/2jt ~ 354 MHz. It is seen that there is a large gain feature when the frequency difference between the probe and dressing fields is approximately equal to -Q'^, indicating the onephoton stimulated emission process between the states |+,«) ^^ |-,« + 1) shown in fig. 12a. For a higher probe beam power that is sufficiently strong to induce the two-photon stimulated emission process | + , « + ! ) ^^ \-,n - I) shown in fig. 12b, a new feature appears in the gain spectrum at approximately -\^'^, as seen in fig. 13b. Carefiil inspection of the data reveals that the feature represents reduced absorption rather than gain, which they attributed to the pseudoabsorption effect arising from the dispersion of the dressed atoms, as discussed above. The dashed line in the figure shows the theoretical predictions for the experiment taking into account the dispersion, dressing-field inhomogeneities, the standing-wave probe-field pattern within the cavity, and the barium isotopes. It is seen that the agreement between the measurements and predictions is very
248
Two-photon lasers
[4, § 6
good. From their analysis, they infer that they would have observed an increase of the probe-beam transmission of approximately 8% in the absence of dispersive effects (pseudo-absorption). These observations represents the first measurement of continuous-wave twophoton stimulated emission in the optical part of the spectrum and support the conjecture of Lewenstein, Zhu and Mossberg [1990] that dressed atoms can be used to achieve two-photon lasing. More recently, Lange, Agarwal and Walther [1996] observed enhanced two-photon decay fi-om dressed Rydberg atoms in the microwave part of the spectrum, suggesting that the general dressed-atom approach can also be of use for developing a new class of two-photon masers. I also note that the so-called 'Doppleron' resonances observed in the velocity distribution of atoms moving in an intense optical standing wave (Bigelow and Prentiss [1990], Tollett, Chen, Story, Ritchie, Bradley and Hulet [1990]) can be interpreted in terms of multi-photon stimulated emission, suggesting that a two-photon laser could be realized using cold-atom and CQED methods. In addition, Manson, Changjiang Wei and Martin [1996] have observed multiphoton resonances in the driven nuclear magnetic resonance response within the ground-state hyperfine levels of the nitrogen-vacancy center in diamond, which may be useful in realizing dressed-state masers in a solid material.
6.2. Realization of the continuous-wave two-photon laser The first observation of continuous-wave two-photon lasing was reported by Gauthier, Wu, Morin and Mossberg [1992] using an experimental arrangement very similar to that used by Zhu, Wu, Morin and Mossberg [1990] to measure two-photon amplification. To achieve lasing, the cavity losses were reduced and the number of dressed atoms in the cavity was increased. The cavity losses were reduced by replacing the enhancement cavity by a confocal cavity with higherreflectivity mirrors (w^ = 66|jim, L^ = 5 cm, and K = 5.3xl0^rads"^). The optical finesse for this resonator was 1800, corresponding to a g of 6.4x 10^. In addition, the barium atomic beam was replaced by one that could produce higher atomic number densities and had a larger diameter so that A^ :^ 2x10^ atoms were present in the cavity mode, although at the cost of increasing the residual Doppler width to ^40 MHz (to be compared to the 19 MHz natural width of the transition and the 1.7 MHz cavity linewidth). The possibility of obtaining twophoton lasing is indicated by the fact that N > A^th given above. Gauthier, Wu, Morin and Mossberg [1992] conducted two preliminary experiments to characterize the dressed-atom laser system. In one experiment,
4, §6]
The dressed-state two-photon laser
249
Cavity-Pump Detuning A^n^ Fig. 14. Dressed-state two-photon laser, (a) Power emitted from the cavity as a function of the cavity frequency showing the range of one-photon lasing. (b) Transmission of an intense probe beam through the cavity showing the two-photon gain feature at zlc = <^c ~ <^d = ~2^d- ^^^^ Gauthier, Wu, Morin and Mossberg [1992].
the optical power emitted from one end of the cavity was measured as the cavity resonance frequency co^ was tuned through the dressed-atom resonances. As seen in fig. 14a, it was found that normal one-photon dressed-atom lasing occurred when cOc ~ ct^d - -^d ^^^ ^^^^ ^^ lasing occurred when (o^'^ co^ - \Q'^ (the predicted frequency of maximum two-photon gain). This measurement confirmed that the one-photon gain feature was spectrally removed from the region where high two-photon gain was expected and hence it would not obscure two-photon lasing. The other experiment measured the gain of the system directly by observing the transmission of an intense laser beam as it passed through the cavity in the presence of the dressed atoms, as shown in fig. 14b. The pseudo-absorption present in the experiment of Zhu, Wu, Morin and Mossberg [1990] was not a factor in this experiment because the probe-laser frequency was scanned through the dressed-atom-cavity resonance and the maximum transmission was recorded. A pronounced, intensity-dependent gain feature was observed when (Dc ^ a^d - |i2^ that coincides with the predicted location of the two-photon gain feature. Note that the tail of the one-photon gain feature overlaps somewhat with the two-photon gain feature; they estimated that -35% of the laser beam amplification was due to one-photon stimulated emission. It appears that this non-ideal behavior did not seriously disrupt the lasing behavior, as described below, which is consistent with the prediction by Zakrzewski, Lewenstein and Mossberg [1991] that two-photon lasing should suppress the coherence needed for one-photon lasing. Based on the discussion in the previous sections, two-photon lasing should not occur in this system until both a sufficient number of inverted atoms and a sufficient number of photons are in the cavity. To observe two-photon lasing, they adjusted the cavity frequency to the peak of the observed two-photon gain feature
250
[4, §6
Two-photon lasers
-10
0
10
20
30 40
50
-10
0
10
20
30 40
50
Time (/xsec)
Fig. 15. Dressed-state two-photon laser turns on. Experimentally observed temporal evolution of the power emitted from the cavity for (a) weak and (b,c) strong injected fields, (d) Complex oscillations in the generated light for a detuned two-photon laser. From Gauthier, Wu, Morin and Mossberg [1992].
shown in fig. 14b and injected trigger pulses (duration -^l^iiis) into the laser resonator. The time-resolved cavity output power just prior to and following the injected pulse was repeatedly recorded for various trigger-pulse powers. As shown in fig. 15a, the number of injected trigger photons (^inj ^ 3.4x10"*) was insufficient to initiate two-photon lasing, and the cavity output power decayed to zero after the injected pulse was turned off For higher injected powers, the cavity output power remained high after the trigger pulse was turned off as shown in fig. 15b. They estimated that there were ~1.4x 10^ intercavity photons once the laser was turned on and the trigger pulse had left the cavity, which is in reasonable agreement with the predicted value of ^th ^ 9.3x10"* photons. Spiking behavior was observed while the injected field was present, a behavior that is apparently not accounted for in the theory of two-photon lasers. Figure 15c shows the observed two-photon laser behavior over a longer time scale for the case when a stronger trigger field was injected, which appears to suppress the initial spiking in the photon number. Finally, they observed complex temporal evolution of the power generated by the laser when the dressing field was tuned closer to the atomic resonance, as shown in fig. 15d. They proposed that this behavior might be due to the beating of distinct transverse cavity modes, an instability driven by competition between the two-photon and off-resonant onephoton gain processes, or due to instabilities driven by ac Stark shifts induced by the lasing field.
4, §7]
251
The Raman two-photon laser
The trigger-induced transition to a state of nonzero cavity output power is entirely consistent with the expected threshold behavior of a two-photon laser as discussed in § 3 (see fig. 4, above), demonstrating conclusively that twophoton lasing was indeed achieved. To date, no additional experiments have been performed to fully characterize the stability properties or the quantum-statistical properties of the dressed-state two-photon laser.
§ 7. The Raman two-photon laser Continuous-wave two-photon amplification and lasing are also possible in driven multi-level atoms, which opens up the possibility of identifying the generic properties of two-photon lasers and operation on different states of polarization. For example, Concannon, Brown, Gardner and Gauthier [1997] considered laser beam amplification in a laser-driven thermal vapor of potassium atoms when the dressing-laser frequency was tuned in the vicinity of the 4Si/2 ^-> 4Pi/2 transition. They demonstrated that two-photon gain arises in the system by a process they called two-photon Raman scattering, shown schematically in fig. 16a. In this process, intense dressing (dashed arrows) and probe (solid arrows) fields stimulate the atom to make a transition from the initial state \g) to the final state \g') by absorbing two photons from the dressing field (frequency o^d) and adding two new photons to the probe field (firequency w) via virtual intermediate states. Energy conservation requires that O) = O)^- \Agg', where hAgg/ is the energy difference between \g) and \g'). In the experiments, the states \g) and \g') corresponded to the 4Si/2 {F = 1) and 4Si/2 (F = 2) hyperfine states, respectively, where Aggf/2jt = 462 MHz for ^^K. To obtain continuous-wave twophoton amplification based on this stimulated emission process, a steady-state b)
-600-500-400-300-200-100 0 100 probe-pump detuning (MHz)
Fig. 16. Raman two-photon scattering, (a) Scattering diagram showing the Raman two-photon process, (b) Gain spectrum for an intense probe beam propagating through a vapor of laser-driven potassium atoms. From Concannon, Brown, Gardner and Gauthier [1997].
252
Two-photon lasers
[4, § 7
imbalance must exist between the states \g) and |g') so that Ng > Ng', which is accomphshed by optical pumping of the atom by the intense dressing field. Note that this process is similar to the multi-photon scattering diagram shown in fig. 11 for the perturbative explanation of two-photon amplification in driven two-level atoms. It is possible to understand the origin of the two-photon amplification process using the dressed-state basis for the driven three-level atom, but such an analysis is not all that much simpler than using the bare-atom basis. Concannon, Brown, Gardner and Gauthier [1997] pointed out that «-photon Raman scattering processes can occur in this system for probe-beam fi-equencies a> = a>d - Agg'/n, n = 1,2,3,.... Poelker and Kumar [1992] extensively studied the one-photon Raman process (n = 1) in a laser-driven sodium vapor, while Hemmerich, Zimmermann and Hansch [1994] and Cataliotti, Scheunemann, Hansch and Weitz [2002] observed multi-photon Raman scattering in cooled rubidium atoms trapped in the potential wells of a three-dimensional optical lattice. Trebino and Rahn [1987] and Agarwal [1988] investigated a related multiphoton parametric wave-mixing process in laser-driven sodium atoms. In the experiment, potassium vapor was contained in a 7-cm-long evacuated pyrex cell with uncoated, near-normal-incidence optical windows heated to a temperature of ISO^'C which produced a number density of approximately 10^^ atoms/cm^. Because they used natural-abundance potassium, gain and absorption features were also observed due to scattering fi*om "^^K where Agg' = 254 MHz. The dressing laser beam was linearly polarized, collimated to a diameter of 150 |jim (intensity FWHM) as it passed through the cell, had a power of 850 mW at the entrance to the cell, and was tuned approximately 2.4 GHz to the low-frequency side of the Dl transition [4Si/2 (F = 2) -^ 4Pi/2 (F = 1)] occurring near A = 769.9 nm. The probe beam was collimated to a diameter of 65 |xm and had a polarization orthogonal to that of the pump field, which resulted in the maximum two-photon gain. The probe beam nearly copropagated with the pump beam (crossing angle 12mrad) so that the two-photon Raman scattering process was nearly Doppler-fi-ee while minimizing the effects of parametric wave mixing that could potentially compete with the two-photon gain process. Figure 16b shows the probe-beam gain spectrum for a probe beam intensity that was high enough to induce the multi-photon Raman processes. They observed a narrow, intensity-dependent gain feature at a; c^ 0)(\ \\g' [{(JO - co^yijt :^ -231 MHz], which they attributed to the two-photon Raman scattering process shown in fig. 16a. The continuous-wave two-photon gain was as large as 30%! Also apparent in fig. 16b is a feature at ft> ~ Wd - ^^g'/3 [((o- w^yijt ~ -154 MHz], corresponding to 5% three-photon gain. Subsequent research by Brown [1999] using laser-driven potassium atoms in
4, §7]
The Raman two-photon laser
a) detector
y0
atoms optical p u m p ^ ^
t
Bz
253
b)
probe
Raman pumpC-^
Ma-2
-1
Fig. 17. Raman two-photon scattering using an atomic beam of potassium atoms (optically pumped into the state |g22)) and a mutually orthogonal geometry, (a) Experimental setup, (b) Scattering diagram depicting the Raman two-photon process taking into account the hyperfine Zeeman sublevels of the potassium 4Si/2 and 4Pi/2 states. From Pfister, Brown, Stenner and Gauthier [1999].
a vapor cell revealed that the probe beam experienced significant beam steering due to a nonlinear waveguide structure created by the interaction of the atoms and the dressing laser field. It was suspected that the probe beam was guided precisely along the propagation direction of the dressing laser field even when the crossing angle outside the vapor cell was set fairly large, opening the possibility that parametric multi-wave mixing effects (Trebino and Rahn [1987]) might be responsible for some of the observed laser-beam amplification. Therefore, the vapor-cell experiments were abandoned and an atomic beam apparatus was constructed (Brown [1999]) so that a mutually orthogonal experimental geometry could be used to suppress wave-mixing effects, similar to the geometry employed by Zhu, Wu, Morin and Mossberg [1990] and Gauthier, Wu, Morin and Mossberg [1992] in the laser-driven barium experiments. Pfister, Brown, Stenner and Gauthier [1999] demonstrated two-photon amplification in laser-driven potassium atoms using the mutually orthogonal geometry shown in fig. 17a. The interactions are somewhat more complex because the magnetic sublevels of the potassium hyperfine states have to be taken into account for this geometry, as shown schematically in fig. 17b to lowest order in perturbation theory. Laser-beam amplification occurred when two circularly polarized dressing-field photons were annihilated and two linearly polarized probe photons were created as the atom underwent a transitionfi-omthe |g22) to the |g20) Zeeman sublevels. The atomic states are denoted by \aFaMa), where a = g for the potassium 4^Si/2 and a = e for the 4^Pi/2 levels, and F and M are the quantum numbers for the total angular momentum and its projection along
254
Two-photon lasers
[4, § 7
the z quantization axis (taken as the propagation direction of the dressing field). The necessary inversion between the states |g22) and |g20) was maintained using auxiliary optical pumping beams that continuously transferred population from all hyperfine states into |g22). In their experiment, the atoms are produced by an atomic beam with a halfangle divergence of 30 mrad, giving rise to a residual Doppler width of 30 MHz, a diameter of 2.5 mm and an atomic number density of 2 x 10^ ^ atoms cm~^ in the interaction region. The atoms were dressed with a circularly polarized laser beam propagating along the quantization axis with an intensity of 25 W cm~^ and its frequency tuned to the blue side of the \glM) \elM) transition by 522 MHz. A weak uniform magnetic field {B^ ^ 10~^'^^ was applied in the interaction region to overwhelm stray magnetic fields. A z-rolarized probe beam collimated to a radius Wo = 90 [im interacted with the dressed atoms and experienced intensity-dependent gain for sufficiently high probe beam intensities. They measured the two-photon gain directly (without the use of an enhancement cavity), a difficult experiment because the two-photon signal was as small as 10~^, while, on the other hand, the probe power was high enough to saturate typical silicon detectors or the associated electronic preamplifiers. They solved these problems by using high-power photodiodes in a difference-detection setup. The probe beam was split into two beams of equal power, one of which (the signal beam) was sent through the vacuum chamber and the driven atoms, while the other (reference) beam was sent directly to its detector. The power of each beam was converted to a current by a photodiode that had a linear response for powers up to several tens of milliwatts (Hamamatsu S3994), the two photocurrents were subtracted at their mutual junction, and the resulting difference current was converted to a voltage. Figure 18a shows the probe-beam gain spectrum, where several gain and absorption features are evident. They identified the origin of all resonances by measuring their spectral location and their dependence the probe beam power. The small feature located at a probe-pump detuning of co c:^ co^- \Aggf = 232 MHz was attributed to the two-photon Raman process shown in fig. 18b. The continuous-wave two-photon amplification was only about ~ 1.5x10"^! Note that the gain features are very narrow and comparable to the residual Doppler width of the resonance, so that the various resonances were highly spectrally isolated, in contrast to the behavior of the dressed-state two-photon laser (see fig. 14). To further demonstrate that the feature was due to a genuine two-photon process, they measured its dependence on the probe-beam intensity, as shown in fig. 18b. A clear linear increase of the gain was observed for powers up to 3 mW (corresponding to an intensity of 24 W cm~^), for which a saturation plateau
4, §7]
The Raman two-photon laser
0
200
400
600
255
1
Probe-pump detuning (MHz)
2
3
4
5
Probe beam power (mW)
Fig. 18. Raman two-photon amplification, (a) Experimentally observed gain spectrum for an intense probe laser beam propagating though laser-driven potassium atoms in the mutually orthogonal geometry, (b) Laser beam amplification as a fianction of the input probe power. Compare to fig. 7. From Pfister, Brown, Stenner and Gauthier [1999].
occurred at a maximum peak height of 4x 10~^. The observed linear dependence below saturation is a central result of their experiment: it demonstrates directly that photons are emitted two at a time. They noted that the height of the feature is not the net two-photon gain because it overlaps with the wing of a large absorption feature due to transitions from |g22) to |e22). They estimated that the maximum gain was about 50% of the peak height. Using the simple mode for two-photon amplification (see eq. 3.18), they estimated that G(2) = 9xlO-^cmW-i and /sat = 36Wcm-2. The solid line is the fit of this simple model to their observations. In a subsequent study, Femandez-Soler, Font, Vilaseca, Gauthier, Kul'minskii and Pfister [2002] investigated theoretically laser-beam amplification in driven potassium atoms using a semi-classical description of the interaction that accounted for most of the hyperfine level structure (transitions involving states with M < 0 were neglected). They determined a complex 'gain' coefficient
H
Pe22,g22 +
/ ^ i = 0,\
E
^eji,gki
(7.1)
j,k=l,2
where U is the unsaturated gain parameter and (3 is proportional to the probebeam Rabi frequency. Equation (7.1) was obtained by summation of all the one-photon atomic coherences pui induced on the transitions u = \eFM) -^ I ~ \gF'M). The imaginary part of eq. (7.1), or the gain factor G, represents the relative increase in the probe-field amplitude per unit of time, and its real part is proportional to the induced change in refiractive index experienced by the probe beam.
256
[4, §7
Two-photon lasers
-80 -60 -40 -20
0
20
40
60
80
Fig. 19. Predicted gain spectrum (solid line) for an intense laser beam propagating through a collection of laser-driven potassium atoms. Crosses are the experimentally observed points. From Femandez-Soler, Font, Vilaseca, Gauthier, Kul'minskii and Pfister [2002].
The solid curve in fig. 19 shows the gain factor as a function of the probepump detuning Ap for the conditions reported in the experiments described above with no adjustable parameters except for an overall vertical scale factor. The experimental trace is shown as crosses in the figure (note that the vertical scale is offset for clarity). It is seen that there is very good agreement between the predictions of the model and the experiments, indicating that the model will be usefiil for making new predictions for two-photon amplification and lasing in laser-driven potassium atoms. Interestingly, they found that interferences between different quantum pathways were constructive for the two-photon process and that the interferences helped to reduce the ac Stark shifts of the two-photon transition. Continuous-wave two-photon lasing in laser-driven potassium atoms was first observed by Pfister, Brown, Stenner and Gauthier [2001], who combined the experimental apparatusfi^omthe measurements of amplification with a low-loss optical resonator. They used a linear (standing-wave) cavity consisting of two high-reflectivity ultra-low-loss mirrors with radius of curvature 5 cm set close together at a distance of Lc = 1-5 cm so that it is operated in a transverse-modedegenerate sub-confocal configuration (wo = 66 |im). Pinholes were placed in fi*ont of each mirror to suppress higher-order transverse modes that were not degenerate with the lower-order ones due to spherical aberrations and, with these pinholes, they measured an optical finesse of c::^ 1.5x10"^, resulting in K = 2AxlO^ s~^. Within the cavity mode volume, they estimated that there were 7x10^ atoms. The dressing laser beam had an intensity of ~ 25 Wcm~^ and
4, §7]
The Raman two-photon laser
257
0
I Q. CD C
o o
SI Q. I
o
Fig. 20. The Raman two-photon laser turns on. Experimentally observed temporal evolution of the total power emitted from the cavity for (a) small and (b-d) large numbers of injected photons. The dashed vertical lines indicate the length of the injected pulse. From Pfister, Brown, Stenner and Gauthier [2001].
its frequency was tuned to the blue side of the \g\M) ^^ \e2M) transition by 512MHz. They determined the location of the two-photon resonance by injecting a probe beam into the cavity and measuring its transmission. To initiate lasing, they set the cavity frequency to the peak of the observed two-photon transition frequency, momentarily blocked the dressing beam to quell any pre-existing two-photon lasing, and measured the power of the light emitted from the two-photon laser resonator using a avalanche photodiode. In the absence of the probe-laser beam, they observed essentially no light emitted from the resonator, indicating that quantum fluctuations did not provide a sufficient number of photons in the cavity to satisfy the lasing criteria. To trigger lasing, they injected 1.2-(is-long pulses of light into the resonator; fig. 20a shows the case when a below-threshold pulse was injected into the resonator. The dashed vertical lines indicate the duration of the injected pulse; they estimated that ^inj = 1.8x10"^ photons were injected into the cavity. Note that there was significant spiking in the power transmitted through the cavity, even though the input pulse was smooth, similar to that shown in fig. 20 for the dressed-state two-photon laser. For slightly higher trigger-pulse powers (fig. 20b, ^inj = 3.4x 10"^), the cavity photon number grows substantially, displaying considerable spiking, and remains high for a few microseconds, but
258
Two-photon lasers
[4, § 7
was still insufficient to initiate lasing, indicating that they were very close to satisfying both two-photon laser threshold criteria. The two-photon laser turned on for ^inj = 5.2x10'* (fig. 20c), where the power emitted by the cavity rose to approximately 0.2 [iW and remained at this value, corresponding to an average cavity photon number of 3.5x10^ and an intracavity intensity of 41.3 Wcm~^ at the cavity waist ^. This intercavity intensity is slightly higher than the two-photon saturation intensity measured for this process in the amplification experiments described above, indicating that the laser runs fiilly saturated near the threshold condition as discussed in §3. A measurement by Brown [1999] of the transverse profile of the beam emitted by the laser showed that it appeared to be a lowest-order Gaussian mode. Figure 20d shows the power emitted ft'om the resonator on a longer time scale for a larger injected number of photons and during a different experimental run but under essentially identical conditions. The experimental data of fig. 20 give convincing evidence that the device is indeed a two-photon laser. These observations are consistent with theoretical analysis of the Raman two-photon laser by FemandezSoler, Font, Vilaseca, Gauthier, Kul'minskii and Pfister [2002], who find that the laser emission line shape is frequency-pushed due to the ac Stark effect and has the form of a closed curve composed of a stable and an unstable branch. Since the optical resonator was highly isotropic and the magnetic sublevels of the potassium atom support multi-photon transitions for more than just a linearly polarized cavity field, Pfister, Brown, Stenner and Gauthier [2001] investigated the polarization characteristics of the generated light. They placed a linear polarizer oriented in the z-direction in the beam and found that the two-photon laser displayed polarization instabilities even though the total emitted power was nearly constant on the time scale of the instability. It is seen in fig. 21a that the state of polarization underwent very regular oscillations of period 0.11 |JIS, with a 50% depth of modulation. Note that the oscillations commenced even while the injected trigger pulse was present. Similar dynamical behavior was observed for all polarizer orientations, suggesting that the state of polarization is elliptical with an ellipticity of 0.5 and a precessing major axis. They also found that the laser dynamics were quite sensitive to the applied magnetic field: an increase
The photon numbers and intercavity intensities given here are different from the values given in Pfister, Brown, Stenner and Gauthier [2001]. Errors were found in the determination of these quantities from the measured power of the beam emitted from the cavity. Here, q = Pout^hcOcYc and / = SPQ^I/VIJTWI, which takes into account the fact that the cavity is symmetric and an equal power is emitted from the opposite side of the cavity.
The Raman two-photon laser
7]
259
0.12 CO > -
0 g Q. ^
1 S8
0.08 0.04 \ 0.00 3 time (fis)
Fig. 21. Polarization instabilities in the Raman two-photon laser. Temporal evolution of the light emitted from the cavity for (a) small and (b) larger applied static magnetic field strengths. From Pfister, Brown, Stenner and Gauthier [2001].
by as little as 0.5 Gauss was sufficient to generate a much more complicated pattern. Figure 21b shows the complex, possibly chaotic behavior they observed for a magnetic field strength of 2.0 G. They suggested two possible mechanisms that might be responsible for the observed instability. One arises fi-om the multiple frequency-degenerate final lasing states and interference between multiple quantum pathways for different states of polarization for the emitted cavity photons, as shown schematically in fig. 22. As the laser turns on and begins to saturate the pathway shown in fig. 22a (since they injected a z-polarized trigger field), an x-polarized field can begin to grow on the unsaturated pathways shown in fig. 22b and fig. 22c, which is enhanced by the existing z-polarized cavity field due to singly stimulated emission. Another possible mechanism is the presence of standing waves; it is well known that counterpropagating laser beams in conjunction with a tensor nonlinear optical interaction can give rise to polarization instabilities with a reduced instability threshold (Gauthier, Malcuit and Boyd [1988]). They suggested that application of a strong magnetic field to lift the degeneracy of the different final states might suppress the instability. Also mentioned is the possibility that the laser might produce polarization-entangled twin beams of light for precision measurement and quantum communication applications. These experiments demonstrate that multi-level structure in a two-photon gain medium gives rise to novel behavior in the state of polarization of the generated beam. Further experiments are needed to fully explore the quantum-statistical and nonlinear dynamics behavior of this new type of quantum oscillator.
260
[4, §8
Two-photon lasers (a)
(b)
\
^
le>2
le>2
\COH
©
\ CO
±L_±
ig>
-2
\
ig>
-1 (d) le>2
ig> 0 M„ Fig. 22. Raman two-photon scattering in optically pumped and laser-driven potassium atoms showing the possible quantum pathways connecting initial and final lasing state for different states of polarization supported by the cavity. From Pfister, Brown, Stenner and Gauthier [2001].
§ 8. Quantum-statistical and nonlinear dynamical properties As demonstrated in the previous sections, the two-photon laser is a highly nonlinear system that displays unusual properties in comparison to other quantum oscillators. For this reason, it has attracted considerable theoretical attention since the original proposals of Prokhorov [1965] and Sorokin and Braslau [1964]. Describing the complete body of theoretical research on twophoton lasers is beyond the scope of this chapter and deserves a review in itself To give a flavor of the excitement for this work, I highlight the areas concerning the quantum-statistical properties of the light generated by the laser and its nonlinear dynamical properties, describing some of the early research and a few recent results. 8.1. Quantum-statistical properties and squeezing Since the two-photon stimulated emission and absorption processes shown in
4, § 8]
Quantum-statistical and nonlinear dynamical properties
261
fig. 1 require two incident photons to induce the atom to make a transition to the lower state, it is expected that the quantum-statistical properties of the incident light will affect the transition rate. Also, it is expected that light passing through two-photon amplifiers or absorbers, or generated by a twophoton laser, will have interesting properties. Early work considered the case when thermal radiation or light generated by a normal one-photon laser interacts with a two-photon transition. A one-photon laser operating well above threshold is fairly well characterized by a coherent state, where the photon-counting probability distribution is Poissonian, and the dispersion of the quadrature amplitudes is equal and at the minimum value allowed by the uncertainty principle (see Ch. 18 of Mandel and Wolf [1995]). Thermal radiation, on the other hand, is characterized by Bose-Einstein statistics and hence displays photon bunching. It was predicted by Lambropoulos, Kikuchi and Osborn [1966] and later demonstrated by Shiga and Imamura [1967] that the two-photon absorption rate is highest for a thermal field because the probability for two photons to be incident simultaneously at an atom is higher for bunched light. Similarly, he predicted that a beam of light traveling through an unsaturated two-photon amplifier will experience greater gain when its statistics are thermallike (Lambropoulos [1967]). Surprisingly, he found that fluctuations of a thermal beam of light tended to decrease whereas the fluctuations tended to increase for a coherent state, which was attributed to the presence of both spontaneous emission as well as singly stimulated emission. He thus concluded that a two-photon amplifier is not coherent in the sense that it tends to increase the fluctuations of the field amplitudes for a coherent input. These predictions were later confirmed by McNeil and Walls [1974] using a different approach. Fully quantum-mechanical modelling of the degenerate single-mode twophoton laser was first undertaken by McNeil and Walls [1975], who determined the stationary density operator for the light field in the cavity and found that the photon-distribution functions are narrower than that for normal one-photon lasers. To simplify the analysis, they assumed that photons are coupled out of the cavity two-at-a-time so that detailed balance could be invoked when solving the equations describing the probability flow [For the simplified model described in § 3, the photon decay term in eq. (3.2) would be -Kq^ rather than -Kq for such output coupling of photons.] Unfortunately, this assumption totally changes the threshold characteristics of the two-photon laser so that the discontinuous behavior shown in fig. 3 is replaced by a smooth turn-on of the laser, analogous to a second-order phase transition. Bulsara and Schieve [1979] obtained similar results using a semiclassical stochastic approach, and Nayak and Mohanty [1979] generalized the theory to take into account Doppler broadening, but both studies
262
Two-photon lasers
[4, § 8
also considered the unphysical situation when detailed balance is maintained. When McNeil and Walls [1975] relaxed the assumption of detailed balance, they predicted that the laser should produce coherent light obeying Poisson statistics using a perturbative solution to the problem, which was later confirmed using a g-ftinction approach (Gortz and Walls [1975]). In a later study, Zubairy [1980] found quite the opposite using non-perturbative techniques: the relative fluctuations of the number of photons in a two-photon laser are larger than that for a coherent state. In view of the earlier work, it was quite surprising when Yuen [1975] predicted that an ideal degenerate single-mode two-photon laser operating far above threshold should produce a squeezed state of the electromagnetic field (see also Yuen [1976]). In particular, he found that the field was best described by a so-called 'two-photon coherent state', which is a minimum-uncertainty state but with the unusual properties that the variance of one quadrature of the field can be made arbitrarily small (squeezed) at the expense of increasing the fluctuations in the other quadrature. Such behavior has no analogy in classical physics and hence it is thought of as a purely quantum-mechanical effect. Squeezing is predicted to occur when the Hamiltonian has a quadratic form so that it has contributions depending on the square of the photon creation and annihilation operators. When the laser operates far above threshold, he reasoned that the operators describing the raising and lowering of the atomic excitation could be treated as c-numbers and the resulting Hamiltonian would give rise to squeezing. Rowe [1978] arrived at a similar conclusion independently. Soon thereafter, it was shown that two-photon quantum states had the potential for improving the noise characteristics of optical communication channels (Hirota [1977] and Yuen and Shapiro [1978]). Note that a squeezed field can display subor super-Poissonian statistics depending on the phase of the squeezing operator (see Ch. 21 of Mandel and Wolf [1995], for example), so that its existence is not necessarily contradictory with the earlier findings described above. Yuen's predictions resulted in renewed interest in two-photon lasers for testing the foundations of quantum optics in a situation where the light-matter interaction is highly nonlinear, even though the originally envisioned applications for tunable high-power lasers were no longer as relevant. While there was great general interest in squeezed states, there arose considerable controversy as to whether a squeezed state is generated by a twophoton laser. Lugiato and Strini [1982] and Reid and Walls [1983] developed a more accurate model of the atomic-pumping and cavity-loss mechanisms and found that there was no squeezing in the resonantly tuned laser with an injected signal due to the effects of spontaneously emitted photons. Their model was
4, § 8]
Quantum-statistical and nonlinear dynamical properties
263
based on earlier work by Sczaniecki [1980]. Reid, McNeil and Walls [1981] extended this model and found that the two-photon laser displays extreme photon bunching just above threshold, consistent with the findings of Bandilla and Voigt [1982], Herzog [1983] and Cheng and Haken [1988]. On the other hand, Hu and Sha [1991] predicted that squeezing should occur when the laser is tuned away fi:om the two-photon resonance. In addition, Ashraf and Zubairy [1989] found that the spontaneous-emission contribution to the laser linewidth is twice as large as that for a normal one-photon laser, but that the gain contribution to the diffusion coefficient is independent of the mean photon number. While the degenerate laser appears not to generate quantum beams of light, Swain [1988] and Zubairy [1982] found that there exist quantum correlations between the photons in the two modes of a non-degenerate two-photon laser, which, they suggest, might be of use in precision measurement and laser gyroscopes. All of these earlier models are based on an 'effective Hamiltonian', where the atom possesses only two energy levels connected by a two-photon transition. In a series of papers, Wang and Haken [1984] described how to obtain the effective Hamiltonian from a microscopic one that considers the interaction of the fields with the upper and lower energy levels as well as the intermediate states, such as the simplified energy-level diagram shown in fig. 6a for a threelevel atom. Unfortunately, they dropped the terms representing the ac Stark shift (see eq. 3.17) in their analysis. Boone and Swain [1989, 1990] demonstrated that the ac Stark shifts make an important contribution to the off-diagonal density matrix elements describing the non-degenerate two-photon laser, implying that predictions for the linewidth, fi-equency shifts, squeezing spectrum, and stability boundaries are incorrect when using effective Hamiltonians. They found that the effective-Hamiltonian approach is valid near the laser threshold for determining the diagonal density matrix elements, but fails far above threshold because power broadening and shifts of the transitions increase the contribution of the offresonance one-photon processes. On the other hand, the effective-Hamiltonian approach can never correctly describe the off-diagonal elements. One prediction of their work is that the laser linewidth is a factor of two larger than that expected using an effective Hamiltonian. Similar conclusions were made by Lu [1990] for degenerate operation of the laser. Other research incorporating the effects of the intermediate levels includes studies of the photon-number probability distribution (Zhu and Li [1987]), the correlation between the two modes of nondegenerated lasers when the atoms are injected with an initial atomic coherence (Lu, Zhao and Bergou [1989], Majeed and Zubairy [1991, 1995]), and the emission spectrum in the absence of atomic and cavity damping (Nasreen and Razmi [1993]). Bay, Elk and Lambropoulos [1995] has found that ac Stark shifts
264
Two-photon lasers
[4, § 8
can actually enhance the quantum nature of the light generated by the two-photon laser, leading to the generation of sub-Poissonian photon statistics when the laser is tuned away from exact resonance with the transition. This research demonstrates that the predictions for the quantum properties of the two-photon laser are quite sensitive to the details of the interaction, including off-resonant intermediate states. For this reason, it is important to incorporate these details when modeling experimental devices. For example, the intermediate states are properly accounted for in theoretical studies of the two-photon micromaser (Davidovich, Raimond, Brune and Haroche [1987] and Davidovich, Raimond, Brune and Haroche [1988]), the dressed-state twophoton laser (Zakrzewski, Lewenstein and Mossberg [1991], Zakrzewski and Lewenstein [1992], Zakrzewski, Segal and Lewenstein [1992]), and the Raman two-photon laser (Femandez-Soler, Font, Vilaseca, Gauthier, Kul'minskii and Pfister [2002]).
8.2. Nonlinear dynamical properties The two-photon laser also has fascinating nonlinear dynamics properties because it operates in the saturated regime (a source of optical nonlinearity) even at threshold. The most obvious difference between one- and two-photon lasers is the behavior of the first laser threshold, as illustrated in fig. 3. As discussed in §4, both Sorokin and Braslau [1964] and Prokhorov [1965] realized that the laser needed a sufficient inversion and photon density to switch from the off to the on state. The first discussion of the analogy between the two-photon laser threshold and a first-order phase transition familiar from condensed matter physics appeared in papers by Ito and Nakagomi [1977], who investigated the thermodynamic limit in quantum stochastic processes occurring in lasers, and by Butylkin, Kaplan, Khronopulo and Yakubovich [1977], who studied a rateequation model of the laser. Later, Sczaniecki [1980] demonstrated that all m-photon lasers with m > \ display a first-order phase transition as long as the resonator has a one-photon loss mechanism. Recall, from the discussion above, that a two-photon loss mechanism changes the two-photon laser threshold to that of a second-order phase transition. It is also found that a lowest-order perturbative analysis of the two-photon laser can lead to erroneous predictions about the threshold behavior (Songen [1987]). The first systematic studies of the stability properties of the two-photon laser were undertaken independently by groups at the Universitat Stuttgart and the University of Arizona. Wang and Haken [1984] used a microscopic model for
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the light-matter interaction, but dropped the effects of ac Stark shifts from their analysis so that it was equivalent to the effective-Hamiltonian approach described above. They worked in the regime of a very good cavity so that the atomic variables could be adiabatically eliminated and they assumed the intracavity field to be weak enough so that terms related to gain saturation could be expanded in a Taylor series. This is a somewhat dubious approximation given that the laser operates in the moderately saturated regime even at the first laser threshold; I find that this approximation underestimates the photon growth rate as a function of pump rate far above threshold. Even with this limitation, they correctly predicted that the off-state is always stable and the laser threshold is characterized by a first-order transition. For the case when a continuous-wave field is injected into the cavity from an auxiliary source, the off-state is always unstable for sufficiently high pump rate. Also, they found that there are only small differences between the threshold condition for lasers in the ring or standing-wave geometries, homogeneously or inhomogeneously broadened transitions, and degenerate or non-degenerate lasers. From a different perspective, Ovadia and Sargent [1984] studied the stability of degenerate two-photon lasers by determining when 'sidemodes' build up due to an instability. Sidemodes are new fields that build up in the laser cavity symmetric in frequency about the primary laser frequency and appear as a periodic modulation in the light generated by the laser. For the case of a truly single-mode cavity, they can build up because anomalous dispersion allows all three frequencies to have the same wavelength and hence be resonant with the mode (called a single-wavelength instability). They can also build up in multimode cavities so long as there exists a cavity mode that occurs in the vicinity of high sidemode gain (multi-wavelength instability). Ovadia and Sargent [1984] did not include the effects of the ac Stark shifts in their analysis, but they did allow for arbitrary atomic and cavity decay rates. For a cavity tuned precisely to the transition frequency, their analysis showed the first laser transition is analogous to a first-order phase transition, and the onstate is always stable for a good cavity (K < yy). For a moderately good cavity [Y\\ < K < (yy + |y±)], they found a very surprising feature of the two-photon laser: It immediately displays sidemode instabilities at the first laser threshold and only becomes stable at high pump rates. This contrasts normal one-photon laser behavior: these are usually stable above the first laser threshold and become unstable only for pumping far above threshold. Note that the existence of instabilities requires that Y±_ > yy, which can arise from collisional dephasing of the atomic coherence, for example. They also found that the laser readily displays instabilities when the cavity resonance is tuned
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[4, § 8
away from the transition frequency, even for a good cavity. In addition, the threshold for instabilities is lowered significantly by including the ac Stark effect (Ovadia, Sargent and Hendow [1985]), which gives rise to intensity-dependent dispersion and phase-conjugate scattering of the laser field off the two-photon coherence. In a later series of studies from the Universitat Stuttgart group, they also found that the laser can display self-pulsing instabilities near the laser threshold that stabilize only far above threshold (Ning and Haken [1989a]). Near the second laser threshold where it makes a bifurcation from self-pulsing to stable lasing, they used the slaving principle and normal-form techniques to arrive at a nonlinear theory governing the laser dynamics (Ning and Haken [1989b]). Their analysis reveals that the biftircation can be of the normal Hopf type (super-critical), displaying a period-doubling route to chaos, or it can be a co-dimension-two Hopf type (sub-critical), displaying two limit cycles. Related phenomena have been found by Yang, Hu and Xu [1994] for the case when an external field is injected into the laser. In addition, they showed that the selfpulsing behavior can be characterized through use of a geometrical phase (Ning and Haken [1992a,b]). Recent research has more fiilly explored, for example, the dependence of the instability boundaries on the effects of off-resonant one-photon amplification (Roldan, de Valcarcel and Vilaseca [1993]), tristability in good-cavity lasers with an injected signal (Urchueguia, Espinosa, Roldan and de Valcarcel [1996]), standing waves occurring in Fabry-Perot lasers (Espinosa, Vilaseca, Roldan and de Valcarcel [2000]), departures from the mean-field approximation (Abdel-Aty and Obada [2001]), and specific models of the two-photon maser (Davidovich, Raimond, Brune and Haroche [1988]), the dressed-state two-photon laser (Zakrzewski, Lewenstein and Mossberg [1991], Zakrzewski and Lewenstein [1992]), and the Raman two-photon laser (Femandez-Soler, Font, Vilaseca, Gauthier, Kul'minskii and Pfister [2002]). Especially interesting is the novel behavior displayed by wide-aperture two-photon lasers where transverse effects are important. Vilaseca, Torrent, Garcia-Ojalvo, Brambilla and San Miguel [2001] predicted that the laser will spontaneously form two-dimensional bright localized structures in a dark background. The bright spots, called cavity solitons, arise from the two-photon emission and are embedded in a background of emission due to off-resonant one-photon gain. Figure 23 shows the predicted intensity and phase for such a laser in a regime where there is turbulent motion of solitons that appear and disappear spontaneously. They suggest that these coherent structures could be usefril for all-optical parallel signal transmission and processing.
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Fig. 23. Predicted two-photon cavity solitons in an active optical medium, (a) Intensity and (b) phase of the transverse profile of the beam emitted irom the two-photon laser in the turbulent regime. From Vilaseca, Torrent, Garcia-Ojalvo, Brambilla and San Miguel [2001].
§ 9. Future prospects The two-photon laser is a unique class of quantum oscillator that has an unusual threshold behavior, displays nonlinear dynamical instabilities, and is predicted to produce beams of light with interesting quantum statistical properties. Unfortunately, achieving two-photon lasing has proven to be rather difficult and there is not yet a detailed comparison of its operating characteristics with the numerous theoretical predictions. I hope that my research group will remedy this situation in the near future using the Raman two-photon setup described in § 7. I also anticipate that researchers will begin to explore the strong atom-cavity coupling regime where a single atom passing through a cavity can be induced to emit two or more photons at a time on demand (see Bertet, Osnaghi, Milman, Aufifeves, Maioli, Brune, Raimond and Haroche [2002] and Kuhn, Hennrich and Rempe [2002]), thereby giving us the ability to 'quantum engineer' states of the electromagnetic field.
Acknowledgments I gratefully acknowledge very enjoyable collaborations on two-photon laser research with William Brown, Hope Concannon, Juanjo Fernandez-Soler, Josep Lluis Font, Jeff Gardner, Alexandar Kul'minskii, Maciej Lewenstein, Steve Morin, Thomas Mossberg, Olivier Pfister, Michael Stenner, Ramon Vilaseca, Qilin Wu, and Yifu Zhu, and fruitful discussions with Robert Boyd, Michael
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Raymer, Peter Sorokin, and John Thomas. I thank Lorenzo Narducci for encouraging me to write this review, and the long-term financial support of the US Army Research Office and the National Science Foundation, especially the current grant No. PHY-0139991.
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E. Wolf, Progress in Optics 45 © 2005 Elsevier Science B. V. All rights reserved
Chapter 5
Nonradiating sources and other "invisible'' objects by
Greg Gbur Department of Physics and Astronomy, Free University, Amsterdam, The Netherlands
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Contents
Page § 1. Introduction
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§ 2. Nonradiating sources
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§ 3. The inverse source problem
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§ 4. Nonscattering scatterers
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§ 5. One-dimensional localized excitations
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§ 6.
Moving charge distributions and radiation reaction
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§ 7.
Conclusions
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Acknowledgements
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References
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"Here is a solid body which we touch, but which we cannot see. The fact is so unusual that it strikes us with terror. Is there no parallel, though, for such a phenomenon? ... It is not theoretically impossible, mind you, to make a glass which shall not reflect a single ray of light - a glass so pure and homogeneous in its atoms that the rays from the sun will pass through it as they do through the air, refracted but not reflected ..." "That's all very well, Hammond, but these are inanimate substances. Glass does not breathe, air does not breathe. This thing has a heart that palpitates - a will that moves it - lungs that play, and inspire and respire." from What Was It?, by Fitz-James O'Brien (1828-1862) (Wagner and Wise [1994])
§ 1. Introduction
The possibility of invisible objects has intrigued both scientists and nonscientists for well over a century. Though only the animate variety seems to capture the imagination of fiction writers, the existence of any such objects has important practical and physical consequences. In this article we will distinguish between two types of "invisible" objects. The first class consists of scatterers (objects with an inhomogeneous index of refraction) which do not scatter incident plane waves for one or several directions of incidence ^. The second class consists of primary radiation sources (radiating atoms, for instance, or time-varying charge-current distributions) which do not, in fact, radiate power. It is the latter class with which this article will be primarily concerned with. Such nonradiating sources ^ have had a colorful history. Their origins lay in the theory of the extended rigid electron, initiated by Sommerfeld [1904a,b] and others^ in the early 1900s. Evidently Ehrenfest [1910] was the first researcher to recognize explicitly that radiationless motions of such extended charge
These so-called 'nonscattering scatterers' will be discussed in § 4. ^ In three-dimensional radiation problems, the term nonradiating source is used to describe sources which generate no power and produce nofieldoutside their domain of support. For one-dimensional wave problems such as will be discussed in § 5, the term nonpropagating excitation has come to be used. A good description of classical electron models is given by Pearle [1982]. 275
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Nonradiating sources and other "invisible" objects
[5, § 1
distributions are possible. Much later, Schott [1933] demonstrated that it is possible for a rigid charged spherical shell to move in a periodic orbit without radiating. Soon after, Schott [1937] ftirther showed that, under certain additional constraints, such a charged shell may move periodically in the absence of external forces. Bohm and Weinstein [1948] extended Schott's treatment of radiationless modes and self-oscillating modes to other spherically symmetric charge distributions. Goedecke [1964] later demonstrated that there exists at least one asymmetric, spinning, extended charge distribution which does not radiate. Many of these early authors speculated that nonradiating charge distributions might be used as models for elementary particles. Schott suggested that such objects might provide a stable model of the neutron and possibly of other atomic nuclei; Bohm and Weinstein suggested that a nonradiating source might explain the muon as an excited self-oscillating state of the electron. Goedecke suggested that such distributions might lead to a "theory of nature" in which all stable particles or aggregates are described as nonradiating charge-current distributions. In more recent years, the existence (or nonexistence) of nonradiating sources has been shown to be of fundamental importance to the solution of the inverse source problem^. An early paper by Friedlander [1973] explored the mathematical properties of electromagnetic nonradiating sources and discussed some circumstances under which the inverse problem is unique. At about the same time, Devaney and Wolf [1973] investigated monochromatic nonradiating classical current distributions. Not long after, Bleistein and Cohen [1977] explicitly demonstrated that the existence of nonradiating sources implies the nonuniqueness of the inverse source problem. Nevertheless, controversy over this result lingered for some time^. At nearly the same time that the inverse source problem was found to be nonunique for monochromatic sources, the same problem came to be investigated for partially coherent sources. Hoenders and Baltes [1979] developed a criterion for defining nonradiating stochastic sources, and determined a mathematical method to construct such sources. A few years later Devaney and Wolf [1984] obtained a simpler criterion for nonradiating stochastic sources. LaHaie [1985]
^ The inverse source problem may be loosely defined as the problem of determining the spatial structure of a source from measurements of the field radiated by that source. It will be discussed in §3. ^ For a particularly heated argument on the question of uniqueness, see Devaney and Sherman [1982] and the comments which follow it. The argument concerned the uniqueness of an integral equation solution by Bojarski [1982].
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277
explicitly investigated the inverse source problem for partially coherent sources and determined that it is, in general, nonunique. Two papers, one by Devaney [1979] and another by LaHaie [1986], suggested that the inverse source problem is, however, unique for quasi-homogeneous sources. A paper by Gbur [2001a] put these suggestions on firmer theoretical ground and described a method of performing the inversion. No doubt because of the general nonuniquess of the inverse source problem, most papers on nonradiating sources since the early 1980s have focused on the mathematical properties of such sources. Of these, the works of Kim and Wolf [1986] and Gamliel, Kim, Nachman and Wolf [1989] have provided some of the most worthwhile results. Much recent research^ has focused on the description of the "radiating" and "nonradiating" parts of a source (Devaney and Marengo [1998], Hoenders and Ferwerda [1998, 2001], Marengo and Ziolkowski [1999]). The mathematical existence of nonradiating sources also leads to a number of unusual effects for partially coherent sources. Gbur and Wolf [1997] demonstrated that it is possible to mathematically construct a partially coherent source that generates a fully coherent field, and Gbur and James [2000] showed similarly that it is possible to construct an unpolarized source that produces an (almost) fiilly polarized field. In this article we will review some of the most important results regarding nonradiating sources and their curious kin. Section 2 describes the basic theory of such sources. Section 3 discusses the inverse source problem and how the existence of nonradiating sources implies its nonuniqueness. Section 4 discusses so-called nonscattering scatterers and their relation to nonradiating sources. Section 5 discusses localized excitations in one-dimensional problems, the socalled nonpropagating excitations. Section 6 discusses radiationless motions of rigid charge distributions and the connection with radiation reaction. Finally, some closing remarks are made in § 7.
§ 2. Nonradiating sources A vast amount of information has been amassed about the properties of nonradiating sources. This section describes the basic concepts involved, and presents some of the most important mathematical theorems regarding such sources.
^ It should be mentioned that it is unclear how useful such a decomposition might actually be.
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2.1. Monochromatic nonradiating sources We consider a scalar source Q{r, t), confined to a domain D, which produces a field U(r, t). Q(r, t) may represent an acoustical wave source or, with a slight modification to the theory, an electromagnetic source. The field and the source are related by the scalar wave equation, V^ U{r, 0 - ^ ^
U{r, t) = -4jiQ(r, t).
(2.1)
Let us suppose that the source Q(r, t), and hence the field U(r, /), is monochromatic, i.e. that e ( r , 0 = Re{^(r)e-'^^'},
(2.2a)
U{r,t) = Re{w(r)e-^^'^'} ,
(2.2b)
where q(r) and u(r) are, in general, complex fiinctions of position, and Re denotes the real part. On substitution of eqs. (2.2) into the wave equation (2.1), the wave equation reduces to the inhomogeneous Helmholtz equation, V\(r)
+ k^u(r) = -AJiq{r\
(2.3)
where (2.4)
^=^.
c
The solution to this equation is well known to be (Jackson [1975], section 6.6, or Papas [1988], section 2.1)
u{r)= / q{r'y. dV'. JD \r-r'\
(2.5)
To determine the conditions for a source q{r) to be nonradiating, it is useful to examine the field in the far zone. Far away from the source, ^\k\r-r'\
^\kr
\r-r\
—e''*'"r
{kr -^ oo),
(2.6)
5, § 2]
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279 u{rs)
source ^(r) Fig. 1. Notation relating to monochromatic radiation sources.
where r = |r|, and s = r/r is a unit vector in the direction of r (see fig. 1). On substituting fi-om eq. (2.6) into eq. (2.5), it is found that the field in the far zone of the source may be expressed in the form piAr
u{rs) "r «{iny • —
(2.7)
~q{ks),
where ~q{K)-
1
'dV
(2.8)
JD
is the three-dimensional Fourier transform of the source distribution q{r). The fiinction q(ks) is often referred to as the radiation pattern of the source. From eq. (2.7), we obtain the following theorem: Theorem 2.1. A source will be nonradiating, i.e. it will not produce any power in the far zone of the source, if and only if q{ks) = 0 for all directions s.
(2.9)
Condition (2.9) is the most-often mentioned requirement that a nonradiating source distribution q{r) must satisfy. It states that the three-dimensional Fourier transform of the source distribution must vanish on a sphere of radius equal to the wavenumber k of the incident radiation. As a simple example^, we consider a homogeneous spherical source of radius a, i.e. f QQ when r < a, ^('•)=r ^ 0 when r > a.
(2.10)
^ This example was described in detail by Kim and Wolf [1986], though Carter and Wolf [1981] first pointed out that a homogeneous spherical source can produce radiationless solutions.
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+ Fig. 2. Two pieces of the nonradiating source q(r) and their arrangement.
On substituting from eq. (2.10) into eq. (2.9) and carrying out the integration, the nonradiating condition (2.9) takes on the simple form
Mka) = 0,
(2.11)
where yi(A:) is the first-order spherical Bessel fimction. A homogeneous spherical source is therefore nonradiating if and only if ^ is a zero of the first-order spherical Bessel function. The origin of the nonradiating phenomenon may be described as an unusual interference effect involving the fields generated by every element of the source. We may see this explicitly for the homogeneous sphere by considering it to consist of two pieces arranged concentrically, a sphere q\(r) of radius a\ and a spherical shell q2(r) of inner radius a\ and outer radius a2 (see fig. 2). Let us assume that ka2 is the first zero of the first spherical Bessel fiinction, 7i(jc). It can be seen by use of eq. (2.7) that the far-zone fields of the individual pieces are given by ^ikr
Mi(r) = qo-
r
Iji^k
(2.12a)
j\{kax\
,i)tr
uiir) = ^0-
IJl^k
J'^^'^'M^'^^'^
-ux{r\
(2.12b)
where the last equation was simplified by the fact that kaj is a zero of the function j \ . Because ka2 is the first zero of the fimction j \ , it is clear from eq. (2.12a) that the individual fields u\ and U2 are nonzero even though their sum is zero, i.e., the fields u\ and W2 destructively interfere in the far zone of the source. It is to be noted, however, that the choice of a\ is arbitrary, and the above argument is valid for any a\ satisfying a\ < «2- It is not quite appropriate, therefore, to speak of the field of piece 1 of the source destructively interfering with the field of piece 2 of the source; the nonradiating effect is produced by the mutual interference of the fields from all points within the source domain.
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281
We will now briefly describe a few results from the theory of monochromatic nonradiating sources which are of particular interest. Theorem 2.2. A necessary and sufficient condition for a source distribution to be nonradiating is that the equation f q(/)Mk\r-/\)d'/
=0
(2.13)
JD
be satisfied for all values ofr (Bleistein and Cohen [1977], Friedlander [1973]).
We first show that eq. (2.13) follows from the nonradiating condition (2.9). To see this, we multiply eq. (2.9) by e^^*'', and integrate the resulting equation over all real directions s. This gives /
/ q{r') e^'<'-''^ dV' dO = 0,
J{4JI)
(2.14)
JD
where dQ is an element of solid angle, and the O-integration is over the complete 4jr solid angle. By using the well-known identity (Mandel and Wolf [1995], footnote on p. 123), 7o(^|i--/|)=-^ /
e^'<'-''HQ,
(2.15)
4 ^ J {Alt)
eq. (2.13) follows immediately. To demonstrate that eq. (2.9) follows from eq. (2.13), we substitute from eq. (2.15) into eq. (2.13), which results again in eq. (2.14). Let us multiply this equation by its complex conjugate, giving the new equation //
/ / ^*(''') ^(^'') e-^^^'-^'-'-'^e^^^''-^''-''''^ dV' d\" dQ' dQ" = 0.
J J {An) JJD
(2.16) We choose r to lie in the x,y-plane of some Cartesian coordinate system. Let (s^^,Sy,s'^) and (s'^,Sy,s'^) represent the components of 5' and s'\ respectively, in this coordinate system. Equation (2.16) may then be expressed in the form
//
It ^*(''')^^"'"'qir")Q-^'"-'"Q-^U-0-M-s'y')y]
dV' dV'' dQ' dQ" = 0.
Jj{4ji) JJD
(2.17)
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Nonradiating sources and other "invisible" objects
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We now integrate with respect to r over the entire x,7-plane; using the Fourier representation of a Dirac delta function, 1 1"^ — / e^'-^^doc = d{a-b\ ^^
(2.18)
J-oo
equation (2.17) becomes -\ks"-r"
I(4JT)JJD' JJ(4jr) JJD
'
-
'
(2.19)
X d(ks'^ - ks'^) 6(ks[^ - ks'^) dV' d V ' dQ' &Q" = 0. Integrating over all directions of 5' equates the x and 7 components of the vectors s' and s'^ Because s' and s" are unit vectors, this implies that s' = s'\ Then, using eq. (2.8), eq. (2.19) may be further simplified to /
\q(ks'')\^ dQ'' = 0.
(2.20)
The integrand of eq. (2.20) is non-negative. This equation therefore implies that q{ks) must satisfy eq. (2.9), and Theorem 2.2 follows. It is interesting to note that Theorem 2.2 implies that the retarded and advanced fields^ of a nonradiating source are equal. From the definition of 70 it follows that 2ikr On substituting from this equation into eq. (2.13), the latter equation may be rewritten as
This expression demonstrates that the retarded and advanced fields of a nonradiating source q(r) are equal. Theorem 2.3. The field of a nonradiating source vanishes everywhere outside the domain of the source^. See Jackson [1975], section 6.6 for a discussion of retarded and advanced fields. ^ This result was first proven in its present form by Friedlander [1973], Theorem 3.1, although Amett and Goedecke [1968] earlier proved the comparable result for moving charge distributions. The theorem was also indirectly stated by Miiller [1955]. A similar result relating to the field scattered by a scattering potential of finite support was described by Wolf and Nieto-Vesperinas [1985]. We follow closely the derivation of the latter.
5, §2]
283
Nonradiating sources
> z
Fig. 3. Regions 7^^ and 7i defined in the proof of Theorem 2.3. Im a
Im a
4> f
Re a
-^2+1^
-%
> ^• >R e a
Fig. 4. Complex paths of integration C^ and C~
To prove this result, we return to the solution to the inhomogeneous Helmholtz equation given by (2.5), i.e. (2.23) Let us choose a direction to be the z-axis, and denote by IV and IZ' the regions to the right and left of the source domain D with respect to the z-axis, respectively (see figure 3). Now we may expand the Green's fiinction in the regions IZ^, 1Z~ by use of the Weyl representation (Mandel and Wolf [1995], section 3.2.4), ^k\r-r'\
^j^
\r-¥'
, / diS / dasinae^t^^-^'-^'^^, 2 ^ J-n Jc±
nn
(2.24)
where Sx = sin a cos /?,
Sy = sin a sin )S,
cos a.
(2.25)
and the integration over a is taken on the complex curve C^ if r lies in K^ and on the curve C~ if r lies in 1Z~ (see fig. 4).
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Nonradiating sources and other "invisible" objects
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On substituting from the Weyl expansion (2.24) into eq. (2.23), we may express the field u{r) in the regions IV and 7l~ in the form
[ d/? /
da sin a A(a, 13) e^'',
(2.26)
where A(a, 13)= f q{r') e"'^^ ^' dV^
(2.27)
JD
Let us compare eq. (2.27) with the nonradiating condition (2.9). It is seen that if the source is nonradiating, then A{a,p) = Q for all
0 ^ a ^ ^;r,
0^I3<2JT.
(2.28)
Equation (2.28) demonstrates that if the source is nonradiating, then the fiinction A(a,p) vanishes over a two-dimensional region of (a,jS) space. We will use this property shortly. The function A(a,p) may be expressed explicitly in the form A(a, 13)= q{r')exp{-iA:[(sin acosiS)x' + (sin a s i n ^ ) / + (cos a)z']] d?r'. JD
(2.29) Let us consider the behavior of this function for all complex values of a, ^. Because the domain D is of finite extent, it can be shown that, for any /?, A{a,p) is an entire analytic function of a, and that, for any a, A(a,P) is an entire analytic fiinction of 13. It then follows from a theorem of analysis in several complex variables (Dickson and Osgood [1914], p. 143) that A(a,P) is an entire analytic function in two complex variables, a and j3. It therefore cannot vanish over a continuous region of dimensionality equal to or greater than that of a surface, unless it vanishes identically. We have seen, though, that if the source is nonradiating, eq. (2.28) is satisfied, and A(a,l3) therefore vanishes for all values of a, jS. It follows from eq. (2.26) that the field vanishes identically in the halfspaces 1Z^ and 7l~. We may repeat this process with different choices of the z-axis; in this way, it is possible to demonstrate that the field vanishes up to a convex surface enclosing the source domain. If the source domain has concavities within it, analytic continuation methods can be used to demonstrate that the field vanishes also within these concavities (see Colton and Kress [1983], Theorem 3.5). Hence Theorem 2.3 follows. We have shown that the field of a nonradiating source vanishes everywhere outside the domain of the source. Some authors have suggested that nonradiating
5, § 2]
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285
sources might be probed by the methods of near-field optics (Courjon, Banier, Girard and Vigoureux [1993]). Theorem 2.3 demonstrates that this is, in fact, not possible - nonradiating sources possess neither evanescent nor homogeneous external fields. Moreover, it is clear that the radiated field must contain both evanescent and homogeneous waves, if it does radiate: this result was proven in the context of scattering theory by Wolf and Nieto-Vesperinas [1985], Theorem IV Thus purely evanescent radiation sources are not possible, at least for sources of finite domain. Furthermore, our derivation has shown that the radiation pattern of a source in the far zone cannot vanish over a continuous finite solid angle of directions unless it vanishes identically. Thus a source must radiate in "almost all" directions (in the sense of measure theory) or not radiate at all. Theorem 2.4. A bounded nonradiating source distribution q{r) of finite support and the field u{r) that it generates are related by the inhomogeneous Helmholtz equation (2.3), subject to the boundary conditions u(r)l^s
=0,
(2.30a)
du(r) = 0, (2.30b) dn res where S is the boundary of the source domain and d/dn denotes differentiation along the outward normal (Gamliel, Kim, Nachman and Wolf [1989], Kim and Wolf [1986]). This is perhaps the most valuable theorem regarding nonradiating sources, for it gives an easy-to-apply method of constructing such sources ^^ for any domain: one only needs to determine a fiinction/(f) which is continuous, has a continuous first derivative, and satisfies the boundary conditions (2.30). The fiinction/(r) is then the field of a nonradiating source, and the source itself can be readily determined by the use of the inhomogeneous Helmholtz equation (2.3), as q{r) = -^{^^
+ k^)f{r).
(2.31)
Theorem 2.4 also demonstrates the existence of nonradiating sources for any reasonably well-behaved, connected source domain, for it is always possible
It is interesting to note that Ehrenfest [1910] suggested a similar construction for time-dependent electromagnetic sources.
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Nonradiating sources and other "invisible " objects
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to create a function f(r) which satisfies the continuity requirements and the boundary conditions (2.30). Nonradiating cubes, pyramids, and toroids are therefore derivable, and exphcit expressions may be found for their source distributions. Because of the complexity of the derivation, we will not explicitly prove Theorem 2.4; its proof follows fi-om Theorem 2.3 andfi-omthe continuity of the field and its gradient. We will, however, show that it is true for the homogeneous source given by eq. (2.10) ^^ The field within the homogeneous source can be found by the use of the multipole expansion (Jackson [1975], p. 742), i/tlr-r'l
^^
'
^—-y = ^mkY, Y,j>{kr<)h^^\kr.) >/ '(^,07 l^C^', A
(2-32)
where 7/ is the /th-order spherical Bessel function, h] is the /th-order spherical Hankel function of the first kind, and Y[^ are the spherical harmonics. The variables r< and r> denote the smaller and larger of the distances r = |r| and r' = \r'\, respectively. By substituting fi-om eqs. (2.32) and (2.10) into (2.5), the field within the homogeneous sphere [when the nonradiating condition (2.11) is satisfied] is readily found to be u(r) = ^
I ^ oos[k(a -r)]-^
sm[k(a - r)] - 1 1 ,
(2.33)
and its normal derivative is - ^
= - ^ I - — cos[k{a -r)]+ — sm[k{a - r)] (2.34) + —^ sm[k{a - r)] + - cos[A:(fl - r)] >. kr^ r
On setting r = am eqs. (2.33) and (2.34), it is clear that Theorem 2.4 is satisfied. 2.2. Electromagnetic nonradiating sources With slight modification, the resuhs demonstrated in the previous section also apply to electromagnetic sources and fields (Goedecke [1964], Devaney and Wolf
^^ This derivation follows from Kim and Wolf [1986].
5, § 2]
Nonradiating sources
287
[1973], Abbott and Griffiths [1985]). For a monochromatic current distribution j(r) localized to a domain D, the space-dependent parts of the electric field E{Rs) and the magnetic field H{Rs) in a direction s and at a distance R in the far zone of the source are given by (Devaney and Wolf [1974], eqs. (4.4), (4.5) and (4.19)) E{Rs)
=-(2;r)3-(sx[sxj(^s)])—, c ^ ^ R ik e^^^ H(Rs) = {IJtf-s xj{ks) , c R where
(2.35a) (2.35b)
^W-(2^//'-')e--^'dV'
(2.36)
is the three-dimensional spatial Fourier transform of the current density. From eqs. (2.35) it should be evident that the current distribution will not radiate if j{ks) = 0
for all directions s.
(2.37)
This nonradiating condition is comparable to that of the scalar radiation source, eq. (2.9), and theorems about nonradiating electromagnetic sources which are analogues to those of the scalar case may be derived (Devaney and Wolf [1973]). The vectorial nature of the electromagnetic problem introduces another class of sources which do not generate power, as we now show. Let us suppose that the current density of the source has the form j(r) = rf(rl
(2.38)
where/(r) (r = |r|) is a spherically symmetric, continuous function. The current is then purely radial, and the sphere might be described to be "pulsating". On substituting from eq. (2.38) into eq. (2.36), and using the fact that re-^'- = iVKe-^%
(2.39)
we may express the spatial Fourier transform of the current density as (Iny
''V
(2.40)
= iVKf(K), where/(AT) is the three-dimensional Fourier transform of/(r). Therefore jiks) = iVKfiK)\K=k. = i ( ^ / ( ^ ) U = * ) *•
(2.41)
It can be seen that s x](ks) = sx[s
xj(ks)] = 0;
(2.42)
from eqs. (2.35) it is evident that the electric and magnetic fields vanish in the far zone of the current density. In fact, by use of symmetry arguments, it can
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Nonradiating sources and other "invisible" objects
[5, § 2
be shown that the magnetic and electric fields must vanish everywhere outside the domain of the source (Ehrenfest [1910]). Although this behavior is very similar to that of the scalar nonradiating sources described previously, pulsating spheres do not radiate because they produce purely longitudinal fields, and such nonradiation is an entirely electromagnetic effect. Indeed, the nonradiating condition (2.37) need not be satisfied. Such examples are extremely pathological and will not be considered further.
2.3. Partially coherent nonradiating sources In the previous sections the properties of monochromatic nonradiating sources were elucidated; we now broaden these results and consider the properties of partially coherent nonradiating sources ^^, for which the source fiinction Q(r,t) may be a random fiinction of time t. We assume these fluctuations to be stationary, at least in the wide sense (Mandel and Wolf [1995], section 2.2). The mutual coherence function of the source distribution, defined by the expression rQ(rur2,T)={Q\rut)Q(r2,t^T)),
(2.43)
the angular brackets denoting ensemble averaging, describes the correlation of the source fluctuations at pairs of points ri and 1-2 within the source at times t and ^ + r. It will be more convenient, however, to employ the spacefi*equency representation. For this purpose we consider the cross-spectral density ^0(^1,^2, ^ ) of the source distribution, defined as the Fourier transform of the mutual coherence fiinction, 1 Z*^ WQ(rur2,co)=—J rg(ri,r2, r)e^"Mr.
(2.44)
We may also define a cross-spectral density Wu(r\,r2, co) of the field U(r, t) in a similar manner. In the far zone of the source domain, the cross-spectral density
*^ For a thorough description of classical coherence theory and the coherence concepts discussed in this section, see Mandel and Wolf [1995]; a shorter description is given in Bom and Wolf [1999], chapter 10.
5, § 2]
Nonradiating sources
289
of the field and that of the source may then be shown to be related by the formula (Mandel and Wolf [1995], section 4.4.5)
WuiRxSx^RiSi, (o) =
^^e'^R^-^^)WQ{-ksxM2,
(o\
(2.45)
K\K2
where s\= s\ = 1, and WQ{K,,K2,a))=
- i - /
/ Wgir,,r2,co) ^'^^^^^^^^ dVi dV2
(2.46)
is the six-dimensional spatial Fourier transform of the cross-spectral density. From eq. (2.45), it seems that the following condition is necessary and sufficient for a source to be nonradiating at afi*equencyco: WQ{-ks\,ks2,(o) = 0
for all real unit vectors 5i,52.
(2.47)
A simpler condition exists, however, as stated in the following theorem: Theorem 2.5. A partially coherent source is nonradiating at a given frequency co if its cross-spectral density, WQ(ri,r2,(i>\ satisfies the condition (Devaney and Wolf [1984]) WQ(-ks,ks,(o) = 0 for alls.
(2.48)
This theorem follows immediately by noting that the radiant intensity J(s, w) of a source, given by the formula J{s, (JO) = {IJtf Wgi-ks, ks, CO),
(2.49)
represents the rate at which the source radiates energy at firequency co, per unit sohd angle, into the far zone (Mandel and Wolf [1995], section 5.2.1). Thus a source which satisfies eq. (2.48) has a zero radiant intensity and does not radiate atfirequencyco. It can be shown that the nonradiating condition (2.47) follows from Theorem 2.5 if we use the non-negative definiteness property (Mandel and Wolf [1995], section 4.3.2) of the cross-spectral density, i.e. the condition that for
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Nonradiating sources and other "invisible" objects
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any well-behaved complex function/(r), the cross-spectral density satisfies the inequality /
/ ^^(ri,r2,ft;)/*(ri)/(r2)dVi dV2 ^ 0.
(2.50)
JD JD
We consider the following particular choice of/(r): f{r) = ae^''' + be^''',
(2.51)
where a and b are arbitrary complex constants. On substituting from eq. (2.51) into the non-negative definiteness condition (2.50) and using eq. (2.46), condition (2.50) takes on the form \a\^ WQ{-ks\ ,ks\,(jj)+\b\^ WQ{-ks2, ks2, (o) -\-a*bWQ(-ksi,ks2,co)-\-b''aWQ(-ks2,ks\,aj)
^ 0.
(2.52)
We may express this condition in a matrix notation as
[a* Z>*]
WQ(-ks\, ks\, CO) Wgi-ksx, ks2, of) ^ 0. [ WQ(-kS2, kS\, CO) WQ(-kS2, kS2, CO) \ [b \
(2.53)
A necessary condition for this inequality to be satisfied is that the determinant of the matrix be non-negative. Furthermore, by using the fact that the cross-spectral density is a Hermitian function (Mandel and Wolf [1995], section 4.3.2), i.e. that WQ(rur2,co)=W^(r2,ruCO%
(2.54)
it is not difficult to show that WQ(-ks2, ksi, CO) = W^{-ksx, ks2, CO).
(2.55)
By use of eq. (2.55), the requirement that the determinant of the matrix in inequality (2.53) be non-negative takes on the form WQ{-ksx,ksu CO) WQ(-ks2,ks2, co) > | WQ(-ksuks2, co)\^.
(2.56)
It is clear from inequality (2.56) that if eq. (2.48) is satisfied, then eq. (2.47) follows. Relatively few papers have been written about partially coherent nonradiating sources, no doubt because the underlying mathematics are significantly more complicated than that of monochromatic sources. Several papers have been
5, § 3]
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written on the inverse source problem for partially coherent sources (LaHaie [1985], Devaney [1979], LaHaie [1986]). Nonradiating partially coherent electromagnetic sources have also been investigated (Baltes and Hoenders [1978], Hoenders and Baltes [1980]). Some other results regarding nonradiating partially coherent sources, which are analogous to those established for monochromatic nonradiating sources, may be proven by the use of a modal expansion of the cross-spectral density; some such work is described by Gbur and Wolf [1997].
§ 3. The inverse source problem In the most general sense of the term, an inverse problem may be said to be the determination of the "cause" of a phenomenon from measurements of the phenomenon itself (the "effect") ^^. Every inverse problem is based upon a direct problem whose solution represents the usual "cause - effect" sequence of events. Examples of inverse problems include the determination of atomic structure from measurements of atomic spectra (spectroscopy), the determination of crystal structure from X-ray scattering experiments (X-ray crystallography), and the determination of the characteristics of earthquake faults from measurements of seismic waves (seismology). One might even describe criminal investigations as inverse problems - the determination of a "cause" (the criminal and his motive) from measurements of the "effect" (evidence at the crime scene). A large number of inverse problems that have been investigated are said to be ill-posed (in the sense of Hadamard [1902], who originally introduced the concept). For our purposes, it suffices to say that ill-posedness refers to two difficulties: nonuniqueness of the solution of the problem and large errors in the solution of the problem which follow from small errors in the data (in which case the problem is said to be ill-conditioned). To explain these concepts more precisely, let us suppose that the direct problem may be represented by an operator A acting on an "object"y^. Then the "image" of the object,^, may be represented by the equation^ = 4 ^ • The inverse problem is characterized by the inverse operator J~^ acting upon the image^. Nonuniqueness of the inverse
^^ An excellent and very readable introduction to inverse problems is given by Bertero and Boccacci [1998].
292
Nonradiating sources and other "invisible " objects object space
A
[5, § 3
image space nonuniqueness
ill-conditioning
Fig. 5. Demonstration of a direct problem whose inverse problem is ill-posed. Nonuniqueness corresponds to two different objects which have the same image. Ill-conditioning corresponds to two significantly different objects which have nearly the same image.
image space «*^K^pSffA
Jioiiuniqueness
£ ^ ^ ^ ^ g | S | ill-conditioning
physical solutions (prior knowledge) Fig. 6. Demonstration of the use of prior knowledge. To deal with nonuniqueness, the solution is chosen that fits the additional physical constraint on the problem. To deal with ill-conditioning, a physical solution is chosen whose image (i) is sufficiently close to the measured image (ii).
problem and the amplification of errors are illustrated in fig. 5. If the problem is ill-conditioned, a slight change in the image (due to noise, for instance) can result in a drastic change in the reconstructed object. If the problem is nonunique, even a noise-fi-ee image cannot be used to determine the object, because multiple objects can produce the same image. This difficulty is intimately related to nonradiating sources, as we shall see. The general method used to deal with ill-conditioning and nonuniqueness of inverse problems is the application of prior knowledge in determining a solution. Prior knowledge is additional information, determined independently of the image, used to restrict the range of possible solutions. Such prior knowledge usually follows from knowledge of the physics of the problem, such as the size of the object being imaged. Its appUcation is illustrated schematically in fig. 6. Nonuniqueness is, in principle, solved by isolating a single solution that satisfies the additional physical constraints. Ill-conditioning is, in principle, overcome by finding an acceptable physical solution which produces an image sufficiently close to the image which is measured. It should be noted that a particular piece
5, § 3]
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293
of prior knowledge may be insufficient to construct uniquely the solution; the usefulness of specific prior knowledge must be analyzed within the context of the particular problem. In optics, inverse problems are usually divided into two broad classes: (1) the determination of the scattering properties of an objectfi*ommeasurements of the field scattered by that object, and (2) the determination of the properties of a radiation source firom measurements of the field radiated by that source. These problems are generally referred to as the inverse scattering problem and the inverse source problem, respectively^^. Solutions to inverse scattering problems have met with great success in recent years, most significantly in medical imaging. Foremost among these methods is computed tomography (CT), in which the attenuation of X-rays through an object is measured for a number of directions of incidence. By a non-trivial manipulation of the attenuation data, an image of a two-dimensional slice of the object may then be constructed^^. Computed tomography is based upon a geometrical model of the propagation of radiation. When diffraction effects of scattering must be taken into account, the method known as diffraction tomography may be used, in which measurements of the scattered field for multiple directions of incidence may be used to reconstruct the object of interest ^^. The inverse source problem, in contrast, has received little attention, no doubt because of its nonuniqueness. It would seem reasonable, then, to simply probe the structural information of an object by scattering experiments and ignore the source properties entirely, but there are several circumstances in which this is not usefiil or possible: (1) The object may not be accessible to scattering experiments, as in astronomy or seismology. In this case the only information available is the field radiated by the source. (2) The object may only be accessible for scattering experiments for a limited range of directions of incidence, as in geophysical inverse problems, for which the objects of interest are buried. The equivalence of the inverse source problem and the inverse scattering problem for a single
^^ It should be mentioned that the term inverse source problem is used to describe a variety of problems; see Baltes [1978]. We will speak exclusively about inverse problems involving threedimensional, primary radiation sources. ^^ The first CT machine was constructed by Hounsfield [1973]. He received the 1979 Nobel Prize in Physiology or Medicine jointly with A.M. Cormack (who developed the theory) for 'the development of computer assisted tomography'. For more information on computed tomography see, for example, Herman [1980]. ^^ The theoretical foundations of diffraction tomography were developed by Wolf [1969, 1970]. A recent review of the field is given by Wolf [1996].
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Nonradiating sources and other "invisible " objects
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direction of incidence will be discussed in §4. (3) The source of interest may be a subset of some larger scattering medium. For instance, in single-photon emission computed tomography (SPECT) ^^, a radioactive isotope is injected into a patient's body and its position is determined by solving a particular inverse source problem. The nonuniqueness of the inverse source problem is easily understood as follows. Consider a monochromatic source q{r) e~^^^ confined to a domain D. As we have seen in § 2.1, by measurement of the field in the far zone of the source we may determine the spatial Fourier transform of this distribution, q(K), for all values \K\ = k. Now consider a source q'{r) = q{r) + qNR(r), where qm(r) is a nonradiating source. The spatial Fourier transform of the source q'(r) will be q\ks) = q(ks) + qNR(ks) = q{ks\
(3.1)
which is identical to that of the source q{r). Therefore the sources q and q' produce the same field in the far zone of the source; it followsfi*omTheorem 2.3 that their fields are identical everywhere outside the source domain. Therefore no measurements of the field outside the source can distinguish between the sources q and q'\ consequently the inverse source problem for monochromatic sources is nonunique. A similar argument may be made for partially coherent sources. If Wgirx ,r2,0)) is the cross-spectral density of a source Q, the cross-spectral density of its field will be identical to that of a source whose cross-spectral density is given by WQ{r\,r2,cci)+ ^NR(''I,''2,<^), where ^NR is the cross-spectral density of a nonradiating partially coherent source. The uniqueness or nonuniqueness of an inverse problem can often be determined by a simple dimensional argument. Referring again to an "object" and an "image", if the dimensionality of the image is equal to or greater than that of the object, the inverse problem is most likely unique. If the dimensionality of the image is lower than that of the object, the inverse problem is certainly nonunique. The latter case corresponds to a loss of information in the direct problem - the image does not contain enough information to reconstruct the object. For instance, in the monochromatic inverse source problem, the "object" (the source) is a three-dimensional fiinction, while the "image" (far-zone field measurements) is projected over a two-dimensional solid angle. For the inverse source problem for partially coherent sources, the
^^ For details of SPECT, and a similar method, positron emission tomography (PET), see Kak and Slaney [1988], section 4.2.
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295
"object" (the source cross-spectral density) is a six-dimensional function, while the "image" (far-zone field cross-spectral density measurements) is a fourdimensional fiinction^^. Because the inverse source problem is nonunique, we must use some sort of prior knowledge to obtain a unique inversion. More precisely, we must limit ourselves to solving the inverse problem for some class of sources which have distinguishable radiation patterns. Let us consider the class of spatially incoherent sources which may be expressed in the form^^ WQ{rur2. (D) = Igin) (5(1-2 - ri),
(3.2)
where 5(^2 - n) is the three-dimensional Dirac delta function, and Igir) is a measure of the intensity of the source. By use of eq. (2.45), the cross-spectral density of the field at points Rsi and Rs2 in the far zone is found to be Wu(RsuRs2, CO) =
^-J-lQ[k{s2-s,)l
(3.3)
where IQ is the three-dimensional Fourier transform of the source intensity. Let us consider the possibility of the existence of a nonradiating source of the form (3.2). To be nonradiating, the fimction /g[A:(52 - si)] must vanish for all directions 5i, 52; in particular IQ{K) must vanish for all \K\ ^ 2k. However, it is known that because IQ{K) is the Fourier transform of a fiinction of finite support, it is the boundary value of an entire analytic function in three complex variables (see Fuks [1963], p. 352). It follows then that IQ{K) cannot vanish over any three-dimensional region of JT-space unless it vanishes identically - in which case lQ{r) is itself identically zero. Therefore nonradiating incoherent sources do not exist. It follows that the inverse problem for such sources must be solvable. This can be readily seen to be true, for by use of the measurements of IQ{K) for all \K\ ^ 2k, we may perform a Fourier inversion of this function to reconstruct a low-pass filtered version of the source intensity, i.e. ^e'W= /
/e(^)e^''d3r.
(3.4)
J\K\^2k
Our dimensional argument for the uniqueness of inverse problems holds in this case, for the "object" (source cross-spectral density) is effectively a threeAn inverse problem which is unique is the determination of a finite primary planar source (twodimensional) from measurements of the radiation pattern (also two-dimensional). The nonexistence of finite primary planar nonradiating sources was proven by Friberg [1978]. This problem was first considered by Devaney [1980].
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Nonradiating sources and other "invisible" objects
[5, § 3
dimensional function, while the "image" (field cross-spectral density) is a fourdimensional function. In retrospect, the uniqueness of the inverse source problem for incoherent sources may seem obvious because, as mentioned previously, nonradiating sources arise from destructive interference of the field generated by different points in the source. Incoherent sources do not produce such destructive interference. Such sources are an extreme, and unphysical, approximation to realistic sources, however ^^. Another candidate class of sources for which the inverse source problem may be unique are so-called quasi-homogeneous, or globally incoherent sources (see Mandel and Wolf [1995], section 5.2.2), for which the width of the spectral degree of coherence function is much smaller than any distance over which the intensity varies appreciably. Several authors have discussed the inverse problem for quasi-homogeneous sources ^^ Devaney [1979] suggested that complete knowledge of the spectral degree of coherence of the source allowed reconstruction of the source's intensity. LaHaie [1986] extended this work by suggesting that limited knowledge of the spectral degree of coherence is sufficient. More recently, Gbur [2001a] hypothesized that in many cases knowledge of only the width of the spectral degree of coherence function is adequate to make a fair reconstruction. It is worth mentioning that several authors (Porter and Devaney [1982], Devaney and Porter [1985], Marengo, Devaney and Ziolkowski [2000]) have demonstrated that the inverse source problem can be made unique by requiring that the source distribution has "minimum energy", i.e., by choosing the solution that minimizes the quantity
dV'.
(3.5)
JD
It should be cautioned, however, that it is unclear what classes of physical sources, if any, would satisfy this constraint. It is also unclear whether the quantity E in eq. (3.5) even represents a physical energy.
^^ One obvious unphysical feature of a delta-correlated source is its infinite source intensity. This is discussed by Beran and Parrent [1964], section 4.4. ^^ Baltes and Hoenders [1978] were the first to suggest that nonradiating quasi-homogeneous sources do not exist.
5, § 4]
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297
§ 4. Nonscattering scatterers Up to this point we have been considering only "invisible" objects of the second type mentioned in the introduction: time-fluctuating sources which do not radiate. In this section we will consider the first type of invisible objects: those which do not scatter fields incident upon them. It will be shown that the existence of such "nonscattering scatterers" is related to the existence of nonradiating sources.
•\\
scatterer
incident field ^,(r,^)
•/ / scattered field ?7,(r,^)
Fig. 7. Arrangement relating to the scattering of a monochromatic field firom a scattering object.
We consider a monochromatic electromagnetic field with time dependence Q-io)t incident upon a linear, isotropic, nonmagnetic medium occupying a finite domain D (seefig.7). If the index of refi-action n{r, co) varies sufficiently slowly over space (see Bom and Wolf [1999], section 13.1), the Cartesian components of the electromagnetic field are uncoupled to good approximation and each component satisfies the scalar equation V^U(r, CO) + kW(r, co) U(r, co) = 0,
(4.1)
where U(r, co) is the spatial dependence of a component of the total field, incident and scattered. This equation can be rewritten in the form V^ U(r, CO) + k^ U(r, co) = -4jtF(r, co) U(r, co),
(4.2)
where F(r,co)=^k^[n\r,co)-l]
(4.3)
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Nonradiating sources and other "invisible" objects
[5, § 4
is known as the scattering potential of the medium. Equation (4.2) may be simplified further; let us write the total field in the form U{r,a))=U,{r,w)+U,{r,(o\
(4.4)
where U\ is the field incident on the scatterer and U^ is the scattered field. We will consider only incident fields which satisfy the homogeneous Helmholtz equation, i.e. V^ Ui{r, 0)) + k^ Ui(r, o)) = 0.
(4.5)
On substitution of eq. (4.4) into eq. (4.2), and using eq. (4.5), we find the following partial differential equation for the scattered field: V^ ^s('', CO) + k^ Us(r, (v) = -4;rF(r, co) U(r, (o).
(4.6)
This equation should be compared with eq. (2.3) of §2.1. It is seen that the scattered field satisfies the inhomogeneous Helmholtz equation, with the source term given by q(r, CO) = F{r, (o) U(r, w).
(4.7)
The scattered field may therefore be expressed in integral form as ^s(i', co)= [ F(r\ (o) U(r\ (o)^^ d'r\ JD \r-r'\
(4.8)
It is important to note that this equation does not readily give the scattered field, because Us is present in both sides of the equation. However, if the scattering potential is sufficiently weak, the scattered field will be small compared to the incident field; we may then approximate U(r, (o) in eq. (4.8) by Ui(r, (o), and arrive at the expression t/s(r, co)= f F(r\ co) U,(r\ co)^^
d'r\
(4.9)
This approximation for the scattered field is known as the^r^^ Born approximation, or often simply as the Born approximation (Bom and Wolf [1999], section 13.1.2). We are now in a position to investigate the existence of objects which do not scatter fields incident upon them. On comparison of eq. (4.9) with the equation
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Nonscattering scatterers
299
for the field of a radiation source, eq. (2.5), it is clear that, for a given incident field t/i, we may treat the weak scattering problem as a radiation problem with a source given by the expression q{r, 0)) = F(r, (o) Ui(r, (o).
(4.10)
In particular, let us examine the scattered field in the far zone of the scatterer. In this case, the approximation given by eq. (2.6) may be used, and the scattered field in the far zone, in a direction s and at a distance R, may be expressed in the form QikR
Us(Rs,a))^
p
-— / F(r\a))Ui(r\cD)Q-'''"'' ^ JD
dW
(4.11)
From this equation a fundamental theorem regarding nonscattering scatterers follows immediately: Theorem 4,1. A weakly scattering object with a scattering potential F(r) will be nonscattering for a given incident field U[(r) if
L
F(r\ (D) Ui(r\ (O) Q'^'-'' dV' = 0 for all s.
(4.12)
Because of this equivalence between these nonscattering scatterers and nonradiating sources, all results that apply to nonradiating sources also apply to scatterers which are nonscattering for one direction of incidence. For instance, the scattered field of a nonscattering scatterer will vanish everywhere outside the domain of the scattering object, as follows from the use of Theorem 2.3. Furthermore, the solution of the inverse source problem is equivalent to solving the inverse scattering problem for a single direction of incidence. Such invisible scatterers were apparently first described by Kerker [1975], who demonstrated that certain compound dielectric ellipsoids will not scatter a field incident fi-om certain directions. Later work by Devaney [1978] demonstrated that there exist weak scatterers which do not scatter incident plane waves for any finite number of directions of incidence ^^. More recently. Wolf and Habashy [1993] demonstrated that weak scatterers which are nonscattering for
-^^ Much later, Hoenders [1997] showed that such results hold even when the field propagation is described by the equation of radiative transport or a diffusion equation.
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Nonradiating sources and other "invisible " objects
[5, § 4
Fig. 8. The Ewald spheres of reflection (dashed spheres) and the Ewald Umiting sphere, 2 L .
all directions of incidence do not exist ^^. Because of the importance of this latter theorem for inverse scattering theory, we will prove it explicitly: Theorem 4.2. Within the accuracy of the first Born approximation, there is no medium occupying a finite region of space which is a nonscattering scatterer for all directions of incidence. Let us consider the incident fields to be plane waves, t/i(r,a>)=^oe^^*«^
(4.13)
On substituting this formula into eq. (4.11) for the scattered field in the far zone, we find that UsiRs, CO) = {2jTf —F[fc(s - so), CO],
(4.14)
where F(K, CO) = - ^
f F(/, co)^-^''
dV
(4.15)
{ijty Jc, is the three-dimensional Fourier transform of the scattering potential. For a fixed direction of incidence 5o, measurements of the field in the far zone of the scatterer for all possible directions of scattering s can be used to determine those Fourier components of F(K, (o) which lie on a sphere of radius k, centered on JST = ksQ (see fig. 8). This sphere, introduced originally in the theory of X-ray scattering by crystals (James [1948]), is known as the Ewald sphere of
^^ This same paper refers to a theorem by Nachman [1988] which demonstrates that this result is true for all scatterers, not just weak scatterers.
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301
reflection. For the scatterer to be nonscattering for all directions of incidence, the Fourier transform of F must therefore vanish on Ewald spheres of reflection for every direction of incidence and consequently it must vanish within a sphere of radius \K\ ^ 2k (known as the Ewald limiting sphere). However, because F{K, co) is the Fourier transform of a function of finite support, it is the boundary value of an entire analytic function in three complex variables (see Fuks [1963], p. 352). It follows then that F(K, co) cannot vanish over any three-dimensional region of JT-space unless it vanishes identically - in which case F(r, co) is itself identically zero. Therefore Theorem 4.2 is proven, and a scatterer can only be nonscattering for a finite number of directions of incidence. This result is important for the solution of the inverse scattering problem. Theorem 4.2 demonstrates that there are no truly invisible weak scatterers; measurements of the scattered field for enough directions of incidence will provide some information about the scattering object. Let us recall the dimensional argument given in § 3: if the dimensionality of the object and image are the same, the inverse problem is likely to be unique. In this case, the object (the scattering potential) is three-dimensional, and the image (the data obtained from field measurements for all directions of incidence and scattering) is three-dimensional, so the inverse problem is unique. As mentioned earlier, when a nonscattering scatterer is illuminated by an incident field from a direction for which it is nonscattering, the scattered field is entirely contained within the region of the scattering object. It may be said that the scattered field is localized to the domain of the scatterer. In recent years, much attention has been given to the subject of light localization, and we now discuss briefly the similarities and differences between this phenomenon and the properties of the nonscattering scatterers already mentioned. The localization of light ^"^ has been investigated as an analogue of socalled Anderson localization (Anderson [1958]) of electrons in disordered material. When the material through which the light (or electron) propagates is highly scattering and weakly absorbing, the propagation of the light may be described as a diffusion process. It has been demonstrated, both theoretically and experimentally, that if the mean free path of the light is of the order of a wavelength (i.e. the scattering is sufficiently strong in the material), diffusion in the system is impossible and the field is localized within the scattering material.
^^ A discussion of such light localization possiblities is given by John [1991]. Experimental observation of such localization has been claimed both for microwaves (Dalichaouch, Armstrong, Schultz, Platzman and McCall [1991]) as well as for visible light (Wiersma, Bartolini, Lagendijk and Righini [1997]).
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Nonradiating sources and other "invisible " objects
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Fig. 9. Schematic illustration of the localization of light. If the atom at the origin O is a radiating source, the waves radiated by the source along the two opposing paths will have the same phase and will interfere constructively. If the mean free path of the light is sufficiently short (of the order of a wavelength), the contribution of such closed loop paths will dominate and the light will tend to remain about the source.
This localization may be understood to arise from the constructive interference of the field arising from closed paths within the scatterer (see fig. 9). Such localization of light should not be confiised with the nonscattering scatterers described in this section and the nonradiating sources described earlier. The most striking distinction is that light localization of the Anderson type only exists in materials which are strongly scattering. As we have seen above, scatterers may be nonscattering (for a given direction of incidence) even for scatterered fields which satisfy the first Bom approximation. Furthermore, we have seen that nonscattering scatterers (and nonradiating sources) arise from a complete destructive interference of the outgoing radiation, whilst the localization of light arises from the constructive interference of fields returning to their point of origin. Despite these differences, some authors have confiised the two types of localization (see, for instance, Rusek, Orlowski and Mostowski [1996], section III, particularly eq. (21)). For completeness, we mention one more class of supposedly invisible objects that has been discussed in the literature. We consider spherical scatterers of uniform complex refractive index, for which the imaginary part of the index is negative (such active objects may be considered to be a good model for a gain medium pumped to saturation). Alexopoulos and Uzunoglu [1978] have shown that, under certain circumstances, the extinction cross section for scattering of incident plane waves upon such objects vanishes. Such objects may be considered invisible, as the incident field appears to pass undisturbed through them. However, as pointed out by Kerker [1978], the scattered field may in fact be quite large - the loss of the incident field due to scattering is counteracted by an amplification of the unscattered incident field by the active object. Such
5, § 5]
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303
objects are therefore only invisible in a limited sense and not truly invisible as are the nonscattering scatterers described earlier in this section.
§ 5. One-dimensional localized excitations Despite nearly a century of theoretical work on the subject, no one has yet produced a three-dimensional source which is nonradiating or even approximately so. This is no doubt in large part due to the difficulty of constructing a macroscopic, three-dimensional primary source distribution. However, some recent research by Berry, Foley, Gbur and Wolf [1998] has demonstrated that the nonradiating phenomena may also manifest in waves in one-dimensional systems, such as vibrations on a string. These localized excitations have been given the name nonpropagating excitations, and would no doubt be easier to implement in practice. Here we briefly review some of the main results. We consider an infinitely long flexible string under tension T and with mass per unit length ^, undergoing small displacements y{x,t) from the equilibrium position, driven by a force density/(x, t) (force per unit length) and localized in the region a ^ x ^ b (see figure 10). The displacement obeys the wave equation (see, for instance, Morse and Uno Ingard [1986], chapter 4)
^~df—"^^^^^^'^^^
^^-^^
Restricting ourselves to simple harmonic driving forces, /(x,0 = Re{/(x)e--'},
(5.2)
the steady-state solution^(x, t) of eq. (5.1) will have the same time dependence, >;(x,O^Re{X^)e-"^},
(5.3)
and eq. (5.1) then reduces to the one-dimensional inhomogeneous Helmholtz equation. d'yix) , ,2 + k'y{x) = q(x), dx^
(5.4)
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Nonradiating sources and other "invisible" objects
v(xt-)
•
[5, § 5
^^""''^
i 11 i I
->- X
Fig. 10. Illustrating the notation for a vibrating string. The string is shown oscillating with the nonpropagating excitation given by eq. (5.14), with « = 4.
where k is the wave number,
k = -, o=J-,
(5.5)
and ?W = - - ^ .
(5.6)
We will call q(x) the effective force density, or simply \hQ force density. The outgoing solution of eq. (5.4) is well known to be (see, for example, Arfken [1985], sections 16.5, 16.6) 7 W = ^ y q{x')e'\'-^'\dx'.
(5.7)
For displacements to the right (jc > b) and left (x < a) of the region of the applied force, y{x) reduces to piAx
y(^)\R = ^J
pb
^'(^Oe-'^'cbc'
(5.8)
and Q-ikx
pb
X^)IL = ^ ^ / q{x')e'''^'.
(5.9)
It is apparent from eqs. (5.8) and (5.9) that the excitations will vanish everywhere outside the force region a ^ JC ^ ^ if ^(A:) = 0,
^(-A:) = 0,
(5.10)
with k given by eq. (5.5), and q{k) is the Fourier transform of the force density, i.e. qiK)=^l
qix)c-^dx.
(5.11)
Nontrivial force densities that satisfy eq. (5.10) will generate displacements of the string only within the region of the applied force, and will not produce
5, §5]
305
One-dimensional localized excitations
i[ 0.8 0.6 0.4 0.2
x/l
0
(a) Fig. 11. (a) Force density distribution q{x) which produces (b) nonpropagating excitations, with An = Qo/(nJC/l)^, k = nJt/l, for « = 1 and n = 2.
any displacement outside it. As mentioned earlier, we will refer to such a phenomenon as a nonpropagating excitation. As a simple example of a nonpropagating excitation, let a = -I, b = I, I > 0, and let the force be constant throughout this domain, i.e. go when |x| ^ /, q(x) =
(5.12) 0
when 1x1 > /.
This source is the one-dimensional analogue of the nonradiating source example given by eq. (2.10). Upon substituting from eq. (5.12) into eq. (5.11) and requiring that the two conditions (5.10) be fulfilled, we find that there will be nontrivial solutions if and only if kl = nJt,
(5.13)
where n= 1,2,.... This result shows that a constant localized force distribution within the region -I ^ x ^ I produces a nonpropagating excitation only for certain special values of ^/. Using this result in the general expression (5.7) for the displacement, one readily finds that Qo y(x) = { inJi/lf
l - ( - l ) " c o s nJtx when \x\ ^ /, 0
(5.14)
when Ixl > /.
This displacement, along with the associated force density, is shown in fig. 11 for the cases n = \ and n = 2. A simple interpretation of the nonpropagating effect for this example was given by Denardo [1998]. We note that the displacement y(x) given by eq. (5.14) is continuous everywhere on the string,
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Nonradiating sources and other "invisible" objects
[5, § 6
in particular at the boundary of the region of appHed force. One can readily verify that the first derivative (iy{x)/dx is also continuous everywhere on the string. This observation leads directly to an analogue of Theorem 2.4. In fact, analogues to most theorems of three-dimensional nonradiating sources can be applied to nonpropagating excitations. The relative simplicity of the one-dimensional wave problem allows one to evaluate more realistic models of nonpropagation. Gbur, Foley and Wolf [1999] examined the existence of nonpropagating excitations on strings of finite length, strings with damping, and strings with narrow-band driving forces, and showed that in all cases, nonpropagating effects still occur. It is not difficult to show that nonpropagating excitations can be exhibited in two-dimensional wave systems as well. In the Ph.D. thesis of the present author (Gbur [2001b], section 3.7), such two-dimensional excitations were examined and it was shown that they possess properties similar to their one- and threedimensional counterparts. It should be clear fi-om the discussion of this section that nonradiating effects are a general feature of wave-like systems which generate interference and are not limited to electromagnetic or acoustical sources; Marengo and Ziolkowski [1999] have compiled a short list of systems which might be expected to exhibit such phenomena.
§ 6. Moving charge distributions and radiation reaction Up to this point we have considered charge-current distributions whose radiation is due to some sort of internal oscillation of charges. However, the original results in the theory of nonradiating sources were concerned with radiationless periodic translations of rigid charge distributions. In this section we demonstrate that radiationless motions of rigid charge distributions exist ^^, and we relate this result to the theory of nonradiating sources that we have seen so far. Let us consider a rigid three-dimensional charge distribution of total charge e and translational motion characterized by the position vector ^{t). We assume
^^ The method of derivation closely follows that discussed by Goedecke [1964].
5, § 6]
Moving charge distributions and radiation reaction
307
that the motion of the charge distribution is nonrelativistic (§(0
(6.1)
where/(r) is the distribution of charge within the object, satisfying the constraint that
/
f{r)d'r=\,
(6.2)
V
and the distribution is assumed to be locaHzed within a volume V. The current density of such a moving charge distribution is given by the expression Kr,t) = emf{r-m)-
(6-3)
Let us assume that the distribution is moving periodically with period T. Because the motion of the distribution is periodic, and because the distribution is of finite volume V, we may expand the current and charge densities in the following mixed Fourier integral/series representation: CO
„
j{r,t)= Y. //«We'<^"-""'M3^, oo
(6.4a)
.
p(r,0= ^
—K'UK)Q'^'"'-''"'^d^K,
(6.4b)
where cOn = 2jtn/T. The Fourier coefficients of y and p are related here by the use of the continuity equation. It is to be noted that the « = 0 term of eq. (6.4b) represents the net charge of the source distribution. We now define a scalar potential 0(r, t) and a vector potential A(r, t) in the usual way In the Lorenz gauge, these potentials have the form (Jackson [1975], chapter 6)
A{rj) =- r^
0(r,O = / ^ ^ I
\
,; ^dV,
, - ^dV.
(6.5a)
(6.5b)
308
[5, §6
Nonradiating sources and other "invisible " objects
Far from the region within which the source is moving, at a distance r and in a direction specified by a unit vector f, these potentials take on the forms (6.6a) (6.6b) We may substitute from eqs. (6.4) into eqs. (6.6) to express the potentials in terms of periodic oscillations as
^(^^')« ^; X^ JJd'Kd'r'J„iK)sxpkKr'-a}„{t (l>{rr, t)^-J2
r—r•r )
— / A ^ ^
I
(6.7a)
'''' ) I. (6.7b)
These expressions may be simplified by the use of the Fourier representation of the delta function, i.e. (6.8) and eqs. (6.7) become
A(rr,t)
c {2jtf c
V —y„(^„r)e--"', ^-^ r
(6.9a)
^ e'''"' J2 ^rj„{k„r)c-""'',
(6.9b)
^—'
r
where A:„ = (On/c. The electric field is given by the expression E(rr, t) = -V(l>{rr, t) -
\dA{rr,t) c dt '
(6.10)
On substituting from eqs. (6.9) into eq. (6.10), neglecting all terms which decrease more rapidly than 1/r as r -^ oo, and using the elementary vector
5, § 6]
Moving charge distributions and radiation reaction
309
identity a x {b x c) = b{a - c)- c(a • b\ we find that the far-zone electric field is given by the expression Eirtt) = -^—^ c
V ikn [f X [r xUKm ^-^
e-^"«^
(6.11)
r
Expression (6.11) indicates that the distribution will not radiate if the individual current contributions 7„(A:„r) = 0 for all n, and all directions of observation r. This condition should be compared with the nonradiating condition (2.35a) for a monochromatic current distribution. We would like to express the electric field in terms of the structure of the rigid charge distribution, / ( r ) , and its motion, described by the position vector ^{t). To do so, we first note that the coefficients 7„(J5r) of eq. (6.11) are related to the current density yX''? 0 W the formula
On substitution from eq. (6.3) into (6.12), it is not difficult to show that UK)=^f(K)J
e-"*^-««|(Oe'"'"'d?,
(6.13)
where
We may use this result to express the electric field far from the source in the form Eirr,t) = -
^
f ; ^ J ( ^ „ f ) ^ e - - " ' jW^^'Xf
x [f x | ( / ) ] ) e - " ' ' At'.
« = -oo
(6.15) Equation (6.15) leads to the following remarkable theorem: Theorem 6.1, A rigid charge distribution ef(r) undergoing periodic motion will not produce any radiation if the condition f(knf) = 0
(6.16)
is satisfied fi)r all positive integers n and for all directions of observation r, independent of the precise path of the distribution.
310
Nonmdiating sources and other "invisible" objects
[5, § 6
This result was first noted by Schott [1933], who demonstrated it for the particular example of a charged spherical shell of radius a = mcT/1, where T is the period of oscillation and m is any positive integer. Later, Bohm and Weinstein [1948] determined other radiationless, spherically symmetric charge distributions, and Goedecke [1964] demonstrated that radiationless motions exist even for some asymmetric, spinning charge distributions. Several comments about these radiationless modes are in order. First, it is to be noted that, in general, the field will not be identically zero far awayfi*omthe moving charge; there will exist static electric and magnetic fields throughout space [produced by the « = 0 terms of eqs. (6.4)]. Second, it is to be noted that, for the case of the spherical shell, the diameter of the shell is always equal to or greater than the distance that light may travel in one period T of oscillation. This suggests that the radius of the "orbit" of the charge distribution is always less than the spatial extent of the distribution - the motion of the distribution may be described more as a "wobble" than an "orbit". It has been speculated, though not rigorously proven, that this is a general feature of such radiationless motions (Goedecke [1964]). If we focus our attention on a single frequency a;„ of the radiated field, the moving charge will not radiate at that particular frequency if f(knf') = 0
for all r
(6.17)
This condition is formally identical to the nonradiating condition for a monochromatic distribution, eq. (2.9). If it is satisfied, the spectrum of the radiated field will not contain the fi-equency (On, even though the source motion contains that frequency. The existence of such spectral changes implies that it is not possible in certain circumstances to determine the motion of a charge distribution by measurements of the radiation it produces. For instance, if/(a;ir) = 0 for all r, the lowest frequency measured in the field will be a^ = 2a)i. This would lead one to suspect that the distribution is moving with period T/2, instead of the true period T. As described in section 3, we see that the existence of nonradiating objects implies the nonuniqueness of an inverse problem. The existence of radiationless motions has also played a significant role in the study of radiation reaction. Numerous authors (Bohm and Weinstein [1948], Erber [1961], Erber and Prastein [1970]) have pointed out that certain rigid charge distributions can oscillate not only without radiation, but without an external force acting upon them - the motion is maintained by the action of the particle's electromagnetic field upon itself. The condition for nonradiation described in Theorem 6.1 plays an important role in such oscillations.
5]
Acknowledgements
311
One might wonder if the aforementioned results hold when the particle is moving at relativistic speeds. Very little work has been done in generalizing radiationless motions to the relativistic domain, at least in part due to the inconsistency of the concept of rigid charge distributions with relativity theory. However, Prigogine and Henin [1962] have provided one generalization of extended electron theory in which self-oscillating modes are possible. Pearle [1977] has shown that Schott's rigid, uniformly charged spherical shell moving in a relativistically invariant manner does not have any radiationless motions ^^.
§ 7. Conclusions In this article we have reviewed the research on extended oscillating charge distributions which do not radiate. Such nonradiating sources possess many beautiful and nonintuitive mathematical properties and continue to be a field of active research. Beyond being objects of physical interest in and of themselves, we have seen that their existence (or nonexistence) is crucial to the solution of the inverse source problem. One topic absent from this review is experimental work confirming the nonradiating effect; indeed, apparently none has been done as of yet. The introduction of the idea of nonpropagating excitations discussed in § 5 will hopefiilly lead to progress along these lines. It is hoped that this article will bring some attention to an intriguing topic of classical optics and stimulate new investigations into the "physics of invisibility".
Acknowledgements The author would like to extend his thanks to Professor Emil Wolf for usefiil discussions. Many thanks also to Professor Taco D. Visser who translated an early German reference on nonradiating sources and who also allowed the author to spend part of his work days finishing this manuscript.
^^ Other motions of extended charge distributions may also be nonradiating. Meyer-Vemet [1989] suggested that it is possible to create Cerenkov sources which do not radiate.
312
Nonradiating sources and other "invisible" objects
[5
References Abbott, T.A., and DJ. Griffiths, 1985, Acceleration without radiation. Am. J. Phys. 53, 1203-1211. Alexopoulos, N.G., and N.K. Uzunoglu, 1978, Electromagnetic scattering from active objects: invisible scatterers, Appl. Opt. 17, 235-239. Anderson, P.W., 1958, Absence of dififlision in certain random lattices, Phys. Rev 109, 1492-1505. Arfken, G., 1985, Mathematical Methods for Physicists, 3rd Ed. (Academic Press, San Diego, CA). Amett, J.B., and G.H. Goedecke, 1968, Electromagnetic fields of accelerated nonradiating charge distributions, Phys. Rev 168, 1424-1428. Baltes, H.P, ed., 1978, Inverse Source Problems in Optics (Springer, Berlin). Baltes, H.P, and B.J. Hoenders, 1978, A nonradiating linear combination of quasihomogeneous current correlations, Phys. Lett. A 69, 249-250. Beran, M.J., and G.B. Parrent, 1964, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, NJ). Berry, M., J.T. Foley, G. Gbur and E. Wolf, 1998, Nonpropagating string excitations. Am. J. Phys. 66, 121-123. Bertero, M., and P. Boccacci, 1998, Introduction to Inverse Problems in Imaging (Institute of Physics Publishing, Bristol). Bleistein, N., and J.K. Cohen, 1977, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys. 18, 194-201. Bohm, D., and M. Weinstein, 1948, The self-oscillations of a charged particle, Phys. Rev. 74, 1789-1798. Bojarski, N.N., 1982, A survey of the near-field far-field inverse scattering inverse source integral equation, IEEE Trans. Antennas Propag. AP-30, 975-979. Bom, M., and E. Wolf, 1999, Principles of Optics, 7th Ed. (Cambridge University Press, Cambridge). Carter, W.H., and E. Wolf, 1981, Correlation theory of wavefields generated by fluctuating, threedimensional, primary, scalar sources - I. General theory. Optica Acta 28, 227-244. Colton, D., and R. Kress, 1983, Integral Equation Methods in Scattering Theory (Wiley, New York). Courjon, D., C. Banier, C. Girard and J.M. Vigoureux, 1993, Near field optics and light confinement, Ann. Physikl, 149-158. Dalichaouch, R., J.P Armstt-ong, S. Schultz, PM. Platzman and S.L. McCall, 1991, Microwave localization by two-dimensional random scattering. Nature 354, 53-55. Denardo, B., 1998, A simple explanation of simple nonradiating sources in one dimension - Comment on 'Nonpropagating string excitations', by M. Berry, J.T. Foley, G. Gbur and E. Wolf [Am. J. Phys. 66(2), 121-123 (1998)], Am. J. Phys. 66, 1020-1021. Devaney, A.J., 1978, Nonuniqueness in the inverse scattering problem, J. Math. Phys. 19,1526-1531. Devaney, A.J., 1979, The inverse problem for random sources, J. Math. Phys. 20, 1687-1691. Devaney, A.J., 1980, A new approach to emission and transmission CT, in: Proc. 1980 Ultrasonics Symposium, ed. B.R. McAvoy (IEEE Press, New Jersey). Devaney, A.J., and E.A. Marengo, 1998, A method for specifying non-radiating, monochromatic, scalar sources and their fields. Pure Appl. Opt. 7, 1213-1220. Devaney, A.J., and R.P. Porter, 1985, Holography and the inverse source problem, II: Inhomogeneous media, J. Opt. Soc. Am. A 2, 2006-2011. Devaney, A.J., and G.C. Sherman, 1982, Nonuniqueness in inverse source and scattering problems, IEEE Trans. Antennas Propag. AP-30, 1034-1037. Devaney, A.J., and E. Wolf, 1973, Radiating and nonradiating classical current distributions and the fields they generate, Phys. Rev D 8, 1044-1047. Devaney, A.J., and E. Wolf, 1974, Multipole expansions and plane wave representations of the electromagnetic field, J. Math. Phys. 15, 234-244.
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Kerker, M., 1975, Invisible bodies, J. Opt. Soc. Am. 65, 376-379. Kerker, M., 1978, Electromagnetic scattering from active objects, Appl. Opt. 17, 3337-3339. Kim, K., and E. Wolf, 1986, Non-radiating monochromatic sources and their fields. Opt. Commun. 59, 1-6. LaHaie, I.J., 1985, Inverse source problem for three-dimensional partially coherent sources and fields, J. Opt. Soc. Am. A 2, 35-45. LaHaie, I.J., 1986, Uniqueness of the inverse source problem for quasi-homogeneous, partially coherent sources, J. Opt. Soc. Am. A 3, 1073-1079. Mandel, L., and E. Wolf, 1995, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge). Marengo, E.A., A.J. Devaney and R.W. Ziolkowski, 2000, Inverse source problem and minimumenergy sources, J. Opt. Soc. Am. A 17, 34-45. Marengo, E.A., and R.W Ziolkowski, 1999, On the radiating and nonradiating components of scalar, electromagnetic, and weak gravitational sources, Phys. Rev. Lett. 83, 3345-3349. Meyer-Vemet, N., 1989, Nonradiating sources: the subtle art of changing light into black. Am. J. Phys. 57, 1084-1089. Morse, P.M., and K. Uno Ingard, 1986, Theoretical Acoustics (Princeton University Press, Princeton, NJ). Miiller, C , 1955, Radiation patterns and radiation fields, J. Rat. Mech. Anal. 4, 235-246. Nachman, A.I., 1988, Reconstructions from boundary measurements. Annals Math. 128, 531-576. Papas, C.H., 1988, Theory of Electromagnetic Wave Propagation (Dover Publications, New York). Pearle, P., 1977, Absence of radiationless motions of relativistically rigid classical electron. Found. Phys. 7, 931-945. Pearle, P., 1982, Classical electron models, in: Electromagnetism: Paths to Research, ed. D. Teplitz (Plenum Press, New York) pp. 211-295. Porter, R.P, and A.J. Devaney, 1982, Holography and the inverse source problem, J. Opt. Soc. Am. 72, 327-330. Prigogine, I., and F. Henin, 1962, Motion of a relativistic charged particle, Physica 28, 667-688. Rusek, M., A. Orlowski and J. Mostowski, 1996, Localization of light in three-dimensional random dielectric media, Phys. Rev E 53, 4122-4130. Schott, G.A., 1933, The electromagnetic field of a moving uniformly and rigidly electrified sphere and its radiationless orbits, Phil. Mag. 15, 752-761. Schott, G.A., 1937, The uniform circular motion with invariable normal spin of a rigidly and uniformly electrified sphere, IV, Proc. R. Soc. A 159, 570-591. Sommerfeld, A., 1904a, Zur Elektronentheorie. I. Allgemeine Untersuchung des Feldes eines beliebig bewegten Elektrons, Nachr. Akad. Wiss. Gottingen, Math.-Phys. Klasse, pp. 99-130. Sommerfeld, A., 1904b, Zur Elektronentheorie. II. Grundlagen fur eine allgemeine Dynamik des Elektrons, Nachr. Akad. Wiss. Gottingen, Math.-Phys. Klasse, pp. 363-439. Wagner, P C , and H. Wise, eds, 1994, Great Tales of Terror and the Supernatural (Random House, New York). Wiersma, D, P. Bartolini, A. Lagendijk and R. Righini, 1997, Localization of light in a disordered medium. Nature 390, 671-673. Wolf, E., 1969, Three-dimensional structure determination of semi-transparent objects from holographic data. Opt. Commun. 1, 153-156. Wolf, E., 1970, Determination of the amplitude and phase of scattered fields by holography, J. Opt. Soc. Am. 60, 18-20. Wolf, E., 1996, Principles and development of diffraction tomography, in: Trends in Optics, ed. A. Consortini (Academic Press, San Diego, CA) pp. 83-110.
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Wolf, E., and T. Habashy, 1993, Invisible bodies and uniqueness of the inverse scattering problem, J. Mod. Opt. 40, 785-792. Wolf, E., and M. Nieto-Vesperinas, 1985, Analyticity of the angular spectrum amplitude of scattered fields and some of its consequences, J. Opt. Soc. Am. A 2, 886-890.
E. Wolf, Progress in Optics 45 © 2003 Elsevier Science B. V All rights reserved
Chapter 6
Lasing in disordered media by
Hui Cao Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
317
Contents
Page § 1. Introduction
319
§ 2. Random laser with incoherent feedback
320
§ 3. Random laser with coherent feedback - Experiment
332
§ 4.
Random laser with coherent feedback - Theory
358
§5.
Interplay of localization and amplification
362
§ 6.
Applications of random lasers
364
Acknowledgment
366
References
367
318
§ 1. Introduction Both localization and laser theory were developed in the 1960s. Light scattering was traditionally considered detrimental to laser action because such scattering removes photons from the lasing mode of a conventional laser cavity. However, in a strongly scattering gain medium, light scattering plays a positive role: (i) multiple scattering increases the path length or dwell time of light in the active medium, thus enhances light amplification by stimulated emission; (ii) recurrent light scattering provides coherent feedback for lasing oscillation. Lasing in disordered media has been a subject of intense theoretical and experimental studies. It represents the process of light amplification by stimulated emission with feedback supplied by disorder-induced scattering. There are two kinds of feedback: one is intensity or energy feedback, the other is field or amplitude feedback. The former feedback is incoherent and non-resonant, the latter is coherent and resonant. Based on the feedback mechanisms, random lasers are classified into two categories: (1) random lasers with incoherent (or nonresonant) feedback, also called incoherent random lasers, (2) random lasers with coherent (or resonant) feedback, also called coherent random lasers. This article summarizes the research works on both types of random lasers, with an emphasis on our recent experimental study of random laser with coherent feedback. In an amplifying random medium, lightwaves are multiply scattered and amplified. The relevant length scales that describe the scattering process are the scattering mean free path 4 and the transport mean free path /t. The scattering mean free path 4 is defined as the average distance that light travels between two consecutive scattering events. The transport meanfi*eepath k is defined as the average distance a wave travels before its direction of propagation is randomized. These two length scales are related:
(cos 6) is the average cosine of the scattering angle, which can be found fi:om the differential scattering cross section. Rayleigh scattering is an example of (cos 0) = 0 or /t = 4, while Mie scattering may have (cos 6) ^ 0.5, or 4 ?^ 24. 319
320
Lasing in disordered media
[6, § 2
Light amplification by stimulated emission is described by the gain length /g and the amplification length /amp- The gain length is defined as the path length over which the intensity is amplified by a factor e. The amplification length is defined as the (rms) average distance between the beginning and ending point for paths of length /g: /amp=\/^. 3
(1.2)
In the limit without scattering, /amp = /g- The amplification length /amp and the gain length /g are the analogues of the absorption length /abs and the inelastic length k that describe absorption. A random medium is characterized by its dimensionality d and size L. There are three regimes for light transport in a three-dimensional (3D) random medium: (i) ballistic regime, 1 ^ A; (ii) diffusive regime, Z > /t > A; (iii) localization regime, k^ - k = 1, with k^ the effective wave vector in the random medium) (John [1991]). § 2. Random laser with incoherent feedback 2.1. Laser with scattering reflector The two essential components of a laser are the gain medium and the cavity. The gain medium amplifies light through stimulated emission, and the cavity provides positive feedback. A simple laser cavity is a Fabry-Perot cavity made of two mirrors in parallel. Light returns to its original position afi:er traveling one round trip between the mirrors. The requirement of constructive inference determines the resonant firequencies, namely kU + 01 + 02 = 2nm ,
(2.1)
where k is the wave vector, L^ is the cavity length, 0i and 02 represent the phases of the reflection coefficients of the two mirrors, and m is an integer. Only light at the resonant fi-equencies experiences minimum loss and spends a long time in the cavity. The long dwell time in the cavity facilitates light amplification. When the optical gain balances the loss of a resonant mode, lasing oscillation occurs in this mode. The threshold condition is Rx R2 e^^^^ = 1 ,
(2.2)
where Ri and R2 represent the reflectivities of the two mirrors, and g is the gain coefficient.
6, § 2]
Random laser with incoherent feedback
321
In 1966, Ambartsumyan and colleagues realized a different type of laser cavity that provides non-resonant feedback (Ambartsumyan, Basov, Kryukov and Letokhov [1966]). They replaced one mirror of the Fabry-Perot cavity with a scattering surface. Light in the cavity suffers multiple scattering: its direction is changed every time it is scattered. Thus light does not return to its original position after one round trip. The spatial resonances for the electromagnetic field are absent in such a cavity. The dwell time of light is not sensitive to its frequency. The feedback in such a laser is used merely to return part of the energy or photons to the gain medium, i.e., it is energy or intensity feedback. The non-resonant feedback can also be interpreted in terms of "modes". When one end mirror of a Fabry-Perot cavity is replaced by a scattering surface, escape of emission from the cavity by scattering becomes the predominant loss mechanism for all modes. Instead of individual high-2 resonances there appear a large number of low-g resonances which spectrally overlap and form a continuous spectrum. It corresponds to the occurrence of non-resonant feedback. The absence of resonant feedback means that the cavity spectrum tends to be continuous, i.e., it does not contain discrete components at selected resonant frequencies. The only resonant element left in this kind of laser is the amplification line of the active medium. With increasing pumping intensity, the emission spectrum narrows continuously towards the center of the amplification line. However, the process of spectral narrowing is much slower than in ordinary lasers (Ambartsumyan, Kryukov and Letokhov [1967]). Since many modes in a laser cavity with non-resonant feedback interact with the active medium as a whole, the statistical properties of laser emission are quite different from those of an ordinary laser. As shown by Ambartsumyan and colleagues, the statistical properties of the emission of a laser with non-resonant feedback are very close to those of the emission from an extremely bright "black body" in a narrow range of the spectrum (Ambartsumyan, Basov and Letokhov [1968]). The emission of such a laser has no spatial coherence and is not stable in phase. Because the only resonant element in a laser with non-resonant feedback is the amplification line of the gain medium, the mean frequency of laser emission does not depend on the dimensions of the laser but is determined only by the center frequency of the amplification line. If this frequency is sufficiently stable, the emission of this kind of laser has a stable mean frequency. Ambartsumyan, Basov, Kryukov and Letokhov [1967] proposed using the method of non-resonant feedback to produce an optical standard for length and frequency. To realize this, they built continuous gas lasers with non-resonant feedback based on the scattering surface (Ambartsumyan, Bazhulin, Basov and Letokhov [1970]).
322
Lasing in disordered media
[6, § 2
2.2. Photonic bomb In 1968, Letokhov took one step further and proposed self-generation of light in an active medium filled with scatterers (Letokhov [1968]). When the photon mean fi-ee path is much shorter than the dimension of the scattering medium, the motion of photons is diffusive. Letokhov solved the diffusion equation for the photon energy density W{r, t) in the presence of a uniform and linear gain: ^ffi^=DV2fr(r,0+ffF(F,0,
(2.3)
where v is the transport velocity of light inside the scattering medium, /g is the gain length, and D is the diffusion constant given by D=Y-
(2.4)
The general solution to eq. (2.3) can be written as fr(F,0 = ^ a „ f ' „ ( r ) e x p
(2.5)
*^„(F) and Bn are the eigenfiinctions and eigenvalues of the following equation: V'W„(r) + BlW„{r) = 0
(2.6)
with the boundary condition that 'f'^ = 0 at a distance Ze firom the boundary. ZQ is the extrapolation length; usually it is much smaller than the dimension of the scattering medium and can be neglected. Hence, the boundary condition becomes that W„ = 0 3i the boundary of the random medium. The expression for W(r,t) in eq. (2.5) changes fi-om exponential decay to exponential increase in time upon crossing the threshold DB]-^=0,
(2.7)
where B\ is the lowest eigenvalue. If the scattering medium has the shape of a sphere of diameter L one has Bn = 2jtn/L, and the smallest eigenvalue is Bi = 2JV/L. If the scattering medium is a cube with sides of length L, the smallest eigenvalue is ^i = \^Jt/L. Regardless of the shape of the scattering medium.
6, § 2]
Random laser with incoherent feedback
323
the lowest eigenvalue B\ is on the order of \/L. When B\ ^ \/L is substituted into eq. (2.7), the threshold condition predicts a critical volume / / / ^^^^
F„«L^«(^^J
.
(2.8)
With fixed gain length /g and transport mean fi-ee path /t, once the volume V of the scattering medium exceeds the critical volume Fcr, W{r,t) increases exponentially with t. This can be understood intuitively in terms of two characteristic length scales. One is the generation length Zgen, which represents the average distance a photon travels before generating a second photon by stimulated emission. Zgen can be approximated by the gain length /g. The other is the mean path length Zpat that a photon travels in the scattering medium before escaping through its boundary. Zpat ~ vL^/D. When V > Fcr, ^pat ^ ^gen- This means that on average every photon generates another photon before escaping from the medium. This triggers a "chain reaction", i.e. one photon generates two photons, these two photons generate four photons, etc. Thus the photon number increases with time. This is the onset of photon self-generation. Because this process of photon generation is analogous to the multiplication of neutrons in an atomic bomb, this device is sometimes called a photonic bomb. In reality the light intensity will not diverge (there is no "explosion") because gain depletion quickly sets in and /g increases. Taking into account gain saturation, Letokhov calculated the emission linewidth and the generation dynamics. If the scattering centers are stationary, the limiting width of the generation spectrum is determined by the spontaneous emission. Otherwise, the Brownian motion of the scattering particles leads to a random variation (wandering) of the photon frequency as a result of the Doppler effect on the scattering particles. He also predicted damped oscillation (pulsation) in the transient process of generation. Letokhov's predictions were experimentally confirmed twenty-six years later (Genack and Drake [1994]). 2.3. Powder laser In 1986 Markushev and colleagues reported intense stimulated radiation from Na5Lai_;cNdx(Mo04)4 powder under resonant pumping at low temperature (77 K) (Markushev, Zolin and Briskina [1986]). When the pumping intensity exceeded a threshold, the Nd^+ emission spectrum was narrowed to a single line, and the duration of the emission pulse was shortened by approximately four orders of magnitude. Later on, they reported similar phenomena in a wide range of
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[6, § 2
Nd^+-activated scattering materials, including LaaOs, La202S, Na5La(Mo04)4, LasNbOy, and SrLa2W07 (Markushev, Ter-Gabrielyan, Briskina, Belan and Zolin [1990]). The powder was pumped by a 20-ns g-switched laser pulse. When the pump energy reached a threshold (in the range 0.05-0.1 J/cm^), a single emission pulse of 1-3 ns was observed. With a further increase in pump energy, the number of emission pulses increased to three or four. At a constant pumping intensity, the number of pulses, their duration, and the interval between them were governed by the properties of the materials. The emission spectrum above the threshold was related to the nature of the particles. In a powder of polycrystalline particles there is only one narrow emission line at the center of the luminescence band, while in a powder of single crystals the emission spectrum consists of several lines in the range of the luminescence band (Ter-Gabrielyan, Markushev, Belan, Briskina, Dimitrova, Zolin and Lavrov [1991]). In all cases, the spectral width of the emission lines above the threshold is on the order of 0.1 nm. The observed emission is very much like laser emission. Because the particle size (~10(im) was much larger than the emission wavelength, Markushev, Ter-Gabrielyan, Briskina, Belan and Zolin [1990] speculated that individual particles served as effective resonators, and that lasing occurred in the modes formed by total internal reflection at the surface of a particle. However, there might be some weak coupling between the neighboring particles. In a mixture of two powders with slightly shifted luminescence bands [e.g. Na5Lai_2Ndz(Mo04)4 powder with significantly different Nd^^ concentrations], the emission wavelength depended on the relative concentrations of the components in the mixture and the excitation wavelength (Ter-Gabrielyan, Markushev, Belan, Briskina and Zolin [1991]). Briskina and colleagues set up a model of coupled microcavities to interpret the experimental result (Briskina, Markushev and Ter-Gabrielyan [1996], Briskina and Li [2002]). They treated the powder as an aggregate of active optically coupled microcavities and calculated the modes formed by total internal reflection (in analog to the whisperinggallery modes). They found that, due to the optical coupling, the quality factor of a coupled-particle cavity in the compact powder could be higher than that of a single-particle cavity. To confirm their model, they measured the spot of laser-like radiation fi-om a powder of Al3Nd(B03)4 and NdPsOn (Lichmanov, Briskina, Markushev, Lichmanova and Soshchin [1998]). The minimum spot size was 20-30 |im. The particle size was between 4 and 20 |uim, thus the laser-like radiation was from a single particle or a few particles. Later, a powder laser was realized with non-resonant pumping at room temperature (Gouedard, Husson, Sauteret, Auzel and Migus [1993], Noginov, Noginova, Egarievwe, Caulfield, Venkateswarlu, Thompson, Mahdi and Os-
6, § 2]
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troumov [1996]). The gain materials were extended from Nd^+-doped powder to Ti:sapphire powder (Noginov, Noginova, Egarievwe, Caulfield, Cochrane, Wang, Kokta and Paitz [1998]), Pr^^-doped powder (Zolin [2000]), and pulverized LiF with color centers (Noginov, Noginova, Egarievwe, Caulfield, Venkateswarlu, Williams and Mirov [1997]). Although the material systems are different, the observed phenomena are similar: (i) drastic shortening of the emission pulse and spectral narrowing of the emission line above a pumping threshold; (ii) damped oscillation of the emission intensity under pulsed excitation; (iii) drifting of the stimulated emission frequency and hopping of emission line from one discrete frequency to another within the same series of pulses. Gouedard, Husson, Sauteret, Auzel and Migus [1993] analyzed the spatial and temporal coherence of the powder laser. From the contrast of the near-field speckle pattern, they concluded that the powder emission above the threshold is spatially incoherent. This result was explained by incoherent superposition of uncorrected speckle patterns. Their time-resolved measurement showed the speckle pattern changed rapidly in time. The estimated coherence time was ~10ps, indicating low temporal coherence of the powder emission. Noginov, Egarievwe, Noginova, Caulfield and Wang [1999] performed quantitative measurements of the longitudinal and transversal coherence using interferometric techniques. Using a Michelson (Twyman-Green) interferometer, they found a longitudinal coherence time for Ndo.5Lao.5Al3(B03)4 powder (ceramics) emission of 56 ps at a pumping energy twice that of the threshold. This value corresponds to a linewidth of 0.7 A, in agreement with the result of direct spectroscopic linewidth measurement. They also examined the transversal spatial coherence using Young's double-slit interferometric scheme. The transversal coherence was not noticeable when the distance between two points on the emitting surface was approximately 85 |bim. In spite of the detailed experimental study of powder laser, the underlying mechanism was not fiilly understood. Gouedard, Husson, Sauteret, Auzel and Migus [1993] speculated that the grains of the powder emit collectively in a subnanosecond pulse with a kind of distributed feedback provided by multiple scattering. Auzel and Goldner identified two processes of coherent light generation in powder: (i) amplification of spontaneous emission by stimulated emission, (ii) synchronized spontaneous emission, namely superradiance and superfluorescence (Auzel and Goldner [2000], Zyuzin [1998], Zyuzin [1999]). Noginov, Noginova, Egarievwe, Caulfield, Venkateswarlu, Thompson, Mahdi and Ostroumov [1996] noticed the role played by photon diffusion in stimulated emission when comparing the powder laser with the single crystal laser. The diffusive motion of light led to a long path length of emission in the powder
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[6, § 2
and helped to reduce the threshold. Wiersma and Lagendijk [1997a] proposed a model based on light diffusion with gain. They considered a pump pulse and probe pulse incident on a powder slab. The active material was approximated as a four-level (2,1,0', 0) system with the radiative transition from level 1 to level 0' and the pumping from level 0 to level 2. Fast relaxation from level 2 to level 1 and from level 0' to level 0 made both level 2 and level 0' nearly unpopulated, thus the population of level 1 can be described by one rate equation. The whole system was described by three diffusion equations for the energy densities of the (green) pump light Woif, t), the (red) probe light ^R(r, t) and the (amplified) spontaneous emission WA(r, t), and one rate equation for the population density 7Vi(r, Oof level 1: ^
dt
^ = DV^ WG{r, t) - o,^,v[Nt - Nr(r, t)] Wdn
dt dt dNi{r,t) dt
t) + - / G ( F , / ) , IQ
/R
= DV^FFA(F, 0 + ae„(;M(F, t)fVA(F, t) + -Nx{r, t) , o^'6so[N, - N\ (r, t)] Wair, t) - o.^vNi (r, tW^if, t) +
W/,{r,t)]--N,{r,t),
(2.9) where Oabs and o^^ are the absorption and emission cross sections, XQ is the lifetime of level 1, lQ{r,t) and lR(r,t) are the intensities of the incoming pump and probe pulses, IQ and /R are the transport mean free paths at the pump and probe frequencies, and Nt is the total concentration of four-level atoms. Wiersma and Lagendijk solved the above coupled nonlinear differential equations numerically. Their simulation result reproduced the experimental observation of transient oscillation (spiking) of the emission intensity under pulsed excitation. In the slab geometry, the critical volume predicted by Letokhov is reduced to the critical thickness Lcr = ii^fTJ^ = Jtl^mp- For the fixed slab thickness L, there exists a critical amplification length /cr = L/JT. In the beginning of the pump pulse, the average amplification length /amp decreases due to increasing excitation level. Once /amp crosses /cr, the gain in the sample becomes larger than the loss through the boundaries, and the system becomes unstable. This leads to a large increase of the amplified spontaneous emission (ASE) energy density. The characteristic time scale corresponding to the build-up of ASE is Ig/u. The large ASE energy density will de-excite the system again, which leads to an increase
6, § 2]
Random laser with incoherent feedback
327
of/amp. This de-excitation continues as long as the large ASE energy density is present. The characteristic time scale on which the ASE energy density diffuses out of the medium through the front or rear surface is given by L^/D. On one hand, an overshoot of the excitation takes place because the de-excitation mechanism needs some time to set in. On the other hand, once the ASE has built up considerably, the ASE energy density can disappear only slowly due to the presence of multiple scattering, which leads to an undershoot below the threshold. These two processes result in transient oscillations of the outgoing ASE flux. The oscillations are damped because the increase of /amp during the de-excitation is opposed by re-excitation owing to the presence of pump light. Therefore the system reaches the equilibrium situation /amp = kr = L/jt after a few oscillations. The experimental phenomena are reproduced both by models based on light diffusion and by models based on intra-particle resonances. It is hard to tell whether the feedback in powder laser is provided by multiple scattering or total internal reflection, because the gain medium and scattering elements are not separated in the powder. Lawandy, Balachandran, Gomes and Sauvain [1994] separated the scattering and amplifying media in liquid solutions. This separation allowed the scattering strength to be varied independently of the gain coefficient, and facilitated a systematic study of the scattering effect on feedback.
2.4. Laser paint In 1994, Lawandy, Balachandran, Gomes and Sauvain [1994] observed laserlike emission from a methanol solution of Rhodamine 640 perchlorate dye and Ti02 microparticles. The dye molecules were optically excited by laser pulses and served as the gain medium. The Ti02 particles, with a mean diameter of 250nm, were scattering centers. The (input-output) plot of the peak emission intensity versus the pump energy exhibited a well-defined pumping threshold for the slope change. At the same threshold, the emission linewidth (fiill width at half maximum) collapsed rapidly from 70 nm to 4 nm, and the duration of emission pulses was shortened dramatically fi-om 4 ns to 100 ps. The threshold behavior suggested the existence of feedback. The relatively broad and featureless emission spectrum above the threshold indicated that the feedback was frequency-insensitive or non-resonant. In the solution, a feedback mechanism based on morphology-dependent resonance could be ruled out because the gain was outside the scatterer and individual scatterers were too small to serve as morphology-dependent resonators. It was found experimentally
328
Lasing in disordered media
[6, § 2
that the threshold was reduced by more than 2 orders of magnitude when the density of scattering particles was increased from 5 x 10^ to 2.5 x 10^^ cm~^ at a fixed dye concentration of 2.5 x 10"^ M (Sha, Liu and Alfano [1994]). The strong dependence of the threshold on the transport mean free path revealed that the feedback was related to scattering (Lawandy and Balachandran [1995], Zhang, Cue and Yoo [1995a], Balachandran and Lawandy [1995]). However, multiple scattering or light diffusion is negligible unless the smallest dimension of the scattering medium is much larger than the transport mean free path. Experimentally, when a spatially broad pump pulse was incident on a dye cell, a disk-shaped amplifying region was formed near the front window (Wiersma, van Albada and Lagendijk [1995a]). The thickness of the disk was determined by the penetration depth Zpen of the pump light. In Lawandy's experiment, Zpen was close to the transport mean free path. However, the actual sample thickness (i.e., the thickness of the entire suspension) was much larger than the transport mean free path. Hence, light transport in the suspension was diffusive. Nevertheless, the emitted photons could easily escape from the thin amplifying region. Part of them escaped outwards through the front surface, the rest went deeper into the unpumped region of the suspension. After multiple scattering (or random walk), some of these photons returned to the active volume for more amplification. This return process provided energy feedback. When scattering was stronger, the return probability was higher, thus the feedback was stronger. However, incomplete feedback (less than 100% return probability) gave rise to loss. The lasing threshold is at the point where the photon loss rate is balanced by the photon generation rate in the amplifying region. On one hand, the total amount of gain or amplification is the product of the amplification per unit path length and the path length traveled through the amplifying volume. The frequency dependence of the amplification per unit path length gives the highest photon generation rate at the peak of the gain spectrum. On the other hand, owing to the weak frequency dependence of the transport mean free path, the feedback is nearly frequency independent within the gain spectrum, and so is the loss rate for photons. Therefore, with an increase of the pumping rate, the photon generation rate in the spectral region of maximum gain is the first to reach the photon loss rate, while outside this frequency region the photon generation rate is still below the loss rate. Then the photon density around the frequency of gain maximum builds up quickly. The sudden increase of photon density near the peak of gain spectrum results in an collapse of the emission linewidth. A model based on the ring laser in the random phase limit was proposed by Balachandran and Lawandy [1997] to quantitatively explain the experimental
6, § 2]
Random laser with incoherent feedback
329
data. The amplifying volume was approximated as a sharply bounded disk with homogeneous gain coefficient. In a Monte Carlo simulation of the random walk of photons, they calculated the return probabilities Ra and Rti of photons to the gain volume after being launched inward and outward, and the average total path length Lpat. The threshold gain gthr is determined by the steady-state condition RnRt2^'^^'''
= 1.
(2.10)
This condition is analogous to the threshold condition of a ring laser. Note that a typical ring laser has a second condition on the round-trip phase shift: ^^pen = 2jrm, which determines the lasing frequencies. In the scattering medium the phase condition can be ignored, because the diffusive feedback is nonresonant, i.e., it only requires light to return to the gain volume instead of to its original position. In fact, the probability of emitted light returning to its original position is so low in the diffusive regime that the interference effect on the feedback is negligible. Therefore this kind of laser is a random laser with non-resonant or incoherent feedback. It is also called laser paint or photonic paint (Lawandy [1994], Wiersma and Lagendijk [1997b]). The discovery of Lawandy et al. triggered many experimental and theoretical studies that can be summarized as follows. (1) Lasing threshold. The dependence of lasing threshold on the dye concentration and the gain length was investigated by Zhang, Cheng, Yang, Zhang, Hu and Li [1995]. Usually the threshold was reached at the point at which the pump transition was bleached. Such bleaching increased the penetration depth of the pump and consequently led to a longer path length for the emitted light within the gain region, which resulted in a reduced threshold (Siddique, Alfano, Berger, Kempe and Genack [1996]). The influence of the excitation spot diameter on the threshold was also examined (van Soest, Tomita, Sprik and Lagendijk [1999]). In a suspension of Ti02 scatterers in Sulforhodamine B dye, the threshold pump intensity increased by a factor of 70 when the excitation beam diameter approached the mean free path. This is because the large pump beam spot produced a large amplifying volume: the emitted light can travel a long path inside the active region and experiences more amplification before escaping. After the light went into the passive (unexcited) region, there was a large probability that it would return to the amplifying region because of the large pumped area. For a small excitation beam diameter, the emitted light would very likely leave the active volume after a short time, with a small chance of returning. This gave larger photon loss rate and higher threshold. The amplification by stimulated emission was found to be the strongest when the absorption length
330
Lasing in disordered media
[6, § 2
of the pump light and the transport mean free path had approximately the same magnitude (Beckering, Zilker and Haarer [1997]). A critical transport mean free path was identified for each beam diameter, below which the threshold was almost independent of the mean free path (Totsuka, van Soest, Ito, Lagendijk and Tomita [2000]). These results can be explained in terms of the spatial overlap of the gain volume and the diffusion volume. By solving the coupled rate and diffusion equations, Totsuka et al. calculated the spatial distribution of the excited state population (gain volume) and the spatial spreading of the trajectory for the luminescence light (diffusion volume). For gain volumes smaller than the diffusion volume of the luminescence light, the amplification is not efficient as the light then propagates mostly through the gainless region. For gain volumes larger than the diffusion volume of the luminescence light, the excitation pulse energy is not used efficiently for amplification. There exists an optimum condition under which the pulse energy is used most efficiently for stimulated emission. (2) Emission spectra. The stimulated emission spectrum was shifted with respect to the luminescence spectrum. This spectral shift was explained by a simple ASE model accounting for absorption and emission at the transition between the ground and first singlet excited states of the dye (Noginov, Caulfield, Noginova and Venkateswarlu [1995]). Bichromatic emission was produced in a binary dye mixture in the presence of scatterers (Zhang, Cue and Yoo [1995b]). The dye molecules were of the donor-acceptor type, and the energy transfer between them gave double emission bands. The relative intensity of stimulated emission of the donor and the acceptor depended on the scatterer density in addition to the pumping intensity and the concentration of the dyes. The narrowlinewidth bichromatic emission was also observed in the single dye solution with scatterers at large pumping intensity or high dye concentration (Sha, Liu, Liu and Alfano [1996], Balachandran and Lawandy [1996]). John and Pang [1996] explained the bichromatic emission in terms of singlet and triplet transitions of the dye molecules. Using physically reasonable estimates for the absorption and emission cross sections for the single and triplet manifolds and the singlettriplet intersystem crossing rate, they solved the nonlinear rate equations for the dye molecules. This leads to a diffusion equation for the light intensity in the scattering medium with a nonlinear intensity-dependent gain coefficient. Their model could account for most experimental observations, e.g. the collapse of the emission linewidth at a specific threshold pump intensity, the variation of the threshold intensity with the transport mean free path, and the dependence of peak emission intensity on the transport mean free path, the dye concentration, and the pump intensity. (3) Dynamics. One surprising result about the dynamics of stimulated emission
6, § 2]
Random laser with incoherent feedback
331
from colloidal dye solutions is that the emission pulses can be much shorter than the pump pulses when the pumping rate is well above the threshold. For instance, 50-ps pulses of stimulated emission were obtained from a colloidal solution excited by 3-ns pulses (Sha, Liu and Alfano [1994]). The shortest emission pulses were ~20ps and produced by 10-ns pump pulses (Siddique, Alfano, Berger, Kempe and Genack [1996]). Berger, Kempe and Genack [1997] modeled the dynamics of stimulated emission from random media using a Monte Carlo simulation of the random walk of pump and emitted photons. They tracked the temporal and spectral evolution of emission by following the migration of photons and molecular excitation as determined purely by local probabilities. Their simulation results revealed a sharp transition to ultrafast, narrow-linewidth emission for a 10-ps incident pump pulse and a rapid approach to steady state for longer pump pulses. Using a different approach, van Soest, Poelwijk, Sprik and Lagendijk [2001] also studied the dynamics of stimulated emission. They numerically solved the coupled diffusion equations for the pump light and the emitted light and the rate equation for the excited population. Their simulation result illustrated that the slow response of the population, compared to the light transport, starts a relaxation oscillation at the threshold crossing. (4) ^ factor. Compared with the traditional laser theory, a spontaneous emission coupling factor, ^, was introduced for the random laser (van Soest and Lagendijk [2002]). In a conventional laser, ^ is defined as the ratio of the rate of spontaneous emission into the lasing modes to the total rate of spontaneous emission. Its value is determined by the overlap in wave-vector space between the spontaneous emission and the laser field. In conventional macroscopic lasers, the spontaneous emission is isotropic while the cavity modes occupy small solid angles. The directional mismatch contributes to small jS-value (less than 10~^). In the scattering medium, the diffusive feedback is non-directional, thus the spatial distinction between lasing and nonlasing modes vanishes, and the only criterion is the spectral overlap of the spontaneous emission spectrum with the lasing spectrum. This gives a large jS-value (~0.1). (5) Control and optimization. Liquid solutions are awkward to handle, e.g. the sedimentation of scattering particles in the solvent causes instability. Therefore, liquid solvents were replaced by polymers as host materials by Balachandran and Lawandy [1996]. Polymer sheets containing laser dyes and Ti02 nanoparticle scatterers were made with the cell-casting technique. The lasing in solid dye solutions was similar to that in liquid dye solutions, even though the different embedding environments affect the fluorescence characteristics of the dye (Zacharakis, Heliotis, Filippidis, Anglos and Papazoglou [1999]). Many techniques developed for traditional lasers were exploited to optimize and
332
Lasing in disordered media
[6, § 3
control random lasers. For example, external feedback was introduced to control the lasing threshold, de Oliveira, McGreevy and Lawandy [1997] placed a mirror close to the high-gain scattering medium, and measured the spectral line shapes of the emitted light as a function of the distance between them. The main effect of the feedback from the mirror is to increase the lifetime of the photons inside the pump region, resulting in a reduction of the threshold pump energy. The injection-locking technique was also utilized to control the emission wavelength. Introducing a seed into the optically pumped scattering gain medium resulted in an intense isotropic emission with wavelength and linewidth locked to the seeding beam properties (Balachandran, Perkins and Lawandy [1996]). Moreover, multiple narrow-linewidth emission was obtained by pumping one laser paint with output from another laser paint (Martorell, Balachandran and Lawandy [1996]). Recently, temperature tuning was employed to switch random lasers on and off. Liquid crystal was infiltrated into macroporous glass, and the diffusive feedback was controlled through the change of the refractive index of the liquid crystal with temperature (Wiersma and Cavalieri [2001]). In a different approach, a mixture with a lower critical solution temperature, which can be reversibly transformed between a transparent state and a highly scattering colloid by a small temperature change, was used to tune the lasing threshold with temperature (Lee and Lawandy [2002]).
§ 3. Random laser with coherent feedback - Experiment In 1998, we demonstrated a different kind of lasing process in disordered semiconductor powder and polycrystalline films (Cao, Zhao, Ong, Ho, Dai, Wu and Chang [1998], Cao, Zhao, Ho, Seelig, Wang and Chang [1999]). The feedback is supplied by recurrent light current. It is coherent and resonant in contrast to the diffusive feedback. Frolov and colleagues observed similar lasing phenomena in luminescent jr-conjugated polymer films, organic dye-doped gel films, opal crystals saturated with polymer, and laser dye solutions (Frolov, Vardeny, Yoshino, Zakhidov and Baughman [1999], Frolov, Vardeny, Zakhidov and Baughman [1999], Yoshino, Tatsuhara, Kawagishi and Ozaki [1999]). This kind of laser is called a random laser with coherent (or resonant) feedback. 3.1. Lasing oscillation in ZnO powder Random lasing with coherent feedback has been realized in many kinds of random media. Although the materials are different, the phenomena are similar.
6, § 3]
Random laser with coherent feedback - Experiment
333
In this section, the behavior of coherent random laser is illustrated in one of the random media used in our experiment: ZnO powder. The ZnO nanoparticles were fabricated either by physical vapor synthesis or wet chemical reaction. The particles were polydisperse with average particle size ~100nm. The ZnO particles were either deposited onto ITO-coated substrate by electrophoresis or simply cold-pressed to form a pellet. The sample thickness varied from 10|im to 1 mm. The filling factor was around 50%. The transport mean free path k was characterized in a coherent backscattering (CBS) experiment (Kuga and Ishimaru [1984], van Albada and Lagendijk [1985], Wolf and Maret [1985]). ZnO has a direct bandgap of 3.3 eV Strong absorption makes it very difficult to measure k at photon energies equal to or above the bandgap. To avoid absorption, the probe photon energy was set at 3.0eV The probe beam was the second harmonics (A = 410 nm) of a mode-locked Tiisapphire laser (76 MHz repetition rate, 200 fs pulse width). The angular width (frill width at half maximum, FWHM) A0 of the backscattering cone is determined by the transport mean free path l{.
Ae^''"f-^\
(3.1)
KQlf
where HQ is the effective refractive index of the ZnO powder sample, kQ is the effective wave vector of the probe light in the sample, and R is the difftisive reflectivity of the sample-air interface. From A6, we estimated /t ~ A. Such short transport mean free path indicates very strong optical scattering in the ZnO powder. Note that the probe frequency was lower than the ZnO emission frequency, k for the emitted light should be shorter than the measured value, because the refractive index of ZnO increases as the photon energy approaches the bandgap. The ZnO samples were optically excited by the third harmonics (A = 355 nm) or the fourth harmonics (A = 266 nm) of a pulsed Nd:YAG laser (10 Hz repetition rate, 20 ps pulse width). The pump beam was focused to a spot or a stripe on the sample surface. Electrons in the ZnO valence band absorbed pump photons and jumped to the conduction band. They subsequently relaxed to the bottom of the conduction band, followed by radiative decay. The spectrum of ZnO emission was measured by a 0.5-meter spectrometer with a liquid-nitrogen-cooled CCD array detector. The spectral resolution was about 1.3 A. Simultaneously, the spatial distribution of the emitted light intensity at the sample surface was imaged by a ultraviolet (UV) microscope onto a UV-sensitive CCD camera. The amplification of the microscope was about 100 times. The spatial resolution
334
Lasing in disordered media
[6, § 3
was ^0.3 |im. A bandpass filter was placed in fi-ont of the microscope objective lens to block the pump light. Figure 1 shows the evolution of the emission spectrum with pump intensity. At low excitation intensity, the spectrum consisted of a single broad spontaneous emission peak. As the pump power increased, the emission peak became narrower owing to preferential amplification atfi-equenciesclose to the maximum of the gain spectrum. When the excitation intensity exceeded a threshold, discrete narrow peaks emerged in the emission spectra. The linewidth of these peaks was less than 2 A, which was more than 30 times smaller than the linewidth of the ASE peak below the threshold. When the pump intensity increased further, more sharp peaks appeared. The fi-equencies of the sharp peaks depended on the sample position. As we moved the excitation spot across the sample, the frequencies of the sharp peaks changed. This phenomenon suggests that the discrete spectral peaks resultfi-omspatial resonances for light in the ZnO powder, and such resonances are related to the local configuration of ZnO particles. Since individual ZnO nanoparticles are too small to serve as morphology-dependent resonators, the possibility of intra-particle resonances formed by total internal reflection at the particle surface can be ruled out. Thus the origin of the spatial resonances lies in the inter-particle scattering. Due to very strong scattering, recurrent light scattering events arise, i.e., after multiple scattering light returns to a scatterer fi-om which it is scattered before. The interference of the return light is constructive only at certain frequencies. Therefore the requirement for constructive interference of backscattered light selects the resonant fi'equencies. Figure 2 is a plot of the integrated emission intensity versus the excitation intensity. A threshold behavior was observed: above the pump intensity at which discrete spectral peaks emerged, the emission intensity increased much more rapidly with excitation intensity. Figure 3 shows the emission patterns below and above threshold. Below threshold, the spatial distribution of the spontaneous emission intensity was smooth across the excitation area. Due to the variation of pump intensity across the excitation spot, the spontaneous emission in the center of the excitation spot was stronger. Above threshold, as soon as the discrete spectral peaks emerged, spatially separated regions of intense radiation appeared in the image of the emitted light distribution on the sample surface. Each region consisted of a few bright spots with sizes between 0.3 and 0.6 [xm. This phenomenon can be explained as follows: due to local variation of particle density and spatial distribution, there exist small regions of stronger scattering. Light can be trapped in these regions through multiple scattering and interference. For a particular configuration of ZnO nanoparticles, only light at certain frequencies can be confined, because the interference effect is
6, §3]
335
Random laser with coherent feedback - Experiment
25000 c CO
CO
c
CD
c
"co c 0
c
c CD -I—»
c
375
380
385
390
395
400
Wavelength (nm) Fig. 1. Spectra of emission from ZnO powder when the excitation intensity is (from bottom to top) 400, 562, 763, 875, and 1387kW/cm2.
336
Lasing in disordered media
•e
[6, §3
1200
800
400
E LJJ
300
600
900
1200
1500
Excitation Intensity (kW/cm^)
Fig. 2. Integrated intensity of emission from ZnO powder versus excitation intensity.
(b)
(a)
tm
10 micron
10 micron
Fig. 3. Spatial distribution of emission intensity in ZnO powder. Incident pump pulse energy: (a)5.2nJ;(b) 12.5 nJ.
frequency sensitive. In a different region of the sample, the particle configuration is different, thus light at different frequencies is confined. In other words, many resonant cavities are formed by recurrent scattering and interference. Incomplete trapping of light gives rise to cavity loss. When the optical gain reaches the cavity loss, laser oscillation occurs in the cavity modes, that gives discrete lasing peaks in the emission spectrum (fig. 1). Due to the presence of a large number of defects in the ZnO particles, non-radiative recombination of the excited carriers is significant below the lasing threshold. Above the lasing threshold, the fast stimulated emission process makes the radiative recombination of the excited carriers dominate over the non-radiative recombination. This results in a rapid increase of emission intensity (fig. 2). The temporal evolution of emission was measured by a streak camera (Soukoulis, Jiang, Xu and Cao [2002]) with a temporal resolution of 2ps. Figure 4 shows the time traces of emission (a) below, (b) just above and
6, §3]
Random laser with coherent feedback - Experiment
50
100
337
150 200 250 300 350
Time (ps) Fig. 4. Temporal evolution of the emission intensity from ZnO powder: (a) below the lasing threshold; (b) just above the lasing threshold; (c) well above the lasing threshold. The curves are shifted vertically for clarity, the maximum intensity of each curve is normalized to 1.
(c) well above threshold. Below threshold, the decay time of the emission was 167 ps. When the pump intensity exceeded the threshold, the emission pulse was shortened dramatically. The initial decay of emission intensity was very fast, with decay time 27 ps. After ~50ps, the fast decay was replaced by a slow decay The later decay time was 167 ps, which was equal to the decay time below the threshold. The initial fast decay was caused by rapid stimulated emission, and the later slow decay resulted from spontaneous emission and non-radiative recombination. As the pump intensity was increased ftirther, the initial stimulated emission became much stronger than the later spontaneous emission. The drastic shortening of the emission pulses provides additional evidence for lasing in the ZnO powder. Next we investigated the dynamics of individual lasing modes. The emission from the sample was directed to the entrance slit of a 0.5-m spectrometer. The output port of the spectrometer was connected to the streak camera whose entrance slit was perpendicular to that of the spectrometer. By combining the spectrometer with the streak camera, we were able to separate different lasing modes and obtain the time trace of each lasing mode. Figure 5 is a spectraltemporal image taken by the streak camera. The horizontal axis is the time, and
338
Lasing in disordered media
[6, §3
391 -
"2^,. bb 389 '^ 388 S>
^—
387 -
386 50
100
150
200
Time (ps) Fig. 5. Spectral-temporal image of ZnO emission taken by the steak camera. Incident pump pulse energy is 4.5 nJ.
the vertical axis is the wavelength. We noticed that lasing in different modes was turned on and off at different times. The unsynchronized temporal behaviors of these lasing modes suggest that they originate from different cavities with different quality factors. When the pump intensity was near the lasing threshold, relaxation oscillation was observed for some of the lasing modes. The oscillation period varied from mode to mode. This again indicates that the lasing modes are from different cavities with different photon lifetimes. Finally the quantum statistical property of laser emission from the ZnO powder was probed with the spectrometer-streak camera setup (Cao, Ling, Xu, Cao and Kumar [2001]). The streak camera operated in the photon counting mode. To measure the photon statistics of a single lasing mode, we drew a box around the center of an emission peak on the spectral-temporal image (fig. 5). One side of the box corresponds to the wavelength interval AA = 0.12nm, the other side is the time interval A^ = 3.9 ps. The number of photons inside this box was counted for each pulse. After collecting photon count data for a large number of pulses, the probability P{n) for n photons to be within the wavelength interval AA and the time interval A/ was obtained. Because the sampled radiation field was within a frequency interval Av ^ cAA/A^, its relaxation must occur on a time scale longer than 1/Av. We set the sampling time A^ < \/Av so that it was shorter than the coherence time of the radiation field. From another point of view, because Av • A^ = 0.95 < 1, the counting area corresponded to a single electromagnetic (EM) mode. For single-mode coherent light, the photon number distribution P{n) satisfies the Poisson distribution: Pin)
(3.2)
6, §3]
Random laser with coherent feedback - Experiment
339
M
100
Q_
CL
Fig. 6. Black columns: measured photon count distribution of emission from ZnO powder. Dark (light) grey columns: B-E (Poisson) distribution for the same count mean. Incident pump intensity: (a) 1.0, (b) 1.5, (c) 3.0, (d) 5.6 times the threshold intensity where discrete spectral peaks appear.
where {n) is the average photon number. For single-mode chaotic Hght, the photon number distribution P{n) satisfies the Bose-Einstein (B-E) distribution: P(n) =
{nY [1+ («)]«+!*
(3.3)
Note that the above distribution holds only for a single mode. For multimode chaotic light, the photon number distribution approaches the Poisson distribution. From the measured P(n), we obtained the normalized second-order correlation coefficient G2 as G, = 1 +
{(Anf) - in) in)
(3.4)
For the Poisson distribution G2 = 1, while for the B-E distribution G2 = 2. Figure 6a shows the measured P{n) of ZnO emission at the threshold where
340
Lasing in disordered media
2.0-
I
1.8-
\
1.6-
I
1.4-
I
[6, §3
O i
1.2-
3=
I X
z 1.0-
•
r
I
10 p
th
Fig. 7. Second-order correlation coefficient G2 as a function of the ratio of the incident pvimp intensity /p to the threshold intensity /th.
discrete spectral peaks appear. The measured photon count distribution was almost identical to the B-E distribution of the same count mean. The value of Gi was 1.94. As we increased the pump intensity, the photon statistics of ZnO emission started to deviate from the B-E statistics. As shown in fig. 6b, when the pump intensity was 1.5 times the threshold, the measured photon count distribution was between the B-E distribution and the Poisson distribution, Gi became 1.51. When the pump intensity increased to 3 times the threshold, the photon count distribution of ZnO emission got closer to the Poisson distribution (fig. 6c), G2 reduced to 1.19. Eventually, when the pump intensity was 5.6 times the threshold, the photon count distribution was nearly identical to the Poisson distribution (fig. 6d); the corresponding G2 was 1.06. Figure 7 is a plot of the second-order correlation coefficient G2 versus the pump intensity. As the pump intensity increased, G2 decreased gradually from 2 to 1. Because we took only a finite number of pulses in the measurement, the rms error in G2 was equal to {2/K{nY)^^^, with K the number of pulses. The sampling error for G2 was calculated and plotted for each data point in fig. 7. Figures 6 and 7 illustrate that the photon statistics of the emitted light from the ZnO powder changes continuously from B-E statistics at threshold to Poisson statistics well above threshold. The photon statistics of a random laser with coherent feedback are similar to that of a traditional laser, but very different from the photon statistics of a random laser with incoherent feedback (Zacharakis, Papadogiannis, Filippidis and Papazoglou [2000]). In a random laser with coherent feedback, the
6, § 3]
Random laser with coherent feedback - Experiment
341
photon number fluctuation in each mode is quenched by gain saturation well above the threshold. For a random laser with incoherent feedback, only the fluctuation of the total number of photons in all modes of laser emission is suppressed by gain saturation, while the photon number fluctuation in a single mode is not affected. This difference will be explained in the next section.
3.2. Transition between two types of random lasers The previous section illustrates that the behavior of a random laser with coherent feedback is quite different from that of a random laser with incoherent feedback. To understand the difference, we studied the transition between them by varying the amount of scattering in the gain medium (Cao, Xu, Chang and Ho [2000]). The random media used in this experiment are Rhodamine 640 dye solutions containing ZnO nanoparticles. The advantage of the solutions is that the scattering length can be varied continuously through the change of particle density in the solutions. The fi-equency-doubled output (A = 532 nm) of a pulsed Nd:YAG laser (10 Hz repetition rate, 20 ps pulse width) was used to excite the dye molecules in the solution. The emission spectrum was captured after a single pump pulse. Figure 8 shows the evolution of the emission spectra with pump intensity for a particle density of ~ 2.5 x 10^ ^ cm"^. The dye concentration was fixed at 5 x 10"^ M. When the pump intensity crossed a threshold, a drastic spectral narrowing occurred. As shown in the insets of fig. 8, once above the threshold, the emission linewidth collapsed to ~5nm. Simultaneously, the peak emission intensity increased dramatically. This phenomenon was identical to what Lawandy, Balachandran, Gomes and Sauvain [1994] had observed. It corresponded to lasing with nonresonant feedback occurring in the colloid. Next the ZnO particle density was increased to 5x10^^ cm~^ while the dye concentration was kept constant. Figure 9 reveals two thresholds. As the pump intensity increased, it reached the first threshold where the emission spectrum quickly narrowed to about 5nm. As the pump intensity increased further, it reached the second threshold where discrete spectral peaks with linewidths less than 0.2 nm emerged. The phenomenon above the second lasing threshold was similar to that in ZnO powder. It corresponded to lasing with coherent feedback. Finally, the ZnO particle density was increased to IxlO^^cm"^. As shown in fig. 10, with an increase of the pump intensity, the discrete spectral peaks appeared before the collapse of the emission linewidth. This indicates that the
342
Lasing in disordered media
[6, §3
c (D
0.35
^
0.25
0
c g CO
0.15
E LU
0.12
3. c B0
0.08 Y 0.06 h
0.04 h
600
605
610
615
620
wavelength (nm) Fig. 8. Spectra of emission from Rhodamine 640 dye solution containing ZnO nanoparticles. ZnO particle density is -3x10^^ cm~^. Incident pump pulse energy (bottom to top): 0.68, 1.5, 2.3, 3.3, 5.6 ^J. Upper inset: emission intensity at peak wavelength versus pump pulse energy. Lower inset: emission linewidth versus pump pulse energy.
6, §3]
343
Random laser with coherent feedback - Experiment
CO
g CO CO
E LU
CO
0
CO
"E LU
CO
0.15
c CD
LU
0.05
600
604
608
612
616
620
Wavelength (nm) Fig. 9. Spectra of emission from Rhodamine 640 dye solution containing ZnO nanoparticles. ZnO particle density is ~6x 10^^ cm~^. Incident pump pulse energy (bottom to top): 0.74, 1.35, 1.7, 2.25, 3.4 fxJ.
344
[6, §3
Lasing in disordered media
CO
CO
c 0
c g w "E LU
c
CO
c c g CO
X3
co CO
c
CD
600
605
610
615
620
Wavelength (nm) Fig. 10. Spectra of emission from Rhodamine 640 dye solution containing ZnO nanoparticles. ZnO particle density is ~1 x 10^^ cm~^. Incident pump pulse energy (bottom to top): 0.68, 1.1, 1.3, 2.9 ^uJ.
6, § 3]
Random laser with coherent feedback - Experiment
345
threshold for lasing with coherent feedback becomes lower than the threshold for lasing with incoherent feedback in a strong scattering medium. The above phenomena can be understood in terms of the eigenmodes of Maxwell's equations in a random medium. Owing to the finite size of the random medium, the eigenenergies are complex numbers, whose imaginary parts represent the decay rates. Through the coupling to the outside reservoir (i.e., to the EM modes outside the random medium), the eigenmodes interact with each other. Photons can hop from one mode to another through scattering at the boundary. In the delocalization regime, the average decay rate of an eigenmode is larger than the mean fi-equency spacing of adjacent modes. Hence, the eigenmodes are spectrally overlapping, giving a continuous emission spectrum. When scattering is weak, the eigenmodes are strongly coupled. Due to photon exchange among the modes, the photon loss rate for a set of interacting modes is much lower than that for a single mode. In an active random medium, when the optical gain for a set of interacting modes at the frequency of gain maximum reaches the loss of these coupled modes, the total photon number in these coupled modes builds up. This process is lasing with incoherent (or nonresonant) feedback. The quick increase of photon number at the fi-equency of gain maximum results in sudden spectral narrowing (fig. 8). Well above threshold, gain saturation quenches the total photon number fluctuation. However, strong coupling of the eigenmodes (e.g., photon hopping among the modes) prevents stabilization of the photon number in a single mode. With an increase in the amount of optical scattering, the dwell time of light in the random medium increases. The decrease of the coupling of the modes to the outside reservoir weakens the interaction between the modes. When the optical gain increases, it first reaches the threshold for lasing in a set of coupled modes at thefi*equencyof gain maximum. As the optical gain increases further, it reaches the loss of a single mode with long lifetime. Then, lasing occurs in a this mode. A further increase of optical gain leads to lasing in more low-loss modes. Laser emission from these modes gives discrete peaks in the emission spectrum (fig. 9). When the scattering strength increases fiirther, the decay rates of the eigenmodes and the coupling among them continue to decrease. There are a small number of eigenmodes with extremely long lifetime and nearly decoupled from other modes. The threshold gain for lasing in these individual modes becomes lower than the threshold gain for lasing in a set of coupled modes. Then lasing with coherent feedback occurs first (fig. 10). Because of weak coupling of the lasing modes, the photon number fluctuation in each mode is quenched by the gain saturation effect well above the threshold.
346
Lasing in disordered media
[6, § 3
3.3. Characteristic length scales for random lasers In spite of their different feedback mechanisms, the relevant length scales for coherent random lasers are the same as those for incoherent random lasers: the transport mean free path, the gain length, the size of the random medium and the gain volume. A detailed study on the dependence of the lasing threshold and the number of lasing modes on these length scales has been reported by Ling, Cao, Burin, Ratner, Liu and Chang [2001]. The random media used in this study were PMMA sheets containing Rhodamine 640 perchlorate dye and TiOi particles. The sample thickness was between 200 and 500 |im. The mean diameter of Ti02 particles was 400 nm. The Ti02 particle density in PMMA was varied from 8x10^^ to 6xl0^^cm"^. The transport mean free path was characterized in a coherent backscattering experiment. The output from a HeiNe laser was used as the probe light, since its wavelength was very close to the emission wavelength of Rhodamine 640 perchlorate dye. To avoid absorption of the probe light, the PMMA samples used in the CBS experiment contained only nanoparticles but not dye. The PMMA sheets containing dye and nanoparticles were optically excited by the second harmonics of a pulsed Nd:YAG laser. The pump beam was focused to a spot of 50 [xm on the sample surface. Figure 11a is a plot of the incident pump pulse energy at the lasing threshold versus the transport mean free path. The dye concentration was fixed at 5xlO~^M. As the Ti02 particle density in the PMMA sheet increases, the transport mean free path decreases, and the lasing threshold also decreases. The strong dependence of the lasing threshold on the transport mean free path confirms the important contribution of scattering to lasing. With an increase in the amount of optical scattering, the feedback provided by scattering becomes stronger. In other words, the resonant cavities formed by recurrent light scattering have lower loss. Hence, the lasing threshold is reduced. Figure l i b shows the number of lasing modes in samples with different transport mean free path at the same pump intensity. The shorter the transport mean free path is, the more lasing modes emerge. This is because in a sample of stronger scattering, more low-loss cavities are formed by recurrent scattering. At a fixed optical gain, there are more cavities in which the loss is balanced by gain, and lasing oscillation occurs. One surprise in fig. 11a is that lasing with resonant feedback occurs in samples of A > A. Although the coherent feedback provided by scattering is rather weak (in other words, the cavities formed by recurrent scattering are lossy), lasing can still occur as long as the dye concentration and the pump intensity are high enough. Another interesting feature in fig. 11 is that when the transport mean free path approaches the optical
6, § 3 ]
Random laser with coherent feedback - Experiment
347
Fig. 11. (a) Incident pump pulse energy at the lasing threshold versus If/X. (b) Number of lasing modes as a function of /^ at fixed 1.0 |UJ incident pump pulse energy.
wavelength, the lasing threshold pump intensity drops quickly, and the number of lasing modes increases dramatically. This result agrees with John and Pang's prediction of a dramatic threshold reduction in the regime /t ^ A of incipient photon localization (John and Pang [1996]). The lasing threshold depends also on the pump area. The PMMA sheet was initially placed at the focal plane of a lens. Then the lens was moved away from the sample to increase the size of the pump beam spot on the sample. The incident pump intensity /thr at the lasing threshold was recorded as a function of the area A of the pump spot on the sample surface. Figure 12 shows the data for two samples with k of 0.9 |im and 9 [im. As the pump area increased, the lasing threshold first decreased, then saturated. There existed a critical pump area above which the lasing threshold was nearly independent of pump area. The shorter the transport mean free path, the smaller the critical area. In order to understand this result, we need to find out how big the lasing modes are.
348
Lasing in disordered media
[6, §3
^-v
"E
5-
X
o
5
^ ^
4-
c
3-
X
(fi
B
X
•
c
X
Q.
E
2-
3. Q C (D •o O
c
^
•
X X
• •
X •
X
•
1-
X X
>
n-
0.00
0.01
0.02
0.03
0.04
0.05
2v
Pump area (mm
Fig. 12. Incident pump intensity at the lasing threshold versus area of pump beam spot on the sample surface. Ij values: circles, 0.9 fxm; crosses, 9 |im.
Unfortunately, it is impossible to measure the distribution of the laser field inside the random medium. Owing to the short penetration length of the pump light, optical gain is confined to a region next to the sample surface. Only the modes located close to the sample surface have spatial overlap with the gain volume and experience amplification. Therefore the lasing modes are located close to the sample surface, and we can probe the spatial distribution of the lasing modes at the sample surface. We developed a spectrally resolved speckle technique to map the spatial profile of individual lasing modes at the sample surface (Cao, Ling, Xu and Burin [2002]). Previous speckle measurement revealed an increase of path length of an external probe light traveling in an amplifying random medium (van Soest, Poelwijk and Lagendijk [2002]). We measured the speckle of coherent laser emission fi-om the random medium. The experimental geometry is shown schematically in fig. 13a. The pump beam was focused by a lens (5 cm focal length) onto the polymer sheet at normal incidence. The pump spot at the sample surface is about 50 |im in diameter. The emission from the sample was collected by the same lens and directed to a 0.5-meter spectrometer with a cooled CCD array detector. The distance fi*om the lens to the spectrometer's entrance slit was nearly equal to the focal length of the lens. Because the polymer sheet was placed at the focal plane of the lens, the emission fi-om the polymer sheet was collimated before entering the spectrometer. The spectrometer imaged its entrance slit onto a two-dimensional (2D) CCD array detector with 1:1 ratio. Figure 13b is part of a spectral image taken by the CCD array detector above
6, §3]
Random laser with coherent feedback - Experiment
(a)
pump
1ens
A
0 ^ ^ " ^ sample
. /
'
/ Spectrometer
->
^
-
349
\PZZ splitter
(b) 0.06 0.04
^ cd ^ (D W)
^
0.02 0.00 -0.02 -0.04 -0.06
620
621
622
623
624
Wavelength (nm) Fig. 13. (a) Schematic experimental geometry for spectrally resolved speckle measurement of laser emission from random medium, (b) Part of spectral image of emission from a polymer containing dye and microparticles. Incident pump pulse energy is 0.98 ^tJ.
the lasing threshold. The horizontal axis is the wavelength, and the vertical axis is the angle. Below the lasing threshold, the emission spectrum had a single broad spontaneous emission peak. The spontaneous emission intensity was invariant with the angle. Above the threshold, discrete lasing peaks emerged in the emission spectrum. For each spectral peak, the emission intensity fluctuated randomly with the angle. The emergence of speckle is a direct evidence of coherent emission from the sample. As shown in fig. 13b, different spectral peaks exhibited different speckle patterns. When the pump intensity was increased fiirther, the contrast of the speckle pattern, i.e., the amplitude of intensity variation, increased as the emission became more coherent. The far-field speckle pattern of a lasing state is determined by its field pattern at the sample surface. The angular distribution of the outgoing field E(q) is the Fourier transform of the field pattern E(x) at the surface, E{q) = / E{x)exp(i2jTqx)dx - /
,
(3.5)
350
Lasing in disordered media
[6, § 3
where q = sin0/A, with 6 the angle between the emission direction and the normal of the sample-air interface, and x represents the transverse coordinate at the sample surface. The angular distribution of the emission intensity I(q) = \E(q)\^ can be obtained experimentally from the far-field speckle pattern. The spatial field correlation function at the sample surface is CE(X) = JE*(x')E(x -\-x^)dx\ According to eq. (3.5), CE(X) and I(q) form a Fourier transform pair, namely, CE(X) = / I(q) exp (-i Ijtxq) dq .
(3.6)
Therefore, the speckle pattern of a lasing siaic ^ivcs us spatial field correlation fianction. From the 2D spectral image in fig. 13b, we obtained the angular dependence I(q) of the emission intensity for the lasing peak at A = 622.8 nm (marked by an arrow). I(q) was normalized: {I(q)) = 1. The discrete Fourier transform of I(q) gave the spatial field correlation fiinction, whose amplitude is plotted in fig. 14a. The spatial resolution bx for CE(X) is determined by the angular range Aq of I(q), i.e., bx = l/Aq. Because the diameter of the lens is larger than the width of the CCD array detector, the maximum collecting angle for the emission is 6^ = arctan(D/2/) = 0.066, where/ = 50mm is the focal length of the lens, and D = 6.6 mm is the width of the CCD array detector. Since 0ni < 1, sin0 ^ 0, and q ^ O/L Then Aq ^ lOJX ^ D/fk ^ 0.21 \imr^ and bx ^ 4.7 (Lim. The spatial range Ax for CE(X) is determined by the angular resolution 6^, i.e.. Ax = l/dq. Because the emitted light is collimated by the lens, the angular resolution depends on the size s of each CCD pixel (s = 26 [im): bq ^ s/fX = 0.832 mm~^ and Ax ^ 1.2 mm. As a result of Fourier-transforming in a finite range of ^ the value of CE(0) is inaccurate and should be discarded. As shown in fig. 14a, the profile of |C£(x)| has some peaks and valleys. It reflects the field pattern E(x). Integration of |C£(x)| gives poc
S(x)=
nAx/2
|C£(x')|dx'~ / Jx
\CEix')\dx'.
(3.7)
Jx
Experimentally |C£(x)| falls to nearly zero when x approaches Ax/2. Hence, we replaced oc by Ax/2 for the upper limit of the integral in eq. (3.7). Figure 14b is a plot of S(x) obtained by integrating |C£(x)| in fig. 14a. S(x) is a smoother fiinction of x than |C£(x)|. The dotted line in fig. 14b represents the fit of *S'(x) with an exponential decay function, ^^(x) = AQ exp(-x//d), where AQ and Id are fitting parameters, ^^(x) fits very well with the exponential decay fiinction.
6, §3]
351
Random laser with coherent feedback - Experiment 0.25-
100
200
300
400
X (jjim)
^ ^
X (^im)
Fig. 14. (a) Amplitude of spatial field correlation function |C^(jc)| for the lasing peak at A = 622.8 nm in fig. 13b (marked by an arrow), (b) Solid line: S(x) obtained by integrating | Q ( x ) | in (a); dotted line: fit with an exponential decay fimction.
with a chi squared of X^ = 0.0026. The decay length is then /d = 131.8 |im. We also fitted S{x) with an algebraic decay function, S{x) = c\/x^^, with fitting parameters c\ and c^. The fit was very bad here, )(} = 0.46. The above speckle analysis was applied to other lasing peaks in fig. 13b with different spatial field correlation fiinctions. Nevertheless, S(x) always fitted well with an exponential decay fiinction, but not an algebraic decay function. The decay length varied from peak to peak. The exponential decay of S(x) suggests that the envelope of the spatial field correlation function falls exponentially with x, namely, |Q(x)| ~ exp(-jc//d).
352
Lasing in disordered media
[6, §3
400
0.000
p
th
Fig. 15. (a) Decay length as a function of pump intensity /p normalized to the threshold value /th(b) Fitting error x^ versus normalized pump intensity /p//th-
We measured the spatial field correlation fiinctions over a wide range of pump intensities,fi*omthe threshold where discrete spectral peaks appear to well above the threshold. Figure 15a is a plot of the decay length /a versus the pump intensity for one lasing mode. The pump intensity /p was normalized to the threshold value /thr where this lasing peak just emerged in the emission spectrum. *S'(jc) was fitted with an exponential decay function. The fitting error x^ is plotted in fig. 15b. Just above the threshold, the exponential fit of S{x) was not good, as indicated by the relatively large value of ;^^. The decay length was /a = 365 (im. With increasing pump intensity, x^ decreased, suggesting the exponential fit became better. The decay length also decreased. Eventually, S{x) fit very well with an
6, § 3]
Random laser with coherent feedback - Experiment
353
exponential decay, and the decay length did not change with the pump intensity anymore. The behavior of the decay length can be explained as follows. When the lasing peak just appears in the emission spectrum, its intensity is lower than the amplified spontaneous emission intensity. The large decay length indicates that the amplified spontaneous emission is spread over a large region. As the pump intensity increases, the emission intensity of the lasing state increases more rapidly than the ASE intensity. When the emission intensity of the lasing state becomes much higher than the amplified spontaneous emission intensity, the wavefiinction of the lasing state dominates the spatial field correlation fiinction. The independence of the decay length on the pump intensity suggests the wavefiinction of the lasing mode does not change with the pumping rate. The decay length of a lasing mode is also independent of the excitation area. We varied the diameter of the pump spot at the sample surface fi-om 50 to 400 |im by adding two lenses and an aperture to the optical path of the pump beam. When the incident pump power was fixed and the excitation area was reduced, some modes with longer decay length stopped lasing, while other modes with shorter decay length started lasing. This is because the smaller modes have more spatial overlap with the gain volume when the diameter of the pump spot is smaller than the decay length. The effective gain for the smaller states is higher, thus it is easier for them to lase. However, some states kept lasing as the pump area was varied. Their decay lengths did not change with the pump area. From the above we conclude that the decay length reflects the spatial extent of a lasing mode. As the transport mean free path decreases, the average decay length gets shorter, and the lasing modes become smaller. Now we can understand the critical pump area observed in fig. 12. When the pump area is smaller than the size of the lasing mode, the lasing state and the excitation volume have incomplete overlap in the transverse direction (parallel to the sample surface). Enlarging the pump area increases the spatial overlap of the gain volume and the lasing mode. The increase of the effective gain for the lasing mode leads to a reduction in the lasing threshold. Once the pump area covers the entire lasing mode, the effective gain for the lasing mode no longer increases with the pump area, thus the lasing threshold does not depend on the pump area any more. In samples with shorter k the lasing modes are smaller, and so are the critical pump areas.
3.4. Micro random laser After improving the spatial resolution of the spectrally resolved speckle
354
[6, § 3
Lasing in disordered media
0
0.1
0.2
0.3
0.4
0.5
0.6
Incident Pump Pulse Energy (nJ) Fig. 16. Spectrally integrated intensity of emission from ZnO cluster versus incident pump pulse energy. Inset: SEM image of the ZnO cluster.
technique, we found that the dimensions of lasing modes can be as small as a couple of micrometers in closely packed ZnO nanoparticles (1 < kj^ < 10). Therefore, when the random system moves toward the localization threshold {kJx = 1 ) , the size of the lasing states approaches the optical wavelength. This result illustrates that strong optical scattering not only supplies coherent feedback for lasing, but also leads to spatial confinement of laser light in a micrometersized volume. Utilizing this mechanism of optical confinement, we fabricated microlasers with disordered media (Cao, Xu, Seelig and Chang [2000]). To fabricate a micrometer-sized random medium, ZnO nanoparticles were agglomerated to form clusters whose size varied fi-om half a micron to a few microns. The inset of fig. 16 shows the SEM image of a typical ZnO cluster. The size of the cluster was about 1.7 |im. It contained roughly 20000 ZnO nanoparticles. The ZnO cluster was optically pumped by the third harmonics of a pulsed NdiYAG laser. The pump light was focused by a microscope objective lens onto a single cluster. Simultaneously we measured the spectrum of emission fi-om the cluster and imaged the spatial distribution of the emitted light intensity across the cluster. At low pump intensity, the emission spectrum consisted of a single broad spontaneous emission peak (fig. 17a). with FWHM 12 nm. The spatial distribution of the spontaneous emission intensity was uniform across the cluster (fig. 17b). When the pump intensity exceeded a threshold, a sharp peak emerged in the emission spectrum (fig. 17c), with FWHM 0.2 nm. Simultaneously, a couple of
6, §3]
Random laser with coherent feedback - Experiment
120
355
-(c)
3
3. ^
80
(/) B
^ c .2 40 w (/) E LU 0
^ ^ ^ 1 .
(e) 600 d ^ >^ •S 400 c 0 ^ c o 200
(/) (ji
E LU n u
,,!,,,,
375
u 1
'
380
'
'
'
1
'
385
Wavelength (nm)
•
•
•
1
'
390
.
1 |uim
Fig. 17. (a,c,e) Spectra of emission from the ZnO cluster shown in fig. 16. (b,d, f) Corresponding spatial distribution of emission intensity in the cluster. Incident pump pulse energy: (a,b) 0.26 nj; (c,d)0.35nJ; (e,f) 0.50nJ.
356
Easing in disordered media
[6, § 3
bright spots appeared in the image of the emitted light distribution in the cluster (fig. 17d). When the pump intensity was increased fiirther, a second sharp peak emerged in the emission spectrum (fig. 17e). Correspondingly, additional bright spots appeared in the image of the emitted light distribution (fig. 17f). The curve of the total emission intensity as a fimction of the pump intensity in fig. 16 exhibits a distinct change of slope at the threshold where sharp spectral peaks and bright spots appear. Well above the threshold, the total emission intensity increases almost linearly with pump intensity. These data suggest lasing oscillation in the micrometer-sized cluster. The incident pump pulse energy at the lasing threshold is ~0.3nJ. Note that less than - 1 % of the incident pump light is absorbed. The rest is scattered. Since the cluster is very small, optical reflection fi*om the boundary of the cluster might have some contribution to light confinement in the cluster. However, the laser cavity is not formed by total internal reflection at the boundary. Otherwise the spatial pattern of laser light would be a bright ring near the edge of the cluster (Taniguchi, Tanosaki, Tsujita and Inaba [1996]). We believe the 3D optical confinement in a micrometer-sized ZnO cluster is achieved through disorder-induced scattering and interference. Since the interference effect is wavelength sensitive, only light at certain wavelengths can be confined in a cluster. In another cluster of different particle configuration, light at different wavelengths is confined. Because optical confinement is not caused by light reflection at the surface of a cluster but by scattering inside the cluster, we achieved lasing in clusters with irregular shapes and rough surfaces. We now compare the micro random laser with other types of microlasers. Over the past decade, several types of microlasers have been developed. The key issue for a microlaser is to confine light in a small volume with dimensions on the order of the optical wavelength. In the vertical cavity surface emitting laser, light is confined by two distributed Bragg reflectors (Jewell, Harbison, Scherer, Lee and Florez [1991]). The microdisk laser utilizes total internal reflection at the edge of a high-index disk to form whispering-gallery modes (McCall, Levi, Slusher, Pearton and Logan [1992]). In the two-dimensional photonic band-gap defect mode laser, light confinement is realized through Bragg scattering in a periodic structure (Painter, Lee, Scherer, Yariv, O'Brien, Dapkus and Kim [1999]). The fabrication of these microlasers requires expensive crystal growth and nanofabrication facilities. In a micro random laser, the optical confinement is achieved through disorder-induced scattering and interference. The fabrication of the micro random laser is much easier and cheaper than that of most microlasers. The frequencies of micro random lasers cannot be well controlled as compared to other types of microlasers. In fact, the lasing
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frequencies, depending on random particle configurations, are fingerprints of individual clusters.
3.5. A different type of random laser cavity In the preceding sections, the formation of random laser cavities was attributed to recurrent scattering and interference effect. This mechanism applies to random media with discrete scatterers and strong short-range disorder. A different type of resonant cavity can be formed by smooth long-range inhomogeneity in a weakly disordered medium (Poison, Raikh and Vardeny [2002]). Vardeny and coworkers conducted extensive experimental studies on random lasing in weakly disordered media such as :/r-conjugated polymer films (Frolov, Vardeny, Yoshino, Zakhidov and Baughman [1999], Poison, Huang and Vardeny [2001]), organic dye-doped gel films (Frolov, Vardeny, Zakhidov and Baughman [1999]), and synthetic opals infiltrated with jr-conjugated polymers and dyes (Frolov, Vardeny, Zakhidov and Baughman [1999], Yoshino, Tatsuhara, Kawagishi and Ozaki [1999], Poison, Chipouline and Vardeny [2001]). They performed a power Fourier transform (PFT) of the laser emission spectrum to obtain the typical length of lasing cavities (Poison, Raikh and Vardeny [2002]). In the weakly disordered polymer films {k > A), the cavity length was much longer than the transport mean free path. Although the PFT of individual random lasing spectra exhibited position-specific multipeak structures, averaging the PFTs over the sample positions did not smear these features; on the contrary, it yielded a series of distinct transform peaks. Moreover, the shape of the averaged PFT was universal, i.e., increasing the disorder and correspondingly reducing /t did not change this shape: the average of the PFT spectra at different k scaled with /t to a universal curve. Poison, Raikh and Vardeny [2002] suggested that large resonators are formed due to the long-range inhomogeneities of the refractive index. A plausible microscopic origin of these inhomogeneities is the fluctuation of the polymer film thickness. Light is trapped in a high-index region, which is much larger than /t and A, by total internal reflection at the boundary of this region. The ability of most random resonators to trap light is suppressed by the short-range disorder. The dramatic consequence of this suppression is that resonators that "survive" the short-range disorder are sparse, and consequently almost identical. Poison et al. developed a theory based only on the assumption that the random resonators are exponentially sparse without specifying the shape of the resonators. They showed that this assumption was sufficient to reproduce the universal shape of the averaging PFT.
358
Lasing in disordered media
[6, § 4
§ 4. Random laser with coherent feedback - Theory Obviously the theoretical models set up for random laser with incoherent feedback cannot be applied to random laser with coherent feedback. What is calculated in those models is light intensity instead of electromagnetic field. As the phase of the light field is neglected, these models do not include the interference effect which is essential to coherent feedback. In order to model a random laser with coherent feedback, we must replace the diffusion equation for light intensity by the Maxwell equations for the electromagnetic field (Cao, Xu, Chang, Ho, Seelig, Liu and Chang [2000]). Recently, several theoretical models have been set up for a random laser with coherent feedback, e.g., time-dependent theory (Jiang and Soukoulis [2000]), collective modes of resonant scatterers (Burin, Ratner, Cao and Chang [2001]), prelocalized modes in diffusive media (Apalkov, Raikh and Shapiro [2002]), an Anderson model (Patra [2003]), etc. In this section, a brief introduction on some of the theoretical work is presented. 4.1. Chaotic laser theory A chaotic cavity laser is a special kind of random laser. A chaotic cavity exhibits chaotic dynamics due to the irregular shape of the cavity and/or scatterers placed at random positions inside the cavity. A typical chaotic cavity has one or a few openings whose dimension is smaller than the wavelength. Photons are trapped inside the cavity long enough to ergodically explore the entire cavity volume. Misirpashaev and Beenakker [1998] calculated the statistical distribution of the lasing threshold in chaotic cavities. The mean value of the pumping rate at lasing threshold is well below the pumping rate needed to compensate the average cavity loss. The average number of non-competing lasing modes is proportional to the square root of the pumping rate. However, mode competition reduces the number of lasing modes because of spatial hole burning. The laser linewidth is enhanced above the Schawlow-Townes value by the Petermann factor K, owing to the non-orthogonality of the cavity modes (Patra, Schomerus and Beenakker [2000], Frahm, Schomerus, Patra and Beenakker [2000], Schomerus, Frahm, Patra and Beenakker [2000]). Beenakker [1998] computed the statistics of the emitted radiation by the input-output relation below the lasing threshold where gain saturation is negligible. The amplified spontaneous emission exhibits excessive noise, which increases with optical gain and diverges at the lasing threshold. The origin of the excessive noise lies in the presence of a large number of overlapping cavity modes and a broad distribution
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of the corresponding scattering strengths. Taking into account gain nonhnearity above the lasing threshold, Hackenbroich, Viviescas, Elattari and Haake [2001] derived the photon statistics of a single-mode chaotic laser. The distribution of the mean photocount over an ensemble of modes changes qualitatively at the lasing transition, and displays up to three peaks above the lasing threshold. Patra [2002] studied the effect of mode competition on noise of a multi-mode chaotic laser. The amount of photon number fluctuation is increased above the Poissonian value by an amount that depends on the number of lasing modes. Despite extensive theoretical studies, experimental work is sparse due to the difficulty in fabricating chaotic cavities with openings smaller than the optical wavelength. A recent wave chaos experiment in a macroscopic open resonator raises hopes for experimental realization of chaotic cavity laser (Dingjan, Altewischer, van Exter and Woerdman [2002]). Nevertheless, most random laser experiments were performed on open systems with strong coupling to the outside reservoir. The emitted light can no longer explore the entire random medium ergodically before escaping through the boundary. Several theories have been developed to deal with such random systems.
4.2. Time-dependent theory The time-dependent theory for random lasers couples the Maxwell equations with the rate equations of electronic population (Jiang and Soukoulis [2000]). The gain medium is a four-level electronic material. Electrons are pumped from level 0 to level 3, then relax quickly (with time constant r32) to level 2. Level 2 and level 1 are the upper and lower levels of the atomic lasing transition at frequency co^. After radiative decay (with time constant r2i) from level 2 to level 1, electrons relax rapidly (with time constant Tio) from level 1 back to level 0. The populations in the four levels (A^s, N2, Ni, NQ) satisfy the following rate equations:
at r32 dN2(r,t) _ N^iKt) ^ E(r,t) dP(r,t) dt 132 ho)^ dt d M ( r , 0 _N2ir,t) E(r,t) dP(r, t) dt
dNoif.t) dt
X2\
hWa
_Ni(r,t) -Pr(t)No(r,t), Tio
dt
N2{r,t) r2i ' Ni(r,t) Tio
.^ J.
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Lasing in disordered media
[6, § 4
where Pr{i) represents the external pumping rate. P{r,t) is the polarization density, obeying the equation ^
^ dr
+Aco,^^
d^
+ colPir,t) =
Yc m
lLl[N,(r,t)-N2(r,t)]Eir,t),
(4.2) where a>a and Ao)^, represent the center frequency and linewidth of the atomic transition from level 2 to level 1. /r = 1/^21, 7c = e^col/6jteomc^, where e and m are the electron charge and mass. P(r, t) introduces gain to the Maxwell equations:
*
.
(4.3)
where B{r, t) = fiH(r, t). The disorder is described by the spatial fluctuation of the dielectric constant e{r). The Maxwell equations are solved with the finitedifference time-domain (FDTD) method (Taflove [1995]) to obtain the electromagnetic field distribution in the random medium. Fourier-transforming E(r, t) gives the local emission spectrum. To simulate an open system, the random medium has a finite size and is surrounded by air. The surrounding air is terminated by strongly absorbing layers, e.g. uniaxial perfectly matched layers that absorb all the light escaping through the boundary of the random medium (Sacks, Kingsland, Lee and Lee [1995]). Within a semiclassical framework, spontaneous emission can be included in the Maxwell equations as a noise current. Jiang and Soukoulis simulated the lasing phenomenon in ID random system with the time-dependent theory. A critical pumping rate exists for the appearance of lasing peaks in the spectrum. The number of lasing modes increases with the pumping rate and the length of the system. When the pumping rate increases even fiirther, the number of lasing modes does not increase anymore, but saturates to a constant value, which is proportional to the system size for a given randomness. This saturation is caused by spatial repulsion of lasing modes that results from gain competition and spatial localization of the lasing modes. This prediction was later confirmed experimentally (Ling, Cao, Burin, Ratner, Liu and Chang [2001]). The time-dependent theory is especially suitable for the simulation of laser dynamics. Soukoulis, Jiang, Xu and Cao [2002] simulated the dynamic response and relaxation oscillation in random lasers. The simulation reproduced most of the experimental observations and provided an understanding of the dynamic response of random laser.
6, § 4]
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361
Vanneste and Sebbah calculated the spatial profile of lasing modes in 2D random media with the above method (Vanneste and Sebbah [2001], Sebbah and Vanneste [2002]). They compared the passive modes of a 2D random system with the lasing modes when gain is activated. In the strong localization regime, the lasing modes are identical to the passive modes without gain. When the external pump is focused, the lasing modes change with the location of the pump, in agreement with the experimental observation. Therefore local pumping of the system allows selective excitation of individual localized modes. Jiang and Soukoulis also showed that a knowledge of the density of states and the eigenstates of a random system without gain, in conjuction with the firequency profile of the gain, can accurately predict the mode that will lase first (Jiang and Soukoulis [2002]). The advantage of the time-dependent theory is that it can simulate lasing in a real sample after specifying the structure and material information. The numerical simulation gives the lasing spectra, the spatial distribution of lasing modes, and the dynamic response that can be compared directly with the experimental measurement. The problem is that simulation of large samples requires too much computing power and the running time cannot be too long. So far, numerical simulations have been carried out only for ID and 2D systems, even though the method can be applied to 3D systems. Furthermore, the simulation must be done for thousands of samples with different configurations before any statistical conclusion can be drawn. 4.3. Analytical approach Analytical approaches were taken to derive the conditions for coherent steadystate laser oscillation in random media (Herrmann and Wilhelmi [1998], Burin, Ratner, Cao and Chang [2002], Jiang and Soukoulis [2002]). In analog to the Fabry-Perot laser, the threshold conditions for coherent lasing in a ID random system include both the steady-state round-trip gain condition and the round-trip phase shift condition. The semiclassical laser theory can be generalized to random lasers (Jiang and Soukoulis [2002]). The slowly varying approximation gives E{r,t) = E(t)^(r) Qxp(-ia)t). The real and imaginary parts of the Maxwell equation for E(r, t) become VV(F) + ii^[e{r) + eox'{r, o))\ (o^m dE(t) dt
( _ ( , . » ) - i ) ^ ,
=0 ,
(4.4)
(4.5,
362
Lasing in disordered media
[6, § 5
where {e) and -{x") represent the spatially averaged dielectric constant and gain. Equation (4.4) determines the lasing frequency (o, the field distribution 0(r), and the quality factor Q. The term eox'if^ ^ ) causes a shift of the lasing frequency from the eigenfrequency of the passive system (the pulling effect). When Q is large, a small gain is needed to reach the lasing threshold. Then X'{r, w) <^ 1, the pulling effect is very weak. The lasing frequency is almost the same as the eigenfrequency, and the wavefunction of the lasing mode is nearly identical to the eigenfunction of the passive system. Equation (4.5) is the timedependent amplitude equation, it sets the threshold condition -{x"{^)) ^ ^^QEquation (4.5) also gives the stable amplitude of the field above the lasing threshold. 4.4. Quantum theory Quantum theory is needed to understand the quantum-statistical properties of random lasers. Standard quantum theory for lasers applies only to quasidiscrete modes and cannot account for lasing in the presence of overlapping modes. In a random medium, the character of lasing modes depends on the amount of disorder. Weak disorder leads to a poor confinement of light and to strongly overlapping modes. Statistics naturally enters the random scattering theory pioneered by Beenakker (Beenakker [1998], Patra and Beenakker [1999], Patra and Beenakker [2000], Mishchenko, Patra and Beenakker [2001]), but that approach is restricted to linear media and cannot describe random lasers above lasing threshold. Hackenbroich, Viviescas and Haake [2002] developed a quantization scheme for optical resonators with overlapping modes. Feshbach's projector technique, previously applied to condensed matter physics, was employed to quantize the electromagnetic field. The electromagnetic field Hamiltonian of open resonators reduces to the well-known system-and-bath Hamiltonian of quantum optics. Upon including an amplifying medium, it could serve as a starting point for the quantum theory of random lasers.
§ 5. Interplay of localization and amplification The study of random lasers is closely related to that of light localization. The interplay of light localization and coherent amplification has attracted much interest and study. The effect of gain on coherent backscattering (weak localization) was studied both theoretically (Zyuzin [1994], Deng, Wiersma and Zhang [1997], Tutov and
6, § 5]
Interplay of localization and amplification
363
Maradudin [1999]) and experimentally (Wiersma, van Albada and Lagendijk [1995b], de Oliveira, Perkins and Lawandy [1996], van Soest, Poelwijk, Sprik and Lagendijk [2001]). The shape of the backscattering cone is determined by the transport distance of light in the medium. The intensity in an amplifying medium grows exponentially with the path length. Consequently, gain enhances the long path lengths that constitute the top of the backscattering cone. Thus, a relatively larger contribution of long paths yields a sharper and narrower cone as compared to the passive medium. The effect of amplification on light reflection and transmission in a random system has been extensively investigated and compared with the effect of absorption (Pradhan and Kumar [1994], Zhang [1995], Zyuzin [1995], Beenakker [1996], Paasschens, Misirpashaev and Beenakker [1996], Burkov and Zyuzin [1996], Burkov and Zyuzin [1997], Freilikher, Pustilnik and Yurkevich [1997], Jiang and Soukoulis [1999a,b]). These studies were carried out mostly in ID or quasi-ID system with time-independent theory. Optical gain is introduced through the imaginary part of the refractive index. Below the lasing threshold, coherent amplification enhances light reflection and suppresses light transmission. It also leads to an increase in the fluctuation of the transmissivity T and reflectivity R. On approaching the lasing threshold, both the mean value and the variance of T and R diverge. Above threshold, the time-independent theory no longer works, and it must be replaced by time-dependent theory. The lasing threshold is related to transmission in ID random system (Burin, Ratner, Cao and Chang [2002]). More specifically, in the localization regime, the lasing threshold is defined by the transmission through the passive system. Thus, the average lasing threshold depends exponentially on the size of the random system. Moreover, the lasing threshold fluctuates strongly from configuration to configuration. So far, numerical simulation on coherent random lasers has been conducted only in ID or 2D random systems (Li, Ho and Soukoulis [2001], Jiang and Soukoulis [2000], Burin, Ratner, Cao and Chang [2001], Vanneste and Sebbah [2001], Patra [2003]). The characteristics of lasing modes in 3D random media are not fully understood. Theoretically, in a 3D random medium with /? > A the eigenmodes are expected to extend over the entire sample. However, experimental measurement shows that the size of individual lasing modes is much smaller than that of the entire sample even when U > A. Thus, the lasing modes do not seem to be extended states. One possible explanation is that there exist some prelocalized modes with anomalously low loss (Apalkov, Raikh and Shapiro [2002]). These modes are very rare in the diffusive regime {kji > !)• They are analogous to the prelocalized electronic states in diffusive conductors
364
Lasing in disordered media
[6, § 6
that are responsible for the long-time asymptotics of the current relaxation (Altshuler, Kravtsov and Lemer [1991]). Since the number of prelocalized modes is much smaller than the number of extended modes, the transport properties are dominated by extended modes. Therefore, it is hard to probe the prelocalized modes in the transmission measurement. However, when optical gain is introduced to a random medium, the prelocalized modes are preferably amplified because of their long lifetime. Photons in the prelocalized modes stay longer in the gain medium, thus they experience more amplification. As optical gain increases further, the prelocalized modes lase first because of their lower decay rates. Once the prelocalized modes lase, their intensities are much higher than those of the extended modes. Thus, they dominate the emission spectrum and the field pattern. Apalkov, Raikh and Shapiro [2002] calculated the likelihood of prelocalized modes and found it to depend crucially on the size of the scatterers, or more precisely, on the correlation radius of the disorder.
§ 6. Applications of random lasers A random laser is a non-conventional laser whose feedback mechanism is based on random scattering, as opposed to the reflective feedback by the mirrors of a conventional laser. This alternative feedback mechanism is important for manufacturing lasers in spectral regimes where efficient reflective elements are not available, e.g., ultraviolet or X-ray lasers. An ultraviolet random laser has already been realized with ZnO powder, polycrystalline films, and nanowire arrays (Cao, Zhao, Ong, Ho, Dai, Wu and Chang [1998], Cao, Zhao, Ho, Seelig, Wang and Chang [1999], Mitra and Thareja [1999], Thareja and Mitra [2000], Huang, Mao, Feick, Yang, Wu, Kind, Weber, Russo and Yang [2001]). Furthermore, the low fabrication cost, sample-specific wavelength of operation, small size, flexible shape, and substrate compatibility of random lasers lead to many potential applications. For example, a thin layer of laser paint can be coated on screws, nuts, bolts and manufactured parts for machine-vision application (Lawandy [1994]). In the area of search and rescue, the painted-on laser may provide a rugged and low-cost method of identification for downed ships, aircrafts, and satellites. In the medical arena, random lasers have potential for application in photodynamic therapy and tumor detection. Micro random lasers may play the crucial role of the active element or miniature light source in integrated photonic circuits (Wiersma [2000]). They can also be used to monitor the flow of liquids by adding a small amount of nanoparticle clusters to the liquid and detecting their laser emission over large flow distances. The specific
6, § 6]
Applications of random lasers
365
wavelength of operation, depending on the configuration of scatterers, makes the random laser suitable for document encoding and material labeling. For some applications, the current threshold of random lasers is too high. Zacharakis, Papadogiannis and Papazoglou [2002] explored two-photon pumping for random lasers to reduce the threshold. Weak two-photon absorption allows deeper penetration of the pump light into the random medium, and results in better confinement of the emitted light (Burin, Ratner and Cao [2003]). We proposed another scheme to reduce the lasing threshold: incorporating some degree of order into an active random medium (Chang, Cao and Ho [2003], Yamilov and Cao [2003]). Shkunov, DeLong, Raikh, Vardeny, Zakhidov and Baughman [2001] observed both photonic lasing and random lasing in dyeinfiltrated opals. However, the random lasing had higher threshold than photonic lasing. We numerically simulated lasing in a random system with variable degree of order. We found that the lasing threshold reaches a minimum at a certain degree of order (Chang, Cao and Ho [2003]). In other words, there exists an optimum degree of order/disorder for lasing, where the lasing threshold is comparable (within a factor of 2) to the threshold of a single-defect photonic bandgap laser. We mapped out the transition from full order to complete disorder, and identified five scaling regimes for the mean lasing threshold versus system size L (Yamilov and Cao [2003]). For increasing degree of disorder, the five regimes are (a) photonic band-edge, l/L^, (b) transitional super-exponential, (c) bandgap-related exponential, (d) diffusive, 1/Z^, and (e) disorder-induced exponential. These predictions remain to be confirmed experimentally. Unlike conventional lasers with directional output, random media can emit laser emission in all directions (Cao, Zhao, Ong and Chang [1999]). The isotropic output of random lasers is useful for some applications like displays, but in other applications directional output is desired. Shukri and Armstrong [2000] observed directional emission in the backward direction of the pump beam in a dye solution. Alencar, Gomes and de Araujo [2001] exploited the enhancement of directionality due to surface plasmon excitation provided by the contact of polymer sample with a thin metallic film. We utilized external feedback to reduce the lasing threshold and control the output direction of laser emission (Cao, Zhao, Liu, Seelig and Chang [1999]). So far, most random lasers have been pumped optically. Some applications, such as flat-panel, automotive and cockpit displays, require electrical pumping. Recently electrically pumped continuous-wave laser action was reported in rareearth-metal-doped dielectric nanophosphors and Nd-doped 6-alumina nanopowders (Williams, Bayram, Rand, Hinklin and Laine [2001], Li, Williams, Rand, Hinklin and Laine [2002]).
366
Lasing in disordered media
[6
Last but not least, research into random lasers also benefits other research fields. For example, adding optical gain to a random medium creates a new path to the study of wave transport and localization. Because the random lasing modes are eigenstates of a random system, the interaction of the lasing modes reflects the interaction among the eigenstates. Recently, we observed the coupling of random lasing modes (Cao, Jiang, Ling, Xu and Soukoulis [2003]). This provided us with inlbr-iidtion on the interaction among eigenmodes of a random system. Such knowledge is crucial for understanding the escape channels for photons in an eigenstate, which is important for localization theory. Moreover, random lasers offer an opportunity to study the interplay between nonlinearity and localization. Nonlinear optical effects are significant in a random laser owing to its high intensity and resonance enhancement. Noginov, Egarievwe, Noginova, Wang and Caulfield [1998] demonstrated second-harmonic generation in a mixture of powders of laser and ft-equency-doubling materials. Our recent study on the dynamic nonlinear effect in a random laser illustrates that the thirdorder optical nonlinearity modifies both eigenfi*equency and eigenwavefiinction of a random system (Liu, Yamilov and Cao [2003]). As Letokhov's work showed, the study of random laser improves our understanding of galaxy masers and stellar lasers whose feedback is caused by scattering (Letokhov [1972, 1996]).
Acknowledgment I wish to thank my coworkers on the study of random lasers. Drs. J.Y. Xu, Y. Ling, YG. Zhao, and Prof Prem Kumar contributed to the experimental work on random lasers. Drs. A.L. Burin, A. Yamilov, B. Liu, S.-H. Chang, Profs. S.T. Ho and M.A. Ratner conducted theoretical studies of random lasers. Prof R.PH. Chang and his students E.W. Seelig, X. Liu and H.C. Ong fabricated ZnO nanoparticles and poly crystalline films. We enjoyed the fruitfiil collaboration with Prof. CM. Soukoulis and Dr. Xunya Jiang on the simulation of random laser. Stimulating discussions are acknowledged with Profs. VM. Letokhov, S. John, Z.V Vardeny, A.Z. Genack, M.E. Raikh, L. Deych, M.A. Noginov, S.C. Rand, CM. de Sterke, D. Wiersma, A. Taflove, A.A. Asatryan, M. Patra, and Ch.M. Briskina. Our research program was partly sponsored by the National Science Foundation through grants ECS-9877113 and DMR-0093949, and by the David and Lucille Packard Foundation, the Alfred P. Sloan Foundation, and Northwestern University Materials Research Center.
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Author index for Volume 45
Anglos, D. 331 Anwar, N. 58 Apalkov, VM. 358, 363, 364 Aravind, RK. 133 Arbert, M. 93 Arfken, G. 304 Arichi, T. 34, 46 Arieli, Y. 197 Arjavalingam, G. 230 Arlt, J. 128-130, 140, 146, 162, 170-173, 175, 192, 193 Armstrong, J.R 301 Armstrong, R.L. 365 Amaut, H.H. 198 Amett, J.B. 282 Amey, S. 57 Asazawa, H. 55 Ashkin, A. 165-167 Ashraf, I. 239, 263 Asobe, M. 60, 74 Aubin, S. 132, 175 Auffeves, A. 267 Auzel, F. 324, 325
Abbott, T.A. 287 Abdel-Aty, M. 266 Abeles, J. 95 Abella, I.D. 211,214 Abgrall, M. 213 Abraham, E.R.I. 193, 195 Abram, R.H. 10, 11 Abramochkin, E. 133 Adams, C.S. 190, 191 Adams, L.E. 60 Agarwal, G.S. 173, 240, 241, 248, 252 Agrawal, G.P. 72 Agrawal, N. 92, 93 Akhmediev, N.N. 156 Aksyuk, V 57 Al-Awfi, S. 187 Alencar, M.A.R.C. 365 Alexander, D.R. 167 Alexanian, M. 239 Alexopoulos, N.G. 302 Alfano, R.R. 328-331 Allen, L. 123, 133-135, 140, 155, 159, 160, 166, 167, 169, 170, 173, 180, 185-187, 199, 218, 224, 225 Allison, I. 147-149, 163 Alphonse, G.A. 95 Altewischer, E. 359 Altshuler, B.L. 364 Ambartsumyan, R.V 321 An, K. 207, 229 Anastassiou, C. 156 Andersen, J.A. 173 Andersen, XK. 71 Anderson, RW. 301 Anderson, Z. 181 Andrekson, RA. 62, 64 Andrews, M.R. 191
B Babiker, M. 123, 133, 180, 185-187 Baby, V 85, 86, 95 Bagini, V 160 Bahns, J.T. 174 Baker, H.J. 10,11 Balachandran, R.M. 327, 328, 330-332, 341 Balmer, J.E. 5, 39 Baltes, H.R 276, 291, 293, 296 Baluschev, S. 188 Balykin, VI. 184 Bandilla, A. 263 Banier, C. 285 Barak, S. 211,212,233 371
372
Author index for Volume 45
Barbosa, G.A. 198 Bamett, S.M. 199 Barr, D.L. 57 Barrett, M.D. 188, 189, 191 Barry, R.A. 60 Bartolini, P. 301 Barton, IP. 167 Basavanhally, N. 57 Basistiy, I.V 140 Basov, KG. 321 Bassett, I.M. 3, 47 Baughman, R.H. 332, 357, 365 Baumann, M. 11, 12 Bay, S. 228, 230, 263 Bayart, D. 59 Bayram, S.B. 365 Bazhenov, VYu. 138, 140 Bazhulin, S.P 321 Beckering, G. 330 Beenakker, C.W.J. 358, 362, 363 Beijersbergen, M.W. 133, 134, 155, 167, 169 Belan, VR. 324 Belanger, PA. 124, 128 Bell, A.S. 188 Beran, M.J. 296 Berger, G.A. 329, 331 Berger, J. 88, 89, 113 Bergou, J. 263 Berman, PR. 185,235 Bems, M.W. 167 Berry, M. 303 Bertero, M. 291 Bertet, P 267 Besse, PA. 93 Bethe, H.A. 210 Bethune, D.S. 231 Bigelow, N.P 191, 248 Bigo, S. 59 Biraben, F. 226, 235 Bishop, D.J. 57 Bjorkholm, J.E. 165, 226 Bleistein, N. 276, 281 Blit, S. 156 Block, S.M. 166 Blow, K.J. 79 Blumenthal, D.J. 95, 113 Boccacci, P 291 Bogart, G.R. 57 Bohm, D. 276, 310 Boie, R. 57
Bojarski, N.N. 276 Bokor, N. 3, 19, 40-43 Bolle, C. 57 Boiler, K.-J. 233, 234 Boncek, R.K. 80, 85 Bongs, K. 192, 195 Boone, A.W 263 Boone, T. 57 Bom, M. 142, 288, 297, 298 Borne, S. 59 Bose, S. 239 Bouchoule, S. 93 Bouwmeester, D. 207 Boyd, R.W. 212, 229, 240, 241, 259 Boyraz, O. 71 Bradley, C.C. 248 Brambilla, M. 266, 267 Branscomb, L.M. 214 Braslau, N. 209, 210, 216-220, 228, 260, 264 Braunlich, P 212 Brecha, R.J. 191 Breit, G. 210,211 Brevik, I. 167 Brignon, A. 41 Briskina, Ch.M. 323, 324 Brown, W.J. 231, 251-253, 255-260 Bnine, M. 219, 235-239, 264, 266, 267 Bryant, PE. 172 Bulsara, A.R. 261 Burger, S. 192, 195 Burin, A.L. 346, 348, 358, 360, 361, 363, 365 Burkov, A.A. 363 Bussac, C. 41 Butylkin, VS. 216, 222, 264 C Cabot, S. 55 Cacciapuoti, L. 131 Cao, C.Q. 338 Cao, H. 332, 336, 338, 341, 346, 348, 354, 358, 360, 361, 363-366 Caraccia-Gross, M. 93 Carman, R.L. 218, 228, 230 Carr, D. 57 Carter, WH. 279 Cataliotti, F.S. 252 Caulfield, H.J. 324, 325, 330, 366 Cavalieri, S. 332
Author index for Volume 45 Centanni, J.C. 55 Chakmakjian, S. 241 Chaloupka, IL. 152, 173 Chan, F.T. 210 Chan, VW.S. 60 Chang, R.RH. 332, 346, 354, 358, 360, 363365 Chang, S. 166, 167, 169 Chang, S.-H. 341, 358, 361, 363, 365 Chang, T.G. 79-81, 86 Chapman, M.S. 188, 189, 191 Charlet, G. 59 Chavez-Cerda, S. 146-149, 160, 163 Chen, J. 248 Chen, Y.F. 157 Cheng, B. 329 Cheng, W.Z. 263 Chiba, K. 14, 15, 45 Chikkatur, A.P. 198 Childs, J.J. 207 Chipouline, A. 357 Choi, K. 135, 136 Chou, M.H. 94,113 Chow, L. 239 Chrissopoulou, K. 212 Christe, S. 132, 175 Christodouhdes, D. 156 Chu, S. 165, 190, 191 Ciampa, N.A. 57 Cirelli, R.A. 57 Clairon, A. 213 Clarke, R.B.M. 188 Clarkson, W.A. 7, 8, 10, 11, 35, 45 Claussen, N.R. 191 Chfford, M.A. 140 Close, J.D. 193 Cochrane, C. 325 Coerwinkel, R.RC. 155 Cohen, J.K. 276, 281 Cohen-Tannoudji, C. 242 Coldwell, C. 60, 79, 84, 106 Collett, M.J. 195 Colton, D. 284 Compagno, G. 242 Concannon, H.M. 219-224, 226, 251, 252 Connolly, J.C. 95 Copeland, G.E. 213 Coquelin, A. 95 Cormier, M. 124 Cornell, E.A. 181, 191
373
Cornish, S.L. 191 Cotter, D. 79, 92 Courjon, D. 285 Courtial, J. 134, 135, 140, 147-149, 159, 163, 173 Cue, N. 328, 330 D Dabarsyah, B. 57 Dagens, B. 93, 95 Dai, J.Y. 332, 364 Dalgamo, A. 211 Dalichaouch, R. 301 Danakis, A. 133 Danielsen, S.L. 93 Dapkus, RD. 356 Dasari, R.R. 207, 229 Davidovich, L. 219, 237, 238, 264, 266 Davidson, N. 3, 5, 17-24, 26, 29, 40-43, 47-50, 132, 152, 153, 156, 157, 163, 175, 188, 190, 191 Davies, D.A.O. 92 Davis, K.B. 158, 188 de Angelis, M. 131 de Araujo, C.B. 365 de Oliveira, RC. 332, 363 de Valcarcel, G.J. 225, 266 Delfyett, RJ. 95 DeLong, M.C. 365 Denardo, B. 305 Denariez-Roberge, M.M. 226 Deng, K.L. 60, 79, 84, 96-98, 106-110 Deng, L. 193 Deng, W. 362 Denk, W. 213 Dettmer, S. 192, 195 Devaney, A.J. 276, 277, 286, 287, 289, 291, 295, 296, 299 Devaud, M. 226 Devreese, XT. 193, 195 Dholakia, K. 128-130, 140, 159, 160, 162, 166, 170-173, 192, 193 Dickson, L.E. 284 Diez, S. 88-91, 105 Dimitrova, O.V 324 Dingjan, J. 359 Dinno, M.A. 198 Dischler, R. 59 Doerr, C.R. 60 Donley, E.A. 181
374
Author index for Volume 45
Doran, N.J. 67 Drake, J.M. 323 Drummond, P.D. 173 Druzhunina, V 180 Du, K. 11,12,45 Dubetsky, B. 185 Duelk, M. 93 Dunn, M.H. 172 Dupont-Roc, J. 242 Durhuus, T. 93 Dumun, J. 136 Durville, F. 32 Dziedzic, J.M. 165-167
Eberly, J.H. 136,218 Efthimiopoulos, T. 212 Egarievwe, S.U. 324, 325, 366 Ehlers, B. 11,12 Ehrenfest, P. 275, 285, 288 Ehrke, H.-J. 91-93 Eidson, W.W. 225, 228, 234, 235 Einstein, A. 208, 210 Eiselt, M. 75 Ekert, A. 207 Elattari, B. 359 Elk, M. 228, 263 Ellis, A.D 65, 79 Enger, J. 170 Engler, H. 175, 189 Erber, T. 310 Ertmer, W. 180, 181, 185, 192, 195 Espinosa, V 266 Eteson, DC. 225, 228 Eugenieve, E.D 156
Falter, S. 45 Fang, S. 155, 185 Fang-Yen, C. 229 Fathi, D. 188 Feder, K.S. 55 Feick, H. 364 Feiste, U. 88, 89, 91, 113 Fejer, M.M. 94,113 Feld, M.S. 207, 229 Feigner, H. 167 Feng, B. 155, 185 Fenichel, H. 135, 136 Femandez-Soler, J.J. 255, 256, 258, 264, 266
Ferrari, G. 191 Ferwerda, H.A. 277 Feugnet, G. 41 Ficek, Z. 173 Filippidis, G. 331, 340 Finn, S.G. 60 Firth, W.J. 217 Fisher, Y. 152 Fjelde, T. 93, 95 Florez, L.T. 356 Foley, J.T. 303, 306 Font, J.L. 255, 256, 258, 264, 266 Foot, C.J. 175 Forrest, S.R. 93 Frahm, K.M. 358 Frahm, R. 57 Franke, D. 92, 93 Franke-Amold, S. 199 Franken, RA. 214 Freilikher, V 363 Frezza, F. 160 Friberg, A.T. 141, 142, 295 Friedlander, FG. 276, 281, 282 Friedman, M. 85, 86 Friedman, N. 132, 153, 157, 175, 188 Friese, M.E.J. 170, 173 Friesem, A.A. 3, 19, 21-24, 49, 50, 156 Frignac, Y. 59 Frolich, D 232 Frolov, S.V 332, 357 Fujimura, M. 94, 113 Fnkaishi, M. 55 Fuks, B.A. 295, 301 Fukuchi, K. 59 Fulton, D.J. 172 Furcinitti, P 225, 228 Furst, W. 92 G Gaborit, F 95 Gahagan, K.T. 166, 168 Galajda, R 172 Gallatin, G.M. 185 Gamliel, A. 277, 285 Gampel, L. 211 Gamper, E. 93 Gao, J.Y 234, 235 Gao, W. 128, 148, 153, 155, 175, 177, 191 Garces-Chavez, V 160 Garcia-Ojalvo, J. 266, 267
Author index for Volume 45 Gardner, J.R. 251, 252 Garrett, C.G.B. 213, 214 Garwin, R.L. 40, 217 Gasparyan, A. 57 Gates, J. 57 Gauthier, D.J. 212, 219-224, 231, 248-253, 255-260, 264, 266 Gbur, G. 277, 291, 296, 303, 306 Gehm, M.E. 198 Geltman, S. 214 Genack, A.Z. 323, 329, 331 George, R. 57 Giacobino, E. 226, 235 Gingras, M. 226 Gini, E. 93 Girard, C. 285 Glesk, I. 60, 75, 76, 78-81, 84-86, 93, 95-98, 100, 103, 106-113 Glownia, J.H. 230, 231 Glushko, B. 158, 173 Gnass, D.R. 231 Gnauck, A, 57 Goedecke, G.H. 276, 282, 286, 306, 310 Goh, C.S. 57 Gokhale, M.R. 93 Goldner, R 325 Goltsos, W.C. 21, 22 Gomes, A.S.L. 327, 341, 365 Goppert-Mayer, M. 210, 214 Gordon, E.B. 217 Goring, R. 29, 30, 32, 45 Gorlitz, A. 198 Gortz, R. 262 Gouedard, C. 324, 325 Gouesbet, G. 167 Gould, H. 213 Gould, P.L. 185 Goy, R 237, 238 Goyal, S. 57 Graf, Th. 5 Granade, S.R. 198 Grattan, K.T.V 58 Greenstein, XL. 211 Grehan, G. 167 Greywall, D.S. 57 Griffith, J.E. 57 Griffiths, D.J. 287 Grimm, R. 128, 174-178, 180, 189 Grischkowsky, D. 229, 231 Gross, H. 59
375
Gruner-Nielsen, L. 55 Grynberg, G. 226, 235, 242 Guillemot, I. 95 Gunning, R 60 Gupta, S. 198 Gurevich, G.L. 216 Gussgard, R. 167 Gustavson, T.L. 198 Gustavsson, M. 93 Gutierrez-Vega, J.C. 146-149, 163 H Haake, F. 359, 362 Haarer, D. 330 Habashy, T. 299 Hackenbroich, G. 359, 362 Hadamard, J. 291 Hagley, E.W. 193 Haist, T. 169 Haken, H. 222, 235, 263, 264, 266 Hall, D.R. 10,11 Hall, XL. 214 Hall, K.L. 60, 88, 89, 113 Hamaide, X-R 59 Hamao, N. 93 Hammes, M. 180 Hamza, M.Y. 239 Haner, M. 60, 62, 64 Hanna, D.C. 7, 8, 10, 11,35,45 Hansch, T.W. 213, 252 Hansen, RB. 55 Harada, Y. 166, 168, 169 Harbison, XR 356 Haroche, S. 219, 235-239, 264, 266, 267 Harris, S.E. 213 Harrison, R.G. 157 Harter, D.X 241 Hasman, E. 17-20, 47-50, 156 Hatakeyama, H. 84, 113 Haus, H.A. 60 He, H. 130, 140, 170 He, L.S. 239 Heatley, H.R. 217 Hechenblaikner, G. 175 Heckenberg, N.R. 124, 130, 138, 140, 170, 173 Heliotis, G. 331 Hellweg, D. 192 Helmerson, K. 193 Hemmerich, A. 252
376
Author index for Volume 45
Hendow, S.T. 266 Heneghan, S.R 174 Henin, E 311 Hennrich, M. 267 Herman, G.T. 293 Herman, R.M. 132 Herrmann, J. 361 Herzog, U. 263 Hess, R. 93 Hess, T.R. 174 Hickey, J.R 57 Hill, A.E. 214 Hill III, W.T. 128, 182 Hinklin, T. 365 Hirano, T. 134, 174, 180, 195 Hirota, O. 262 Ho, K.M. 363 Ho, S.T. 332, 341, 358, 364, 365 Ho, T.L. 195 Hoefer, B. 46 Hoenders, B.J. 276, 277, 291, 296, 299 Hogervorst, W. 130 Holm, D.A. 219 Holzwarth, R. 213 Hope, J.J. 193, 194 Hopkings, S.A. 175 Hoshimiya, T. 222 Hounsfield, G.N. 293 Hsu, L. 55 Hu, G. 263 Hu, Gang 266 Hu, J. 188 Hu, W. 329 Huang, J.D. 357 Huang, M.H. 364 Huggins, H.A. 57 Huignard, J.P. 41 Hulet, R.G. 248 Husson, D. 324, 325 I Idler, W. 59 Imamura, S. 261 Imbert, C. 167 Inaba, H. 222, 356 Indelicato, P. 213 Inguscio, M. 191 Inouye, S. 191 Ippen, E.R 60 Ironside, C.N. 217
Ishaaya, A.A. 156 Ishimaru, A. 333 Ishizuki, H. 113 Islam, M.N. 71 Ito, H. 126, 127, 181, 183, 264 Ito, T. 59, 330 Itoh, M. 113 Iturbe-Castillo, M.D. 146, 147 Iwatsuki, K. 78 Izawa, T. 34, 46
Jackson, D.J. 233-235 Jackson, J.D. 278, 282, 286, 307 Jacob, J.M. 65 Jahn, E. 92, 93 James, D.RV 277 James, R.W. 300 Janz, C. 93 Jaroszewicz, Z. 144—146 Jepsen, K.S. 93 Jessen, RS. 187 Jewell, J.L. 356 Jhe, W. 121, 126, 127, 148, 150, 151, 176, 181-183 Jiang, X. 336, 358-361, 363, 366 Jin, S. 57 Jinno, M. 68, 71 Joergensen, C. 93 Joffe, A. 158, 188 John, S. 301, 320, 330, 347 Jones, D.R. 10,11 Jourdan, A. 59 Judy, A.E 55 K Kaino, T. 74 Kaiser, W. 213, 214 Kak, A.C. 294 Kamatani, O. 65, 96 Kan, C.K. 55 Kane, M. 75, 76 Kang, K.I. 79-81, 86, 96-98, 100, 107, 108 Kanskar, M. 241 Kantor, K. 55 Kaplan, A. 132 Kaplan, A.E. 216, 222, 264 Kapoor, R. 173 Kasamatsu, T. 59 Kasevich, M. 190, 191
Author index for Volume 45 Katharakis, M. 212 Kato, T. 84,113 Katoh, K. 57 Kawagishi, Y. 332, 357 Kawanishi, H. 57 Kawanishi, S. 55, 64, 65, 70, 96, 113 Kazak, N.S. 130 Kelly, A.E. 79 Kempe, M. 329,331 Keren, S. 49, 50 Kerker, M. 299, 302 Kessler, TJ. 152 Ketterle, W. 158, 188, 191, 198 Khaykovich, L. 5, 17-20, 26, 29, 47-50, 152, 153, 157, 163, 175, 188 Kheifets, M.I. 216 Khilo, N.A. 130 Khoshnevisan, M. 32 Khronopulo, Yu.G. 216, 222, 264 Kikuchi, C. 261 Kikuchi, K. 57 Kim, I. 356 Kim, J. 57 Kim, K. 121, 148, 150, 151, 176, 181, 182, 277, 279, 285, 286 Kimble, H.J. 239 Kind, H. 364 Kindmo, T. 167 King, T.A. 130 Kingsland, D.M. 360 Kintzer, E.S. 60 Kip, D. 156 Kirsanov, B.P. 215 Kitamura, N. 166 Kitamura, S. 94 Kitoh, T. 64,113 Kleiner, V 49 Kleinman, D.A. 214 Klemens, F.P. 57 Klimov, VV 185 Kloch, A. 93, 95 Knize, R.J. 189 Kobayashi, T. 14, 15, 45 Kokta, M.R. 325 Kolodner, P. 57 Kolodziejczyk, A. 144-146 Kozuma, M. 193 Kraeplin, A. 46 Krajinovic, V 88, 89 Kraus, J. 57
'ill
Kravtsov, VE. 364 Kress, R. 284 Kristensen, M. 155 Kritchman, E.M. 21 Kryukov, PG. 321 Kryzhanovsky, B. 158, 173 Kuga, T. 134, 174, 180, 195 Kuga, Y. 333 Kuhn, A. 267 Kuhn, R. 128, 162 Kulin, S. 132, 175 Kul'minskii, A. 255, 256, 258, 264, 266 Kumar, B. 57 Kumar, N. 363 Kumar, P 252, 338 Kuppens, S. 180, 181, 185 Kurisu, M. 55 Kwon, N. 182
Labzowsky, L.N. 213 Lacey, J.PR. 91 Lagendijk, A. 301, 326, 328-331, 333, 348, 363 LaHaie, I.J. 276, 277, 291, 296 Lai, W.K. 180, 187 Lai, W.Y 57 Laine, R.M. 365 Lambropoulos, P 212, 228, 230, 261, 263 Lan, YP 157 Landau, L.D. 40 Lange, W. 248 Lankard, J.R. 231 Lapeyere, G.J. 187 Larat, Ch. 41 Laurent, P 213 Lavrov, A.V 324 Lawandy, N.M. 327-332, 341, 363, 364 Leanhardt, A.E. 198 Lee, H.J. 190, 191 Lee, H.S. 135, 136 Lee, J.F. 360 Lee, K. 121, 148, 150, 151, 176, 181, 182, 332 Lee, R. 360 Lee, R.K. 356 Lee, S.S. 166, 167, 169 Lee, T.C. 57 Lee, YH. 356 Leger, J.R. 21, 22
378
Author index for Volume 45
Lembessis, VE. 180, 187 Lemonde, P. 213 Lemer, I.V 364 Letokhov, VS. 184, 185, 217, 321, 322, 366 Levenson, M.D. 214 Levi, A.F.J. 356 Lewenstein, M. 239, 241, 244-246, 248, 249, 264, 266 Lezama, A. 241 Li, B. 365 Li, L.F. 324 Li, Q. 363 Li, X. 263 Li, Y.Q. 198 Li, Z. 329 Liang, Y. 71 Liao, P.P. 226,229,231 Liao, Y. 45 Lichmanov, A.A. 324 Lichmanova, VN. 324 Lichtenwalner, C. 57 Liesen, D. 213 Lieuwen, D. 57 Lifshitz, E.M. 40 Lin, M.T. 57 Lin, W. 93 Lin, Y. 181 Ling, Y 338, 346, 348, 360, 366 Lipeles, M. 211 Littman, M.G. 124, 126 Liu, B. 366 Liu, C. 32 Liu, C.-H. 328,330,331 Liu, F. 330 Liu, J.Q. 57 Liu, X. 346, 358, 360, 365 Logan, R.A. 62, 64, 356 Long, Q. 128, 153, 155, 191 Loosen, P 11,12,45 Lou, J.W. 71 Loy, M.M.T. 229,231 Lu, J. 37, 38, 46 Lu, N. 263 Lu, W. 157 Lucek, J.K. 60, 92, 96 Ludwig, R. 88-91, 105, 113 Lugiato, L.A. 262 Lukasik, J. 213 Liithy, W. 39, 40, 46
M MacDonald, M.P 170-172 Machavariani, G. 156 MacVicar, I. 147-149, 159, 163 Mahdi, M. 324, 325 Maioli, P 267 Mair, A. 198 Majeed, M. 263 Maki, J.J. 229 Malcuit, M.S. 212, 229, 259 Mamaev, A.V 156 Mandel, L. 207, 261, 262, 281, 283, 288-290, 296 Manek, L 128, 174-176, 180, 189 Manning, R.J. 79, 92 Mansfield, W.M. 57 Manson, N.B. 248 Mao, S. 364 Maradudin, A.A. 362 Marengo, E.A. 277, 296, 306 Maret, G. 333 Mark, J. 75 Marksteiner, S. 181 Markushev, VM. 323, 324 Martin, A. 158, 188 Martin, J.PD. 248 Martorell, J. 332 Maruo, H. 37, 39 Marzlin, K.P 187 Masahiro, M. 229 Masalov, A.V 187 Masuda, H. 56 Masuhara, H. 166 Mathason, B.K. 95 Matsubara, H. 56 Matsui, Sh. 34, 46 Matsumoto, T. 68 Matsunaga, T. 60 Mavoori, H. 57 McCall, S.L. 301, 356 McClelland, J.J. 185 McDuff, R. 124, 138, 140 McGloin, D. 166 McGreevy, J.A. 332 Mclntyre, D.H. 188 Mcleod, J.H. 128 McNeil, K.J. 261-263 Meacher, D.R. 176 Melchior, H. 93 Melendes, R. 32
Author index for Volume 45 Merlemis, N. 212 Meschede, D. 207 Messerschmidt, B. 46 Meyer-Vemet, N. 311 Meyerhofer, D.D. 152, 173 Meystre, P. 224-226 Miceli, J.J. 136 Miesner, HJ. 191 Migus, A. 324, 325 Mikkelsen, B. 93 Milam, D. 128, 182 Milman, P. 267 Mineo, N. 57 Minogin, VG. 182 Mirov, S.B. 325 Misawa, H. 166 Mishchenko, E.G. 362 Misirpashaev, T.Sh. 358, 363 Mitra, A. 364 Miyamoto, I. 37, 39 Modugno, G. 191 Mohanty, B.K. 261 Molina-Terriza, G. 198 Moodie, D.G. 60 Moores, J.D. 60 Mori, K. 55, 64 Morie, M. 59 Morin, S.E. 231, 246-250, 253 Morioka, T. 64, 65, 70, 113 Mork, J. 75 Morris, M.D. 57 Morsch, O. 176 Morse, PM. 303 Mortimore, D.B. 66 Moseley, R.R. 172 Moskvin, Y.L. 217 Moslener, U. 175, 180, 189 Mossberg, T.W. 231, 239, 241, 244-250, 253, 264, 266 Mostowski, J. 302 Motamedi, M.E. 32 Movsessian, M.E. 212 Moy, CM. 193, 194 Muller, C. 282 Miiller, G. 207 Muller, O. 167 Munger, C.T. 213 Murai, H. 57 Muratov, V 57 Muthukrishana, A. 199
N Nachman, A.I. 277, 285, 300 Nakagomi, T. 264 Nakaki, T. 181 Nakamura, K. 55 Nakamura, S. 84, 93, 94, 113 Nakatsuka, H. 229 Nakazawa, M. 71, 74, 113 Narducci, L.M. 225, 228, 234, 235 Narum, P 241 Nasreen, T. 263 Nayak, N. 241, 261 Nefiodov, A.V 213 Neilson, D.T. 57 Nelson, L.E. 55 Nesset, D. 79 New, G.H.C. 146-149, 163 Nha, H. 181 Nielsen, T.N. 55 Niering, M. 213 Nieto-Vesperinas, M. 282, 285 Nijander, C. 57 Nikolaus, B. 232-234 Nill, M. 175, 189 Ning, C.Z. 220, 222, 235, 266 Nishi, S. 78 Nishikawa, M. 55 Nitta, I. 95 Noginov, M.A. 324, 325, 330, 366 Noginova, N.E. 324, 325, 330, 366 Noh, H. 121, 148, 150, 151, 176, 181 Nolan, D.A. 71 Novick, R. 211 Nuzman, C. 57 O Obada, A.-S.E 266 O'Brien, J.D. 356 Ogasahara, D. 59 Oh, K. 181 O'Hara, K.M. 198 Ohhira, R. 59 Ohtsu, M. 126, 127, 181, 183 Oka, T. 56 Okabe, Y. 57 Okada, J. 229 Olsson, B.-E. 95, 113 Olsson, N.A. 62, 64, 72 O'Neil, A.T. 135, 147-149, 159, 163 O'Neill, A.W. 75
379
380
Author index for Volume 45
Ong, H.C. 332, 364, 365 Ono, T. 59 Oohara, T. 60, 74 Orlowski, A. 302 Oraios, P. 172 Orozco, L.A. 132, 175 Osbom, R.K. 261 Osgood, W.E 284 Osnaghi, S. 267 Ostroumov, V 324, 325 Ouchi, K. 56 Ovadia, S. 222, 235, 265, 266 Ovchinnikov, Yu.B. 128, 174^178, 189 Ozaki, M. 332, 357 Ozeri, R. 152, 153, 157, 163, 175
Paasschens, J.C.J. 363 Padgett, M.J. 123, 133-135, 140, 146-149, 155, 159, 163, 166, 169, 170, 173, 199 Pack, U. 181 Painter, O. 356 Paitz, J. 325 Pang, G. 330, 347 Papademetriou, S. 241 Papadogiannis, N.A. 340, 365 Papas, C.H. 278 Papazian, R. 57 Papazoglou, T.G. 331, 340, 365 Parameswaran, K.R. 94,113 Pardo, F. 57 Parente, E 213 Park, J.H. 55 Park, S.Y. 55 Parrent, G.B. 296 Passante, R. 242 Patel, N.S. 88, 89 Paterson, C. 141, 143, 144 Paterson, L. 170-172 Patra, M. 358, 359, 362, 363 Patrick, D.M. 65 Patte, P 213 Pau, S. 57 Pearle, P 275,311 Pearton, S.J. 356 Pedersen, R.J.S. 93 Peet, VE. 158, 173 Peker, B. 132, 175 Pendock, G.J. 91 Perkins, A.E. 332, 363
Perrier, PA. 96 Persico, E 242 Petermann, K. 88, 89 Peters, C.W. 214 Petrosyan, D. 230 Petrov, D.V 187 Pfister, O. 231, 253, 255-260, 264, 266 Phillips, J.D. 79 Phillips, W.D. 193 Pieper, W. 92 Pierattini, G. 131 Piestun, R. 123 Pigier, C. 156 Pitcher, D. 60 Platzman, RM. 301 Plunien, G. 213 PochoUe, J.P 41 Poehhnann, W. 59 Poelker, M. 252 Poelwijk, EJ. 331, 348, 363 Poingt, E 95 Pokasov, P 213 Poison, R.C. 357 Poprawe, R. 11, 12, 45 Porter, R.P 296 Possner, T. 29, 30, 32, 45, 46 Poustie, A.J. 79 Power, W.L. 133, 180, 185-187 Prabhu, M. 37,38,46 Pradhan, R 363 Prastein, S.M. 310 Prentiss, M.G. 248 Prigogine, I. 311 Pritchard, D.E. 158, 188, 198 Prokhorov, A.M. 209, 215, 260, 264 Prucnal, PR. 60, 75, 76, 78-81, 84-86, 93, 95-98, 100, 103, 106-113 Prybyla, J.A. 57 Pu, H. 191 Pustihiik, M. 363 Q Qamar, S. 239 Quade, M. 45 R Radic, S. 55 Rahman, B.M.A. 58 Rahn, L.A. 252, 253 Raikh, M.E . 357, 358, 363--365
Author index for Volume 45 Raimond, J.M. 219, 235-239, 264, 266, 267 Ramirez, A. 57 Ramsey, D. 57 Rand, S.C. 365 Ratner, M.A. 346, 358, 360, 361, 363, 365 Rau, L. 95 Rauner, M. 180, 181, 185 Rauschenbach, K.A. 60, 88, 89, 113 Raymer, M.G. 241 Razmi, M.S.K. 263 Rediker, R. 32 Reichert, J. 213 Reicherter, M. 169 Reid, M.D. 262, 263 Rempe, G. 267 Ren, K.F. 167 Renaud, M. 93, 95 Renn, MJ. 181 Righini, R. 301 Riis, E. 188 Rioux, M. 124, 128 Ritchie, W.M. 248 Roati, G. 191 Roberts, J.L. 191 Robertson, D.A. 155, 159 Robinson, EJ. 214 Rodriguez-Dagnino, R.M. 146, 147 Rogers, D.C. 79 Rokni, M. 211,212,233 Roldan, E. 225, 266 Rolston, SX. 132, 175, 181, 193 Roosen, G. 167 Rosamilia, J.M. 57 Rottwitt, K. 55 Rowe, D.J. 262 Rozas, D. 140 Rubinsztein-Dunlop, H. 124, 130, 140, 170, 173 Ruel, R. 57 Runser, R.J. 60, 79, 81, 84-86, 103, 106, 108, 110, 112 Rusek, M. 302 Russo, R. 364 Rychtarik, D. 180 Ryf, R. 57 Ryzhevich, A.A. 130 S Sacks, Z.S. 140, 360 Safiman, M. 156
381
Sahara, A. 71,113 Saito, Y. 14, 15, 45 Sakaki, K. 126, 127, 181, 183 Salamo, G.J. 210 Salomon, C. 213 Salpeter, E.E. 210,211 San Miguel, M. 266, 267 Sankur, H.O. 32 Sanots, J.P. 213 SantarelU, G. 213 Santarsiero, M. 160 Sargent III, M. 219, 222, 224-226, 235, 265, 266 Sarkisyan, D. 158, 173 Saruwatari, M. 64, 65, 70, 78, 113 Sasada, H. 134, 174, 180, 195 Sasaki, J. 84,113 Sasaki,!. 84,113 Sato, S. 166, 168, 169 Sauer, J.A. 188, 189, 191 Sauteret, C. 324, 325 Sauvain, E. 327, 341 Savage, CM. 181, 193, 194 Scheinfein, M.R. 185 Scherer, A. 356 Schettini, G. 160 Scheunemann, R. 252 Schieve, W.C. 261 Schiffer, M. 180, 181, 185 Schlemmer, H. 232 Sehliwa, M. 167 Schmidt, C. 91,113 Schomerus, H. 358 Schott, G.A. 276, 310 Schreiber, P. 29, 30, 32, 45, 46 Schubert, C. 88, 89, 91, 113 Schubert, M. 219 Schuenmann, U. 175, 189 Schultz, S. 301 Schwarz, M. 41 Scifres, D.R. 43 Sczaniecki, L. 263, 264 Sebbah, R 361, 363 Seelig, E.W. 332, 354, 358, 364, 365 Segal, D. 245, 264 Segev, M. 156 Sekiya, K. 59 Selivanenko, A.S. 215, 217 Sengstock, K. 180, 181, 185, 192, 195 Seo, S.W. 109
382
Author index for Volume 45
Set, S.Y. 57 Sha, C. 263 Sha, W.L. 328,330,331 Shake, I. 55, 64 Shamir, J. 123 Shapiro, B. 358, 363, 364 Shapiro, J. 210 Shapiro, J.H. 262 Shea, H.R. 57 Shechter, R. 3, 19 Shen, Y.R. 212 Shepherd, S. 172 Sherman, G.C. 276 Shi, H. 95 Shibuya, Y. 57 Shiga, F. 261 Shimizu, Y 134, 174, 180, 195 Shimoda, T. 84, 113 Shin, S. 108 Shiokawa, N. 134, 174, 180, 195 Shkunov, M.N. 365 Shukri, M. 365 Sibbett, W. 170-172 Siddique, M. 329, 331 Sidorov, A.I. 174, 176 Simkin, D.J. 229 Simoni, A. 191 Simonneau, C. 59 Simpson, J.R. 62, 64 Simpson, N.B. 159, 166, 169, 170 Sinclair, B.D. 172 Slaney, M. 294 Slusher, R.E. 356 Smith, C.R 124, 138, 140 Smith, G.M. 155 Smith, K. 60, 96 Smith, R. 141, 143, 144 Smith, W.V 216, 217, 220 Snadden, M.J. 188 Soding, J. 177, 178 Soff, G. 213 Soh, H.T. 57 Sokoloff, J.R 75, 76, 78, 84, 85, 100, 113 Sommerfeld, A. 275 Sonek, G.J. 167 Sones, B.A. 229 Song, Y 128, 182 Songen, S. 264 Sorokin, RR 209, 210, 216-220, 228, 230, 231,260,264
Soshchin, N.R 324 Soskin, M.S. 138, 140 Soukoulis, CM. 336, 358-361, 363, 366 Southwell, W.H. 32 Spagnola, G.S. 160 Sparbier, J. 233, 234 Spektor, B. 123 SpitzerJr, L. 211 Spreeuw, R.J.C. 133, 167, 169 Sprik, R. 329,331,363 Squicciarini, M. 234, 235 Srivastava, A.K. 55 Stamper-Kum, D.M. 191 Stancil, RC. 213 Stenger, J. 191 Stenholm, S. 214 Stenner, M.D. 231, 253, 255-260 Stentz, A.J. 55 Stewart, B.W. 135, 136 Stocker, J.C. 71 Stone, H.S. 108 Storkfelt, N. 93 Story, J.G. 248 Strasser, T.A. 55 Strickler, J.H. 213 Strini, G. 262 Stroud Jr, C.R. 199, 224, 225, 241 Stubkjaer, K.E. 93 Studenkov, RV 93 Stulz, S. 55 Stwalley, W.C. 174 Sugimoto, T. 84, 113 Sugimoto, Y 93, 94 Suhara, T. 113 Sulhoff, J. 55 Sulhoff, J.W. 55 Sun, Y 55 Suzuki, H. 56 Suzuki, K. 74, 78 Svelto, O. 220, 222, 233 Svoboda, K. 166 Swain, S. 235, 263 Swanson, E.A. 60 Swartzlander Jr, G.A. 140, 166, 168 Sypek, M. 144, 145 T Tabosa, J.W.R. 187 Taflove, A. 360 Tajima, K. 84, 93, 94, 113
383
Author index for Volume 45 Takara, H. 55,64,65,70, 113 Takatsuji, M. 224 Takeda, N. 57 Takekoshi, T. 189 Takushima, T. 57 Tamanuki, T. 84,113 Tamm, C. 124 Tanbun-Ek, T. 62, 64 Taniguchi, H. 356 Tanno, N. 222 Tanosaki, S. 356 Tanoue, T. 56 Tatsuhara, S. 332, 357 Teffeau, K. 57 Teller, E. 210,211 Tempere, J. 193, 195 Tepichin, G.A. 146, 147 Ter-Gabrielyan, N.E. 324 Terano, A. 56 Thareja, R.K. 364 Themistos, C. 58 Thomas, J.E. 198 Thompson, T. 324, 325 Tikhonenko, V 156 Tino, G.M. 131 Tiziani, H.J. 169 Toliver, P. 60, 79, 81, 84, 85, 103, 106, 108-111 Tolk, N. 211 ToUett, J.J. 248 Tomita, M. 329, 330 Tong, W. 85, 86 Toor, A.H. 239 Toptchiyski, G. 88, 89 Torii, Y. 134, 174, 180, 195 Tomer, L. 198 Torrent, M.C. 266, 267 Torres, J.P. 198 Toschek, P.E. 232-234 Totsuka, K. 330 Tran, P 59 Trebino, R. 252, 253 Treusch, H.G. 11,12 Truscott, A.G. 173 Tsubin, R.V 158, 173 Tsujita, K. 356 Tsukada, M. 60 Tucker, R.S. 91 Tung, J.H. 210 Tumbull, G.A. 155
Turunen, J. Tutov, A.V
141, 142 362
U Uchimura, R. 34, 46 Uchiyama, K. 55, 64, 65, 70, 113 Udem, T 213 Ueda, K. 37, 38, 46 Ueno, Y. 84,94,113 Uno Ingard, K. 303 Urchueguia, J.F. 266 Uzunoglu, N.K. 302 V Vaa, M. 93 van Albada, M.P 328, 333, 363 van Berlo, W. 93 van der Veen, H.E.L.O. 133, 134 van Exter, M.P 359 van Soest, G. 329-331, 348, 363 Vanneste, C. 361, 363 Vardeny, Z.V 332, 357, 365 Vasara, A. 141, 142 Vasnetsov, M.V 138, 140 Vaziri, A. 198, 199 Veith, G. 59 Vella, R. 57 Vengsarkar, D.S. 55 Venkateswarlu, P 324, 325, 330 Vigoureux, J.M. 285 Vilaseca, R. 222, 225, 255, 256, 258, 264, 266, 267 Villareal, F.J. 10, 11 Vinas, S.B. 144, 145 Viviescas, C. 359, 362 Vogt, W. 93 Voigt, H. 263 Volke-Sepulveda, K. 160, 170-172 Volostnikov, V 133 W Wada, H. 57 Wagemann, E.U. 169 Wagner, PC. 275 Walls, D.F. 261-263 Walther, H. 207, 248 Wang, B.C. 85, 86, 95 Wang, H. 128, 153, 155, 191 Wang, J.C. 325, 366 Wang, K.K. 174
384
Author index for Volume 45
Wang, Q.H. 332, 364 Wang, S.C. 157 Wang, W 148, 181 Wang, X. 32, 124, 126, 155, 185 Wang, Y. 121, 128, 148, 150, 151, 153, 155, 174-177, 181-183, 185, 188, 191 Wang, Z. 155, 185 Wang, Z.C. 263, 264 Waseda, Y. 166, 168, 169 Wasik, G. 174, 176 Watanage, K. 56 Webb, R.P. 75 Webb, WW. 213 Weber, E. 364 Weber, H.G. 88-91, 105, 113 Weber, H.P. 39, 40, 46 Webster, S.A. 175 Wei, Changjiang 248 Wei, J. 93 Weidemuller, M. 175, 189 Weihs, G. 198, 199 Weinert, CM. 92 Weinreich, G. 214 Weinstein, M. 276, 310 Weis, A. 57 Weiss, CO. 124, 222 Weitz, M. 213, 252 Welford, W.T. 3, 4, 47 Welling, H. 232 Wen, J. 193 Wetter, N.U. 45 White, A.G. 124, 138, 140 White, CD. 57 Wiederhold, G. 219 Wieman, CE. 181, 191 Wiersma, D. 301, 326, 328, 329, 332, 362364 Wiggins, T.A. 132 Wilhelmi, B. 219, 361 Williams, A. 325 Williams, G.R. 365 Winston, R. 3, 4, 47 Wise, H. 275 Wiseman, H.M. 195 Woerdman, J.R 133, 134, 155, 167, 169, 359 Wolf, C 55 Wolf, E. 142, 207, 261, 262, 276, 277, 279, 281-283, 285-291, 293, 296-299, 303, 306 Wolf, RE. 333 Wolfson, D. 93, 95
Wong, WS. 60 Wood, D. 67 Worland, D.Ph. 43 Wright, E.M. 187, 192, 193 Wright, W.H. 167 Wu, J.Y 332, 364 Wu, Q. 231, 246-250, 253 Wu, Y 364 Wynne, J.J. 233-235 X Xie, C 198 Xu, J. 37, 38, 46 Xu, J.Y. 336, 338, 341, 348, 354, 358, 360, 366 Xu, L. 85, 86, 95 Xu, Shushan 266 Xu, X. 182
Yakubovich, E.I. 216, 222, 264 Yakuoh, T. 34, 46 Yamada, E. 71, 74, 113 Yamada, K. 57 Yamagishi, A. 222 Yamaguchi, S. 14, 15, 45 Yamamoto, T. 71, 113 Yamauchi, Y 57 Yamilov, A. 365, 366 Yan, M. 128, 182, 185, 187 Yang, H. 364 Yang, J. 329 Yang, K.H. 174 Yang, R 364 Yang, Shiping 266 Yariv, A. 356 Yatsiv, S. 211,212,233 Ye, X.M. 210 Yeh, J.R. 189 Yekutieli, G. 21 Yin, J. 121, 128, 148, 150, 151, 153, 155, 174-177, 181-183, 185, 187, 188, 191 Yoh, C 57 Yokohama, I. 74 Yoo, K.M. 328, 330 Yoshida, E. 71,113 Yoshino, K. 332, 357 Yotsutangai, M. 55 Young, J.R 213 Yu, B.Y 108-110
Author index for Volume 45 Yu, D. 157 Yuen, H.R 262 Yurkevich, I. 363
Zacharakis, G. 331, 340, 365 Zakhidov, A.A. 332, 357, 365 Zakrzewski, J. 241, 244, 245, 249, 264, 266 Zbinden, H. 39 Zeilinger, A. 198, 199, 207 Zhang, D. 329 Zhang, D.Z. 232-234 Zhang, J. 45 Zhang, W. 187, 193, 328, 330 Zhang, Y 329 Zhang, Z. 362 Zhang, Z.M. 239 Zhang, Z.Q. 363 Zhao, F. 263
385
Zhao, YG. 332, 364, 365 Zhong, W.D. 60 Zhou, D. 60,79, 84, 106, 110 Zhu, S. 263 Zhu, Y 128, 148, 151, 174-176, 181-183, 185, 187, 188, 191, 239, 241, 244, 246-249, 253 Zielonkowski, M. 175, 189 Zilker, SJ. 330 Zimmermann, C. 252 Zinner, M. 185 Ziolkowski, R.W. 277, 296, 306 Zolin, VF. 323-325 Zoller, P. 181 Zozulya, A.A. 156 Zubairy, M.S. 262, 263 Zych, L.J. 213 Zyuzin, A.Yu. 325, 362, 363
Subject index for Volume 45
-, - rectangular-profile hollow (DRHB) 165, 196 -, doughnut hollow 160, 166 -, focused hollow (FHB) 162 -, Hermite-Gaussian (HGB) 133 -, hollow Gaussian 158 -, Laguerre-Gaussian (LGB) 121, 128, 138, 158, 169-171, 198 -, locahzed hollow (LHB) 144, 162 -, Mathieu 121, 146-149, 163, 164, 196 - , - h o l l o w (MHB) 163 -, ring-shaped hollow 126, 127 — shaping, anamorphic 3, 6, 45, 50 — , fiber-bundle method for 39, 40 — , Fourier-transform technique for 21-26 — , micro step-mirror method for 11-14 — , microprism-array technique for 14-17 — , retroreflector technique for 17-20 — , stacked-glass-plates method for 34—36 — , stubbed-waveguide technique for 37-39 — , tapered-reflecting-tubes method for 4044 — , tilted-cylindrical-lens method for 26-32 — , two-grating-arrays method for 32-34 — , - mirror method for 7-11 BGB, see beam, Bessel-Gaussian birefiingence 87 Bloch-Siegert shift 240 Bom approximation 298, 302 Bose-Einstein condensate 121 , blue-detuned waveguide for 191 , manipulation and control 191 , red-detuned ring trap for 192 , waveguide for 192 — distribution 339 — statistics 261
ac Stark shift 226, 237, 244, 245, 250, 265, 266 acetophenone 168, 169 adiabatic rapid passage 229 alkali-metal atom 235 all-optical demultiplexing 100, 107 - packet routing 100 - - s w i t c h 61, 63,92, 111-113 , based on cross-phase modulation 65 , — passive waveguides 93, 94 , cascaded TOAD 84-87 , gain-transparent SOA-based 89-92 , in networks 94-111 , interferometric SOA-based 71-93 , Mach-Zehnder-based 80, 91 , nonlinear loop mirror (NOLM) as 65 , SOA-based Mach-Zehnder 79 , ultra-fast nonlinear interferometer 8789 amplified spontaneous emission 79, 90, 229, 230, 326 Anderson localization 301 atom optics, apphcation of DHBs 173-188 atomic-beam splitter 187 - fimnels 183 - l e n s 184,187 -motors 185 - wave guide 181
B bad-cavity limit 245 beam, Bessel 129, 135, 136, 141-147, 159 -, Bessel-Gaussian (BOB) 121, 159, 160 - dark hollow (DHB) 121, 158, 196 -, double-Gaussian-profile hollow (DGHB) 161 387
388
Subject index for Volume 45 Hermite polynomial 133 Hermite-Gaussian mode 132 HGB, see beam, Hermite-Gaussian hologram, Bragg volume 21 -, computer-generated 121, 130, 138, 140, 141, 160 hyper-Raman scattering 241
cavity quantum electrodynamics 207 chaotic laser theory 358 coherence, spatial 207, 228 -, spectral degree of 296 -, temporal 207 cooling, Doppler 180 -, of neutral atoms 176-180 -, optical-potential evaporative 189 -, Raman 189 -, rf-induced evaporative 191 Cooper pairing 198 cross-spectral density 288-291, 294
I incoherent source 296 inverse scattering problem 293 - source problem 276, 277, 291-296
D detailed balance 261 DGHB, see beam, double-Gaussian-profile hollow diffraction efficiency 21 disordered media, lasing in 319 distributed Bragg reflector 356 Doppler broadening 237, 261 - effect 246 DRHB, see beam, double-rectangular-profile hollow E erbium-doped fiber amplifier 82 evanescent-wave cooling 175 Ewald sphere of reflection 300,301
Fabry-Perot cavity 320, 321 — interferometer 63 FHB, see beam, focused hollow fiber Bragg grating 57 four-wave mixing (FWM) 61, 62 , phase-matched 212 Fresnel approximation 142, 145 - zone plate 21
Gouy phase shift: 133 gradient force 166 group-velocity dispersion
71
H Helmholtz equation, inhomogeneous 283, 285, 298, 303
278,
K Kirchhofif integral 142 Kramers-Kronig relations
89
Laguerre polynomial 133 laser, distributed feedback semiconductor diode 56,60 -, erbium-doped fiber 60 -, gain-switched DFB 60 -, micro-electromechanical systems 57 -, mode-locked semiconductor 60 -, — Ti: sapphire 333 -, powder 323-327 - random 320, 332, 340, 341, 346, 353, 356-359, 361, 362, 364 - r u b y 212 - paint 327-332 - with scattering reflector 320, 321 LGB, see beam, Laguerre-Gaussian LHB, see beam, localized hollow localization of light 301 localized excitation 303 Lorenz gauge 307 M Mach-Zehnder interferometer 80, 92 - modulator 63 - switches 82 magneto-optical trap 132 maser, single-atom 207 Mathieu equation 163 - fiinction 147, 163 Maxwell's equations 359-361 - in random medium 345 MHB, see beam, Mathieu hollow
Subject index for Volume 45 Michelson interferometer 92, 93, 325 Mie scattering 319 mutual coherence function 288 N nanowire arrays 364 near-field optics 285 nonlinearity, third-order 61 nonpropagating excitation 305 nonradiating source 275-291 — , electromagnetic 286-288 — , partially coherent 288 nonscattering scatterers 297-303 O optical axicon 128 -fiber 55,71 - t r a p 121,166,175 - tweezers 165, 170, 173
partially coherent source 276, 277, 289 period-doubling route to chaos 266 phase matching 62 - transition, first-order 264 — , second-order 264 photon localization 347 - number distribution 338, 339 --fluctuation 341,345 photonic bomb 322 Poisson distribution 338, 339 polarization-sensitive optical isolator 87 Porro prism 19
quantum information 207 -interference 211 - noise 230 quasi-homogeneous source 296
Rabi frequency 225, 240, 242, 246 Raman scattering, anti-Stokes 230 random medium 320, 331 Rayleigh length 133 -scattering 189,319 Rydberg state 236
389
Sagnac interferometer 65, 75, 82, 92 second-harmonic generation 214 - order correlation coefficient 339 self-focusing 228 - phase modulation 67 - pulsing instability 266 semiconductor, metal-oxide 55 - laser amplifiers 75 - optical amplifier (SOA) 64 Sisyphus cooling 123 - - , DHB-based 176, 187 - - , PHB-based 177 SOA, see semiconductor optical amplifier spatial hole burning 358 speckle pattern 325, 349 spontaneous emission 207 squeezed state 262 squeezing 260 superconductivity 198 superfluorescence 229, 230
terahertz optical asymmetric demultiplexer (TOAD) 76 third-harmonic generation 62 time-division multiplexing, optical 55, 5961 TOAD, see terahertz optical asymmetric demultiplexer tomography, computed 293, 294 trapping of neutral atoms 121 two-photon absorption 209, 213, 214 — adiabatic inversion 231 — - amplification 224, 225, 231, 233, 246, 252, 254, 261 — - coherence 225, 226, 228 — gain coefficient 227, 229 — - laser 209, 210, 215-218, 220, 222-232, 261-267 , dressed-state 239-251 , effective Hamiltonian approach to 244 , Raman 251-259 — - maser 210, 219, 235-239, 264 — oscillator 215 - - p r o c e s s 208,210-218 — Rabi frequency 236 — spontaneous emission 211, 219, 236 — stimulated emission rate 209, 219
390
Subject index for Volume 45
W wavelength-division multiplexing , dense 55-59 Weyl expansion 284 X X-ray crystallography
291
68
Z ZnO nanoparticles 334, 341-344, 354 - powder 335-340, 364 zone plate 145, 146
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, H. Kubota 7 Dynamic characteristics of visual processes, A. Fiorentini 8 Modem alignment devices, A.C.S. 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, i?. G. Gioyawe/// 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, E 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 2 3 4
Higher order aberration theory, J. Focke Applications of shearing interferometry, O. Bryngdahl Surface deterioration of optical glasses, K. Kinosita Optical constants of thin films, P. Rouard, P. Bousquet
^ Volumes I-XL were previously distinguished by roman rather than by arable numerals. 391
1- 36 37- 83 85-143 145-197
392 5 6 1
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: Kirchhoff'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 functions, 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, J. Picht
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, P. Beckmann Measurement of the second order degree of coherence, M. Erangon, S. Mallick Design of zoom lenses, A". Kzwfl/Y 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 II: electromagnetic theory, F. Kottler
1- 52 5 3 - 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. Pegis 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, S. 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, C.L. Mehta
1- 50 51-131 133-200 201-237 239-294 295-341 343-372 373-440
Contents of previous volumes
393
VOLUME 9 (1971) 1 2 3 4 5 6 7
Gas lasers and their application to precise length measurements, A.L. Bloom 1- 30 Picosecond laser pulses, A.J. Demaria 31-71 Optical propagation through the turbulent atmosphere, J.W. Strohbehn 73-122 Synthesis of optical birefringent networks, E.O. Ammann 123-177 Mode locking in gas lasers, L. Allen, D.G.C. Jones 179-234 Crystal optics with spatial dispersion, VM. Agranovich, VL. Ginzburg 235-280 Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, J Petykiewicz 281-310 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden 311-407
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 light 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, J.A. 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 l^l-TtAA
VOLUME 13 (1976) 1
On the validity of Kirchhoff'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, WM. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G. Schulz, J. Schwider
1-• 25 27-- 68 69- 91 93--167
394
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, L.A. Riseberg, M.J. Weber The ultrafast optical Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, P.J. Vernier Optical fibre waveguides - a review, P.J.B. Clarricoats
1- 46 ATI- 87 89-159 161-193 195-244 245-325 3>TI-A01
VOLUME 15 (1977) 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, T.W. Cole Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, J.E. Sipe
1- 75 11-137 139-185 187-244 245-350
VOLUME 16 (1978) 1 2 3 4 5
Laser selective photophysics and photochemistry, VS. Letokhov Recent advances in phase profiles generation, J.J. Clair, C.I. 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-mechanicalfi-amework,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-fi-ee 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 refi-action, 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 Pefina
1-126 127-203
Contents of previous volumes 3 4
395
Strong fluctuations in light propagation in a randomly inhomogeneous medium, VI. Tatarskii, KU. 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, S. Ushioda 139-210 Principles of optical data-processing,//./^M^^envecA: 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. Courtis, P. Cruvellier, M. Detaille, M. Saisse Shaping and analysis of picosecond light pulses, C Froehly B. Colombeau, M. Vampouille Multi-photon scattering molecular spectroscopy, S. Kielich Colour holography, R Hariharan Generation of tunable coherent vacuxmi-ultraviolet radiation, WJamroz, B.R Stoicheff
1-61 63-153 155-261 263-324 325-380
VOLUME 21 (1984) 1 2 3 4 5
Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, Z.^. LMg/flto 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, W.C Schieve
1-67 69-216 217-286 287-354 355^28
VOLUME 22 (1985) 1 2 3 4 5
Optical and electronic processing of medical images, D. Malacara Quantum fluctuations in vision, MA. 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^roi;^^a7<2, 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 11-144 145-196 197-270 271-340 341-398
VOLUME 23 (1986) Analytical techniques for multiple scattering fi'om rough surfaces, J.A. DeSanto, G.S. Brown Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka Optical films produced by ion-based techniques, P.J. Martin, R.R Netterfield
1- 62 63-111 113-182
396
Contents of previous volumes
4 Electron holography, A. Tonomura 5 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. Hariharan 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. Take Principles and design of optical arrays, Wang Shaomin, L. Ronchi Aspheric surfaces, G. Schulz
1-190 191-278 279-348 349-415
VOLUME 26 (1988) 1 2 3 4 5
Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals, I.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, E 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, R.J. Cook
1- 86 87-179 181-270 271-359 361^16
Contents of previous volumes
397
VOLUME 29 (1991) 1 Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall 1- 63 2 Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, VD. Ozrin, A.L Saichev 65-197 3 Generation and propagation of ultrashort optical pulses, LP. Christov 199-291 4 Triple-correlation imaging in optical astronomy, G. Weigelt 293-319 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C Flytzanis, E Hache, M.C. Klein, D. Ricard, Ph. Roussignol 321^11 VOLUME 30 (1992) 1 2 3 4 5
Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Eabre 1- 85 Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P Shchepinov 87-135 Localization of waves in media with one-dimensional disorder, VD. Ereilikher, 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. Sundaram 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. Psaltis, Y Qiao Optical atoms, R.J.C. Spreeuw, J.P Woerdman Theory of Comptonfi*eeelectron 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 siHcon: physics, technology and status, B.P Pal 2 Optical neural networks: architecture, design and models, F.T.S. Yu 3 The theory of optimal methods for localization of objects in pictures, LP Yaroslausky 4 Wave propagation theories in random media based on the path-integral approach, M.I. Chamotskii, J. Gozani, VI. Tatarskii, VU. Zavorotny 5 Radiation by uniformly moving sources. Vavilov-Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, VL. Ginzburg 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus
1- 59 61-144 145-201 203-266 267-312 313-361
VOLUME 33 (1994) 1 The imbedding method in statistical boundary-value wave problems, VL Klyatskin 2 Quantum statistics of dissipative nonlinear oscillators, V Pefinovd, A. Luks 3 Gap solitons, CM. De Sterke, IE. Sipe 4 Direct spatial reconstruction of optical phasefi-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, E Wyrowski
1-127 129-202 203-260 261-317 319-388 389-^63
398
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 Wave propagation in inhomogeneous media: phase-shift approach, LP. Presnyakov The statistics of dynamic speckles, T. Okamoto, T. Asakura Scattering of light from multilayer systems with rough boundaries, /. Ohlidal, K. Navrdtil, M. Ohlidal Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss
1-158 159-181 183-248 249-331 333-402
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 1-41 2 Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders 49-128 3 Super-resolution by data inversion, M. Bertero, C De Mol 129-178 4 Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, LA. Apresyan 179-244 5 Photon wave function, /. Bialynicki-Birula 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,/.L. FflZ^^/mi'^/ 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 95-184 185-256 257-343 345-405
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. Mendlovic, 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
399
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 Hght, 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. Mendlovic, 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-\\A 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. Pefina 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
VOLUME 43 (2002) 1 2 3 4
Active optics in modern large optical telescopes, L. Noethe Variational methods in nonlinear fiber optics and related fields, B.A. Malomed Optical works of L.V Lorenz, O. Keller Canonical quantum description of light propagation in dielectric media, A. Luks and V Pefinovd
1- 69 71-193 195-294 295^31
400
Contents of previous volumes
5
Phase space correspondence between classical optics and quantum mechanics, D. Dragoman 6 "Slow" and "fast" hght, R. W Boyd and D.J. Gauthier 7 The fractional Fourier transform and some of its applications to optics, A. Torre
433^96 497-530 531-596
VOLUME 44 (2002) 1 Chaotic dynamics in semiconductor lasers with optical feedback, J. Ohtsubo 2 Femtosecond pulses in optical fibers, EG. Omenetto 3 Instantaneous optics of ultrashort broadband pulses and rapidly varying media, A.B. Shvartsburg, G. Petite 4 Optical coherence tomography, A.E Fercher, C.K. Hitzenberger 5 Modulational instability of electromagnetic waves in inhomogeneous and in discrete media, FKh. Abdullaev, S.A. Darmanyan, J. Gamier
1-- 84 85--141 143--214 215--301 303-366
Cumulative index - Volumes 1-45*
AbduUaev, F.Kh., S.A. Darmanyan, J. Gamier: Modulational instability of electromagnetic waves in inhomogeneous and in discrete media Abeles, R: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical firequencies 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., VL. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.R: Single-longitudinal-mode semiconductor lasers Agrawal, G.R, see Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Radgett, M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefhngent 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. Baby, V, 5ee Glesk, I. Baltes, H.R: On the validity of Kirchhofif's law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, YD. Ozrin, A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-free 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
* Volumes I-XL were previously distinguished by roman rather than by arabic numerals. 401
44, 2, 7, 16,
303 249 139 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 45, 53 13,
1
29, 65 1, 21, 12, 27,
67 217 287 161
402
Cumulative index - Volumes 1-45
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. 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 fimction Bjork, G., see Yamamoto, Y. Bloom, A.L.: Gas lasers and their application to precise length measurements Bokor, N., see Davidson, N. 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. Cao, H.: Easing in disordered media 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., DW. Sweeney: Zone plate coded imaging: theory and applications Chang, R.K., see Fields, M.H. Chamotskii, M.I., 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, W.M. Christov, LP: 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, LI, C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.LB.: 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
53
33, 319 35, 61 18, 257 36, 129 27, 227 36, 1 16, 357 36, 245 28, 87 9, 1 45, 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
17, 85 45, 317 41, 97 16, 289 21, 287 41, 1 32 203
41, 283 37, 345 41, 97 13, 69 29, 199 36, 1 16, 71 14, 327 5, 1 36, 1 15, 187
403
Cumulative index - Volumes 1-45 Colombeau, B., see Froehly, C. Cook, R.J.: 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 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
20, 63 28, 361
Dainty, J.C: The statistics of speckle patterns Dandliker, R.: Heterodyne holographic interferometry Darmanyan, S.A., see AbduUaev, F.Kh. Dattoli, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton free electron lasers Davidson, N., see Oron, R. Davidson, N., N. Bokor: Anamorphic beam shaping for laser and diffuse light De Mol, C , see Bertero, M. De Sterke, CM., J.E. Sipe: Gap solitons Decker Jr, J.A., see Harwit, M. Delano, E., RJ. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.I: 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 function 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 ampHfiers Dutta Gupta, S.: Nonlinear optics of stratified media
14, 17, 44, 31, 42, 45, 36, 33, 12, 7, 9,
Eberly, J.H.: Interaction of very intense light with free 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.-X, 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 Fercher, A.F., C.K. Hitzenberger: Optical coherence tomography Ficek, Z. and H.S. FreedhoflF: 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
20, 1 26, 349 11, 223 20, 1 8, 133 1 1 303 321 325 1 129 203 101 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, 30, 42, 22, 36, 44, 40, 41, 1,
95 1 147 341 1 215 389 1 253
29, 321
404
Cumulative index - Volumes 1-45
Focke, J.: Higher order aberration theory Forbes, G.W., see Kravtsov, Yu.A. FranQon, 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 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 hght 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 diffusion-like models of photon migration in turbid media Gantsog, Ts., see Tanas, R. Gao, W., 5^e Yin, J. Gamier, J., see Abdullaev, FKh. Gauthier, D.J., see Boyd, R.W. Gauthier, D.J.: Two-photon lasers Gbur, G.: Nonradiating sources and other "invisible" objects 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, VL., 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, L: Information processing with spatially incoherent light Glesk, I., B.C. Wang, L. Xu, V Baby, PR. Prucnal: Ultra-fast all-optical switching in optical networks Gniadek, K., J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W: Synthetic-aperture optics Gozani, J., see Chamotskii, M.L Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, VD. Hache, F, see Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, P.: Colour holography Hariharan, P.: 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.
4, 1 39, 1 6, 71 41, 181 40, 389 30, 137 9, 311 42, 325 20, 63 8, 51 41, 283 1, 109 3, 187 34, 35, 45, 44, 43, 45, 45, 18, 13, 17, 30, 31, 9,
333 355 119 303 497 205 273 1 169 85 1 321 235
32, 267 2, 109 24, 389 45, 53 9,281 8, 1 32, 203 12, 233 30, 137 29, 29, 20, 24, 36, 12, 30,
321 1 263 103 49 101 205
405
Cumulative index - Volumes 1-45 Hasman, E., see Oron, R. Heidmann, A., see Reynaud, S. Hello, R: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Herriot, D.R.: Some applications of lasers to interferometry Hitzenberger, C.K., see Fercher, A.R Homer, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images
42, 325 30, 1 38, 85 10, 289 6, 171 44, 215 38, 343 10, 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 Jaeger, G., A.V Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection Jamroz, W., B.R Stoichefif: Generation of tunable coherent vacuum-ultraviolet radiation Javidi, B., XL. Homer: Pattern recognition with nonlinear techniques in the Fourier domain Jones, D.G.C., see Allen, L.
5, 247 3, 29 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 Tumnen, J. Kurizki, G., A.E. Kozhekin, T. Opatmy, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities
5, 1 37 257 43 195 26 105 41 97 20 155 42 1 4 85 28 87 29 321 33, 1 34, 1 30 205 7 1 3 1 4 281 6 331 42 93 26 227 29 65 36 179 39 1 1 211 40 343
Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves
14. 47 11 123
38, 343 9, 179
42
93
406
Cumulative index - Volumes 1-45
Lederer, R, see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and appUcations 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 Luks, A., see Pefinova, V Luks, A., see 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, VL Mallick, S., see Frangon, 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 Mainfi-ay, G. Marchand, E.W.: Gradient index lenses Martin, PL, 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, F, 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 refi-action 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
41, 483 16, 119 6, 1 16, 1 39, 373 8, 343 41, 97 5,287 38, 263 40, 271 35, 61 21, 69
41, 419 33, 129 40, 115 43. 295
28, 87 32, 313 22, 1 33, 261 6, 71 41, 483 42, 93 43, 71 2, 181 13, 27 25, 1 41, 97 32, 313 11, 305 23, 113 22, 145 21, 1 15, 77 8, 373 38, 263 40, 271 30, 261 36, 1 27, 227 7, 231 17, 279 19, 45 31, 1 35, 355 1, 31
Cumulative index - Volumes 1-45
407
MoUow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence Miirata, K.: Instruments for the measuring of optical transfer functions Musset, A., A. Thelen: Multilayer antireflection coatings
19, 1 5, 199 8, 201
Nakwaski, W., M. Osinski: 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.R, see Martin, RJ. Nishihara, H., T Suhara: Micro Fresnel lenses Noethe, L.: Active optics in modem large optical telescopes
38, 41, 25, 34, 23, 24, 43,
165 97 1 249 113 1 1
34, 41, 34, 25, 44, 34, 15, 44, 7, 39, 42,
249 181 249 191 1 183 139 85 299 63 93
Ohlidal, L, 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 Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback Okamoto, T, T. Asakura: The statistics of dynamic speckles Okoshi, T: Projection-type holography Omenetto, KG.: Femtosecond pulses in optical fibers 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.L, see Ostrovskaya, G.V Ostrovsky, Yu.L, VP Shchepinov: Correlation holographic and speckle interferometry Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, VD., 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, ^., 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. Pefina, I : Photocount statistics of radiation propagating through random and nonlinear media
42, 325
35, 61 38, 165 22, 197 22, 197 30, 87 24, 165 33, 319 29, 65 39, 291 32, 1 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67 37, 57 41, 97 18, 127
408
Cumulative index - Volumes 1-45
Pefina, J., see Pefina Jr, J. Pefina Jr, J., J. Peiina: Quantum statistics of nonlinear optical couplers Pefinova, V, A. Luks: Quantum statistics of dissipative nonlinear oscillators Pefinova, V, A. Luks: Continuous measurements in quantum optics Pefinova, V, see Luks, A. Pershan, PS.: Non-linear optics Peschel, T., see Etrich, C. Peschel, U, see Etrich, C. Petite, G., see Shvartsburg, A.B. 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 Prucnal, PR., see Glesk, L Psaltis, D., see Casasent, D. Psaltis, D., Y. Qiao: Adaptive multilayer optical networks
41, 41, 33, 40, 43, 5, 41, 41, 44, 9, 5, 31, 41,
Qiao, Y., see Psaltis, D.
31, 227
Raymer, M.G., LA. Walmsley: The quantum coherence properties of stimulated Raman scattering Renieri, A., see Dattoh, 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, E: 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., J.L. 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.
359 359 129 115 295 83 483 483 143 281 351 139 1
27,315 34, 159 45, 53 16, 289 31, 227
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,
65 1 87 259 1 419 49 389
Cumulative index - Volumes 1-45
409
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, X: Advanced evaluation techniques in interferometry Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework Sergienko, A.V, see Jaeger, G. Sharma, S.K., DJ. Somerford: Scattering of light in the eikonal approximation Shchepinov, VR, see Ostrovsky, Yu.I. Shvartsburg, A.B., G. Petite: Instantaneous optics of ultrashort broadband pulses and rapidly varying media Sibilia, C, see Mihalache, D. Simpson, J.R., see Dutta, N.K. Sipe, J.E., see Van Kranendonk, J. Sipe, IE., see De Sterke, CM. 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, XC. Sodha, M.S., A.K. Ghatak, VK. Tripathi: Self-focusing of laser beams in plasmas and semiconductors Somerford, D.X, see Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices Soskin, M.S., M.V Vasnetsov: Singular optics Spreeuw, R.XC., XP. Woerdman: Optical atoms Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Stoicheff, B.P, see Jamroz, W. Strohbehn, XW.: 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, P.W. 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.
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.X, R.Q. Twiss: Michelson stellar interferometry
25 191 23 63
35, 197 14, 195 17, 163 13, 93 25, 349 13, 93 28, 271 10, 89 16, 413 42, 277 39, 213 30, 87 44, 143 27, 227 31 189 15 245 33, 203 10, 229 39 373 12 53 6 211 10 165 10 45 21 355 13 39 27 42 31 5 37 20 9
169 213 109 219 263 145 345 325 73
2 1 19 45 24 1 31 1 12 1 21 287 8 133
35 355 17 239
410
Cumulative index - Volumes 1-45
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 Mikaelian, 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 DattoH, G. Torre, A.: The fractional Fourier transform and some of its applications to optics Tripathi, VK., see Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering Turunen, J., M. Kuittinen, F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.
40, 77 18, 32, 5, 26, 7, 8, 7, 18, 23, 31, 43, 13,
204 203 287 1 231 201 169 1 183 321 531 169
2, 131 40, 343 17, 239
Upatnieks, J., see Leith, E.N. Upstill, C , see Berry, M.V Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids
6, 1 18, 257 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, LA., see Raymer, M.G. Wang, B.C., see Glesk, I. Wang Shaomin, L. Ronchi: Principles and design of optical arrays Weber, M.L, see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weiss, G.H., see Gandjbakhche, A.H. Welford, WT: Aberration theory of gratings and grating mountings Welford, WT: Aplanatism and isoplanatism Welford, WT, 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.
28, 45, 25, 14, 29, 34, 4, 13, 27,
181 53 279 89 293 333 241 267 161
39, 10, 17, 27, 31,
63 89 163 161 263
15, 6, 37, 42, 14,
245 259 57 219 245
33, 261 39, 63
Cumulative index - Volumes 1-45
411
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, R, see Bryngdahl, O. Wyrowski, R, see Bryngdahl, O. Wyrowski, R, see Turunen, J.
40,
1
1, 10, 28, 33, 40,
155 137 1 389 343
Xu, L., see Glesk, I.
45, 53
Yamaguchi, I.: Pringe 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., 5ee Carriere, J. Yin, J., W Gao, Y Zhu: Generation of dark hollow beams and their applications Yoshinaga, H.: Recent developments in far infi-ared spectroscopic techniques Yu, RT.S.: Principles of optical processing with partially coherent light Yu, RT.S.: Optical neural networks: architecture, design and models 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, VI. Z\m,Y.,seeYm,]. Zuidema, P., see Bouman, M.A.
22, 271 6, 105 8, 295 28, 28, 32, 41, 45, 11, 23, 32,
87 87 145 97 119 77 221 61
38, 263 40, 32, 18, 45, 22,
271 203 204 119 77
