Progress in Optics, Volume 53

EDITORIAL ADVISORY BOARD G.S. Agarwal Stillwater, USA T. Asakura Sapporo, Japan M.V. Berry Bristol, England C. Bro...

Author: Emil Wolf

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EDITORIAL ADVISORY BOARD G.S. Agarwal

Stillwater, USA

T. Asakura

Sapporo, Japan

M.V. Berry

Bristol, England

C. Brosseau

Brest, France

A.T. Friberg

Stockholm, Sweden

F. Gori

Rome, Italy

D.F.V. James

Toronto, Canada

P. Knight

London, England

G. Leuchs

Erlangen, Germany

P. Milonni

Los Alamos, NM, USA

J.B. Pendry

London, England

J. Peˇrina

Olomouc, Czech Republic

J. Pu

Quanzhou, PR China

W. Schleich

Ulm, Germany

PROGRESS IN OPTICS VOLUME 53

EDITED BY

E. Wolf University of Rochester, N.Y., U.S.A.

Contributors U. L. Andersen, B. Crosignani, E. DelRe, M. R. Dennis, P. Di Porto, R. Filip, G. S. He, U. Leonhardt, M. Martínez-Corral, K. O’Holleran, M. J. Padgett, T. G. Philbin, G. Saavedra

Amsterdam • Boston • Heidelberg • London • New York • Oxford • Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2009 Copyright © 2009 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; e-mail: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/ permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made Library of Congress Catalog Card number: 61-19297 ISBN: 978-0-444-53360-9 ISSN: 0079-6638 For information on all Elsevier publications visit our web site at elsevierdirect.com Typeset by: diacriTech, India Printed and bound in Hungary 09 10 11 12 10 9 8 7 6 5 4 3 2 1

PREFACE

This volume contains articles which present reviews of current research in six areas of classical and quantum optics. Some of them deal with theory, others with experiments. The ﬁrst article by M. Martinez-Corral and G. Saavedra gives an account of modern techniques that have been developed to provide high quality microscope images. The techniques include single photon confocal microscopy, the so-called confocal theta microscopy, standing wave microscopy, and two-photon excitation scanning microscopy. The second article by U. Leonhardt and T. G. Philbin reviews researches concerning the use of so-called meta-materials for application in devices that perform geometrical transformations. They have potential applications to production of perfect lenses and cloaking devices. The article by E. Del Re, B. Crosignani, and P. DiPorto which follows, reviews developments in a ﬁeld which originated about a decade ago, namely photorefractive solitons. They are forerunners in the ﬁeld of optical soliton physics, providing the foundation of a rich ﬁeld in the physics of nonlinear waves. The article traces and brings together various elements that form our present day understanding of the underlying physics. The next article, by Guang S. He, gives an overview of developments in the ﬁeld of stimulated light scattering with intense coherent light. It provides a review of the subject and discusses the principles of the materials and experimental features and applications of various scattering processes. The ﬁfth article by M. Dennis, M. Padgett, and K. O’Holleran presents a comprehensive review of a class of interesting phenomena in the ﬁeld of “singular optics” which originated in the 1970’s: namely of “optical vortices and polarization singularities”. The article presents a review of the development since the subject originated and includes discussions of topics such as the orbital angular momentum of light in the context of singular optics. The concluding article on quantum feedback control of light by U. Andersen and R. Filip is concerned with processes of gaining information about the dynamics of physical systems and about using the information to change the system in real time. Such processes are of v

vi

Preface

importance in various branches of traditional engineering but, more recently they have proved to be of interest in the quantum optics domain. Some examples of such developments are presented. As previous volumes in the series, the present one shows again that optics continues to be a dynamic ﬁeld in which interesting new developments are taking place. Emil Wolf Department of Physics and Astronomy and The Institute of Optics, University of Rochester, Rochester, New York 14627 February 2009

CHAPTER

1 The Resolution Challenge in 3D Optical Microscopy Manuel Martínez-Corral* and Genaro Saavedra*

Contents

1 Introduction 2 Basic Theory for Microscope Imaging 2.1 Three-Dimensional Imaging as the Result of Axial Scanning 2.2 The Virtual 3D PSF 2.3 The Optical-Sectioning Capability 2.4 Metrics for the Optical-Sectioning Capability 3 The High-Numerical-Aperture Approach 3.1 Calculation of the PSF 3.2 Calculation of the OTF 3.3 Metrics for Resolution Improvement 3.4 Sampling Expansion 3.5 Spherical Aberration 4 Optical-Sectioning Microscopy 4.1 Confocal Scanning Microscopy 4.2 Structured Illumination Microscopy 4.3 Axially-Oriented Structured Illumination Microscopy 4.4 Two-Photon Excitation Scanning Microscopy 5 Conclusions Acknowledgments References

1 3 8 9 10 11 18 19 22 24 28 31 33 34 43 53 62 63 64 65

1. INTRODUCTION The spatial resolution of optical microscopes is mainly restricted by diffraction, that is, by their capability to produce a tight diffraction spot when imaging a light point source. Since Abbe (1873) formulated his wave theory for microscopic imaging, the improvement of resolution of optical * Department of Optics, University of Valencia, E46100 Burjassot, Spain Progress in Optics, Volume 53, ISSN: 0079-6638, DOI: 10.1016/S0079-6638(08)00201-1. Copyright © 2009 Elsevier B.V. All rights reserved.

1

2

The Resolution Challenge in 3D Optical Microscopy

microscopes has been the aim of many research efforts. The need for smaller spot sizes has induced fruitful researches not only in the case of optical frequencies (Sales, 1998) but also in other spectral ranges like X-ray microscopy (Miao et al., 2002a) or electron microscopy (Miao et al., 2002b). As pointed out by Gustafsson, Agard, and Sedat (1999), the light microscope remains an irreplaceable research tool for modern biology. Unlike any other microscope, it allows the study of samples in vivo or in their native hydrated environments, a highly speciﬁc labeling of multiple components, and the observation of detailed internal structures of 3D samples. In the search for maximum spatial resolution, emphasis has generally been placed on the use of modern microscope objectives with the highest available numerical aperture (NA) (Blanca and Hell, 2002). However, since lens design is by now a mature technology, the light-collecting angle of commercially available objective lenses is close to its practical maximum, with little if any room for further improvement. Another problem of conventional optical microscopy is the fact that even with the best optical elements it is not possible to obtain sharp images of 3D biological samples, since any image focused at a certain depth in the sample contains blurred information from the rest of the sample. This fact gives rise to 3D images with deteriorated contrast. In the last few years, it has been realized that the classical resolution limits, even though imposed by physical laws, can in fact be exceeded (Gustafsson, 1999). Three particularly important assumptions of Abbe theory are the following: (i) observation takes place in the conventional geometry in which light is collected by a single objective lens; (ii) the excitation light is uniform throughout the sample; (iii) ﬂuorescence takes place through normal, linear absorption, and emission of a single photon. The negation of any these assumptions lead to new basic concepts for resolution extension. The negation of assumption (ii) permitted the design of a new class of optical microscope, which not only allows the improvement of lateral spatial resolution but also the achievement of important optical sectioning. We refer to the single-photon confocal scanning microscope (CSM) in which the monochromatic light emanating from a point source is focused onto a small region of the 3D ﬂuorescent sample by a high-NA objective (Brakenhoff, Blom, and Barends, 1979; McCutchen, 1967; Minsky, 1988; Sheppard and Choudhury, 1977). The ﬂuorescent light emitted by the sample is collected by the same objective, and it passes through a pinhole centered at the conjugated point, and is ﬁnally detected by a large-area sensor. An interesting feature of CSMs is that they deliver an electronic image that is available for digital signal processing (Pawley, 1995). The main advantage of CSMs is their sectioning power due to the light rejection from out-of-focus parts of the sample, but since the diffraction pattern of a focused bright spot

Basic Theory for Microscope Imaging

3

is naturally prolated for any NA, CSM suffers from anisotropic imaging, which in turn limits the effective lateral resolution. As a classical result, the related point-spread function (PSF) is the product of the two PSFs of the illuminating and collecting lenses, thereby providing not only an increase of lateral resolution, but mainly the ability to transmit axial frequencies. Another realization of non-uniform illumination is through patterned illumination techniques. In the so-called structured illumination microscopy, a periodic pattern transverse to the optical axis is projected onto the specimen and a stack of 2D images is recorded after scanning the object axially. By proper decoding procedures, it is possible to improve the lateral resolution, as well as to obtain an important capability of optical sectioning. The advantages of structured illumination over CSM are in imaging speed and light efﬁciency (Heintzmann and Ficz, 2006). The negation of assumptions (i) and (ii) has led to the invention of a new scanning imaging technique, which produces 3D images with quasiisotropic resolution. We refer to 4Pi scanning microscopy, in which the sample is illuminated with a periodic pattern parallel to the optical axis and the ﬂuorescence light is collected by two opposed high-NA objectives (Hell and Stelzer, 1992a). A serious drawback of confocal architectures is photobleaching, which appears since the entire sample is bleached when any single plane is imaged. Another disadvantage of this technique when used in biomedical imaging is its poor depth penetration. To solve these problems, negation of assumption (iii) was invoked in proposing two-photon excitation (TPE) scanning microscopy (Denk, Strickler, and Webb, 1990). This non-linear imaging technique relies on the simultaneous absorption of two photons, whereby a single ﬂuorescence photon is emitted (Göppert-Mayer, 1931). The overall ﬂuorescent light is collected, and ﬁnally, the image is synthesized from the 3D sampling of the object. Two-photon excitation generally uses near-infrared light, which suffers less from absorption and scattering by biological tissues, allowing deeper penetration of the excitation beam. Besides, since photobleaching depends here on the time-averaged square of the intensity distribution, it is restricted to the neighborhood of the imaged plane. Two-photon, or even multiphoton, excitation ﬂuorescence microscopy is even more attractive both because the imaging process can be considered as self-pinholed, due to the square dependence of the up-converted signal, and because contrast is dramatically enhanced by the subsequent side-lobe lowering in the illuminating lens PSF.

2. BASIC THEORY FOR MICROSCOPE IMAGING For an intuitive understanding of the optical principles of microscope imaging, it is convenient to start by analyzing a conﬁguration (Figure 1)

4

The Resolution Challenge in 3D Optical Microscopy

that schematizes a conventional wide-ﬁeld optical microscope by means of a telecentric arrangement (Born and Wolf, 1999a). Take into account that modern microscope objectives are usually designed for inﬁnite conjugate ratio (Juskaitis, 2003). A second lens of higher focal length, the tube lens, is used to provide a real image in the neighborhood of its back focal plane (BFP). The aperture stop of such an arrangement is just the aperture stop of the objective, which is usually placed at its BFP. In Figure 1, we have represented the telecentric system in the simplest form. It consists of two thin converging lenses, ideally of inﬁnite diameter, that are coupled in an afocal manner. To allow the system to be telecentric at both the object and the image side, the aperture stop of amplitude transmittance p(xp ) is placed just at the common focal plane. We calculate the image of 3D samples that are labeled with ﬂuorescent materials. When the 3D ﬂuorescent sample is uniformly illuminated by a monochromatic light beam whose wavelength is within the excitation band of the ﬂuorescent dyes, the 3D intensity distribution is proportional to the 3D function O(x, z) that describes the spatial distribution of ﬂuorescence generation in the sample. The ﬂuorescence radiated by the sample can be considered, in quite good approximation, to be quasi-monochromatic and spatially incoherent. To calculate the image of such 3D intensity

(x0 ,y

0 , z0 )

y p (x p,

L1

y p)

yp

x

L2

xp F1

y9 F F91; 2

x9

f1

(x 9 , y 0 9 ,z 0 9 ) 0

f1

F92

f2 f2

FIGURE 1 Schematic of a telecentric, wide-field optical microscope. The light emanating from the object is collected by the objective (L1 ) and focused by the tube lens (L2 ).

Basic Theory for Microscope Imaging

5

distribution, we start by calculating the intensity distribution in the image of an arbitrary point of the sample, for example that at (x0 , z0 ); see Figure 1. We perform at this stage the analysis in a paraxial context. From the point source emanates a monochromatic, spherical (in fact paraboloidal in the paraxial approximation) wavefront. At the front focal plane (FFP) of L1 , the amplitude distribution of the impinging spherical wave is given by (Goodman, 1996)

U(x, z = 0) = −

1 −ikz0 k e exp −i |x − x0 |2 , z0 2z0

(1)

with wave number k = 2π/λ and writing x = (x, y). Any axial distances involved here and in the forthcoming reasoning are, of course, oriented. This is, for example, the case for the focal lengths in Figure 1; their direction and sign are determined by the orientation of the arrow. In the case of z0 , a negative value corresponds to a point placed to the left of F1 . Note that we are considering only wavefronts that propagate in the positive z-direction. Therefore for positive values of z0 , Eq. (1) represents a virtual distribution of amplitudes (Lukosz, 1967; Ojeda-Castañeda and Gómez-Sarabia, 1989). Next, we make use of the well-known fact that the amplitude distribution at the BFP of a thin lens is related to the amplitude distribution at the FFP through a 2D Fourier transformation, namely (Goodman, 1996)

1 ˜ xp U Up (xp ) = , 0 p(xp ), iλf1 λf1

(2)

˜ is the Fourier transform of U. In this equation, we have multiplied where U the impinging wavefront by the amplitude transmittance of the aperture stop, p(xp ), and have omitted the irrelevant constant phase factor exp(i2kf1 ). The amplitude distribution at the BFP of L2 is obtained from Up (xp ) again through a 2D Fourier transformation, that is,

x x 1 1 ˜ x , 0 ⊗2 p˜ Up = U , U (x , z = 0) = iλf2 λf2 M M λf2

(3)

where ⊗2 represents the 2D convolution product, and M = −f2 /f1 stands for the lateral magniﬁcation of the telecentric system. From Eq. (3) it is clear that telecentric arrangements provide, stricto senso, 2D images, since the output is obtained as the 2D convolution between the uniformly scaled input and a 2D function. That function can then be named the PSF of the imaging system. Note however that such ability is not exclusive to telecentric arrangements, since any single lens can produce 2D images.

6

The Resolution Challenge in 3D Optical Microscopy

The speciﬁc feature of telecentric arrangements becomes apparent when one analyzes their response to any arbitrary light point source within 3D object space. If we introduce Eq. (1) into Eq. (3), we straightforwardly ﬁnd that

M −e−ikz0 p˜ z0

U (x , z = 0) =

k x 2 ⊗2 exp −i |x − x0 | , λf2 2z0

(4)

where x0 = Mx0 and z0 = M2 z0 are, respectively, the transverse and axial coordinates of the conjugate of the object point source through the telecentric system (see Figure 1). It is remarkable that both the lateral and the axial magniﬁcation are independent of the coordinates of the point object. The amplitude distribution at any arbitrary plane within the image space, for example at a distance z from the BFP of L2 , is obtained after 2D convolution between the function U (x , z = 0) and the amplitude PSF of free-space propagation, namely

k 2 eikz exp i |x | . iλz 2z

(5)

After straightforward mathematical manipulation, we obtain

U (x , z ) = Mp˜

x k eik(z −z0 ) 2 |x ⊗2 , exp i − x | 0 λf2 z − z0 2(z − z0 )

(6)

which can be rewritten as

U (x , z ) = M h (x − x0 , z − z0 ),

(7)

k 2 x eikz ⊗2 exp i |x | h (x , z ) = p˜ λf2 z 2z

(8)

where

is just the 3D amplitude distribution in the image space generated by a monochromatic point source placed just at F1 . Let us now extend our analysis to the whole 3D ﬂuorescent sample, whose spatial distribution of intensity is given by O(x, z). Note that in general one should take into account that the radiation emitted by distant object points might be scattered or reﬂected by nearer object points.

7

Basic Theory for Microscope Imaging

Such radiation blockage will, of course, affect the intensity distribution in the image and, unless the object is an aerial image, will always be present. However, we will assume that the ﬁrst-order Born approximation holds (Born and Wolf, 1999b) so that the multiple scattering and depletion of the incident beam are negligible. In such case, the image intensity can be summed over the elemental slices that constitute the 3D object, because the ﬁrst Born approximation assumes that the principle of superposition is valid. Then,

∞

I (x , z ) = M

2

2 O(x0 , z0 )h (x − M x0 , z − M2 z0 ) d2 x0 dz0 ,

(9)

−∞

or, in terms of (x0 , z0 ),

1 I (x , z ) = 2 M

∞ 2 x0 z0 , 2 h (x − x0 , z − z0 ) d2 x0 dz0 O M M

−∞

2 1 x z , 2 ⊗3 h (x , z ) . = 2O M M M

(10)

This equation demonstrates that a paraxial telecentric arrangement, when imaging 3D spatially incoherent objects in absence of radiation blockage, has the property of 3D linearity and stationarity. The proof is the fact that the 3D image is obtained as the 3D convolution between a uniformly scaled copy of the object and a 3D function. That function is naturally recognized as the 3D intensity PSF of the telecentric system. The intensity PSF is then obtained as the squared modulus of the function in Eq. (8), which in the integral form is

ikz

∞

h (x , z ) = iλe

p(xp ) exp −i

−∞

k z 2f22

|xp |

2

2π exp −i xp x d2 xp . λf2

(11) Although we are analyzing an incoherent imaging process, it can easily be shown that telecentric arrangements have the property of 3D linearity and stationarity in the case of coherent imaging as well. Thus, we can call the function h (x , z ) the amplitude PSF. Based on the above, one can state that telecentric systems have the ability to produce, stricto senso, 3D images. This capability is exclusive to telecentric arrangements, since focal arrangements produce in the image space

8

The Resolution Challenge in 3D Optical Microscopy

3D intensity distributions in which both the magniﬁcation and the impulse response depend on the axial coordinate.

2.1. Three-Dimensional Imaging as the Result of Axial Scanning Note that the matrix sensors, such as charge-coupled devices (CCD) or complementary metal-oxide semiconductors (CMOS), usually employed for recording the images provided by optical microscopes are 2D. This implies that the intensity distribution calculated in Eq. (10) does not correspond to any real experimental situation, since no 3D matrix sensor is available for recording the 3D image. Instead, in actual microscopy, a 2D matrix sensor is placed just at the BFP of the tube lens, and then, a stack of 2D images is recorded while stepping the object through the in-focus plane. According to Eq. (9), for the primary axial position of the object, the intensity distribution at the sensor plane, z = 0, is

I0 (x )

∞

2 = I x ,z = 0 = M O(x0 , z0 )h x − Mx0 , −M2 z0 d2 x0 dz0 . (12) −∞

2

By stepping the sample an axial distance −zS , the object plane of former axial coordinate zS is now in focus, and the intensity at the detector plane is

∞

I (x , zS ) = M

2 O(x0 , z0 + zS )h x − Mx0 , −M2 z0 d2 x0 dz0 . (13) −∞

2

This can be rewritten after performing the mapping

z = M 2 zS ,

z0 = M2 (z0 + zS ),

and

x0 = Mx0 ,

(14)

with the result

1 I (x , z ) = 2 M

=

∞ x0 z0 , |h (x − x0 , z − z0 )|2 d2 x0 dz0 O M M2

−∞

1 x z , ⊗3 |h (x , z )|2 . O M M2 M2

(15)

We have now shown that the same 3D convolution as in Eq. (10) can be obtained in a realistic imaging experiment. In this hybrid experiment, the 3D image is constructed by computer from a stack of 2D images

Basic Theory for Microscope Imaging

9

recorded when axially scanning the sample. At this point, we deﬁne as “wide-ﬁeld” those microscopes that use a 2D matrix sensor for the acquisition, sometimes after axial scanning, of the image.

2.2. The Virtual 3D PSF If one reads Eq. (10) from left to right, one can understand the imaging process as the result of two successive phenomena, that is, the uniform scaling inherent in typical geometrical-optics imaging, and the diffractive effects due to the limited size of the apertures. The diffractive effects are introduced into the equation by the convolution with the intensity PSF. However, this form of “reading” the 3D imaging is not comfortable for microscopists, since the properties of the image seem to be associated with the characteristics of the tube lens. For this reason, it could be convenient to calculate the 3D image by switching the order of the successive phenomena. Thus, one can ﬁrst calculate the virtual intensity distribution in the object space, which already incorporates the diffraction effects, and then the uniformly scaled copy of the virtual object. The virtual intensity distribution is calculated as

Iv (x, z) = M2 I (M x, M2 z).

(16)

After scaling Eq. (10) accordingly, we obtain the following expression for the virtual object:

Iv (x, z) = O(x, z) ⊗3 |h(x, z)|2 ,

(17)

where

eikz h(x, z) = z

∞

p(xp ) exp −i

−∞

kz 2f12

|xp |

2

2π exp −i xp x d2 xp λf1

(18)

will be called the virtual amplitude PSF. We have found, then, that the image formation capability of a microscope is fully characterized by the intensity PSF, |h |2 , or equivalently by its virtual conjugate in the object space, |h|2 . The latter is more intuitive for microscopists, because it relates the imaging properties of a microscope with the characteristics of the objective. Consequently, the forthcoming analysis will be performed in terms of the virtual intensity PSF, that is, in terms of the intensity distribution obtained when a microscope objective is illuminated by a monochromatic plane wave; see Figure 2. In what follows, we will use Eqs. (17) and (18) to describe the imaging features of optical microscopes, but simplifying notation by omitting the subscripts from the focal length and the intensity.

10

The Resolution Challenge in 3D Optical Microscopy

yp y

xp

x F

f f

F⬘

FIGURE 2 The imaging properties of an optical microscope are mainly determined by the focusing characteristics of the objective.

2.3. The Optical-Sectioning Capability From the aforementioned statements, one could believe that telecentric arrangements are especially qualiﬁed for 3D imaging, since they have 3D linearity and stationarity. However, as explicitly shown in Eq. (13), at any transverse section of the 3D image, the sharp image of the in-focus section of the 3D object is accompanied by the blurred images of out-of-focus sections of the 3D object. This fact signiﬁcantly affects the contrast in the 3D image. At this point, we introduce a new concept for the evaluation of the ability of a system to form 3D images: the optical-sectioning capability is the ability of an imaging system to provide sharp, high-contrast 2D images of the different transverse sections of a 3D object (Wilson, 1990). To evaluate the optical-sectioning capability of telecentric systems, we start by analyzing the imaging features in the spectral domain. Since we are dealing with 3D signals, we can calculate the spectrum of the image by 3D-Fourier-transforming Eq. (17), namely

˜ I(Q) =

I(R) exp(−i2πQ · R) d3 R,

(19)

image

where R = (x, y, z), while Q = (u, v, w) represents the triplicate of spatial frequencies. For later use, we deﬁne u = (u, v). Next, we focus our attention on the axial-frequency content of the image,

˜ 0, w) = I(0,

I(x, z) exp(−i2πwz) d2 x dz.

(20)

image

Taking into account that total power,

∞ I(x, z) d2 x = P, −∞

(21)

11

Basic Theory for Microscope Imaging

must be conserved when varying z, we ﬁnd that

˜ 0, w) = P δ(w), I(0,

(22)

where δ is de Dirac delta function. Equation (22) shows that, independently of the nature of the object and of the pupil function, I˜ is singular at Q = (0, 0, 0). In other words, the axial frequencies of 3D objects are not transmitted by wide-ﬁeld optical microscopes, and therefore, they are absent from the 3D image. Now, we can reﬁne the deﬁnition of the optical-sectioning capability: it is the degree of efﬁciency in transferring the different axial-frequency components of the 3D object. Consequently, we can state that conventional microscopes do not have optical-sectioning capability. It could be, however, surprising to a microscopist to hear about this lack of optical-sectioning capability, as different sections of a probe can be viewed with reasonable image quality simply by changing the axial position of the slide. The response is that, while it is possible to see the 2D sections of 3D objects, these sections have very poor contrast; see Figure 3a. The lack of optical sectioning becomes apparent when one tries to obtain images of structures oriented in the axial direction. Such structures are totally absent from the image; see Figure 3b.

2.4. Metrics for the Optical-Sectioning Capability To work with a general formulation that does not depend on the particular parameters of the microscope, it is usual to deﬁne a set of normalized, dimensionless coordinates. To this end, we take into account that pupil functions have compact support, and we identify by rm the outermost radius of the pupil. In such case, we can rewrite Eq. (18) as

1 h(x, z) =

2 i2πz w0 iλrm e

p(xp ) exp −i πz|xp |2 exp −i 2πxp x d2 xp ,

−1

(23) where the normalized coordinate at the pupil plane is deﬁned as xp = xp /rm . The transverse and axial normalized coordinates within the focal volume are deﬁned as

x=

rm x λf

and

z=

2 rm z. λf 2

(24)

2 . There is also a phase factor that depends on the parameter w0 = f 2 /rm

12

The Resolution Challenge in 3D Optical Microscopy

(a)

y L1

) yp p (xp, yp

x

L2

xp y⬘

F1

f1 f1

F⬘1⬅

F2

x⬘

f2 f2

F⬘2

(b) y L1

yp

x

y ) p (xp, p

L2

xp

y⬘

F1

f1 f1

F F⬘1⬅ 2

x⬘

f2 f2

F⬘2

FIGURE 3 Conventional microscopes provide images in which (a) transverse sections are reproduced, albeit with poor contrast, but (b) axial structure is lost.

In the very typical case in which the pupil function has rotational symmetry, the amplitude PSF can be rewritten as

1 h(r, z) =

2 i2πz w0 i2πλrm e

p(rp ) exp −iπz r2p J0 2π r rp rp drp ,

(25)

0

where rp = rp /rm and r = (rm /λf )r, r being the radial coordinate at the focal volume. As an example, Figure 4 plots the square modulus of Eq. (25) for the case of a circular clear aperture. We can recognize the typical Airy-disk proﬁle in the section z = 0. Along the optical axis one

13

Basic Theory for Microscope Imaging

Normalized lateral coordinate: x

2

1

0

⫺1

⫺2 ⫺3

⫺2

0 1 ⫺1 Normalized axial coordinate: z

2

3

FIGURE 4 Meridian section of the intensity PSF of a wide-field microscope in the typical case of a clear circular aperture.

can recognize the expected sinc2 variation corresponding to the circular aperture.

2.4.1. The Integrated Intensity Function The optical-sectioning capability of an imaging system can be evaluated from its intensity PSF through the calculation of a function known as the integrated intensity (Sheppard and Wilson, 1978), which is deﬁned as

∞ |h(x, z)|2 d2 x.

Iint (z) =

(26)

−∞

This function evaluates, section by section, the total power in the 3D image of a point. The integrated intensity function tells us how out-of-focus parts of 3D objects contribute to the 2D image of the in-focus section. Taking into account the power-conservation law, it is clear that the integrated intensity is constant in wide-ﬁeld microscopy and therefore that all sections of the 3D object contribute with the same weight to the in-focus 2D image. To have a heuristic understanding of the integrated intensity, let us perform a conceptual experiment. As drawn in Figure 5, in the experiment, a laminar ﬂuorescent layer is uniformly illuminated and axially scanned in the object space. When the layer is at arbitrary distance z0 from the objective, the ﬂuorescence-generation function is

O(x, z) = δ(z − z0 ),

(27)

14

The Resolution Challenge in 3D Optical Microscopy

t escen Fluor r laye

y L1

yp p (x p, yp

x

)

L2

xp

y 9Sensor

F1

z0

f1 f1

F F91; 2

x9

f2 f2

F9 2

FIGURE 5 Conceptual experiment to define the integrated intensity function. The thin fluorescent layer is axially scanned and a stack of 2D images is captured with the sensor.

where a normalization factor has been omitted. The 2D image (the virtual object indeed) at the detector plane is given by the function

I(x, z) = δ(z − z0 ) ⊗3 |h(x, z)|2 ,

(28)

evaluated at z = 0, namely

I(x, 0) =

|h(x0 , z0 )|2 d2 x0 ,

(29)

which is just the formula of the integrated intensity. Consequently, we can understand the integrated intensity function as the response of the imaging system to an axially scanned ﬂuorescent layer. Heuristically, the opticalsectioning capability is the ability of the imaging system to determine the axial position of the layer. In wide-ﬁeld microscopes, this function is constant, which implies that these microscopes cannot discriminate the axial position of the layer.

2.4.2. The 3D OTF The function of an optical microscope is to provide magniﬁed images of objects in which the details are too ﬁne to be seen by the naked eye or to be resolved by a matrix image sensor. The ﬁne details of the object correspond to the high spatial frequencies. The efﬁciency with which such periodic components are transmitted to the image depends on the optical system. The function that accounts for this efﬁciency is the so-called optical

Basic Theory for Microscope Imaging

15

transfer function (OTF), which will be denoted here as H(u, w). It can be calculated as the 3D Fourier transform of the intensity PSF. Naturally, since the intensity PSF is the square modulus of the amplitude PSF, the OTF can be obtained as the self-correlation

˜ w) ∗3 h(u, ˜ w). H(u, w) = h(u,

(30)

Although we are dealing with spatially incoherent imaging processes, ˜ w) as the coherent transfer function (CTF) we can identify the function h(u, of the system, which is calculated as the Fourier transform of the amplitude PSF, namely

∞

˜ w) = h(u,

h(x, z) exp{−i2π(x u + z w)}d2 x dz.

(31)

−∞

The normalized frequencies are related to the actual ones through

u=

λf u rm

and

w=

λf 2 w. 2 rm

(32)

Note that the normalization factor ρc = rm /λf is just the radial cutoff frequency of the system. By substituting Eq. (23) into Eq. (31), we obtain

˜ w) = h(u,

∞

p xp d2 xp

−∞

∞ × −∞

∞

exp −i2πx u + xp d2 x

−∞

1 2 dz, exp −i2πz w − w0 + xp 2

(33)

where some irrelevant factors have been omitted. It can immediately be found that

˜h(u, w) = p(u) δ w − w0 + 1 |u|2 . 2

(34)

The 3D CTF of a telecentric imaging system is conﬁned onto a parabolic shell, as shown in Figure 6. Although the shell is axially symmetric, the value of the CTF at any point of the shell is given by the pupil function p(u), which in general may not be axially symmetric.

16

The Resolution Challenge in 3D Optical Microscopy

v

w0

u

w 0.5

FIGURE 6 The 3D CTF is confined onto a shell of a paraboloid of revolution about the w axis. The shell is axially shifted by w 0 .

To obtain the OTF, we perform the self-correlation of the parabolic shell. To simplify calculations, we consider the case of an axially symmetric pupil function in which case the OTF is axially symmetric as well. Therefore, it is not necessary to perform the correlation along all the transverse Cartesian frequencies, but only along the positive values in one Cartesian direction, namely

H(ρ, w) = H u+ , ρ, 0, w =

˜h α − ρ , β, γ − w 2 2 ρ w ∗ ˜ × h α + , β, γ + dα dβ dγ. 2 2

(35)

Taking into account Eq. (34) we obtain

ρ ρ p α − , β p∗ α + , β δ(w + αρ)dα dβ 2 2 = P(ρ, α) δ(w + αρ)dα,

H(ρ, w) =

(36)

where

ρ ρ ∗ P(ρ, α) = p α − , β p α + , β dβ 2 2

(37)

is just the projection onto the α axis of the product of two radially symmetric pupil functions mutually displaced by a distance ρ along the mentioned axis (Frieden, 1966). By solving the integral in Eq. (36), we obtain

1 w H(ρ, w) = P ρ, − . ρ ρ

(38)

Basic Theory for Microscope Imaging

17

In the case of the circular aperture, the function P(ρ, α) is simply given, for each value of ρ, by the projection onto α of the common area of two circles (see Figure 7), namely

⎫ ⎧ ⎨ ρ 2⎬ , P(ρ, α) = 2 Re 1 − |α| + ⎩ 2 ⎭

(39)

and therefore (Sheppard and Gu, 1992)

⎫ ⎧ 2 ⎨ |w| ρ 2 ⎬ . + H(ρ, w) = Re 1− ⎩ ρ ρ 2 ⎭

(40)

In Figure 8, we have represented the above equation on a meridian section. Note that the OTF is symmetric about the axial-frequency axis. As one could have easily predicted from Eq. (22), the OTF exhibits a singularity at the origin. The OTF is a compact-support function conﬁned to the region deﬁned by the parabolic curves

w = ±ρ(1 − ρ/2).

␤

␤ 5 12(␣ 1 ␳/2)

(41)

2

1 ␣ ␳

FIGURE 7 The function P(ρ, α) is obtained after projecting onto the α axis the product of two circle functions mutually displaced by a distance ρ.

OTF

w Missing cone

0.5

1.0

2.0

FIGURE 8 Meridian section of the 3D OTF of a circular aperture.

u

18

The Resolution Challenge in 3D Optical Microscopy

It is emblematic that there exists, in the neighborhood of the origin, a cone in which the OTF is zero. This cone is known as the missing cone (Streibl, 1984). Neither the axial frequencies nor the oblique frequencies included in the missing cone are transferred to the 3D image. In other words, no depth information is present in the image of 3D samples. This explains, in the OTF context, the lack of optical sectioning of conventional wide-ﬁeld microscopes when imaging 3D objects.

2.4.3. Relation Between the Axial OTF and the Integrated Intensity We have shown that the optical-sectioning capability of an imaging system can be investigated through two different functions: the integrated intensity and the axial component of the 3D OTF. In fact, both functions constitute different representations of the same information, since they are a Fourier-transform pair. To show this, we take into account that the OTF is the Fourier transform of the 3D intensity PSF,

H(u, w) =

|h(x, z)|2 exp{−i2π(u · x + w z)}d2 x dz,

(42)

and particularize to the axial frequencies,

|h(x, z)|2 exp{−i2πw z} d2 x dz

H(0, w) = =

Iint (z) exp{−i2πw z}dz.

(43)

In conventional microscopy, the integrated intensity is constant, and therefore, the axial OTF is a delta function centered in the origin. Only zero-order axial frequencies are transferred to the image.

3. THE HIGH-NUMERICAL-APERTURE APPROACH Up to now we have considered imaging systems for which the paraxial approximation holds, and therefore spherical wavefronts have been approximated by parabolic wavefronts. One should take into account, however, that in actual microscopy the aim of obtaining images with high spatial resolution makes it necessary to use microscope objectives with the maximum available NA, which cannot be approximated as a thin lens. On the contrary, an accurate analysis should take account of the principal surfaces of the microscope objective, as shown in Figure 9. The back principal surface is, as in the paraxial case, a plane surface (S2 in Figure 10).

19

The High-Numerical-Aperture Approach

Princ

ipal s

y

urfac

es ) yp p (xp, yp

x

L2

xp

F1

y9 D CC

F 19 ≡ F2

f1

x9

f1 f2 f2

F92

FIGURE 9 An actual optical microscope is accurately schematized through a telecentric arrangement with the objective represented by its principal surfaces. S2

S1

p (rp) P1

ss fq

P R

␪ F

f

f

FIGURE 10 When a high-NA objective is illuminated by a plane wave, the amplitude transmittance of the aperture stop is projected onto the spherical principal surface.

The front principal surface, S1 , is a sphere of radius f centered at the front focal point. As already stated, in most high-NA microscope objectives, the aperture stop is inserted just at the BFP. From this sketch, one can understand that a microscope objective, ideally free of aberrations, transforms an impinging monochromatic plane wave into a truncated spherical wavefront centered at the focal point, F. The amplitude transmittance of the aperture stop is mapped onto S1 .

3.1. Calculation of the PSF As stated by McCutchen (1963), the objective, such as a cookie cutter, chops out a chunk of the spherical wavefront, which can be regarded as a Huygenian source. The amplitude at any point in the vicinity of the focus is calculated by integrating contributions from this source, taking into account their relative phases. To calculate the amplitude distribution in

20

The Resolution Challenge in 3D Optical Microscopy

the neighborhood of the focus, we proceed by making use of the ﬁrst equation of Rayleigh–Sommerfeld (Born and Wolf, 1999c), which reconstructs the amplitude distribution in the vicinity of the focus as the superposition of the secondary spherical wavelets that originated at the spherical wavefront, namely

h(R) = −

i λ

U(R1 )

eiks 2 d S. s

S1

(44)

The positions of a typical point, P1 , of the wavefront, and of observation point, P, respect to F are given by vectors R1 and R, respectively. Vectors qˆ and sˆ are, respectively, the unit vectors in such directions. The amplitude of any secondary wavelets is given by

U(R1 ) = p(R1 )

e−ikf , f

(45)

where p/f is the amplitude of the Huygenian source. The factor exp(−ikf ) is required to move the zero of phase from the Huygenian source, where it would otherwise be, to the geometric focus. In the vicinity of the focus, we can approximate

ˆ ≈s−f qR

and

d2 S ≈ f 2 d2 ,

(46)

where d2 is the solid angle that d2 S subtends at F. Then, Eq. (44) can be rewritten as

h(R) = −

i λ

ˆ ˆ eikqR p(q) d2 .

(47)

The above equation constitutes the so-called Debye scalar integral representation of strongly focused ﬁelds (Debye, 1909) and expresses the ﬁeld as a coherent superposition of monochromatic plane wavefronts. The directions of propagation of the wavefronts fall inside the geometrical cone deﬁned by the focus and by the projection of the pupil function onto the spherical principal surface. Since in most objective lenses the amplitude transmittance of the aperture stop has axial symmetry, it is convenient to express the positions in the reference sphere in terms of a set of spherical coordinates centered at the focus:

qˆ = (−sin θ cos ϕ, −sin θ sin ϕ, cos θ) ,

(48)

21

The High-Numerical-Aperture Approach

and

d2 = sin θ dθ dϕ.

(49)

Besides, we express the position of point P in terms of a set of cylindrical coordinates centered again at the focus:

R = (r cos ψ, r sin ψ, z).

(50)

Therefore, the amplitude distribution in the focal volume can be written as

i h(r, ψ, z) = − λ

2πα p(θ, ϕ) exp{−ikr sin θ cos(ϕ − ψ)} 0 0

ikz cos θ

×e

sin θ dθ dϕ.

(51)

Assuming axial symmetry for the pupil amplitude transmittance, the focal amplitude has axial symmetry as well,

2π h(r, z) = −i λ

α

p(θ) J0 (kr sin θ) eikz cos θ sin θ dθ.

(52)

0

Again in this case, one commonly uses normalized coordinates, deﬁned by

r=

r sin α λ

and

z=

2z sin2 α/2 λ

(53)

so that we can write the amplitude distribution in the focal volume as

α 2π sin θ z exp iπ 2 h(r, z) = −i p(θ) J0 2πr λ sin α sin α/2 × exp −i2πz

sin2 θ/2 sin2 α/2

0

sin θ dθ.

(54)

We must recall at this point that energy consideration should be taken into account in the projection of the incident plane wavefront onto the emanated spherical wavefront; see Figure 10. Most microscope objectives are designed to fulﬁll the aplanatic condition, also known as sine condition,

22

The Resolution Challenge in 3D Optical Microscopy

to produce images with transverse invariance (Gu, √ 2000; Sheppard and Gu, 1993). In this case, an apodizing factor g(θ) = cos θ should be included in the integrand of Eq. (54). This equation is exact for scalar waves and is a good approximation for light if the NA is small enough that different parts of the arriving wavefront do not have their polarizations signiﬁcantly twisted relative to one another on the way to focus. In many cases, it is useful to express Eq. (54) in terms of the normalized radial coordinate at the pupil plane. In case of aplanatic systems, the angular and the radial coordinates are related through rp = sin θ/ sin α. In that case

2π h(r, z) = −i λ

1

p rp J0 2πr rp

0

⎧ ⎪ ⎨

× exp −i2πz ⎪ ⎩

⎫ ⎬ 1 − r2p sin2 α ⎪ sin2 α/2

sin2 α rp drp . ⎪ ⎭ 1 − r2p sin2 α

(55)

3.2. Calculation of the OTF The calculation of the 3D OTF can easily be performed by taking into account that its coherent counterpart, the 3D CTF, is related with the amplitude PSF through a 3D Fourier transform, namely

h(x, z) =

˜ w) exp{i2π(u · x + wz)}d2 u dw. h(u,

(56)

The normalized frequencies are related with the actual ones through the scaling

u=u

λ sin α

and w = w

λ 2 sin2 α/2

.

(57)

The amplitude PSF can be rewritten as

∞ h(x, z) =

˜ ei2πQ·R d2 |Q|2 d|Q|, h(Q)

(58)

0

where R = (x, z) and Q = (u, w). From this equation, one can derive the Debye integral of Eq. (47), as expressed in normalized coordinates,

The High-Numerical-Aperture Approach

23

u

sin ␣

cos ␣ 1 w

FIGURE 11 The 3D CTF of a high-NA microscope is confined onto a spherical shell of unit radius.

provided that

˜ ˆ δ |Q|2 − 1 . h(Q) = p(q)

(59)

Then, the 3D CTF is conﬁned onto the surface of a sphere, which, when expressed in normalized frequencies, has unit radius; see Figure 11. This is the result reported by McCutchen (1963) in his mythical paper in which he stated that the amplitude distribution in the focal volume is obtained as the 3D Fourier transform of the Huygenian source. Therefore, the Huygenian source is the CTF of the system. In the usual case of an axially symmetric pupil function, the 3D CTF is (Sheppard et al., 1994).

˜h(Q) = h(ρ, ˜ w) = p(ρ) δ w − 1 − ρ2 . 1 − ρ2

(60)

Naturally, the 3D OTF is obtained, as in the paraxial case, through the 3D self-correlation of the 3D CTF, as illustrated in Figure 12a. Similarly to the paraxial result, the high-NA OTF has a doughnut structure in which the most remarkable feature is the existence of a cone of nontransmitted spatial frequencies; see Figure 12b. Note that the mathematics that describe the high-NA focalization and imaging are different from the those describing the paraxial phenomena. However, the two situations are conceptually equivalent. By this, we mean that in both cases, the 3D intensity PSF exhibits a sharp central peak surrounded by low sidelobes. Also, in both cases, the 3D OTF has a doughnut shape in which the missing cone is responsible for the lack of optical-sectioning capability of conventional microscopes. As in the paraxial case, the high-NA integrated intensity can

24

(a)

The Resolution Challenge in 3D Optical Microscopy

(b) OTF

12cos a w

Missing cone

2sin α u

FIGURE 12 (a) The 3D OTF in the high-NA case is obtained through the self-correlation of a spherical shell of unit radius. (b) Also, in the high-NA case, the 3D OTF has a doughnut structure with a missing cone. For the calculation we set α = 3π/8.

be calculated through the 1D Fourier transform of the axial component of the 3D OTF, and therefore it is constant. We can conclude that, although paraxial equations do not provide an accurate description of the imaging properties of optical microscopes, they are relatively simple and easy to handle, and provide good conceptual ideas about the functioning of optical microscopes.

3.3. Metrics for Resolution Improvement Resolution is the key feature of optical microscopes. The ability of optical microscopes to provide sharp images of the ﬁnest details of samples has been usually evaluated through the OTF or, alternatively, through the intensity PSF by applying the two-point resolution Rayleigh criterion. In past years, an important research effort has been addressed at developing the so-called PSF-engineering techniques (Ando, 1992; Barakat, 1962; de Juana et al., 2003; Dorn et al., 2003; Mills and Thompson, 1986; OjedaCastañeda, Andrés, and Díaz, 1986; Sherif and Török, 2004; Toraldo di Francia, 1952). Such techniques aim to modify the shape of the intensity PSF so that it is narrowed in the transverse and/or in the axial direction. Here, we concentrate on the engineering of the axial PSF. We start by particularizing Eq. (54) to points in the optical axis,

α 2π z exp iπ 2 h(0, z) = −i p(θ) λ sin α/2 × exp −i2πz

sin2 θ/2 sin2 α/2

0

sin θ dθ.

(61)

25

The High-Numerical-Aperture Approach

Following McCutchen’s approach, Eq. (61) can be converted into a 1D Fourier transform provided that one performs the non-linear mapping

ζ=

sin2 θ/2 2

sin α/2

−

1 2

q(ζ) = p(θ).

and

(62)

After the mapping, we ﬁnd that

0.5 h(0, z) =

q(ζ) exp(−i2πzζ) dζ,

(63)

−0.5

where some irrelevant pre-multiplying factors have been omitted. Note that, apart from a scaling and shifting, the non-linear mapping is the same as suggested by McCutchen,1 and therefore the function q(ζ) is nothing but the projection of p(θ) on the optical axis, as illustrated in Figure 13. It is remarkable that independent of the value of the NA, the projection of a clear circular aperture onto the optical axis has a rectangular form, and therefore q(ζ) = rect(ζ). Thus, for any NA, the axial intensity PSF is

sin2 πz = sinc2 (z). (πz)2

|h(0, z)|2 =

S2

(64)

S1

p (rp)

␪ p (␪)

f

q (␨)

F

f

FIGURE 13 Independently of the numerical aperture of the imaging system, a clear circular aperture projects as a rectangle on the optical axis.

1 In his paper, McCutchen (1963) suggested the non-linear mapping = cos θ , which allows the projection

of the Huygenian source onto the optical axis.

26

The Resolution Challenge in 3D Optical Microscopy

The actual form of the axial PSF, expressed in actual spatial coordinates, is obtained after undoing the coordinate mapping of Eq. (53). Note from this mapping that the higher the NA the narrower the axial PSF. Next, we consider the case in which one inserts a diffractive element, usually called a pupil ﬁlter, in the aperture stop with the aim of engineering the axial PSF. To estimate the ability of such ﬁlter to improve the axial resolution, one needs an analytical tool for the easy evaluation of the width of the central lobe of the PSF without needing to compute all the focal intensity. Since the parabolic term in the power-series expansion for the axial PSF dominates within the central peak, an interesting method for evaluating the resolution is one originally suggested in a paraxial context by Sheppard and Hegedus (1988). They deﬁned the gain in axial resolution as the ratio between the parabolic intensity fall-off provided by the ﬁlter and that provided by a circular aperture. In the case of pupil ﬁlters whose amplitude transmittance is real, if we expand the normalized axial PSF in power series up to second order, we ﬁnd that

|h(0, z)|2

z ≈ 1 − 4π IN (z) = 2 1/σ |h(0, 0)|

2 ,

(65)

where

m2 − m0

σ=

m1 m0

2 (66)

is the standard deviation of the function q(ζ), and

0.5 q(ζ) ζ n dζ

mn =

(67)

−0.5

represents its nth statistical moment. Now, one can deﬁne the gain in axial resolution as

GA =

σF , σC

(68)

where the subscripts C and F correspond, respectively, to the circular aperture and the pupil ﬁlter. Obtaining the gain in transverse resolution is not as simple as obtaining its axial counterpart. This is because, as pointed out by McCutchen (1963),

The High-Numerical-Aperture Approach

S2

27

S1

p (rp)

qF (␨) ␪ F p (␪)

f

f

FIGURE 14 The lateral PSF is obtained through the 1D Fourier transform of the projection of the pupil onto one axis perpendicular to the optical axis.

the shape of the function obtained as the projection of the circular aperture onto an axis perpendicular to the optical axis (see Figure 14) depends on the NA of the objective. The fall-off in intensity of the squared modulus of the 1D Fourier transform of such projection is the reference ﬁgure for the calculation of the gain in transverse resolution. Therefore, the gain strongly depends on the value of α. Proceeding in the same way as in the axial case, we ﬁnd, after straightforward calculations, that

GT =

2(1 − cos α)/3 (3 + α) − GA , (3 + α) − (1 − cos α)/3

(69)

where we have considered the simplest case of q(ζ) being an even function. Imaging systems for which the paraxial approximation holds have GT ≈ 1. The gains in resolution have constituted an interesting tool for the design of many pupil proﬁles for the improvement of the lateral and/or the axial resolution of imaging systems. Among these, we single out the so-called shaded-ring (SR) pupil ﬁlters (Martínez-Corral et al., 2003). As we show below, SR ﬁlters have the ability to narrow the central lobe of the axial PSF. This narrowing is produced at the expense of only an insigniﬁcant deterioration of the lateral resolution and a small enlargement of the axial sidelobes. SR ﬁlters simply consist of a purely absorbing ring with constant transmittance, centered on the objective-lens aperture (see Figure 15). Depending on the width of the ring, different degrees of axial-peak compression can be achieved. The value of the axial gain in resolution is

GA =

1 − ημ3 , (1 − ημ)

(70)

28

The Resolution Challenge in 3D Optical Microscopy

q (␨) ␩

␮/2

⫺0.5

0.5

FIGURE 15 1D projection and actual 2D transmittance of a SR pupil filter.

with the parameters η and μ identiﬁed in Figure 15. Note that all the pairs (η, μ) fulﬁlling Eq. (70) for a given value of GA, correspond to ﬁlters with the same axial gain but different sidelobe energies, as illustrated in Figure 16. In this ﬁgure, we have plotted, ﬁrst, several curves for different values of GA. Every point of a curve corresponds to a different (η, μ) pair. The leftmost point of a curve corresponds to the dark-ring ﬁlter (Blanca and Hell, 2002; Martínez-Corral et al., 1995). For an intensity simulation, we selected GA = 1.20. The minimum of the curve (marked with a circle in the ﬁgure) corresponds to the SR ﬁlter with μ = 0.67 and η = 0.66. In terms of the sidelobe energy, the selected SR ﬁlter is 25% better than the opaque-ring one.

3.4. Sampling Expansion Point-spread function engineering constitutes an interesting tool for improving the performance of optical microscopes. Apart from the gains in resolution, which provide an easy method for estimating the superresolving abilities of pupil ﬁlters, other analytical tools are useful for the fast computation of the 3D PSF. Sampling expansions of the PSF have been used in the past for the computation of 2D or even 3D diffraction patterns (Barakat, 1980; Jacquinot and Roizen-Dossier, 1964; Li and Wolf, 1984; Piestun, Spektor, and Shamir, 1996), but always within the frame of the paraxial approximation. To be able to use this tool for calculating 3D PSF of optical microscopes, it is necessary to extend the Landgrave and Berriel-Valdos (1997) approach to a non-paraxial context.2 To this end, we start by rewriting Eq. (54) as

2π h(r, z) = −i λ

1 p(cos θ) J0 cos α

× exp i2πz

sin θ 2πr sin α

cos θ 2 sin2 α/2

d(cos θ).

2 Preliminary results in this direction were reported by Arsenault and Boivin (1967).

(71)

The High-Numerical-Aperture Approach

(a)

2.4

29

GA ⫽ 1.15 GA ⫽ 1.20 GA ⫽ 1.25

Sidelobe-to-peak ratio

2.0

GA ⫽ 1.30

1.6

1.2

0.8

0.4 0.2

(b)

0.4 0.6 0.8 Transmittance parameter: ␮

1.0

1.0

Axial intensity PSF

0.8

0.6

0.4

0.2

0.0 ⫺4

0 2 ⫺2 Normalized axial coordinate: z

4

FIGURE 16 (a) Sidelobe-to-peak energy ratio for families of SR filters with the same axial gain. (b) Axial PSFs corresponding to the SR filter with GA = 1.20 (solid curve) and to a circular aperture (dashed curve).

Let us now suppose that the kernel of the above transformation,

K(θ; r, z) = J0

sin θ 2πr sin α

exp i2πz

cos θ 2 sin2 α/2

,

(72)

30

The Resolution Challenge in 3D Optical Microscopy

can be expanded in a Fourier series as

K(θ; r, z) =

∞

fm (r, z) K(θ; 0, m).

(73)

∞ 2π fm (r, z) h(0, m). λ m=−∞

(74)

m=−∞

In that case,

h(r, z) = −i

Following the Landgrave and Berriel-Valdos approach, now the problem consists in ﬁnding the coefﬁcients of the kernel expansion. To solve this problem, we notice the following orthogonal property,

1

K(θ; 0, m) K ∗ (θ; 0, m ) d(cos θ) = (1 − cos α) δm,m ,

(75)

cos α

which permits us to ﬁnd that

1 fm (r, z) =

cos α

K(θ; r, z) K ∗ (θ; 0, m) d(cos θ) 1 − cos α

=

hC (r, z − m) , 1 − cos α

(76)

where hC (r, z) is the 3D PSF corresponding to the circular aperture. Finally, we obtain

h(r, z) = −i

∞ 2π hC (r, z − m) h(0, m). λ(1 − cos θ) m=−∞

(77)

This important formula, which represents the non-paraxial form of the axial sampling theorem (Arsenault and Boivin, 1967), indicates that the 3D amplitude PSF of an optical microscope with a pupil ﬁlter inserted into the objective aperture results from the coherent superposition of an inﬁnite number of axially shifted PSFs that correspond to the circular aperture. The shifts are equal to integer numbers. The weighting-factor set of this superposition is obtained by sampling the axial PSF of the pupil ﬁlter in the axial nulls of a circular-aperture PSF.

The High-Numerical-Aperture Approach

31

3.5. Spherical Aberration In many microscopy realizations, the specimen is embedded in a medium that does not match the refractive index of the immersion liquid. This fact creates important phase distortion in the, otherwise spherical, wavefronts emitted by the ﬂuorophores. Consequently, the PSF is also distorted. In our formalism, the phase distortions are incorporated by modifying the Huygenian source by the term exp{−i2πW(θ)} so that

α h(r, z) =

p(θ) exp[−i2πW(θ)] J0 0

× exp −i2πz

sin2 θ/2 sin2 α/2

sin θ 2πr sin α

sin θ dθ,

(78)

where we have omitted some constant and a phase factor external to the integral. To evaluate the phase distortions, we can view, as explained in Section 2.2, this diffraction process from the opposite direction, that is, as the case in which a high-NA objective is illuminated by a monochromatic plane wave and the corresponding emerging beam is focused deeply through a planar interface between two media of different refraction index. In that case, we can assume that each plane-wave component of the ﬁeld emerging from the objective obeys Snell’s law, n1 sin θ = n2 sin θ , when refracted at the interface. The resulting ﬁeld is reconstructed as the superposition of refracted plane waves. In Figure 17, a plane-wave component is represented through a light-ray normal to the wavefront. The phase delay suffered by the rays is proportional to the optical path difference (Török et al., 1995),

W(θ) =

1 d

n2 cos θ − n1 cos θ . [n1 l1 (θ) − n2 l2 (θ)] = λ λ

(79)

Following the classical approach of Sheppard and Cogswell (1991), we expand this expression into power series of sin(θ/2), up to fourth order. We obtain

! 2 n 2n1 d W(θ) = (n1 − n2 ) 1 + sin2 (θ/2) + 2(n1 + n2 ) 13 sin4 (θ/2) . λ n2 n2 (80)

32

The Resolution Challenge in 3D Optical Microscopy

n1 n2 A <1 5AF

␪9

F

␪ B

C

<2 5 AC d

FIGURE 17 Scheme for the evaluation of the phase distortions occurring in tight focusing through a planar interface.

The amplitude distribution in the neighborhood of the focus is then given by

α

h r, z ; w40 =

p(θ) J0 0

sin θ 2πr sin α

× exp i2π w40

sin4 (θ/2) sin4 (α/2)

sin2 (θ/2) − z 2

sin (α/2)

! sin θ dθ, (81)

In this equation, we have omitted an irrelevant constant phase factor and have deﬁned the reduced axial coordinate z = z − w20 , where

d n1 w20 = 2 (n2 − n1 ) sin2 (α/2) λ n2

(82)

is the defocus coefﬁcient that accounts for the shift suffered by the focus of the wave (in the scheme of Figure 17 the focus is shifted from F to B). The phase distortions are proportional to the factor sin4 (θ/2). This kind of proportionality is classically associated with spherical aberration (Mahajan, 1991) so that we can introduce the spherical aberration coefﬁcient deﬁned as

w40 = 2

n2 d 2 n2 − n21 13 sin4 (α/2). λ n2

(83)

33

Lateral coordinate: x

Optical-Sectioning Microscopy

2

w40 5 0

w405 21.0

w40 5 22.0

w40 5 23.0

1 0 21 22

26 24 22 0

2

4

26 24 22 0

2

4

26 24 22 0

2

4

26 24 22 0

2

4

Reduced axial coordinate: z9

FIGURE 18 Intensity distribution in the focal spot corresponding to different values of w40 .

This parameter evaluates, in units of wavelength, the phase distortion at the border of the pupil. As an example of the impact of the spherical aberration induced by the refractive-index mismatch, Figure 18 plots the squared modulus of Eq. (81). The calculations were performed with the following parameters: λ = 0.633 μm, NA= 1.4, n1 = 1.52 (oil), and n2 = 1.33 (water). The penetration distances d = 0, 10.1, 20.2, and 30.3 μm give rise to the following values for the spherical-aberration coefﬁcient: w40 = 0, −1.0, −2.0, and −3.0. The ﬁgure clearly illustrates the strong inﬂuence of refractive-index mismatch on the shape of the focal spot. More accurate calculations should take into account polarization effects and the inﬂuence of the transmission coefﬁcients of the interface (Haeberlé et al., 2003; Sheppard and Török, 1997; Török and Varga, 1997). However, the basic principles presented here can be demonstrated without these coefﬁcients. Among the solutions reported for compensating spherical aberration, one of the simplest is altering the tube length at which the microscope operates (Sheppard and Gu, 1991). When the microscope is used to form the image of 2D samples, the spherical aberration is compensated by use of the correction collar, which is commercially available in modern high-NA objectives (Schwertner, Booth, and Wilson, 2005). However, when imaging 3D samples, the amount of spherical aberration is proportional to the scanning depth and therefore a technique able to dynamically adjust the correction is necessary. An optimum solution could be in the use of adaptive optics (Booth et al., 2002). Another solution, which is much simpler from a technological implementation point of view, is the use of diffractive optical elements designed to confer the microscope an important robustness against spherical-aberration variations (Escobar et al., 2006).

4. OPTICAL-SECTIONING MICROSCOPY As stated in the preceding section, conventional wide-ﬁeld optical microscopes can be used to obtain images of 3D samples. Since no 3D matrix image sensors are available, the 3D images are acquired by recording on a

34

The Resolution Challenge in 3D Optical Microscopy

2D sensor a stack of 2D images produced in an axial scan of the object. The 2D images are stored in a computer and displayed with the help of imageprocessing software. However, such 3D images do not have optical sectioning, since conventional microscopes do not transmit the axial frequencies. Since modern biology, medicine, material science, and other ﬁelds of technology require the acquisition in-vivo of sharp, high-resolution images of 3D samples, there is a need for the implementation of microscopical techniques that have optical-sectioning capability. In the past few years, many new imaging architectures have been invented and developed with surprising optical-sectioning capabilities. Some of these are based on 3D scanning of the sample and therefore on storing point-by-point the intensity information about the sample. This is the case for confocal scanning microscopy (Corle and Kino, 1996; Masters, 1996), two-photon scanning microscopy (Diaspro, 2002; Masters, 2003), or 4Pi scanning microscopy (Hell and Stelzer, 1992b). Other techniques can produce optical sectioning after 1D scanning. This is the case for structured illumination microscopy or the selective-plane illumination microscopy (SPIM) (Huysken et al., 2004). The present section is devoted to these optical-sectioning imaging techniques.

4.1. Confocal Scanning Microscopy Let us start by giving a heuristic explanation of the principles of confocality; later, we will give a more rigorous mathematical explanation. A CSM is composed of two identical subsystems: the illumination system and the collection system. As shown in Figure 19, the monochromatic light emanating from a laser source is collimated, expanded, and focused into the 3D sample with a high-NA objective. The tightly focused beam illuminates all parts of the sample within the illumination cone. The small region surrounding the focus of the beam receives very high illumination density. If the illumination wavelength is within the excitation band of the ﬂuorescent dye, the sample will emit ﬂuorescent light whose spatial distribution of intensity is proportional to the illumination intensity. Since the collection system is adjusted so that the focus of the objective lens is conjugated with a pinhole, the light emanating from such point is collected with the maximum efﬁciency, as we see in Figure 19a. In contrast, as shown in Figure 19b, parts of the sample outside the in-focus plane receive a smaller illumination density, reducing the probability of ﬂuorescent emission. In addition, since these parts are not conjugated with the pinhole, the light is collected with very low efﬁciency. Consequently, the probability of collection of light from out-of-focus planes is much smaller than that from in-focus planes. The pinholed detection confers confocal systems their proverbial optical-sectioning capability.

Optical-Sectioning Microscopy

Large-area detector

(a)

35

Large-area detector

(b)

Pinhole

Pinhole

Laser

Laser

Dichroic beam splitter

Dichroic beam splitter

Objective lens

Objective lens

Focus In-focus plane

Out-of-focus plane

FIGURE 19 (a) Functional scheme of a confocal scanning microscope. (b) In the case of out-focus planes, the probability of fluorescent excitation and of light collection is much smaller.

Note that in principle the confocal system only provides one scalar value that is proportional to the ﬂuorescent intensity emitted by a small region of the object, but any image of the object. In fact, a confocal system does not provide images in the conventional form. The confocal images are obtained after a synchronized scanning of the sample and detection. The 3D matrix built with the intensities acquired after the 3D scanning of the sample constitutes a 3D image, which is stored in a computer for further display. To give a more rigorous explanation of confocal principle, let us consider the scheme drawn in Figure 20 where, for the sake of simplicity, we have depicted the confocal system in transmission geometry. Note that the confocal setup is composed by the arrangement in cascade of two telecentric systems like that shown in Figure 9. The illumination system focuses the monochromatic light emanating from the laser source onto a small region of the object. The intensity distribution of the illumination pattern, |hill (x, z)|2 , is just given by the squared modulus of Eq. (54). Provided that the wavelength of the illumination beam is within excitation band of the ﬂuorescent dye, the 3D distribution of emitted ﬂuorescent intensity is

I1 (x, z) = |hill (x, z)|2 O(x, z),

(84)

36

The Resolution Challenge in 3D Optical Microscopy

Colle

ction

Illum

inati

FIGURE 20

syste

m

on sy

stem

Scheme of a confocal scanning microscope working in transmission mode.

where, again, O(x, z) is the function that accounts for the concentration of ﬂuorescent emitters. Note that the wavelength of the ﬂuorescence light is slightly greater than that of the illuminating beam. This difference in wavelengths, which depends on the selected ﬂuorescence dye, will be incorporated into our formalism through the so-called Stokes shift, which is deﬁned as

ε=

λill . λﬂ

(85)

The collection system is, again, a telecentric conventional microscope and therefore provides in the neighborhood of the tube-lens BFP the 3D image of the object described by Eq. (84). As we already explained in Section 2, the characteristics of the image can be studied by performing the 3D convolution between the object and the virtual 3D PSF. Here, the intensity distribution of the virtual object is

Iv (x, z) = I1 (x, z) ⊗3 |hﬂ (x, z)|2 .

(86)

Let us ﬁrst consider the case that one places a large-area detector at z = 0 and therefore collects all the emitted light. Note that this plane corresponds virtually to the FFP of the microscope objective or actually to the BFP of the tube lens. The overall intensity captured by the detector is

IT =

Iv (x, z = 0)d2 x.

(87)

By substituting Eqs. (84) and (86) into Eq. (87), and expressing the 3D convolution in its integral form, we ﬁnd that

"

IT =

|hill (α, γ)|2 O(α, γ) d2 α dγ

# |hﬂ (x − α, −γ)|2 d2 x .

(88)

Optical-Sectioning Microscopy

37

The term in brackets is just the integrated intensity of the collection system, which, as shown in Section 3, is always constant. Therefore, the above equation can be rewritten as

IT =

|hill (α, γ)|2 O(α, γ)d2 α dγ.

(89)

As we see, the result of the above integral is a scalar that is proportional to the overall ﬂuorescence emitted by the illuminated region of the sample. In other words, we only have information about a speciﬁc region of the object. If we repeat this measurement but shift the object to a point of coordinates (xs , zs ), we obtain

|hill (α, γ)|2 O(xs + α, zs + γ) d2 α dγ.

IT (xs , zs ) =

(90)

As does Eq. (89), the above equation provides only a scalar that is proportional to the overall ﬂuorescence emitted by the region of the sample under the illumination cone, which now is centered at (xs , zs ). If we repeat this process for all points of the object, or at least we sample the object in a subset of points, and store the acquired intensity values, then Eq. (90) can be seen as a 3D convolution, namely

IT (x, z) = |hill (x, z)|2 ⊗3 O(−x, −z).

(91)

If we compare this equation with Eq. (17), we ﬁnd that the scanning non-confocal microscope we have just described has the same imaging properties as the conventional wide-ﬁeld optical microscope. Now, we return to the confocal scheme in Figure 20, but consider the case in which we collect only the light passing through the pinhole placed at the center of the tube lens BFP. To evaluate this, one only has to particularize Eq. (86) for (x = (0, 0), z = 0), that is,

Iconf = Iv (0, 0, 0) =

|hﬂ (−α, −γ)|2 |hill (α, γ)|2 O(α, γ)d2 α dγ.

(92)

Naturally, as in the case of the non-confocal set-up, this system does not provide an image, but only a single intensity value, which is proportional to the area of the product of two probability density functions: the probability of generation of ﬂuorescence—which is given by the product between the illumination PSF and the function O(x, z)—and the probability of collection of ﬂuorescent light. Note that this scalar value depends similarly on the two subsystems, the illumination system and the collection system. If we

38

The Resolution Challenge in 3D Optical Microscopy

now proceed in the same form as in the non-confocal case, that is, if we build a 3D function with the intensity values acquired in 3D scanning of the object, we ﬁnd the function

Iconf (xs , zs ) =

|hﬂ (−α, −γ)|2 |hill (α, γ)|2 × O(xs + α, zs + γ) d2 α dγ.

(93)

This equation is the integral representation the following 3D convolution:

% $ Iconf (x, z) = |hill (x, z)|2 |hﬂ (−x, −z)|2 ⊗3 O(−x, −z),

(94)

Note that Eq. (94) is for a process that, stricto senso, is a 3D imaging process, since Iconf (x, z) is obtained as the convolution between an uniformly scaled version of the object intensity and the PSF of the confocal system. Thus, we can write the PSF of the confocal instrument as

PSFconf (x, z) = |hill (x, z)|2 |hﬂ (−x, −z)|2 .

(95)

Illumination and collection systems play symmetrical roles in the imaging properties of CSMs. This symmetry confers the confocal systems an additional degree of freedom, because one can act in different, and independent, ways on the two pupil apertures, and therefore on the two individual PSFs. This degree of freedom can be made use of in improving the resolution by PSF-engineering techniques. Our above calculations have been based on transmission geometry for the confocal arrangement. However, CSMs usually have reﬂection geometry, like that in Figure 19. In this case, since the function |hﬂ |2 is even with respect to the axial coordinate z, the expression for the confocal PSF is

PSFconf (x, z) = |hill (x, z)|2 |hﬂ (x, z)|2 .

(96)

Naturally, for confocal microscopes with axially symmetric pupil apertures, there are no differences between the transmission and reﬂection PSFs. The formalism presented here has the advantage that one works with the same kind of functions as in the conventional microscopy. Therefore, one can normalize the PSFs in the same way and make use of the same type of representation like the PSF maps, the integrated intensity, or the 3D OTF. We use then the following coordinates, which are normalized with respect

39

Optical-Sectioning Microscopy

to the wavelength of the illumination beam

x=

x sin α λill

and

z=

2z sin2 α/2 . λill

(97)

In this case

PSFconf (x, z) = |hill (x, z)|2 |hﬂ (εx, εz)|2 .

(98)

Normalized lateral coordinate: x

Naturally, if the two pupil apertures have axial symmetry, the above equation is easily expressed in cylindrical coordinates. To illustrate the utility of the confocal principle, we have depicted in Figure 21 the intensity PSFs corresponding to a conventional and to a CSM. To calculate the plots, we have assumed that the two microscopes use the same objective (NA = 1.20, water), with circular aperture, and the same ﬂuorescent dye (ε = 0.8). There is an important gain in resolution in the confocal case. The central lobe has narrowed by 20% in the lateral direction and by 18% in the axial direction. Note however that this gain in 3D resolution does not justify the use of confocal systems instead of conventional ones. One should take into account that in practical set-ups this theoretical improvement cannot be reached due to the ﬁnite size of the pinhole and to the importance of noise due to the low signal captured. Thus, the complexity of confocal systems makes them not too recommendable if one only wants to obtain some gain in lateral resolution. The most important feature of the confocal PSF is the fact that the central lobe is surrounded by outer lobes that have almost vanished due to the product of illumination and collection PSFs. This vanishing stems from the fact that confocal systems are not conservative systems, since the energy collected from any transverse plane is not constant. This strongly inﬂuences the form of the integrated intensity, as shown in Figure 22.

(a) 2

(b)

(c)

1 0 ⫺1 ⫺2 ⫺3 ⫺2 ⫺1

0

1

2

3 ⫺3 ⫺2 ⫺1 0 1 2 3 ⫺3 ⫺2 ⫺1 Normalized axial coordinate: z

0

1

FIGURE 21 Intensity PSFs corresponding to (a) the illumination system, (b) the collection system, and (c) the overall confocal microscope. Note that (a) also represents the PSF of a conventional microscope.

2

3

40

The Resolution Challenge in 3D Optical Microscopy

1

Integrated intensity

0.8

0.6

0.4

0.2

0 23

22

21 0 1 Normalized axial coordinate: z

2

3

FIGURE 22 Integrated intensity function for a CSM. The parameters for the simulation were λill = 350 nm, λfl = 440 nm (Coumarin 400), and NA = 1.2 (water).

The curve in Figure 22 is a good representation of the main feature of confocal scanning microscopy, in fact the reason for its existence: the optical-sectioning capability. As we see, only the signal emanating from the confocal plane is detected with maximum efﬁciency. This efﬁciency falls off sharply for points in the vicinity of the confocal plane and vanishes for planes only a few micrometers away. In other words, almost no signal is collected from out-of-focus planes and therefore sharp images of the sections that compose the 3D sample can be obtained optically. As with conventional microscopy, the calculation of the 3D OTF constitutes a very useful tool for the understanding of the imaging features of confocal microscopes. The confocal OTF is obtained by performing the 3D correlation between the illumination and collection OTFs, namely

Hconf (u, w) = Hill (u, w) ∗3 Hﬂ

u w , , ε ε

(99)

where the expressions for the individual OTFs are just those deduced in Section 3.2, and represented, for the case of a circular aperture, in Figure 12. In Figure 23, we present the confocal OTF for the case where the circular aperture is the pupil function in both the illumination and the collection arms. Note that if one inserts a pair of pupil ﬁlters in the confocal setup, the form of the OTF changes, but the limits of the function, represented in

41

Optical-Sectioning Microscopy

w (11´)(12 cos ␣)

2(11´)sin ␣

u

FIGURE 23 The 3D OTF of a confocal scanning microscope. For the calculations we set α = 3π/8.

the ﬁgure by the white curve, remain unchanged. If we now compare the confocal OTF with the conventional one, we ﬁnd the following important facts: (i) the lateral frequency cut-off has augmented by a factor 1 + ε; (ii) the maximum axial frequency that is transmitted by the systems has augmented by the same factor; (iii) the really important feature of the confocal system is the fact that the missing cone has been ﬁlled and therefore the system has real optical-sectioning capability. The value of the cut-off frequency cannot be used, strictly, as the ﬁgure of merit for the evaluation of the resolution of an optical system. See, for example, Figure 24, which presents the lateral OTFs for a conventional and a confocal microscope. The cut-off frequency has augmented by a factor 1 + ε in the confocal case. However, the actual gain in resolution is much smaller because the confocal OTF falls off much faster than the conventional one. Anyway, although the values of the frequency cut-off do not constitute a strict ﬁgure of merit for the resolution, they can be used to give a reliable evaluation of the relation between the lateral and the axial resolution of CSMs.3 We deﬁne the ratio, A/T , between the axial and the lateral resolution as the quotient between the lateral cut-off frequency and the axial one, namely

A/T =

sin α uc . = wc sin2 α/2

(100)

We see from Figure 25 that the axial resolution is always poorer than the lateral one. Even for the highest available NA(α ≈ 3π/8), the axial resolution is

3 Note that strictly speaking one can use the term “axial resolution” only in case of imaging systems that

transmit the axial frequencies. In conventional microscopes, it is possible, of course, to calculate the axial PSF. However, since such systems do not transmit the axial frequencies, they do not have any axial resolving power.

42

The Resolution Challenge in 3D Optical Microscopy

1

Conventional

Normalized lateral OTF: Hconf (u,0)

Confocal 0.8

0.6

0.4

0.2

0 0.0

0.2

0.4

0.6 0.8 1.0 1.2 1.4 Normalized lateral frequency: u

1.6

1.8

2.0

FIGURE 24 Comparison between the lateral OTFs of a conventional and a confocal scanning microscope. As in previous calculations, we have assumed the value ε = 0.8 for the Stokes shift.

20

Resolution ratio: ΛA/L

15

10

5

0

FIGURE 25 angle.

0

␲/8

␲/4 Aperture angle: ␣

3␲/8

␲/2

Representation of the resolution ratio for different values of the aperture

Optical-Sectioning Microscopy

43

still about three times poorer than the lateral one. This difference between axial and lateral resolution leads to an anisotropic 3D-imaging quality. Several attempts have been made to reduce this image anisotropy. Among them, we mention a very ingenious technique consisting of creating a standing wave by the interference of two opposing wavefronts, as done in the so-called standing-wave microscopy and in 4Pi-confocal microscopy. An alternative technique to reduce the anisotropy is based on the use of pupil ﬁlters for narrowing the central lobe of the axial PSF. The drawback of using such ﬁlters in conventional imaging is that the narrowing of the PSF main peak is obtained at the expense of higher sidelobes. However, due to the factorial nature of the confocal PSF, in confocal architectures, this collateral effect is overcome. In the past few years, several authors have proposed the use of purely absorbing or complex-transmittance pupil ﬁlters to improve the resolution of confocal microscopes (Boyer, 2002; Neil et al., 2000; Sheppard, 1999). The ﬁlters described in these papers, which are mainly designed to improve the transverse resolution of confocal microscopes, are inserted into the illumination path to produce an illumination PSF that is superresolving but with high sidelobes. Confocality causes the sidelobes to be strongly reduced in the confocal PSF. As an example of the utility of this technique, we refer to the application of the SR pupil apertures described above (Ibáñez-López et al., 2005). In particular, if one inserts the SR ﬁlters selected in Figure 16 (μ = 0.67 and η = 0.66) into the pupil aperture of the illumination arm of the confocal system, and the circular aperture in the collection arm, one obtains an important improvement (12.0% in terms of the full-width at the half maximum (FWHM)) in axial resolution and only a small worsening (1.8%) in lateral resolution (see Figure 26). Note that there are two basic reasons for not using pupil ﬁlters in the detection arm. First, due to the pinholed detection, the light efﬁciency of collecting system is very poor and ﬂuorescent light losses should be avoided. The second, and even more important, reason is that due to the mismatch between illumination and collection wavelengths, the collection PSF is about 25% wider than the illumination PSF so that narrowing the collection PSF would have an insigniﬁcant inﬂuence on the confocal PSF width. The optical-sectioning capability of an imaging system is better evaluated in terms of the integrated intensity. Calculations not shown here demonstrate that the SR ﬁlter narrows the integrated intensity by 12.5%.

4.2. Structured Illumination Microscopy The CSMs described above possess a unique optical-sectioning capability due to their ability to reject the light emanating from out-of-focus sections of

44

The Resolution Challenge in 3D Optical Microscopy

(a)

T

1 v 2 an s r

0 rd o co

se er

m)

0 e: z (m nat i ord

e: at

in 1

x

m

(m

1

21

l co

Axia

)

(b)

Tr

1 v 2 ns a se

er

21 at 0 din or

co

e:

x

i

oord

(m

m

1 )

1 Axial c

) 0 : z (mm e t a n

FIGURE 26 (a) Numerically evaluated 3D PSF of a confocal instrument with two circular apertures; (b) same as (a), but with the selected SR filter in illumination. The parameters for the calculation were as follows: λill = 350 nm, λfl = 440 nm, and NA = 1.2 (water).

the 3D samples. However, conventional wide-ﬁeld microscopes still have some advantages over confocal ones. These advantages result from the fact that practical biological samples are often too weakly ﬂuorescent to yield a usable signal if pinhole diameter is too small. On the other hand, for studies of dynamic objects, the imaging speed is of major importance. The required acquisition time cannot be achieved if a pixel-by-pixel scan has to be performed. It is therefore highly desirable to perform 3D imaging with parallel pixel acquisition. The ﬁrst proposal for surpassing the diffraction limit of resolution by combining extended patterned illumination with wide-ﬁeld imaging was due to W. Lukosz and collaborators who, in the 1960s, proposed a novel method for extending the optical bandwidth of transmitted spatial frequencies (Lukosz, 1966; Lukosz and Marechand, 1963).

45

Optical-Sectioning Microscopy

) t (x,y (x,y ) G

L1

yp p (xp ,yp )

L2

xp y 9 Image

F1 F91 ≡

f1

plane x9

F2

G9(x,

y)

f1 f2 f2

F92

M 2z

0

FIGURE 27 Scheme of the original Lukosz’s setup. The essential feature of such an arrangement is the insertion of two gratings, G and G , in conjugate planes of object and image spaces.

4.2.1. Decoding by a Matched Detection Mask The scheme proposed by Lukosz4 is illustrated in Figure 27, which presents a telecentric imaging system in which a 2D object of amplitude transmittance t(x) is illuminated by a monochromatic plane wave. The light beam emerging from the 2D object impinges on a 1D grating of amplitude transmittance

n G(x) = Cn exp i2π x . p n=−∞ ∞

(101)

As in previous sections, we can analyze this diffraction phenomenon in terms of the virtual amplitude distribution that, after diffraction through the grating, is generated virtually at the object plane. Such a virtual amplitude distribution can be understood as the pattern that by free-space propagation reconstructs the amplitude distribution at the grating plane. To calculate the virtual pattern, u(G) (x), one simply needs to apply in cascade the Fresnel diffraction formula, namely

! eikz0 k k e−ikz0 2 exp i |x| exp −i |x|2 G(x) ⊗ u(G) (x) = t(x) ⊗ iλz0 2z0 −iλz0 2z0 ∞ n2 n n = (102) Cn exp iπλz0 2 exp i2π x t x + λz0 , y . p p p n=−∞ 4 A similar scheme was proposed by Grimm and Lohmann (1966).

46

The Resolution Challenge in 3D Optical Microscopy

The virtual pattern consists of a series of properly weighted, properly shifted copies of the object transmittance. Besides, each copy is illuminated by a properly inclined plane wave. The cornerstone of Lukosz’s method is to work with object ﬁelds limited enough to avoid overlapping between the copies. To make the forthcoming calculations easier, it is convenient to operate in the spatial frequency domain. To this end, we perform the 2D Fourier transform of the above equation to obtain ∞

n2 exp −iπλz0 2 u˜ (G) (u) = p n=−∞

n n ˜t u − , v exp i2πλz0 u . (103) p p

Taking into account the conjugation relation between the object and the image plane [see Eq. (3)], the frequency content at the image plane is

u˜ (G) (u) = M u˜ (G) (Mu) pˆ (u) ∞

n2 = M pˆ (u) exp −iπλz0 2 p n=−∞ n × exp i2πλz0 Mu , p

n t˜ Mu − , Mv p (104)

where pˆ (u) = p(−λf2 u). From Eq. (104), we see that the spectral content at the image plane consists of a set of shifted copies of the conventional-image spectrum. This allows the transmission of high spatial frequencies that otherwise are blocked by the aperture stop, pˆ (u), but all these frequency components are overlapped within the pupil area. Now, a decoding procedure is needed that allows the un-shifting of the spectral copies so that they no longer overlap, thereby obtaining a new image containing all the transmitted spatial frequencies. In the Lukosz scheme, the decoding is done by means of a second 1D grating, G (x) = G(x/M), which is a magniﬁed copy of the ﬁrst one, and that is placed just at the conjugate of G(x) (see Figure 27). Again, an observer placed behind the second grating can see at the image plane a virtual diffraction pattern whose frequency content is given by

u˜ (G)(G’) (u)

∞

n2 = exp M 2 p2 n =−∞ n × exp i2πλz0 u , Mp −iπλz0

u˜ (G)

n ,v u− Mp

(105)

Optical-Sectioning Microscopy

47

where z0 = M2 z0 . After substituting Eq. (104) into Eq. (105), and with some straightforward mathematical manipulation, we obtain

n Cn Cn−n exp i2πλz0 Mu p n =−∞ n=−∞ n n2 n × exp −iπλz0 2 pˆ u − , v t Mu − M , Mv , Mp p p (106)

u˜ (G)(G’) (u) = M

∞

∞

which can be rewritten as

u˜ (G)(G’) (u)

n =M pˆ n (u) exp i2πλz0 Mu p n=−∞ ∞

n n2 ˜ × exp −iπλz0 2 t Mu − M , Mv , p p

(107)

where ∞

pˆ n (u) =

C C n

n =−∞

n−n

n ˆp u − ,v . Mp

(108)

The term n = 0 of the series corresponds to the central image of the object [see Eq. (102)], whose spectral content is

u˜ o(G)(G’) (u) = M

∞ n =−∞

n , v t˜ (Mu, Mv). Cn C−n pˆ u − Mp

(109)

If we now perform the inverse Fourier transform of Eq. (109), we obtain the virtual amplitude distribution at the image plane, namely

uo(G)(G’) (x) =

x 1 t ⊗ hS (x), M M

(110)

where

∞ x n x . hS (x) = p˜ Cn C−n exp i2π λf2 Mp n =−∞

(111)

48

The Resolution Challenge in 3D Optical Microscopy

We have just found that Lukosz’s decoding method allows to transfer the grating modulation from the object to the PSF. This approach to structured illumination requires the use of a matched detector grid and working with object ﬁelds small enough to avoid imaging overlapping. Although this approach has been followed by some active groups (Zalevsky and Mendlovic, 2004; Zalevsky, Mendlovic, and Lohmann, 1999), up to now no convincing results have been obtained in a high-NA context. Other methods for removing the illumination grid pattern are based on computational reconstruction, as we will show next.

4.2.2. Decoding by Computational Reconstruction We now follow a scheme similar to that by Lukosz, but we extend the reasoning to the more general case dealing with 3D ﬂuorescent samples. The 1D grid pattern can be produced either by imaging a 1D grating onto the object plane or by the interference of two collimated beams (Cragg and So, 2000; Frohn, Knapp, and Stemmer, 2000; Gustafsson, 2000; Heintzmann and Cremer, 1998), as illustrated in Figure 28. The intensity illumination pattern can be written as

G(x) = 2 [1 + cos(2πAx + φi )] ,

(112)

CCD

Microscope tube lens

Monochromatic, collimated beam

Beam splitter Aperture stop

1D grating

Illumination pattern

Microscope objective 3D object

FIGURE 28 The interference between two coherent, collimated beams in the object space creates the patterned illumination.

49

Optical-Sectioning Microscopy

where A = 2 sin σ/ sin α, with 2σ the angle between the interfering beams. For the illuminated sample, the ﬂuorescence is proportional to the excitation intensity G(x) times the density of dye molecules O(x, z). In this case, the 3D intensity distribution in the image is given by

I(x, z) = 2 [1 + cos(2πAx + φi )] O(x, z) ⊗3 |h(x, z)|2 .

(113)

To obtain an image with increased resolution, it is necessary to transfer the grid pattern from the object side to the PSF side. This can be done by computational reconstruction after recording, at any axial scanning step, three images I0 , I1 , and I2 , obtained by adjusting the illuminating-beam phase differences to φi = 0, π/2, and π, respectively.5 By linear combination of such intensities, one can build the following functions:

A0 (x, z) =

1 [I0 (x, z) + I2 (x, z)] = O(x, z) ⊗3 |h(x, z)|2 , 2

1 [(1 − i) I0 (x, z) + 2iI1 (x, z) − (1 + i)I2 (x, z)] 4 1 = exp(i2πAx) O(x, z) ⊗3 |h(x, z)|2 , 2

(114)

A1 (x, z) =

1 [(1 + i) I0 (x, z) − 2iI1 (x, z) − (1 − i)I2 (x, z)] 4 1 = exp(−i2πAx) O(x, z) ⊗3 |h(x, z)|2 . 2

(115)

A2 (x, z) =

(116)

Taking into account the following property of the convolution product,

&

' & ' exp(i2πax) f (x) ⊗ g(x) = exp(−i2πax) f (x) ⊗ exp(−i2πax) g(x) , (117)

one can build the following synthetic 3D image

IS (x, z) = A0 (x, z) + exp(i2πAx) A1 (x, z) + exp(−i2πAx) A2 (x, z) = O(x, z) ⊗3 |hs (x, z)|2 ,

(118)

where

|hS (x, z)|2 = |h(x, z)|2 cos2 (πAx) . 5 The value of φ can easily be modiﬁed at will by laterally displacing the 1D grating. i

(119)

50

The Resolution Challenge in 3D Optical Microscopy

As easily understandable from Figure 28, the maximum value for the grating normalized frequency is A = 2 (corresponding to σ = α). In Figure 29, we have plotted the 3D PSF corresponding to such structured illumination case, which can be seen as a high-frequency sinusoidal 1D pattern modulated by the conventional PSF. The lateral resolution is strongly improved in the x direction, where the central lobe is much narrower than its conventional counterpart. Note however that since the sidelobes are very high, the width of the central lobe is not a good ﬁgure of merit for the evaluation of resolution. Instead, it is preferable, and much more illustrative, to analyze the imaging process in the frequency domain. The synthetic OTF is obtained as the 3D Fourier transform of the synthetic PSF, that is

" #

1

1 HS Q = H Q ⊗3 δ Q + δ(u − A, v, w) + δ(u + A, v, w) . (120) 2 2 In Figure 30, we have represented, in grayscale, two sections of the 3D synthetic OTF. The strong improvement in lateral resolution is achieved only along the x direction. To achieve the same improvement along other lateral directions, additional triplets of images, with the grating oriented in the proper direction, are needed. On the other hand, much higher improvements in lateral resolution can be obtained through saturatedpatterned excitation microscopy (SPEM). Speciﬁcally, SPEM takes advantage of the fact that systematic saturation of the ﬂuorophores creates a non-linear relation between the illumination intensity and the excitation

Lateral coordinate: x

2

1

0

21

22 23

22

21 0 1 Normalized axial coordinate: z

2

3

FIGURE 29 Illumination intensity PSF corresponding to structured illumination obtained with α = 3π/8 and σ = α.

Optical-Sectioning Microscopy

(a)

51

v

u

(b) w

u

FIGURE 30 Two views of the 3D OTF obtained under structured illumination. The frequency cutoff is doubled but only in the direction of the grating. The missing cone is still present so that no optical sectioning is achieved.

probability. The non-linearity leads to the generation of higher spatial harmonics in the pattern of emittability so that the range of detectable frequencies representing the object is extended (Heintzmann, Jovin, and Cremer, 2002).

4.2.3. Optical-Sectioning Microscopy Through Structured Illumination As shown in the preceding section, structured illumination methods applied to microscopy permit an imaging process in which the pass band is enlarged as a result of the sum of shifted copies of the conventional microscope OTF. This improvement in lateral resolution is not accompanied by any improvement in optical-sectioning capability. It is easy to realize, however, that a smart use of structured illumination would permit, by using proper values for the grating normalized frequency, A, to ﬁll the missing cone and therefore to simultaneously obtain improvement in lateral resolution and good optical-sectioning capability (Gustaffson, Agard, and Sedat, 2000). As an example of this, Figure 31 shows the synthetic OTF of an structured-illumination microscope designed for achieving optimum optical-sectioning capability, which is obtained for A = sin α(sin σ = 0.5 sin2 α). Much higher improvement in lateral resolution and optical sectioning could be obtained by increasing the number of shifted copies of the OTF that are summed up (Frohn, Knapp, and Stemmer, 2001).

52

The Resolution Challenge in 3D Optical Microscopy

(a)

V

u

(b) w

u

FIGURE 31 Two views of the 3D OTF for a structured-illumination microscope with optical-sectioning capability. The gain in lateral resolution is worse than in the previous example, but the missing cone is filled so that some good optical sectioning is achieved.

A slightly different approach for obtaining optically sectioned images through structured illumination was proposed by Neil, Juskaitis, and Wilson (1997, 1998).6 Speciﬁcally, they proposed, as well, to illuminate the object with a 1D grid pattern and take, at any axial-scanning step, three images with relative phases φi = 0, 2π/3, and 4π/3. The optically sectioned synthetic image is obtained as

Is (x, z) =

(I1 − I2 )2 +(I1 − I3 )2 + (I2 − I3 )2 .

(121)

The decoding can be performed with a single exposure by exploiting the chromatic channels of a color camera, and the moving grid can be replaced with a ﬁxed color grid (Krzewina and Kim, 2006). Structured-illumination optical-sectioning microscopes have recently been produced and commercialized by Zeiss (2008) and by Thales Optem (2008) under the names of ApoTome and OptiGrid. More recently, a new microscope combining optical sectioning by ﬂuorophore excitation using a single light sheet with structured illumination has been proposed (Breuninger, Greger, and Stelzer, 2007).

6 Note that, although for the sake of the clarity we are referring to Neil’s group proposal at this point in

our article, their work was pioneering in optical-sectioning microscopy by structured illumination and was published before other papers that we have already cited above.

Optical-Sectioning Microscopy

53

4.3. Axially-Oriented Structured Illumination Microscopy As stated in Section 4.1, one of the limitations of CSMs is the fact that the axial resolution is about three times worse than the lateral resolution. This fact has prompted many research groups to investigate methods for improving the axial resolution. Speciﬁcally, some interesting techniques have been proposed in the context of using non-uniform excitation light, as we will show in the following.

4.3.1. Standing-Wave Fluorescence Microscopy The basic idea of standing-wave microscopy (SWM) is to produce a ﬂat standing wave of laser light to create a set of excitation nodal planes along the optical axis. This form of exploiting counter-propagating interfering beams for axial resolution was ﬁrst attempted by placing a mirror beneath the sample in an epi-ﬂuorescence microscope (Bailey et al., 1993; Lanni, Taylor, and Waggoner, 1986). Like any conventional ﬂuorescence microscope, SWM uses a high-NA objective and a CCD camera, or an adequate matrix sensor, to acquire a series of wide-ﬁeld images of the sample while stepping the focus plane successively through the sample between images. As in structuredillumination microscopy, at every step of the axial scanning, three wideﬁeld images are recorded after adjusting the phase difference of the interfering beams to φ = 0, π/2, and π. The recorded images are linearly processed to extract the new information and produce a reconstruction with improved axial resolution. The 3D OTF of this hybrid imaging process is given by (Krishnamurti, Bailey, and Lanni, 1996)

"

1

#

1

HSW Q = H Q ⊗3 δ Q + δ u, v, w − K + δ u, v, w + K , 2 2 (122) where K = sin β/ sin2 (α/2), with 2β the angle between the interfering beams. Figure 32 plots an example of the SWM OTF. From the ﬁgure one can easily infer that, although some improvement in axial resolution is obtained by SWM, especially when imaging very thin samples, no opticalsectioning capability is achieved since the axial and surrounding spatial frequencies are not transmitted by the system.

4.3.2. Image Interference Microscopy A very ingenious method for obtaining substantial improvement in the transmission of axial frequencies was suggested by Lohmann (1978). The idea consisted, basically, of illuminating the sample with beams focused by

54

The Resolution Challenge in 3D Optical Microscopy

u

w

FIGURE 32 Numerically evaluated OTF of a SWM. The parameters for the calculation were β = π/2 and α = 3π/8.

CCD

Tube lens Beam splitter

Laser

3D object

Microscope objective

FIGURE 33 Scheme of the I2 M microscope. The sample is illuminated uniformly. The fluorescence light is collected with two opposed objectives.

two opposing microscope objectives. To produce the interference illumination pattern, the objectives should be illuminated with light emanating from the same laser. Then, the two spherical wavefronts interfere in the common focus so that the total Huygenian source is doubled. On the basis of this fundamental concept, some optical-sectioning imaging techniques were proposed, such as the technique named image interference microscopy (I2 M), in which the ﬂuorescence sample is mounted between two opposing high-NA objectives, each of which is focused on the same focal plane within the sample (Gustafsson, Agard, and Sedat, 1995). The sample is illuminated uniformly and scanned axially; see Figure 33. At any scan step two images are obtained through the two objectives. The images are combined by a beam splitter and superposed on a single CCD camera. The optical path lengths of the two image beams are adjusted to be equal, causing the two beams of light emanating from each ﬂuorescent

Optical-Sectioning Microscopy

55

u

sin ␣

1 w

FIGURE 34 The 3D CTF of a I2 M microscope is composed of two opposed spherical shells.

molecule to interfere with each other in the camera. The 3D CTF corresponding to this imaging process is composed, as the Huygenian source, of two opposed spherical shells, as shown in Figure 34. Naturally, the 3D OTF is obtained as the self-correlation of the CTF. In Figure 35, we have represented, in grayscale, a meridian section of the 3D OTF. Since the OTF is a compact support function whose limits depend on the NA, we have calculated the OTF for two different values of α. We see that some regions within the missing-cone area have been ﬁlled and therefore some optical sectioning is achieved. The higher the NA, the bigger is the improvement in optical-sectioning capability. However, the improvement is very modest, because the values of the OTF within the former missing-cone region are very small. This modest gain does not seem to justify the technological complexity of using two opposing high-NAobjectives to form the interference image. The performance of this microscope can be improved if one uses a similar set-up, but employs the two arms also for the illumination. In this way, one allows the counter-propagating plane waves to interfere in the object area to create a standing wave, as in the case of SWM. Further improvement can be obtained if, instead of coherent illumination, one uses an extended, spatially incoherent light source. In this way, if one considers a typical Köhler conﬁguration for the illumination, any point of the extended source is turned into two collimated beams in object space so that the excitation ﬁeld is a superposition of ﬂat standing waves tilted to the optic axis that are mutually incoherent with changing angle of incidence on the focal plane. The illumination OTF of this new microscope, which is called the I5 M microscope (Gustafsson, Agard, and Sedat, 1999), is composed of two line segments plus a point at the origin; see Figure 36. The extension along the axial direction of the I5 M OTF is much wider than any other wide-ﬁeld microscope ever proposed and also wider than that of CSMs;

56

The Resolution Challenge in 3D Optical Microscopy

w

(a) 2

1

2sin ␣

(b)

u

w

2

1 2sin ␣ u

FIGURE 35 The OTF of an I2 M microscope: (a) α = π/4; (b) α = 3π/8.

Optical-Sectioning Microscopy

sin ␣

57

u

w

FIGURE 36

The OTF of the illumination arm of an I5 M microscope.

u

w

FIGURE 37 The OTF of an I5 M microscope calculated for α = 3π/8.

see Figure 37. However, the value of the OTF within some extended regions of its support is very small, which can give rise to loss of information in actual microscope set-ups, in which the noise can be over the transmitted signal and mask the image. To overcome these problems, a new technique has been developed that also belongs to the general class of axially oriented structured illumination microscopes; this technique is known as 4Pi scanning microscopy.

4.3.3. The 4Pi Confocal Scanning Microscope In the 4Pi confocal microscope, twin opposing high-NA objectives are used to coherently illuminate the same point of a ﬂuorescent specimen (Hell and Stelzer, 1992a,b). As we schematize in Figure 38, the two objectives are illuminated normally with light emanating from the same

58

The Resolution Challenge in 3D Optical Microscopy

Laser

Large-area detector Pinhole

3D object

FIGURE 38 Schematic layout of practical implementation of a 4Pi microscope.

laser. Then, the two spherical wavefronts interfere in the common focus so that the total aperture is doubled. The OTF of the illumination arm is obtained as the self-correlation of the CTF, and its form is the same as that of the I2 M microscope; see Figure 35. For collection, the same procedure is applied as for illumination. Thus, the ﬂuorescent light generated at any point of the object is collected by the twin objectives and directed toward the large-area detector, after passing through the pinhole. The OTF of the collection system is the same as that of the illumination system, but scaled by the Stokes shift ε. Naturally, the OTF of the 4Pi confocal arrangement is obtained as the correlation between the illumination and the collection OTFs. From Figure 39, we see that the 3D OTF of a 4Pi microscope is conﬁned within a surface that, for high values of the NA, is close to spherical. Note however that the ideal OTF, in which the slope is the same along any direction passing through the origin, is not achieved. Instead, its value is very low in some regions. Such frequencies are hardly present in the ﬁnal image because they are masked by the noise. Naturally, the higher the NA of the objective, the smaller the low-value zones of the 3D OTF. To have a better understanding of the nature of the 4Pi concept, it is convenient to analyze the system in the spatial domain. The illumination PSF is given by the squared modulus of the sum of the incoming wavefront amplitudes, namely 2 PSF 4Pi ill (r, z) = |hill (r, z) + hill (r, −z)| ,

(123)

where the function hill (r, z) is that of Eq. (54). Under certain circumstances,7 this function is even in z (Wang, Friberg, and Wolf, 1995),

7 When the mapped transmittance is an even function in ζ , the 3D PSF is even in z.

59

Optical-Sectioning Microscopy

u

(a)

w

u

(b)

w

FIGURE 39 The OTF of the 4Pi confocal microscope: (a) α = π/4; (b) α = 3π/8. For the calculations we have considered the case of ε = 0.8.

and we will assume this to hold in the forthcoming reasoning. This alleviates the mathematics and does not imply any loss of generality. Thus,

PSF 4Pi ill (r, z)

2

= 4 cos

π

z sin2 α/2

|hill (r, z)|2 .

(124)

60

The Resolution Challenge in 3D Optical Microscopy

In Figure 40, we have plotted the 1D curve corresponding to the axial component of the 3D illumination PSF. The representation is in terms of the actual axial coordinate z and given for two different values of the NA. From the ﬁgure, we can outline the following features: (a) 4Pi arrangements make sense only in a high-NA context (If the twin objectives are of low NA, the light is focused very weakly, and only a standing-wave interference pattern is obtained); (b) 4Pi microscopy has no analogy in the paraxial context; so one cannot take advantage, as we did above, of paraxial analog to alleviate mathematics in the search for a heuristic explanation; (c) the alignment of the system is a crucial issue. Besides, any axial diphase would produce a displacement of the interference pattern, and therefore a pair of maxima would be obtained instead of one sharp focus. There are two possibilities for the detection. One is to collect the entire ﬂuorescent signal with one of the objectives and direct it to the detector through the pinhole; this corresponds to the so-called 4Pi(b) confocal microscope. The other is to collect the ﬂuorescent light with the two objectives and allow the wavefronts to interfere at the pinhole plane; this corresponds to the 4Pi(c) conﬁguration.8 The 3D PSF of 4Pi(c) is given by the product 4Pi PSF4Pi (r, z) = PSF 4Pi ill (r, z) PSF ﬂ (r, z),

(125)

4Pi where PSF 4Pi ﬂ (r, z) = PSF ill (εr, εz). This PSF is plotted in Figure 41. For this calculation, we ﬁxed the value of the aperture angle to α = 3π/8 and that of the Stokes shift to ε = 0.8. As we see in the ﬁgure, the interference process in the illumination and in the collection allows the production of a 3D PSF that is much narrower in the axial direction than its confocal counterpart and with the same width in the lateral direction. Therefore, a system is obtained whose axial resolution is even better than the lateral one. However, the PSF shows sidelobes in the axial direction that are too high. These secondary maxima reduce the beneﬁt of narrowing the central peak. Some techniques have been proposed to overcome this problem. One is the use of veryhigh-NA objectives (Lang, Engelhardt, and Hell, 2007). This solution has the problem that the working distance is dramatically reduced. Another proposal is the use of TPE ﬂuorescent dyes (Schrader and Hell, 1996). After reduction of the sidelobes (which implies that the gaps in the OTF are ﬁlled), data deconvolution techniques can be used to improve the image quality (Nagorni and Hell, 2001). Additional improvement in sidelobes reduction can be obtained by inserting axially symmetric pupil ﬁlters into

8 There is one more possible conﬁguration, 4Pi(a), in which the sample is illuminated only from one side,

and the ﬂuorescent signal is collected from the two sides.

61

Optical-Sectioning Microscopy

(a)

1.0

Illumination PSF

0.8

0.6

0.4

0.2

0.0 26

(b)

24

22 0 2 Axial coordinate: z /␭

4

6

4

6

1.0

Illumination PSF

0.8

0.6

0.4

0.2

0.0 26

FIGURE 40

24

22

0 2 Axial coordinate: z/␭

The axial PSF of the illumination system: (a) α = π/4; (b) α = 3π/8.

Normalized lateral coordinate: x

62

The Resolution Challenge in 3D Optical Microscopy

2

(a)

(b)

(c)

1 0 21 22 23 22 21

0

1

2

3 23 22 21

0

1

2

3 23 22 21

0

1

2

3

Normalized axial coordinate: z

FIGURE 41 The 3D PSF of a 4Pi microscope results from the product between the illumination and the collection PSFs.

the illumination path (Blanca, Bewersdorf, and Hell, 2001; Martínez-Corral, Caballero, and Pons, 2002). 4Pi optical scanning microscopes have recently been produced and commercialized by Leica Microsystems (2008).

4.4. Two-Photon Excitation Scanning Microscopy A serious drawback of single-photon ﬂuorescence confocal microscopes is photobleaching, which occurs since the entire sample is bleached when any single plane is imaged. Another drawback of the technique when used in biomedical imaging is its poor depth penetration. To solve these problems while keeping the optical-sectioning capability, TPE scanning microscopy has been developed. This non-linear imaging technique, ﬁrst reported in the early 1990s (Denk, Strickler, and Webb, 1990), is based on the simultaneous absorption of two photons with nearly equal wavelengths, following which a single ﬂuorescence photon is emitted (Göppert-Mayer, 1931). The excitation wavelength is typically twice that of the single-photon case. The ﬂuorescence intensity is proportional to the square of the illumination intensity. Therefore, TPE microscopes have the ability to strongly limit the excitation region. The overall ﬂuorescence light is collected by a largearea detector, and the ﬁnal image is synthesized from the 3D sampling of the object. TPE scanning microscopes inherently possess optical-sectioning capability despite the absence of pinholed detection. Good reviews of TPE scanning microscopy antecedents are by Dittrich and Schwille (2001) or by Masters and So (2004). Two-proton excitation scanning microscopes have the following advantages over single-photon confocal instruments: (a) since there is no pinholed detection, there are no constraints on the practical attainment of the resolution predicted by the theory; (b) the absence of a pinhole reduces the sensitivity to misalignment and increases the signal-to-noise ratio; (c) photobleaching is restricted to the close neighborhood of the focusing plane (This is because photobleaching depends here on the time-averaged square of the intensity, which falls off sharply above and below the focal plane); (d) the near-infrared light used for TPE is absorbed and scattered to a lesser extent by tissues, which allows deeper penetration of the excitation beam.

Conclusions

63

The resolving power of a microscope is usually evaluated in terms of its PSF, which in the TPE case is deﬁned as the ﬂuorescence emission distribution generated by a light point. The ﬂuorescence emission is proportional to the probability of simultaneous absorption of two lowenergy photons. Since two statistically independent events have to occur, this probability is proportional to the square of the excitation PSF. Since the excitation wavelength is double that of the single-photon case, the resolving power of TPE microscopes is about half the resolving power of conventional ones.

5. CONCLUSIONS This chapter has presented a thorough discussion of the theoretical principles of 3D microscopy. It has been shown that conventional wide-ﬁeld microscopy is not suitable for providing good-quality images of 3D specimens. On the basis of the paraxial diffraction equations, it has been shown that conventional microscopes, when dealing with 3D ﬂuorescent samples, provide sets of 2D images. These images of the different transverse sections of the 3D object contain, in addition to the sharp image of the infocus section, the blurred images of the rest of the specimen. This is the natural consequence of conservation of energy, inherently associated with waveﬁeld propagation. The paraxial formalism has been generalized in an elegant and very simple way to a non-paraxial context, showing that the equations that govern non-paraxial imaging are similar to those that govern paraxial imaging. The only difference is the integral that provides the 3D PSF. Therefore, all the background acquired in many decades of study concerning paraxial imaging can easily be transferred to the non-paraxial regime. We have deﬁned the metrics for evaluating the optical-sectioning capability of a microscope. The optical sectioning can be evaluated through the axial component of the 3D OTF or, alternatively, through its 1D Fourier transform: the integrated intensity function. These ﬁgures of merit allow us to verify the lack of optical-sectioning capability of conventional microscopes and provide us with hints on how to overcome that lack. One of the solutions comes from the use of confocal scanning arrangements. A CSM gets its optical-sectioning capability from the smart combination of two simple ideas: one is the inhomogeneous illumination of the sample, the other is the rejection of the light emanating from out-of-focus parts of the sample. The illumination pattern is obtained by using a microscope for illumination, while light rejection is achieved by using a second microscope and a pinhole in the collection arm of the system. Although CSMs are well known for their high optical-sectioning capability, they still have the problem of providing 3D images with anisotropic

64

The Resolution Challenge in 3D Optical Microscopy

quality so that the 3D image strongly depends on the specimen orientation. This is because the resolution is always much better in the lateral than in the axial direction. A simple approach for alleviating this problem comes from the use of PSF-engineering strategies. Speciﬁcally, it has been shown that by using a pupil ﬁlter consisting of one SR one can substantially increase the isotropy of the PSF. Another solution, which is more effective but also much more complicated from a technological point of view, is the so-called 4Pi CSM. This technique, which has been successfully commercialized, was born from the smart combination of two well-known principles: the principle of confocal microscopy and that of SWM. The other solution for the lack of optical sectioning is the so-called structured illumination microscopy. As usual in microscopy, this technique resulted from a very smart application of an old principle, which was already published by Lukosz in the early 1970s. From that time, it is known that the heterodyne combination of some images of the same object, obtained in different illumination conditions, can double the lateral resolution of a paraxial imaging system. What is novel are the extension of this principle to high-NA imaging and its use not only in doubling the lateral resolution but also to confer optical-sectioning capability to conventional microscopes. Let us conclude by saying that during many decades in the 20th century, the research in microscopy was concentrated on a single engineering problem: the design of objectives with increasing NA. However, in the last two decades, we have seen an explosion in the research on microscopy. This has been due to the interest in 3D microscopy. This chapter has dealt only with the principles and with the more widespread realizations of 3D microscopy. Other emerging techniques have not been described here but could probably constitute the future of 3D microscopy. We refer, for instance, to the STED (Hell, 2003; Hell and Wichmann, 1994), second- or third-harmonic generation microscopy (Dudovich, Oron, and Silberberg, 2002), or CARS microscopy (Zoumi, Yeh, and Tromberg, 2002). The success of such techniques would transform the microscopy of the 21st century.

ACKNOWLEDGMENTS The authors wish to acknowledge stimulating discussions with Dr.J.Ojeda-Castañeda (Universidad Autónoma San Luis Potosí, México), Dr.P.Török (Imperial College, London), and Dr.C.J.R.Sheppard (National University of Singapore). We acknowledge, as well, the indispensable help with the calculation of the ﬁgures from R.Martínez-Cuenca and I.Escobar (University of Valencia, Spain). We also acknowledge Professor P.Andrés (University of Valencia) for providing us with a solid education in Optics, with research honesty and, in general, with humanity in the relation with collaborators. This work was funded by the Plan Nacional I+D+I (grant DPI2006-8309), Ministerio de Educación y Ciencia (Spain).

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65

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CHAPTER

2 Transformation Optics and the Geometry of Light Ulf Leonhardt* and Thomas G. Philbin†

Contents

1 Introduction 2 Fermat’s Principle 3 Arbitrary Coordinates 3.1 Coordinate Transformations 3.2 The Metric Tensor 3.3 Vectors and Bases 3.4 One-Forms and General Tensors 3.5 Coordinate and Non-Coordinate Bases 3.6 Vector Products and Levi–Civita Tensor 3.7 The Covariant Derivative of a Vector 3.8 Covariant Derivatives of Tensors and of the Metric 3.9 Divergence and Curl 3.10 The Laplacian 3.11 Geodesics and Curvature 4 Maxwell’s Equations 4.1 Geometries and Media 4.2 Transformation Media 4.3 Wave Equation and Fermat’s Principle 4.4 Space–Time Geometry 5 Transformation Media 5.1 Spatial Transformation Media 5.2 Perfect Invisibility Devices 5.3 Perfect Lenses 5.4 Moving Media 5.5 Optical Aharonov–Bohm Effect 5.6 Analog of the Event Horizon 6 Conclusions Acknowledgments References

70 74 80 80 83 85 85 87 88 90 95 98 99 100 106 107 109 112 115 117 118 121 124 129 132 135 142 144 148

* School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK † Max Planck Research Group Optics, Information and Photonics, Günther-Scharowsky Str. 1, 91058 Erlangen, Germany Progress in Optics, Volume 53, ISSN: 0079-6638, DOI: 10.1016/S0079-6638(08)00202-3. Copyright © 2009 Elsevier B.V. All rights reserved.

69

70

Transformation Optics and the Geometry of Light

1. INTRODUCTION Metamaterials are beginning to transform optics and microwave technology thanks to their versatile properties that, in many cases, can be tailored according to practical needs and desires (Caloz and Itoh, 2006; Eleftheriades and Balmain, 2005; Engheta and Ziolkowski, 2006; Halpern, 2007; Krowne and Zhang, 2007; Marqués, Martin, and Sorolla, 2008; Milton, 2002; Sarychev and Shalaev, 2007). Although metamaterials are surely not the answer to all engineering problems, they have inspired a series of signiﬁcant technological developments and also some imaginative research, because they invite researchers and inventors to dream. Imagine there were no practical limits on the electromagnetic properties of materials. What is possible? And what is not? If there are no practical limits, what are the fundamental limits? Such questions inspire taking a fresh look at the foundations of optics (Born and Wolf, 1999) and at connections between optics and other areas of physics. In this chapter, we discuss such a connection, the relationship between optics and general relativity; expressed more precisely, we explore the relationship between geometrical ideas normally applied in general relativity and the propagation of light, or electromagnetic waves in general, in materials (Leonhardt and Philbin, 2006). Farfetched as it may appear, general relativity turns out to have been put to practical use in the ﬁrst working prototype of an electromagnetic cloaking device (Schurig et al., 2006); it gives perhaps the most elegant approach to achieve invisibility (Leonhardt, 2006b; Pendry, Schurig, and Smith, 2006), and general relativity even works behind the scenes of perfect lenses (Leonhardt and Philbin, 2006; Pendry, 2000; Veselago, 1968). The practical use of general relativity in electrical and optical engineering may seem surprisingly unorthodox; traditionally, relativity has been associated with the physics of gravitation and cosmology (Misner, Thorne, and Wheeler, 1973; Peacock, 1999) or, in engineering (van Bladel, 1984), has been considered a complication, not a simpliﬁcation. For example, the global positioning system would not be as accurate as it is without taking relativistic corrections into account that are due to gravity and the motion of the navigation satellites. However, here we are not concerned with the inﬂuence of the natural geometry of space and time on optics, the space–time curvature due to gravity, but rather we show how optical materials create artiﬁcial geometries for light and how such geometries can be exploited in designing novel optical devices. Connections between geometry and optics are nothing new; the ideas we explain here are rather, to borrow a phrase of Sir Michael Berry, “new things in old things.” These ideas are based on Fermat’s principle (Born and Wolf, 1999) formulated in 1662 by Pierre de Fermat, but anticipated nearly a millennium ago by the Arab scientist Ibn al-Haytham and inspired by the Greek polymath Hero of Alexandria’s reﬂections on light almost two

Introduction

71

millennia ago. According to Fermat’s principle, light rays follow extremal optical paths in materials (shortest or longest, mostly shortest), where the length measure is given by the refractive index. Media change the measure of length. This means that any optical medium establishes a geometry (Bortolotti, 1926; Rytov, 1938; Schleich and Scully, 1984): the glass in a lens, the water in a river, or the air creating a mirage in the desert. General relativity has cultivated the theoretical tools for ﬁelds in curved geometries (Landau and Lifshitz, 1995; Misner, Thorne, and Wheeler, 1973). In this chapter, we show how to use these tools for applications in electromagnetic or optical metamaterials. Metamaterials are materials with electromagnetic properties that originate from man-made sub-wavelength structures (Marqués, Martin, and Sorolla, 2008; Milton, 2002; Sarychev and Shalaev, 2007). Perhaps the best-known metamaterials are the materials used in the pioneering demonstrations of negative refraction (Shelby, Smith, and Schultz, 2001) or invisibility cloaking of microwaves (Schurig et al., 2006; see Figure 1) or for negative refraction of near-visible light (Soukoulis, Linden, and Wegener, 2007). These materials consist of metallic cells that are smaller than the

10␮r

␮␪

⑀z

4 3 2 1 0 21

10

20

30 40 [mm]

50

60

z^ ␪^

r^

FIGURE 1 Cloaking device. Two-dimensional microwave cloaking structure (background image) with a plot of the material parameters that are implemented. The cloaking device is made of circuit-board with structures that are about an order of magnitude smaller than the wavelength. The structures are split-ring resonators with tunable magnetic response. The split-ring resonators of the inner and outer rings are shown in expanded schematic form (transparent square insets). (From Schurig et al., 2006. Reprinted with permission from AAAS.)

72

Transformation Optics and the Geometry of Light

(a)

(b)

(c)

10 mm 5 mm

(f) (f)

(e) (e)

(d) (d)

10 mm

(g)

1mm

(h) (h)

1mm

(i) (h) (i)

10 mm

FIGURE 2 Photonic-crystal fibers (PCF). An assortment of optical (OM) and scanning electron (SEM) micrographs of PCF structures. (a) SEM of an endlessly single-mode solid core PCF. (b) Far-field optical pattern produced by (A) when excited by red and green laser light. (c) SEM of a recent birefringent PCF. (d) SEM of a small (800 nm) core PCF with ultrahigh non-linearity and a zero chromatic dispersion at 560-nm wavelength. (e) SEM of the first photonic band gap PCF; its core formed by an additional air hole in a graphite lattice of air holes. (f) Near-field OM of the six-leaved blue mode that appears when (e) is excited by white light. (g) SEM of a hollow-core photonic band gap fiber. (h) Near-field OM of a red mode in hollow-core PCF (white light is launched into the core). (i) OM of a hollow-core PCF with a Kagomé cladding lattice, guiding white light. (From Russell, 2003. Reprinted with permission from AAAS.)

relevant electromagnetic wavelength. Each cell acts like an artiﬁcial atom that can be tuned by changing the shape and the dimensions of the metallic structure. It is probably fair to consider microstructured or photonic-crystal ﬁbers (Russell, 2003) as metamaterials as well (see Figure 2). Here subwavelength structures—airholes along the ﬁber—signiﬁcantly inﬂuence the optical properties of the fused silica from which the ﬁbers are made. Metamaterials have a long history: the ancient Romans invented ruby glass, which is a metamaterial, although the Romans presumably did not know this concept. Ruby glass (Wagner et al., 2000) contains nano-scale gold colloids that render the glass neither golden nor transparent, but ruby,

Introduction

73

depending on the size and concentration of the gold droplets. The color originates from a resonance of the surface plasmons (Barnes, Dereux, and Ebbesen, 2003) on the metallic droplets. Metamaterials per se are nothing new: what is new is the degree of control over the structures in the material that generate the desired properties. The speciﬁc starting point of our theory is not new either. Gordon (1923) noticed that moving isotropic media appear to electromagnetic ﬁelds as certain effective space–time geometries. Bortolotti (1926) and Rytov (1938) pointed out that ordinary isotropic media establish nonEuclidean geometries for light. Tamm (1924, 1925) generalized the geometric approach to anisotropic media and brieﬂy applied this theory (Tamm, 1925) to the propagation of light in curved geometries. Plebanski (1960) formulated the electromagnetic effect of curved space–time or curved coordinates in concise constitutive equations. Electromagnetic ﬁelds perceive media as geometries and geometries act as effective media. Furthermore, in 2000, it was understood (Leonhardt, 2000) that media perceive electromagnetic ﬁelds as geometries as well. Light acts on dielectric media via dipole forces [forces that have been applied in optical trapping and tweezing (Dholakia, Reece, and Gu, 2008; Neuman and Block, 2004)]. These forces turn out to appear like the inertial forces in a speciﬁc space– time geometry. This geometric approach (Leonhardt, 2006a) was used to shed light on the Abraham–Minkowski controversy about the electromagnetic momentum in media (Abraham, 1909, 1910; Leonhardt, 2006d; Minkowski, 1908; Peierls, 1991). Geometrical ideas have been applied to construct solutions of Maxwell’s equations by coordinate transformations (Dolin, 1961) and conductivities that are undetectable by static electric ﬁelds (Greenleaf, Lassas, and Uhlmann, 2003a,b), which were the precursors of invisibility devices (Alu and Engheta, 2005; Gbur, 2003; Leonhardt, 2006b,c; Milton and Nicorovici, 2006; Pendry, Schurig, and Smith, 2006; Schurig, Pendry, and Smith, 2006) based on implementations of coordinate transformations. From these recent developments grew the subject of transformation optics. Here media, possibly made of metamaterials, are designed such that they appear to perform a coordinate transformation from physical space to some virtual electromagnetic space. As we describe in this chapter, the concept of transformation optics embraces some of the spectacular recent applications of metamaterials. Transformation optics is beginning to transform optics. We would do injustice to this emerging ﬁeld if we attempted to record every recent result. By the time this chapter goes to press, it would be outdated already. Instead we focus on the “old things in new things,” because those are the ones that are guaranteed to last and remain inspiring for a long time to come. This chapter rather is a primer, not a typical literature review. We try to give an introduction into connections between geometry and electromagnetism in media that is as consistent and elementary as possible, without assuming much prior knowledge. We begin in Section 2 with

74

Transformation Optics and the Geometry of Light

a brief section on Fermat’s principle and the concept of transformation optics. In Section 3 we develop in detail the mathematical machinery of geometry. Although this is textbook material, many readers will appreciate a (hopefully) readable introduction. We do not assume any prior knowledge of differential geometry; readers familiar with this subject may skim through most of Section 3. After having honed the mathematical tools, we apply them to Maxwell’s electromagnetism in Section 4, where we develop the concept of transformation optics. In Section 5 we discuss some examples of transformation media: perfect invisibility devices, perfect lenses, the Aharonov–Bohm effect in moving media, and analogs of the event horizon. Let’s begin at the beginning, with Fermat’s principle.

2. FERMAT’S PRINCIPLE In a letter dated January 1, 1662, Pierre de Fermat formulated a physical principle that was destined to shape geometrical optics, to give rise to Lagrangian and Hamiltonian dynamics and to inspire Schrödinger’s quantum mechanics and Feynman’s form of quantum ﬁeld theory and statistical mechanics. Fermat’s principle is the principle of the shortest optical path: light rays passing between two spatial points A and B chose the optically shortest path (see Figure 3). In some cases, however, light takes the longest path; in any case, light rays follow extremal optical paths (see Figure 4). The optical path length s is deﬁned in terms of the refractive index n as

s=

B n dl = n dx2 + dy2 + dz2

(1)

A

A

B

FIGURE 3 Fermat’s principle. Light takes the shortest optical path from A to B (solid line), which is not a straight line (dotted line) in general. The optical path length is measured in terms of the refractive index n integrated along the trajectory. The gray level of the background indicates the refractive index. The figure illustrates the ray trajectories involved in forming a mirage.

Fermat’s Principle

75

L

A

A9

B

FIGURE 4 Fermat’s principle: longest optical path. The figure shows a simple example where light takes the longest optical path: light traveling in a straight line through a lens from one focal point A to a point B beyond the other focal point A . To see why the optical path from A to B is the longest, compare the solid line of the actual path with an example of a virtual path, with the dashed and dotted paths, and note that the optical path length between the focal points A and A of the lens is always the same, regardless of whether light travels along a straight line or is refracted in the lens. Therefore, the optical path length taken differs from the virtual path length by the difference between the two short sides and the long side of the triangle from L to A and B. The sum of the two short sides of a triangle is always longer than the long side: light has taken the longest optical path.

in Cartesian coordinates. If the refractive index varies in space—for nonuniform media—the shortest optical path is not a straight line, but is curved. This bending of light is the cause of many optical illusions. For example, picture a mirage in the desert (Feynman, Leighton, and Sands, 1983). The tremulous air above the hot sand conjures up images of water in the distance, but it would be foolish to follow these deceptions; they are not water, but images of the sky. The hot air above the sand bends light from the sky, because hot air is thin with low refractive index and so light prefers to propagate there. Fermat’s principle has profoundly inﬂuenced modern physics, and like most if not all profound discoveries, it has deep roots in the history of science. Fermat was inspired by the Greek polymath Hero of Alexandria’s theory of light reﬂection in mirrors. The Arab scientist Ibn al-Haytham anticipated Fermat’s principle in his Book of Optics (written during his house arrest in Cairo from 1011 to 1021). Fermat’s principle was instantly greeted with objections, because it appears to violate causality—it presumes an idea of destiny. The principle governs the path between A and B if it is known that light travels from A to B; Fermat’s principle shows how the ray’s destiny is fulﬁlled, but it does not explain why the light ray arrives at B and not at some other end point. Wave optics resolves this problem, because a wave emitted at A propagates in all directions (but possibly with greatly varying amplitude). The path of extremal optical length (1) is the place of constructive interference between all possible paths.

76

Transformation Optics and the Geometry of Light

FIGURE 5 Optical conformal mapping (Leonhardt, 2006b). Suppose that an optical medium performs a coordinate transformation from a straight Cartesian grid (left) to curved coordinates in physical space (right). The trajectories of light rays follow the curved coordinates, but this apparent curvature is an illusion and may be used to create optical illusions, for example, invisibility.

We will derive Fermat’s principle later, in Section 4.3. Imagine one illuminates a non-uniform medium with a grid of light rays (see Figure 5); each ray is curved according to Fermat’s principle. With this consideration, the question we pose here is this: is it possible to transform away the curvature of the grid? In this case, the curved path of each ray would appear as a straight line in some transformed space. So, in other words, is it possible that the bending of light is an illusion of choosing the wrong coordinates? Curvature would be an illusion of Cartesian linear thinking. Such transformable media may create optical illusions by themselves; in fact they may create the ultimate illusion, invisibility: if the transformable grid contains a hole, anything inside the hole is invisible. The transformation medium acts as a cloaking device. For simplicity, imagine a two-dimensional situation where the refractive index varies in x and y, and light is conﬁned to the x, y plane. Consider a coordinate transformation from x and y to x and y , respectively. Is Fermat’s principle obeyed in transformed coordinates,

s=

n dx2 + dy2 = n dx2 + dy2

?

(2)

The reader easily sees from

dx =

∂x ∂x dx + dy , ∂x ∂y

dy =

∂y ∂y dx + dy ∂x ∂y

(3)

Fermat’s Principle

77

that dx2 + dy2 is proportional to dx2 + dy2 if

∂y ∂x = , ∂x ∂y

∂y ∂x =− . ∂x ∂y

(4)

In this case the coordinate transformation changes n2 (dx2 + dy2 ) into n2 (dx2 + dy2 ), thus preserving the form (1) of Fermat’s principle. One obtains

n2

∂x ∂x

2

+

2

∂x ∂y

= n2

∂y ∂x

2

+

∂y ∂y

2

= n2 .

(5)

A transformation with the property (4) is known as a conformal transformation in two-dimensional space (Nehari, 1952). Conformal transformations leave Fermat’s principle (1) intact; they correspond to materials with an isotropic refractive index proﬁle. If n = 1, the transformed space is empty; light would propagate along a straight line there: the refractiveindex proﬁle acts as a transformation medium. So far we have discussed light rays. How does the conformal transformation (4) act on light waves? Suppose that both amplitudes ψ of the optical polarization satisfy the Helmholtz equation

ω2 2 ∇ + 2 n ψ = 0, c 2

(6)

where ω denotes the frequency and c the speed of light in vacuum. It is convenient to write the Laplacian ∇ 2 as

∂2 ∂2 ∇ = 2+ 2 = ∂x ∂y 2

∂ ∂ +i ∂x ∂y

∂ ∂ −i . ∂x ∂y

(7)

We obtain from the differential equations (4) of the conformal map the transformation

∂ ∂ −i ∂x ∂y

∂y ∂ ∂x ∂ ∂y ∂ ∂x ∂ + − i − i ∂x ∂x ∂x ∂y ∂y ∂x ∂y ∂y ∂x ∂y ∂ ∂ = +i −i ∂x ∂x ∂x ∂y ∂y ∂x ∂ ∂ = −i −i ∂y ∂y ∂x ∂y

=

(8)

78

Transformation Optics and the Geometry of Light

and, by exchanging i with −i,

∂ ∂ +i ∂x ∂y

= =

∂x ∂y −i ∂x ∂x ∂y ∂x +i ∂y ∂y

∂ ∂ +i ∂x ∂y

∂ ∂ +i . ∂x ∂y

(9)

Furthermore, since

∂ ∂ +i ∂x ∂y

∂x ∂y +i ∂x ∂x

=

∂2 x ∂2 y ∂2 y ∂2 x + i = 0, (10) − + i ∂x∂y ∂x∂y ∂x2 ∂x2

the Laplacian ∇ 2 is transformed into

∂y ∂x ∂y ∂ ∂ ∂x ∂ ∂ +i −i ∇ = +i −i ∂x ∂x ∂x ∂y ∂x ∂y ∂x ∂y 2 2 ∂x ∂y 2 = + ∇ ∂x ∂x 2 ∂x 2 ∂y 2 ∇ . = + (11) ∂y ∂y

2

Consequently, the Helmholtz equation (6) is invariant under conformal transformations if the refractive index is transformed according to Eq. (5). Waves are transformed in precisely the same way as rays. There is an elegant shortcut to the theory of this optical conformal mapping (Leonhardt, 2006b) that allows us to condense the previous calculations in a few lines: complex analysis (Ablowitz and Fokas, 1997; Needham, 2002). Suppose we denote the two-dimensional coordinates by complex numbers

z = x + i y,

z∗ = x − i y.

(12)

From

∂z ∂ ∂z ∂ ∂ ∂ ∂ = + + = , ∂x ∂x ∂z ∂x ∂z∗ ∂z ∂z∗ we obtain

1 ∂ ∂ ∂ = −i , ∂z 2 ∂x ∂y

∂ ∂ ∂ =i − i ∗, ∂y ∂z ∂z

∂ ∂ 1 ∂ + i = ∂z∗ 2 ∂x ∂y

(13)

(14)

Fermat’s Principle

79

and hence

∇2 = 4

∂2 . ∂z∂z∗

(15)

Consider a coordinate transformation described by a function w(z) that depends on z, but not on z∗ ,

x + i y = w(z)

with

∂w = 0. ∂z∗

(16)

Since

2

∂w = ∂z∗

∂x ∂ ∂ ∂y ∂x ∂y +i x + i y = − +i +i , ∂x ∂y ∂x ∂y ∂y ∂x

(17)

the differential equation (4) of conformal maps are naturally satisﬁed; they are the Cauchy–Riemann differential equations of analytic functions (Ablowitz and Fokas, 1997; Needham, 2002; Nehari, 1952). Finally we obtain from

dw 2 2 ∂2 dw dw∗ ∂2

∇

∇ =4 =4 = ∂z∂z∗ dz dz∗ ∂w∂w∗ dz

2

(18)

the relationship (5) between the original and the transformed refractiveindex proﬁle in the Helmholtz equation (6) as

dw

n . n =

dz

(19)

Complex analysis not only simpliﬁes the theory but also provides optical conformal mapping (Leonhardt, 2006b) with a vast resource in calculational tools and geometrical insights (Ablowitz and Fokas, 1997; Needham, 2002; Nehari, 1952). Conformal coordinate transformations represent a special case; most spatial transformations are non-conformal, and we could also envision transformations that mix space and time. Consequently, media that implement such transformations are not subject to Fermat’s principle in the form (1). Furthermore, the Helmholtz equation (6) is only approximately valid (Born and Wolf, 1999). Light should be described as an electromagnetic wave subject to Maxwell’s equations. There are various ways of developing the concepts of transformation media for the general case. In Section 4, we discuss a theory that perhaps plays a similar role to that

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Transformation Optics and the Geometry of Light

played by complex analysis in optical conformal mapping. It will equip the reader with calculational shortcuts and geometrical insights. For this theory we borrow concepts from general relativity, but we do not assume that the reader is familiar with them. The necessary ingredients from differential geometry are derived by elementary means in the following section.

3. ARBITRARY COORDINATES The theory of transformation media requires consideration of Maxwell’s equations in arbitrary coordinates. This means that the natural mathematical language of transformation media is differential geometry, the mathematics that also describes curved spaces and Einstein’s general relativity. Here, we introduce the reader to the mathematics of arbitrary coordinates to the extent necessary to deal with coordinate transformations of Maxwell’s equations. The reader will see that this formalism provides the most transparent way of describing non-Cartesian coordinate systems: even something as apparently familiar as electromagnetism in spherical polar coordinates is much simpler in the language of differential geometry than in the standard treatment found in the electromagnetism textbooks.1

3.1. Coordinate Transformations We deal ﬁrst with spatial coordinates; the extension to space–time coordinates will then be straightforward. Our interest is in writing equations that are valid in an arbitrary spatial coordinate system {xi , i = 1, 2, 3} and in performing an arbitrary transformation to another set of coordinates that we distinguish from the original by a prime on the index: {xi , i = 1, 2, 3}. Throughout, we take as concrete examples Cartesian coordinates {x, y, z} and spherical polar coordinates {r, θ, φ}, related by

{xi } = {x, y, z},

y = r sin θ sin φ,

{xi } = {r, θ, φ}, r = x 2 + y 2 + z2 , θ = tan−1 x2 + y2 /z ,

z = r cos θ,

φ = tan−1 (y/z).

x = r sin θ cos φ,

(20)

1 The introduction to curved coordinates given here is broadly similar to that in Schutz’s excellent text

(Schutz, 1985).

Arbitrary Coordinates

81

At this point we introduce the Einstein summation convention in which a summation is implied over repeated indices; for example

Ai B i ≡

Ai Bi = A1 B1 + A2 B2 + A3 B3 .

(21)

i

This convention allows us to dispense with writing summation signs, which are completely unnecessary. For reasons that will become clear later on, our summations will generally be over a pair of indices in which one index is a subscript and the other is a superscript, as in Eq. (21). We also introduce the Einstein range convention by which a free index (i.e., an index that is not summed over) is understood to range over all possible values of the index, for example

Ai ≡ Ai , i = 1, 2, 3 .

(22)

Together, the summation and range conventions allow an economy of notation such as the following:

R

i

jik

≡

R

i

jik ,

j, k = 1, 2, 3 .

(23)

i

The differentials of our two sets of coordinates, xi and xi , are related by the chain rule:

∂xi i dx , i ∂x

dxi =

dxi =

∂xi dxi , ∂xi

(24)

with similar relations holding for the differential operators:

∂xi ∂ ∂ = , ∂xi ∂xi ∂xi

∂ ∂xi ∂ = . ∂xi ∂xi ∂xi

(25)

We denote the transformation matrices in Eqs. (24) and (25) by

i

i

∂xi = i , ∂x

i i

∂xi = i. ∂x

(26)

Note that primes or unprimed indices in i i and i i do not mean that we simply use different indices: i i and i i are different matrices where we differentiate with respect to different sets of coordinates. The

82

Transformation Optics and the Geometry of Light

reader may verify that for the example (20) the transformation matrices (26) are

⎛

i i

⎞ sin θ cos φ r cos θ cos φ −r sin θ sin φ ⎜ ⎟ = ⎝ sin θ sin φ r cos θ sin φ r sin θ cos φ ⎠ cos θ −r sin θ 0 ⎛x ⎜ r ⎜ ⎜y ⎜ =⎜ ⎜ r ⎜ ⎝z r ⎛

⎞ xz −y ⎟ x2 + y 2 ⎟ ⎟ yz ⎟ x ⎟, 2 2 ⎟ x +y ⎟ ⎠ 2 2 0 − x +y sin θ cos φ

⎜ 1 ⎜ cos θ cos φ ⎜ i =⎜ r ⎜ ⎝ 1 − csc θ sin φ r ⎛

⎞

cos θ

⎟ 1 1 cos θ sin φ − sin θ ⎟ ⎟ r r ⎟ ⎟ ⎠ 1 csc θ cos φ 0 r y r

x r

⎜ ⎜ ⎜ ⎜ xz =⎜ ⎜ r2 x2 + y2 ⎜ ⎜ y ⎝ − 2 x + y2

(28)

sin θ sin φ

i

r2

yz x2 + y 2 x 2 x + y2

(27)

⎞

z r

⎟ ⎟ ⎟ 2 2 x +y ⎟ ⎟. − ⎟ r2 ⎟ ⎟ ⎠ 0

(29)

(30)

From Eqs. (24) and (26), we ﬁnd dxi = i i dxi = i i i j dxj and dxi =

i i dxi = i i i j dxj , which imply

i i i j = δi j ,

i i i j = δi j ,

(31)

where δi j and δi j are the Kronecker delta, the matrix elements of the unity

matrix. Equation (31) states that the matrices i i and i i are the inverse of each other. This property can be deduced directly from the deﬁnitions (26) and the chain rule. The reader may verify the relations (31) for the example (27)–(30).

Arbitrary Coordinates

83

3.2. The Metric Tensor Although we have awarded ourselves the freedom of covering space with any coordinate system we wish, the distances between points in space are invariant—they are the same no matter which coordinates we use to calculate them. The basic quantity is the square of the inﬁnitesimal distance ds between the points xi and xi + dxi . For Cartesian coordinates xi = {x, y, z}, this is given by the three-dimensional Pythagoras theorem:

ds2 = dx2 + dy2 + dz2 = δij dxi dxj ,

(32)

where δij is again the Kronecker delta. For general coordinates xi , the square of the line element ds2 is given by an expression quadratic in the coordinate differentials dxi :

ds2 = gij dxi dxj .

(33)

In Eq. (33), we have introduced the metric tensor gij , the quantity that allows us to calculate distances in space. The metric tensor is always symmetric in its indices,

gij = gji ,

(34)

for the following reason: a matrix can always be written as a sum of its symmetric and antisymmetric parts, and the reader can verify that an antisymmetric part of gij would not contribute to the distance (33). So it only makes sense to consider a symmetric metric tensor. In Cartesian coordinates (32), the metric tensor is the Euclidean metric δij . We can write the relation (33) also in the coordinate system xi , denoting the metric tensor in this system by gi j ; from the invariance of ds we have

j

ds2 = gi j dxi dxj = gij dxi dxj = gij i i j dxi dxj ,

(35)

where we have used Eqs. (24) and (26) in the second line. Equation (35) reveals how the metric tensor changes under a coordinate transformation: j

gi j = i i j gij .

(36)

Writing the metric tensors gi j and gij as matrices G and G, we can display the transformation procedure in the matrix form

G = T G,

(37)

84

Transformation Optics and the Geometry of Light

where denotes the transformation matrix i i deﬁned in Eq. (26), whereas i i is the inverse matrix −1 . Consider a transformation from a Cartesian coordinate system, in which gij = δij , to another Cartesian system, by means of a rotation. As the new coordinates are Cartesian, the transformed metric must also be Euclidean, and Eq. (36) shows that this is the case: j

gi j = i i j δij = δi j

(Cartesian −→ Cartesian),

(38)

where the second equality follows from the fact that rotations are performed by orthogonal matrices with T = −1 . Rotations thus preserve the Euclidean metric. The metric tensor not only characterizes the measure of length in arbitrary coordinates but also turns out to describe the volume element as well. To see this, we represent the Cartesian volume element, dV = dx dy dz, in arbitrary coordinates according to the standard rule

dV = |det | dV .

(39)

From the matrix representation (37) follows

g = (det )2 g,

(40)

where g and g denote the determinants of the metric tensors. Note that g is always positive, because the determinant g of the Euclidian metric is unity. Consequently, we obtain the volume element

dV =

g dV .

(41)

√ Dropping the primes, we note that g dV always describes the volume element, in Cartesian or curved coordinates. Returning to our example of spherical coordinates, we can use the transformation procedure (36) to compute the metric tensor and the volume element: the metric tensor in Cartesian coordinates is gij = δij and the required transformation matrix is expressed in Eq. (27), so we obtain ⎛

1 0 = ⎝ 0 r2 0 0

gi j

⎞ 0 ⎠, 0 2 2 r sin θ

ds2 = gi j dxi dxj = dr2 + r2 (dθ 2 + sin2 θ dφ2 ).

(42)

(43)

Arbitrary Coordinates

85

The volume element (41) is given by the square root of the determinant of the matrix (42); we arrive at the familiar spherical volume element dV = r2 sin θ dr dθ dφ.

3.3. Vectors and Bases The transformation relations (24)–(26) determine the transformation properties of vectors, and of more general objects. The coordinate displacements dxi are the components of a vector in space; therefore, the components of a general vector V will transform in the same way under a change of coordinates:

V i = i i V i ,

V i = i i V i .

(44)

The components V i and V i in (44) refer to an expansion of V in terms of the basis vectors ei and ei associated with each coordinate system:

V = V i ei = V i ei

= i i V i ei .

(45)

The second line in Eq. (45) was obtained by use of Eq. (44), and comparison with the ﬁrst line gives the transformation of the basis vectors:

ei = i i ei .

(46)

In Cartesian coordinates xi = {x, y, z}, the basis vectors are the familiar unit vectors in the x-, y-, and z-directions:

ei = {ex , ey , ez } = {i, j, k}.

(47)

From (46) and (27), we then obtain the basis vectors in spherical polar coordinates as ei = {er , eθ , eφ } with

er = sin θ cos φ i + sin θ sin φ j + cos θ k, eθ = r cos θ cos φ i + r cos θ sin φ j − r sin θ k, eφ = −r sin θ sin φ i + r sin θ cos φ j.

(48)

3.4. One-Forms and General Tensors The expression (33) is the squared length of the vector dxi , and the metric tensor similarly gives the squared length of a general vector:

|V |2 = V · V = gij V i V j = gi j V i V j .

(49)

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Transformation Optics and the Geometry of Light

In Eq. (49), we have used the fact that the length of a vector is an invariant quantity, and the equality of the expressions evaluated in the coordinate systems xi and xi can be explicitly shown from the transformation rules (36) and (44) together with the inverse relations (31). Note that the invariance of Eq. (49) under coordinate transformations works because the vector V i and metric gij are transformed by matrices that are inverse to each other. This inverse relationship of their transformations is to be associated with the fact that V i is an upper-index object, whereas gij is a lower-index object. Each summation in (49) over an upper and lower index, called a contraction, is a coordinate-invariant operation because of this inverse property. It is useful to construct from V i and gij a lower-index quantity Vi that transforms in the manner of gij as follows:

Vi = gij V j .

(50)

The transformation rule for Vi is easily established from those of V i and gij ,

Vi = gi j V j = i i gij V j = i i Vi ,

(51)

so that Vi does indeed transform similarly to gij . The quantities Vi are the components of a covariant vector or one-form. We can view Eq. (50) as lowering the index on the vector V i using the metric tensor, producing the associated one-form Vi . Deﬁning the inverse metric tensor gij by

gij gjk = δik ,

(52)

V i = gij Vj .

(53)

we obtain from relation (50)

Equation (53) can be regarded as raising the index on the one-form Vi using the inverse metric tensor, producing the associated vector V i. Note that the vector V i and one-form Vi are the same only in Cartesian coordinates, where gij = δij . The expression (49) for the squared length of a vector is more compactly written using the one-form Vi : |V |2 = V · V = Vi V i . The scalar product of two vectors V and U is

V · U = gij V i U i = Vi U i = V i Ui = Vi U i = V i Ui ,

(54)

where the coordinate invariance of the scalar product follows from the transformation properties of vectors and one-forms. From the position of

Arbitrary Coordinates

87

the indices on the inverse metric tensor gik , we expect it to transform in a vector-like fashion. We verify this by writing the vector and one-form

j

in Eq. (53) in terms of their transformed values, i i V i = gij j Vj from

j

which follows V i = i i gij j Vj = gi j Vj so that

j

gi j = i i j gij .

(55)

The transformation rule for a tensor with an arbitrary collection of indices is now clear; we give as an example a four-index tensor:

j

Ri j k l = i i j kk l l Ri jkl .

(56)

One can also raise and lower the indices of a general tensor in complete analogy to Eqs. (53) and (50). In this manner, Eq. (52) may be regarded as a raising or lowering operation so that the Kronecker delta δi j is the metric tensor with one index raised or the inverse metric tensor with one index lowered. It is straightforward to introduce bases for general tensors, but we will not pursue this here. The reader should, however, be able to deduce the index positions and transformation properties of any tensor basis, starting with the one-form basis.

3.5. Coordinate and Non-Coordinate Bases The scalar product of two basis vectors is seen from the deﬁnition (54) to be

ei · ej = gij ,

(57) j

since the components (ei )j of a basis vector ei are δ i . The basis vectors (47) in Cartesian coordinates constitute an orthonormal basis. For the spherical polar basis (48), we can compute the dot products using the right-hand sides of Eq. (48) or, much more simply, by using the scalar product (57) and the metric (42) in spherical polar coordinates. We see that the basis vectors are orthogonal to each other, but not all of them are unit vectors:

|er |2 = 1,

|eθ |2 = r2 ,

2

eφ = r2 sin2 θ.

(58)

One can of course easily construct an orthonormal basis eˆ i by rescaling eθ and eφ :

eˆ r = er ,

eˆ θ =

1 eθ , r

eˆ φ =

1 eφ . r sin θ

(59)

88

Transformation Optics and the Geometry of Light

The reader unfamiliar with the material of this section will only have encountered spherical polar coordinates in combination with the orthonormal basis (59). How did we end up with the non-orthonormal basis? The answer is that we let the coordinates induce our basis through their differentiable structure. Recall that the components of a vector were introduced in analogy with the coordinate differentials dxi . The discussion of vector components then immediately speciﬁes the basis as in Eqs. (44)–(46). Such a basis, induced naturally by the coordinates, is called a coordinate basis. The fact that the differentiable properties of the coordinates completely determine the coordinate basis is the reason why the coordinate bases (46) behave exactly like the partial derivative operators in (25).2 The orthonormal basis (59), by contrast, is not induced in a similar manner by any coordinate system—it is a non-coordinate basis (see Schutz, 1985 for more detail). The only coordinate system that induces an orthonormal coordinate basis is the Cartesian system. It might be suspected that it is always simpler to work in an orthonormal basis; in fact, for most purposes a coordinate basis is much simpler, in particular for the manipulations in curvilinear coordinates performed in electromagnetism textbooks. The reason these texts use the more complicated non-coordinate bases is that to exploit the simplicity of coordinate bases requires a little knowledge of tensor analysis.

3.6. Vector Products and Levi–Civita Tensor For describing electromagnetism in arbitrary coordinates, we need to deﬁne the notion of vector products in three-dimensional space. Vector products V × U are antisymmetric, V × U = −U × V , and the vector products of the Cartesian basis vectors are cyclic, i × j = k, j × k = i, and k × i = j. For implementing these properties, we introduce the permutation symbol [ijk] deﬁned by

⎧ ⎪ ⎨ +1 if ijk is an even permutation of 123, [ijk] = −1 if ijk is an odd permutation of 123, ⎪ ⎩ 0 otherwise.

(60)

We deﬁne the Levi–Civita tensor ijk as the tensor whose components in some particular right-handed Cartesian coordinate system are given by the permutation symbol:

ijk = [ijk],

(right-handed Cartesian coordinates).

(61)

2 This is not just a pleasant correspondence; in modern differential geometry, the partial derivative opera-

tors are the coordinate basis vectors.

Arbitrary Coordinates

89

We can ﬁnd the components of the Levi–Civita tensor in any other coordinate system, Cartesian or otherwise, by transforming the expression (61) according to the rule (56); we constructed a tensor by deﬁnition. Speciﬁcally, the Levi–Civita tensor in an arbitrary coordinate system is

j

i j k = i i j kk [ijk] = det(l l )[i j k ] =

[i j k ] det

(62)

using the Leibniz formula for the determinant of l l (Stoll, 1969). As in Section 3.2, det is the determinant of the transformation matrix l l , the matrix inverse of l l . We obtain from Eq. (40)

det = ± g .

(63)

Which sign should we take here? Clearly, the sign in question is the sign of det . If det is negative, the transformation changes the handedness of the coordinate system; so the new system is left-handed (Goldstein, 1980). For example, a transformation that changes the sign of one or all three of the coordinates in the right-handed Cartesian system has det = −1 and results in a left-handed Cartesian system. A general transformation between Cartesian coordinate systems consists of a rotation and a translation, together with possible reﬂections of the coordinates. For any such transformation the Euclidean metric is preserved, so Eq. (38) holds and the transformation matrix is orthogonal; it then follows from Eq. (63) that

det = ±1

(Cartesian −→ Cartesian),

(64)

where the sign is negative if the transformation includes a handedness change. Using the relationship (63), we can now write the Levi–Civita tensor in arbitrary coordinates (62) in terms of the metric; we drop the prime on the indices of the arbitrary coordinate system and obtain

1 ijk = ± √ [ijk], g

(65)

where it is understood that the plus (minus) sign is obtained if the system is right-handed (left-handed). It is now clear that Eq. (61) holds in all right-handed Cartesian coordinate systems, not just in the one we started with. Some or all of the indices of ijk can be lowered according to the prescription (50); if all are lowered, we obtain another simple expression to

90

Transformation Optics and the Geometry of Light

complement the representation (65):

1 1 √ ijk = gil gjm gkn lmn = ± √ gil gjm gkn [lmn] = ± √ g[ijk] = ± g[ijk]. g g (66) The Levi–Civita tensor is completely antisymmetric; this means that when its components are taken with all indices in the upper or lower position, they are antisymmetric under interchange of two adjacent indices, e.g., ijk = − jik . The Levi–Civita tensor is required to compute vector products in an arbitrary coordinate system:

U × V = ijk Uj Vk ei .

(67)

The reader can verify that (67) is the standard vector product in righthanded Cartesian coordinates; the deﬁnitions we have given show that it maintains the same form when transformed to an arbitrary coordinate system. The components of the vector product can be written in terms of Eq. (65) or (66):

(U × V )i = ijk Uj Vk ,

(U × V )i = ijk U j V k .

(68)

We can apply the Levi–Civita tensor for expressing the well-known double vector product in arbitrary coordinates,

A × (B × C) = B(A · C) − C(A · B).

(69)

We obtain from formulas (65) and (66) and the deﬁning properties (60) of the permutation symbol

ijk klm =

j j [ijk] [klm] = δil δm − δim δl .

(70)

k

Contracted with the components of three vectors Aj , Bl , and Cm , this identity generates the double vector product (69). Curls are also computed using the Levi–Civita tensor, but they contain a differentiation and we must learn how to differentiate in arbitrary coordinates.

3.7. The Covariant Derivative of a Vector A scalar ﬁeld in space is a function of the coordinates, and we can take its partial derivative with respect to xi ; in writing the partial derivative, we

Arbitrary Coordinates

91

can introduce another ink-saving device as follows:

∂ ψ ≡ ψ,i . ∂xi

(71)

Here, a comma means partial differentiation, with the following index giving the coordinate with respect to which the derivative is taken. In Cartesian coordinates, the derivatives (71) are of course the components of the gradient vector ∇ψ. It is easy to see, however, from Eqs. (25), (26), and (51) that the expression (71) transforms as a one-form. Consistent with the index being in the lower position, the derivatives (71) are in fact the components of the one-form associated with the gradient vector, and this distinction can only be ignored in Cartesian coordinates where the vector and the one-form have the same components. In a general coordinate system, we must raise the index in Eq. (71) using the inverse metric tensor to obtain the components of the gradient vector, i.e.,

(∇ψ)i = ψ,i ,

(∇ψ)i = gij ψ,j ,

∇ψ = gij ψ,j ei .

(72)

∇ψ is a vector and so it is a coordinate-independent object; the reader may verify that gij ψ,j ei is the same in every coordinate system using the transformation rules for the quantities involved, gij ψ,j ei = gi j ψ,j ei . Note that if we were to use an orthonormal frame the components of the gradient one-form and vector would not be given by Eq. (72); this is easily seen in the case of spherical polar coordinates, where if we replace the coordinate basis (48) in Eq. (72) with the orthonormal, non-coordinate basis (59), the components (72) are rescaled. Thus in a non-coordinate basis, the partial derivatives ψ,i are not the components of the gradient one-form, and this makes such a basis unsuitable for our tensor calculus. Partial derivatives with respect to the coordinates only have a simple tensorial meaning if we use the basis induced by those coordinates. A vector ﬁeld in space V = V i ei consists of a sum of products of scalar ﬁelds V i and basis vector ﬁelds ei ; to differentiate V we must of course use the Leibniz rule:

∂ej ∂V j ∂ V = ej + V j i . i i ∂x ∂x ∂x

(73)

This simple relation represents the most important fact about curvilinear coordinates. In Cartesian coordinates, the basis vectors are constant, and to differentiate a vector we need only to differentiate its components. The coordinate basis vectors for any other coordinate system, however, change

92

Transformation Optics and the Geometry of Light

e1 e2 er e2 er

e␾

e␾

e1

FIGURE 6 Basis vectors change with position. Polar coordinates in two dimensions (spherical polar coordinates with θ = π/2) are depicted on the left. The basis vectors are everywhere orthogonal, but they rotate as one moves through the plane. While er remains a unit vector, the length of eφ , equal to r, varies from zero to infinity. A general two-dimensional coordinate system is shown on the right. Here, the orientations and lengths of both basis vectors vary, as does the angle between them. Since the dot product e1 · e2 does not vanish, the metric tensor is not diagonal in these coordinates (g12 = 0).

in orientation and magnitude as one moves through space (see Figure 6). The rate of change of a vector is itself a vector and so can be expanded in terms of the basis ei ; we therefore have

∂ej ∂xi

= kji ek .

(74)

The 27 quantities kji are called the Christoffel symbols; kji is the kth component of the derivative of ej with respect to xi. Let us pause to consider how kji change under a coordinate transformation. In the transformed system the Christoffel symbols are deﬁned by ∂ej/∂xi = kj i ek , and since we know how the basis vectors and the partial derivative operators transform [Eqs. (46) and (25)], we can deduce the transformation law for the Christoffel symbols. We leave it as an exercise for the reader to show that

j

j

kj i = kk j i i kji − kj,i j i i .

(75)

Note that kji do not obey the transformation law (56) of tensor components; therefore, they do not constitute a tensor.

Arbitrary Coordinates

93

The transformation rule (75) reveals an important property of the Christoffel symbols: they are symmetric in their lower indices, i.e.,

i jk = i kj .

(76)

To prove this, ﬁrst write the second term in the transformation rule (75) explicitly in terms of the partial derivatives (26) and show that it is symmetric in j and i . Now, the Christoffel symbols in any coordinate system can be obtained by transforming from Cartesian coordinates according to the rule (75), but in Cartesian coordinates the kji vanish. This shows that

in the new coordinate system kj i = ki j , but this coordinate system is arbitrary, so (76) holds for any coordinates. Using the Christoffel symbols we write down the derivative (73) of a vector. Insertion of Eq. (74) in Eq. (73) gives

∂ j j V = V ,i ej + kji V j ek = V ;i ej , i ∂x

(77)

where we have deﬁned the quantities

V

j ;i

≡V

j ,i

+

j

ki V

k

,

(78)

which give the components of the derivative of V with respect to xi . Just as the derivative of a scalar (a zero-index tensor) gives a one-form (a oneindex tensor), the derivative (77) of a vector V gives a two-index tensor j V ;i called the covariant derivative of V . The semi-colon in the deﬁnition (78) thus means covariant differentiation of the vector, in which both the components and the basis are differentiated, whereas the comma means differentiation of the components. “Covariant derivative” therefore just means “correct derivative”! Suppose we transport the vector V j from a point xi to its inﬁnitesimally close neighbor xi + dxi without rotating it or changing its length; this is called parallel transport. At the new point, the coordinate basis has changed, but the vector has not changed, so the vector components must vary as well as the coordinate basis. Here it is important that we differentiate correctly: the covariant derivative of the vector along dxi vanishes, but the ordinary derivative of its components will not, unless we are in Cartesian coordinates. Imagine V j and its parallel-transported neighbor as being part of a vector ﬁeld. We use DV j to denote the increment of V j along dxi and introduce an alternative notation for the covariant derivative, as the

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differential operator

∇i V j ≡ V

j ;i

=V

j

j

,i

+ ki V k .

(79)

The increment of V i along dxi is then

DV j = ∇i V j dxi .

(80)

In the case of parallel transport, the vector V j simply remains the same:

DV j = 0

(Parallel transport).

(81)

A non-trivial example of parallel transport is given by Foucault’s pendulum (Hart, Miller, and Mills, 1987): consider a pendulum that is attached to the Earth, but free to oscillate (Foucault suspended one from the dome of the Pantheon in Paris in 1851). The direction in which a pendulum swings follows the rotation of the Earth; it is parallel-transported on the surface of a sphere. The pendulum slowly turns due to the non-Euclidean curvature of the sphere, as we discuss in Section 3.11. j The covariant derivative of a vector V j is the tensor V ;i , so it must transform according to the tensor prescription (56). Note that neither of the two terms on the right-hand side of Eq. (78) separately constitutes a tensor. From the transformation rule of the Christoffel symbols (75), we see that the j transformation of ki V k will not adhere to the tensor transformation (56), j

and the same is true of V ,i , but when they are added together, however, the result does obey the transformation rule (56) of a tensor, as the reader should verify. In standard vector calculus, one is conﬁned to scalars and vectors, and j so one does not encounter the two-index tensor V ;i in its full glory, but only some aspects of covariant differentiation. For example, the divergence j ∇ · V is constructed from the covariant derivative V ;i by contraction of its two indices. To see this note that in Cartesian coordinates the divergence j is V i ,i , which (only) in these coordinates is equal to V i ;i . Now, since V ;i is a tensor, the contraction V i ;i is a scalar, the same in all coordinate systems, just like the dot product (54). We therefore have

∇ · V = ∇i V i = V i ;i = V i ;i .

(82)

Let us return to the example of spherical polar coordinates to see what a set of Christoffel symbols looks like. We can compute the Christoffel

Arbitrary Coordinates

95

symbols from Eqs. (74) and (48), or by transforming from Cartesian coordinates, in which i jk are zero, using the rules (75) and (27)–(30). Clearly, the recipe (74) presents the easier path and we ﬁnd θrθ = φ φr

=

1 , r

1 , r

φ rφ

φ φθ

=

1 , r

= cot θ,

θθr =

1 , r

r θθ = −r,

r φφ = −r sin2 θ,

φ θφ

= cot θ,

θφφ = − sin θ cos θ, (83)

all the other Christoffel symbols vanishing. We can now compute the divergence (82) in spherical polar coordinates, applying the covariant derivative (78):

∇·V = =

∂ r ∂ ∂ φ 2 r V + Vθ + V + V + cot θ V θ ∂r ∂θ ∂φ r ∂ φ 1 ∂ 2 r 1 ∂ (r V ) + sin θ V θ + V . 2 sin θ ∂θ ∂φ r ∂r

(84)

Remember that in Eq. (84) we are using the coordinate basis (48); to ﬁnd the expression in the orthonormal, non-coordinate basis (59) requires an obvious rescaling of the vector components.

3.8. Covariant Derivatives of Tensors and of the Metric Our knowledge of how to differentiate scalars and vectors leads directly to the expressions for the covariant derivatives of more general tensors. Consider the scalar product (54), written in terms of a one-form and a vector Ui V i . Since this is a scalar, the correct derivative is the ordinary partial derivative of a scalar ﬁeld:

(Ui V i ),j = Ui V i ,j + Ui,j V i ,

(85)

where we have employed the Leibniz rule. Let us rewrite this equation in terms of the covariant derivative (78) of the vector:

(Ui V i ),j = Ui V i ;j − Ui i kj V k + Ui,j V i = Ui V i ;j + (Ui,j − kij Uk )V i . (86) The left-hand side of Eq. (86) is a tensor, the gradient one-form of the scalar Ui V i ; so the right-hand side is also a tensor. Now everything after the ﬁnal equality sign, except the quantity in brackets, has already been shown to

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be a tensor; therefore, the quantity in brackets is also a tensor; it is the covariant derivative of the one-form Ui :

Ui;j = Ui,j − kij Uk ,

(87)

or, using the notion of the covariant derivative as a differential operator,

∇j Ui = Ui,j − kij Uk .

(88)

The covariant derivatives depend on the character of the object that is differentiated; vector components (78) are differentiated differently than the components of one-forms (87). The index position of the semi-colon in the deﬁnitions (78) and (87) indicates this better than the differential operators (79) and (88). Note that the covariant derivative obeys the Leibniz rule:

(Ui V i ),j = (Ui V i );j = Ui V i ;j + Ui;j V i .

(89)

The fact that Ui;j is a tensor can also be shown directly by proving that it transforms according to the tensor rule (56), using the known transformation properties of the objects on the right-hand side of (87). One can deduce the expression for the covariant derivative of any tensor by constructing a scalar from it with vectors and one-forms and applying the above procedure. For example, expanding the derivative of Aj i Ui V j , one ﬁnds the covariant derivative of a mixed tensor,

∇k Aj i = Aj i,k − mjk Ami + i mk Aj m .

(90)

The general rule is simple: high indices get a positive sign in front of the Christoffel symbols and low indices a negative sign. An important case is the covariant derivative of the metric tensor itself:

gij;k = gij,k − l ik glj − l jk gil , g

ij ;k

=g

ij

j

,k

+ i lk glj + lk gil .

(91) (92)

A highly signiﬁcant property of the metric tensor now emerges if we consider the fact that, as a tensor, it transforms as j

gi j ;k = i i j kk gij;k .

(93)

It is clear from Eq. (91) that the covariant derivative of the metric vanishes in Cartesian coordinates, where gij = δij and the Christoffel symbols are all

Arbitrary Coordinates

97

zero, but Eq. (93) shows that if gij;k is zero in one coordinate system it is zero in all coordinate systems3 , so we have

gij;k = 0.

(94)

Similar reasoning starting from Eq. (92) shows that

g

ij ;k

= 0.

(95)

It is instructive to verify the property (94) explicitly for spherical polar coordinates using the expressions (91) and (42) and the Christoffel symbols (83). It follows from Eqs. (94) and (95) that Vi;j = ( gik V k );j = gik;j V k + gik V i ;j = gik V i ;j and V i ;j = ( gik Vk );j = gik;j Vk + gik Vi;j = gik Vi;j , which are further examples of the general index lowering and raising operations (50) and (53). Since Vi;j is a tensor, we can in fact raise either of its indices; so we have ;j

Vi = gjk Vi;k .

(96)

Equation (96) deﬁnes what it means to have a covariant-derivative index in the upper position. It is very important that Eqs. (94) and (91) serve to determine the Christoffel symbols in terms of the metric tensor. To see this, we insert the expression (94) in Eq. (91) and write it three times, with different permutations of the indices:

gij,k = l ik glj + l jk gil , gik,j = l ij glk + l kj gil , −gjk,i = −l ji glk − l ki gjl .

(97)

In view of the symmetry (34) of the metric tensor and the symmetry (76) of the Christoffel symbols, the sum of the three lines gives 2gil l jk = gij,k + gik,j − gjk,i . Employing the inverse metric tensor gij , we ﬁnally ﬁnd

i jk =

1 il g ( glj,k + glk,j − gjk,l ). 2

(98)

3 This is a general property of tensors as a consequence of their transformation rule (56): if a tensor vanishes

in one coordinate system it vanishes in all of them.

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Transformation Optics and the Geometry of Light

Equation (98) represents the most economic way of computing the Christoffel symbols in general. Again, it is highly instructive to take the spherical-polar metric (42) and re-calculate the Christoffel symbols (83) using the recipe (98).

3.9. Divergence and Curl We can use the expression (98) for the Christoffel symbols to ﬁnd a very simple formula for the divergence of a vector in arbitrary coordinates. From the deﬁnitions (82) and (78), the divergence is

∇ · V = V i ;i = V i ,i + i ji V j ,

(99)

and inserting (98) gives

∇ · V = V i ,i +

1 il 1 1 g ( glj,i − gji,l )V j + gil gli,j V j = V i ,i + gil gil,j V j , 2 2 2 (100)

as is seen by relabeling summation indices and employing the symmetry of the metric tensor gij and its inverse gij . Then, we use a general property of the determinant of a matrix: the derivative of the determinant g with respect to the matrix element gij gives g gij , where gij is the inverse matrix. [This property follows from Laplace’s formula of expressing the determinant in terms of cofactors and Cramer’s rule for the inverse matrix (Stoll, 1969).] Consequently,

g,j 1 √ 1 ∂g 1 il g gil,j = =√ gil,j = g ,j , 2 2g ∂gil 2g g

(101)

and so we arrive at the simpliﬁed formula for the divergence

1 √ i ∇·V = √ g V ,i . g

(102)

The advantage of this formula compared to the deﬁnition (99) is clear, because it only contains an ordinary partial derivative. It is easy to see from the metric (42) that in spherical polar coordinates the divergence formula (102) gives the previous result (84). Another derivative operation familiar from Maxwell’s equations is the curl of a vector. Like the vector products (67) and (68), the curl ∇ × V is formed using the Levi–Civita tensor. Since the partial derivatives of

Arbitrary Coordinates

99

Cartesian coordinates are covariant derivatives in general coordinates, we have

(∇ × V )i = ijk Vk;j .

(103)

A simpliﬁcation occurs in formula (103), however; from the covariant derivative (87), one ﬁnds that the terms containing the Christoffel symbols cancel. We can therefore, in the case of the curl, use partial derivatives:

(∇ × V )i = ijk Vk,j ,

jk

(∇ × V )i = i Vk,j .

(104)

In this way, we have found convenient expressions for the mathematical ingredients of Maxwell’s equations, the divergence (102) and the curl (104).

3.10. The Laplacian We can use formula (102) to ﬁnd the divergence of the gradient vector (∇ψ)i of a scalar ψ, which will give us the general expression for the Laplacian of a scalar. Note that we must use the gradient vector (∇ψ)i , rather than the gradient one-form (∇ψ)i , since the divergence is deﬁned for vectors. From Eqs. (72) and (102), we get

1 √ ij g g ψ, j . ∇ 2 ψ = ∇ · (∇ψ) = (∇ψ)i ;i = (gij ψ,j );i = √ ,i g

(105)

For spherical polar coordinates, we apply Eq. (42) and easily obtain the well-known result

∂2 ψ ∂ 1 ∂ψ 1 1 ∂ 2 ∂ψ r + 2 sin θ + ∇ ψ= 2 . (106) ∂r ∂θ r ∂r r sin θ ∂θ r2 sin2 θ ∂φ2 2

Any reader who has had the misfortune of having to work out this expression without using formula (105) from differential geometry should now appreciate the power of the machinery we have developed. As a further salutary example, consider the monochromatic wave equation for the electric ﬁeld

∇2 E +

ω2 E = 0. c2

(107)

This is the equation as it is usually written in the textbooks, but the notation in the ﬁrst term is treacherous for the student. When the wave equation (107) is considered in curvilinear coordinates, for example to ﬁnd the radiation modes in a waveguide (Jackson, 1998), the student is apt to think

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that the components (∇ 2 E)i of the vector ∇ 2 E are the Laplacians ∇ 2 (Ei ) of the electric ﬁeld components Ei —this is not true. The partial derivatives of Cartesian coordinates get replaced with covariant derivatives in a i;j general system, so the vector ∇ 2 E has in fact components gjk Ei ;j;k = E ;j , which are not the Laplacians ( gjk Ei ,j );k of Ei . The three Laplacians ( gjk Ei ,j );k are not the components of a vector. The correct wave equation (107) is thus i;j ;j

E

+

ω2 i E = 0. c2

(108)

Expressed in curvilinear coordinates, the wave equation (107) provides three coupled equations for the components Ei . In practice, the electromagnetism textbooks ensure that they only deal with those components of the wave equation (107) for which (∇ 2 E)i is equal to ∇ 2 (Ei ), such as the z-component in cylindrical coordinates, but this only conﬁrms in the student’s mind the false belief that (∇ 2 E)i is the same as ∇ 2 (Ei ), and in the future he or she may come a cropper as a consequence.

3.11. Geodesics and Curvature We are now acquainted with the mathematics of arbitrary coordinates in three-dimensional Euclidean space, but there is an extra bonus for our efforts—we can also deal with curved space. A notion of curvature on a space is naturally induced by the metric tensor: if coordinates exist in which the metric has the Euclidean form (32) the space is ﬂat; otherwise it is curved. Curvature of a three-dimensional space is difﬁcult to visualize, but a familiar curved two-dimensional space is provided by the surface of a sphere. If the sphere has radius a, then from Eq. (43) the metric on the surface is

ds2 = a2 (dθ 2 + sin2 θ dφ2 ).

(109)

There is no transformation to coordinates {x1 , x2 } in which this metric takes the Euclidean form

ds2 = (dx1 )2 + (dx2 )2 .

(110)

Note that the crucial feature of the sphere that prevents its metric being transformed to the Euclidean (110) is not that it is a closed space with a ﬁnite area; we can consider any ﬁnite patch of the sphere, ignoring its global structure, and we would still be unable to ﬁnd a coordinate transformation to Eq. (110). The crucial fact that makes the sphere a curved space

Arbitrary Coordinates

101

is that we cannot form a patch of the sphere from a ﬂat piece of paper without stretching the paper. Consider, in contrast, the surface of a cylinder. A cylinder can be formed by rolling up a ﬂat piece of paper; so this space must be ﬂat. The metric on a cylinder of radius a, in cylindrical polar coordinates with r = a, is

ds2 = dz2 + a2 dφ2 = dz2 + (d(aφ))2 ,

(111)

which is of the Eulidean form (110), proving that it is ﬂat. It is not important that the coordinate φ in (111) is periodic; curvature is a local property of a space, in contrast to its topology. A cylinder and a plane have different topologies, but they have the same curvature, namely zero. How can we quantify the curvature of a space? We ﬁrst need to generalize the notion of a straight line to the case of curved spaces such as the sphere. The key property of a straight line joining two points in ﬂat space is that it is the shortest path between those points. In a curved space we can still construct the shortest line between two points and this is called a geodesic. For a sphere, the geodesics are the great circles, i.e., the circles whose centers are the center of the sphere. A geodesic is a curve xi (ξ) in space, with parameter ξ. The length s of the curve between two points ξ = ξ1 and ξ = ξ2 is the integral of the line element (33),

ξ2 s= ξ1

ξ2 ξ2 dxi (ξ) dxj (ξ) dξ. ds = gij dxi (ξ) dxj (ξ) = gij dξ dξ ξ1

(112)

ξ1

For the curve xi (ξ) to be a geodesic, the length (112) must be a minimum, so the variation δs must vanish when we perform a variation δxi (ξ) of the curve (maintaining the parameter values ξ1 and ξ2 at the endpoints), but this is completely equivalent to the principle of least action in mechanics (Landau and Lifshitz, 1976) with “Lagrangian”

L = gij x˙ i x˙ j ,

x˙ i ≡

dxi (ξ) . dξ

(113)

The geodesic is therefore given by the Euler–Lagrange equations:

1 d ∂L ∂L d 1 j gij x˙ − glj,i x˙ l x˙ j . 0= − i = i dξ ∂x˙ dξ L 2L ∂x

(114)

Equation (114) determines a geodesic curve once an initial tangent vector (direction of the geodesic) dxi (ξ)/dξ is speciﬁed. The parameter ξ is

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arbitrary, but we obtain a much simpler equation if we choose ξ to be the distance s along the geodesic, because in this case L = 1. We calculate

d d ∂L dxi d2 xi dxl dxj g = g = + g ij ij lj,i dξ ∂x˙ i dξ ds ds ds ds2

(115)

and insert this result in the Euler–Lagrange equations (114) that we matrixmultiply with gij from the left. Recalling the expression (98) for the Christoffel symbols, we obtain the simple geodesic equation j k d2 xi (s) i dx (s) dx (s) = 0. + jk ds ds ds2

(116)

In Euclidean space, the geodesic equation (116) always gives the equation of a straight line, whether this line is expressed in Cartesian or curved coordinates, as the reader may verify for the case of the spherical polar coordinates using the Christoffel symbols (83). The reader may also show that the geodesics for the sphere, with metric (109), are the great circles. We ﬁnd a simple interpretation for the geodesic equation (116) using the concept of parallel transport discussed in Section 3.7. Note that the vector dxi /ds appearing in the geodesic equation is the tangent vector to the geodesic curve xi (s). Equation (116) thus means that the covariant increment of the tangent vector along dxi (s) is zero: the tangent vector is parallel-transported along the geodesic curve:

DU i = ∇j U i dxj (s) = 0,

Ui ≡

dxi (s) . ds

(117)

Thus, a geodesic parallel-transports its tangent vector. In ﬂat space, as one moves along a geodesic (straight line), the tangent vectors at all points are parallel. Equation (117) generalizes this property for geodesics in curved space. Geodesics are lines of inertia. The behavior of geodesics in a space can be used to quantify the amount of curvature. In ﬂat space, two geodesics (straight lines) either have a constant separation (parallel lines) or the separation distance changes at a constant rate, i.e., it changes linearly with distance along the lines. The second derivative of the separation with respect to distance along the geodesics is therefore zero. In curved space by contrast, this second derivative does not vanish, a phenomenon called geodesic deviation; it is used to measure the curvature. As an example, consider again the surface of a sphere, depicted in Figure 7. We see clearly that the separation of any two geodesics (great circles) does not change linearly with distance along

Arbitrary Coordinates

103

FIGURE 7 Two geodesics (great circles) on the sphere. The separation as a function of distance from a meeting point is not linear; as one moves along, the geodesics have an “acceleration” toward each other. Tangent vectors to the geodesics are drawn at the points where the separation has reached a maximum. These vectors are parallel in three-dimensional space.

the geodesics: the separation increases from zero to a maximum and then decreases to zero again. At the points where the tangent vectors to the geodesics are drawn in Figure 7, the separation is a maximum and the tangent vectors are parallel as viewed in the three-dimensional space in which the sphere is embedded. The geodesics are parallel at these points, inasmuch as a notion of parallelism can be introduced on the sphere, but these “parallel lines” meet! Thus the postulates of Euclidean geometry do not hold in this space: it is curved. As another example, consider the geodesics on the surface of a cylinder. These are drawn by simply cutting open the cylinder into a ﬂat sheet, drawing straight lines on the sheet, and rolling it up again to reform the cylinder. Clearly the geodesics behave just as in the plane: there is no geodesic deviation (second derivative of separation with respect to distance along geodesics is zero), showing again that a cylinder is a ﬂat space. How are we to compute the geodesic deviation? We need to consider a family of geodesics xi (s, δ), where the parameter δ labels (continuously) the geodesics in the family and for ﬁxed δ the parameter s is the distance along a geodesic. Note that δ is not the distance between geodesics, since two ﬁxed values of δ determine two geodesics that may be moving apart. The vector joining two inﬁnitesimally separated geodesics with equal parameter value

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s is, however, given by V i dδ with

Vi =

∂xi (s, δ) . ∂δ

(118)

Now, what we are interested in is the rate of change of this joining vector as s increases, speciﬁcally its second covariant increment along the geodesic, D2 V i . To compute the geodesic deviation, we use the notation (79) of D and use the relationship

U k ∇k V i =

∂2 xi ∂xk ∂xi ∂xk ∂xi ∂xk ∂xj ∇k = + i jk = ∇k = V k ∇k U i . ∂s ∂δ ∂s ∂δ ∂s ∂δ ∂δ ∂s (119)

Then we calculate D2 V i = U j ∇j (U k ∇k V i )ds2 , regarding the ∇j as operators obeying the Leibniz rule of derivatives (89),

U j ∇j U k ∇k V i = U j ∇j V k ∇k U i = U j V k ∇j ∇k U i + U j ∇j V k ∇k U i = U j V k ∇j ∇k U i + V j ∇j U k ∇k U i = U j V k ∇j ∇k − ∇k ∇j U i + V k ∇k U j ∇j U i (120) = U j V k ∇j ∇k − ∇k ∇j U i , where we applied the geodesic equation (117) in the last step. If the commutator of the covariant derivatives vanishes, the geodesic deviation is zero and the space is ﬂat. What is the meaning of this commutator? Recall the concept of parallel transport. Imagine we move a vector Ai from the point xi to the inﬁnitesimally close neighbor xi + dxi and then again to the next neighbor by another increment dyi ; ﬁnally we close a loop in moving the vector back by −dxi followed by −dyi : the vector would change as (∇j ∇k − ∇k ∇j )Ai dxj dyk . We can imagine any closed loop as consisting of patches of inﬁnitesimal loops, so in a non-Euclidean geometry, a vector transported along a closed loop does not return to itself (see Figure 8). Foucault’s pendulum (Hart, Miller, and Mills, 1987), transported with the rotating Earth on a sphere, does not return to its original oscillation direction after one loop (24 h). To calculate the commutator (∇j ∇k − ∇k ∇j )Ai , we use the covariant derivative (90) of the mixed tensors ∇k Ai and ∇j Ai ﬁrst and then apply the formula (79) for the covariant derivative of a vector. We get, using the

Arbitrary Coordinates

105

FIGURE 8 Parallel transport around a closed loop on the sphere. A vector at the north pole is parallel-transported to the equator along the geodesic to which it is tangent. Since a geodesic parallel-transports its tangent vector, the vector is still tangent to the geodesic at the equator. The vector is then parallel-transported a quarter of the distance around the equator (also a geodesic). Parallel-transport preserves the vector, as much as the curvature allows; so it stays at the same angle to the equator. Finally, the vector is parallel-transported back to the north pole along a geodesic. On return to the north pole, the vector is rotated by 90 degrees with respect to the initial vector. This rotation is completely specified by the Riemann curvature tensor. Note that on a plane (zero Riemann tensor) this kind of operation does not rotate the vector.

symmetry (76) of the Christoffel symbols,

− i lk ∇j Al ∇j ∇k − ∇k ∇j Ai = ∇k Ai + i lj ∇k Al − ∇j Ai

= i lk Al

,j

,j

+ i lj Al ,k + l km Am

,k

− i lj Al − i lk Al ,j + l jm Am ,k = i lk,j − i lj,k + i mj mkl − i mk mjl Al .

(121)

The quantity in brackets on the right-hand side of this formula depends only on the geometry of the space (the metric) and determines the amount of curvature. It transforms as a tensor with four indices (56), because the

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left-hand side of Eq. (121) is a tensor with three indices and the right-hand side contains a contraction over a vector. The space is ﬂat if and only if this quantity is zero, giving zero geodesic deviation. It is one of the most important objects in geometry, the Riemann curvature tensor4 :

Ri jkl ≡ i jl,k − i jk,l + i mk mjl − i ml mjk .

(122)

The Riemann tensor describes both the geodesic deviation

D2 V i = Ri jkl V l dxj dxk

(123)

and the result of loops in parallel transport

∇j ∇k − ∇k ∇j Ai = Ri ljk Al .

(124)

In Euclidean space, no matter how complicated the coordinate system we use, the Riemann tensor will always turn out to be zero (the reader may check this for spherical polar coordinates). On the other hand, for a curved space such as the sphere, the Riemann tensor will not vanish in any coordinate system. In arbitrary coordinates xA = {x1 , x2 }, a sphere of radius a has a Riemann tensor

RA BCD =

1 A δ C gBD − δA D gBC , 2 a

(125)

as can be checked for the case of the coordinates (109). Incidentally, besides the sphere, all readers are familiar with the physical effect of another curved space: the space–time geometry in which they reside, the geometry of gravity (Misner, Thorne, and Wheeler, 1973). Here space–time is curved, not only three-dimensional space. Space–time has a Riemann tensor whose largest components at the surface of the Earth are of the order of 10−23 m−2 . This space–time curvature is the reason the reader does not ﬂoat off into space.

4. MAXWELL’S EQUATIONS After having discussed the mathematical machinery of differential geometry we are now well-prepared to formulate the foundations of

4 The tensor character (56) of Ri

symbols.

jkl

can be veriﬁed using the transformation properties of the Christoffel

Maxwell’s Equations

107

electromagnetism, Maxwell’s equations (Jackson, 1998). In empty space, the Maxwell equations for the electric ﬁeld strength E and the magnetic induction B are ∇·E = ∇×E =−

∂B , ∂t

ρ , ε0

∇ · B = 0,

∇×B=

1 ∂E + μ0 j. c2 ∂t

(126)

We use SI units with electric permittivity ε0 , magnetic permeability μ0 , and speed of light c in vacuum; ε0 μ0 = c−2 . Charge and current densities are denoted by ρ and j. In the following we express Maxwell’s equations in arbitrary coordinates and arbitrary geometries. We show how a geometry appears as a medium and how a medium appears as a geometry. We develop the concept of transformation optics where we use the freedom of coordinates to describe transformation media as elegantly as possible. Furthermore, we generalize transformation optics to space–time geometries. We also return to our starting point, Fermat’s principle.

4.1. Geometries and Media Maxwell’s equations (126) contain curls and divergences. Using the expressions (102) and (104) from differential geometry, we can now write these in arbitrary coordinates: 1 √ i ρ gE = , √ ,i g ε0 ijk Ek,j = −

∂Bi , ∂t

1 √ i g B = 0, √ ,i g

ijk Bk,j =

1 ∂Ei + μ0 j i . c2 ∂t

(127)

This form of Maxwell’s equation is also valid in arbitrary geometries, i.e., in curved space, for the following reason: any geometry, no matter how curved, is locally ﬂat—at each spatial point we can always construct an inﬁnitesimal patch of a Cartesian coordinate system, although these local systems do not constitute a single global grid. For each locally ﬂat piece, we postulate Maxwell’s equations (126), and in writing these equations in arbitrary coordinates we naturally express them in a global frame. Let us rewrite the form (127) of Maxwell’s equations with all the vector indices in the lower position and the Levi–Civita tensor expressed in terms

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of the permutation symbol according to formula (65):

[ijk]Ek,j

√ √ √ gρ g gij Ej = , g gij Bj = 0, ,i ,i ε0 √ ij √ ij ∂ ± g g Bj 1 ∂ ± g g Ej √ , [ijk]Bk,j = 2 + μ0 g ji . (128) =− ∂t ∂t c

In this form, Maxwell’s equations in empty space, but in curved coordinates or curved geometries, resemble the macroscopic Maxwell equations in dielectric media (Jackson, 1998), ∇ · D = ρ, ∇×E =−

∂B , ∂t

∇ · B = 0,

∇×H =

∂D + μ0 j, ∂t

(129)

written in right-handed Cartesian coordinates: Di ,i = ρ, [ijk]Ek,j = −

∂Bi , ∂t

Bi ,i = 0, [ijk]Hk,j =

∂Di + ji . ∂t

(130)

In fact, the empty-space equations (128) can be expressed exactly in the macroscopic form (130) if we replace Bi in the free-space equations with Hi /μ0 , rescale the charge and current densities, and take the constitutive equations Di = ε0 εij Ej ,

Bi = μ0 μij Hj . √ εij = μij = ± g gij .

(131) (132)

Consequently, the empty-space Maxwell equations in arbitrary coordinates and geometries are equivalent to the macroscopic Maxwell equations in right-handed Cartesian coordinates. Geometries appear as dielectric media. The electric permittivities εij and magnetic permeabilities μij are identical—these media are impedance-matched (Jackson, 1998), and εij and μij are matrices—the media are anisotropic. Spatial geometries appear as anisotropic impedance-matched media. The converse is also true: anisotropic impedance-matched media appear as geometries. We easily derive this statement from the constitutive equa√ tions (132): calculate the det ε of εij . The result is det ε = ± g, the factor in front of the metric in the constitutive equations (132). So we obtain

Maxwell’s Equations

109

from the constitutive equations of a geometry the metric tensor of a medium

gij =

εij . det ε

(133)

For general εij = μij , this geometry is curved, but, if and only if the Riemann tensor (122) vanishes, the spatial geometry is ﬂat. In such a case, there exists a coordinate transformation of physical space where Maxwell’s equations are purely Cartesian and where space is ﬂat and empty. The electromagnetic ﬁelds in real, physical space are transformed ﬁelds—the results of coordinate transformations. Media that perform such a feat are called transformation media.

4.2. Transformation Media Transformation media implement coordinate transformations in Maxwell’s equations. Note carefully how this interpretation of Maxwell’s equations (127) works: we write the free-space equations in coordinates that are not right-handed Cartesian, but we then interpret these equations as being in a right-handed Cartesian system with an effective medium (132). This sounds a bit paradoxical, but the way to think of it is to imagine two different spaces as well as two different coordinate systems (see Figure 9). In the

FIGURE 9 Transformation media implement coordinate transformations. The left figure shows the Cartesian grid of electromagnetic space that is mapped to the curved grid of physical space shown in the right figure. The physical coordinates enclose a hole that is made invisible in electromagnetic space (where it shrinks to the point indicated there). Consequently, a medium that performs this transformation acts as an invisibility device. The case illustrated in the figure corresponds to the transformation r = R1 + r (R2 − R1 )/R2 in cylindrical coordinates where the prime refers to the radius in electromagnetic space. The region with radius R1 is invisible; R2 − R1 describes the thickness of the cloaking layer.

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Transformation Optics and the Geometry of Light

ﬁrst space, which we call electromagnetic space, we have no medium and we write the empty-space Maxwell equations in right-handed Cartesian coordinates. We then perform a transformation that gives us a non-trivial effective medium (132), and we interpret the transformed coordinates as being right-handed Cartesian in a new space, physical space, which contains the medium (132). The Cartesian grid in electromagnetic space will deform under the transformation, and this deformed grid shows ray trajectories in physical space. There are two aspects of transformation media that make them highly signiﬁcant. First, we know a good deal about solutions of Maxwell’s equation in vacuum (light rays travel in straight lines, etc.), and to ﬁnd the effect of the medium, we can just take a vacuum solution in electromagnetic space and transform to physical space using the coordinate transformation that deﬁnes the medium: the transformed ﬁelds are a solution of the macroscopic Maxwell equations in physical space. Second, since a transformation medium is deﬁned by a coordinate transformation, we can use this as a design tool to ﬁnd materials with remarkable electromagnetic properties. Some readers may have nagging doubts about the juxtaposition of the mathematical tools of general relativity with the attribution of a physical signiﬁcance to coordinate transformations. For a relativist, coordinate systems have no physical meaning; the geometry of the space is the important thing, which is independent of the coordinate grid one chooses to cover the space, but here we wish to consider materials that, as far as electromagnetism is concerned, perform active coordinate transformations. In this theory, the coordinate transformation is physically signiﬁcant, it describes completely the macroscopic electromagnetic properties of the material, and differential geometry is just as useful for these purposes as it is in general relativity. In our description of transformation media, the starting point of the theory was a right-handed Cartesian system in electromagnetic space; any non-trivial transformation from this system gives an effective medium in physical space. It is, however, often convenient to adjust coordinates to the particular situation under investigation (see Figure 10). We should be allowed to use any coordinates we wish in electromagnetic space. To implement this freedom of coordinates, we generalize the theory. Suppose that we describe electromagnetic space by a curvilinear system such as cylindrical or spherical polar coordinates; any deformation of this system through a coordinate transformation is to be interpreted as a medium, but to describe the electromagnetism in the presence of this medium we employ the original curvilinear grid. Let xi be the curvilinear system in electromagnetic space and we denote its metric tensor by γij . Then in electromagnetic space the empty-space

111

Maxwell’s Equations

FIGURE 10 Transformation media in cylindrical coordinates. The figure shows the transformation of Figure 9 in cylindrical coordinates. These are the coordinates best adapted to this case.

Maxwell equations (127) are 1 √ i ρ 1 √ i γE = , √ γ B = 0, √ ,i ,i γ ε0 γ √ i √ i ∂ γB 1 ∂ γE , [ijk]Bk,j = 2 + μ0 j i , [ijk]Ek,j = − ∂t ∂t c

(134)

where we have made the metric dependence of the Levi–Civita tensor explicit. Now we perform a coordinate transformation and, as before, we interpret the resulting equations as being macroscopic Maxwell equations written in the same (curvilinear) system we started with, but in physical space. We cast the equations (128) in physical space as the macroscopic equations (129) in the curvilinear system with the metric γij : √

[ijk]Ek,j

γ Di

,i

√ ∂ γ Bi , =− ∂t

=

√ γ ρ,

[ijk]Hk,j

√ γ Bi = 0, ,i

√ ∂ γ Di √ + μ0 γ ji . = ∂t

(135)

By the same reasoning as before, we can interpret the free-space equations (128) as macroscopic equations (135) written in the curvilinear system if we rescale the charge and current densities and take the constitutive equations (131) with

√ g ε = μ = ± √ gij . γ ij

ij

(136)

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Transformation Optics and the Geometry of Light

If we wish to implement a certain coordinate transformation, formula (136) gives a simple and efﬁcient recipe for calculating the required material properties in arbitrary coordinates (Leonhardt and Philbin, 2006).

4.3. Wave Equation and Fermat’s Principle Impedance-matched media establish geometries and geometries appear as impedance-matched media. In general, these media are anisotropic, but we still expect that light rays follow a version of Fermat’s principle (1) of the extremal optical path. The metric gij should set the measure of optical path length. So we anticipate that light rays follow a geodesic with respect to the geometry given by the material properties (133). How do we deduce Fermat’s principle from Maxwell’s equations? First we write down the wave equation for monochromatic ﬁelds with frequency ω, the reﬁned version of the Helmholtz equation (6). Any electromagnetic ﬁeld consists of a superposition of monochromatic ﬁelds, if gij does not change in time, which we assume, and also that there are no external charges and currents in the region we consider. We obtain from Maxwell’s equations (127)

ω2 ijk klm Em;l = 2 Ei . ;k c

(137)

Note that we expressed the derivatives as covariant derivatives (semicolons instead of commas) in anticipation of simpliﬁcations to come. The covariant derivative of the Levi–Civita tensor klm is zero, because it is determined by the metric tensor, which has vanishing covariant derivative (Misner, Thorne, and Wheeler, 1973). Consequently, using the notation (79) for the covariant derivatives

ω2 i ijk l m i j i j l m i j j i E = ∇ ∇ E = δ δ − δ δ klm k m m l ∇k ∇ E = ∇j ∇ E − ∇ ∇j E , l c2 (138) where we applied formula (70) for the double vector product. The ﬁrst term in the right-hand side of Eq. (138) resembles the covariant derivative of the divergence, ∇j Ej , that would vanish since the charge density is zero, but covariant derivatives do not commute in general: their commutator is given by the Riemann tensor (124). So, we obtain, lowering the index i,

ω2 Ei = (∇k ∇i − ∇k ∇i ) Ek − ∇ j ∇j Ei , c2

(139)

113

Maxwell’s Equations

and arrive at the wave equation (Piwnicki, 2001, 2002)

∇ j ∇j Ei + Rij gjk Ek +

ω2 Ei = 0, c2

(140)

where Rij denotes the Ricci tensor (Misner, Thorne, and Wheeler, 1973)

Rij = Rkikj .

(141)

The non-Euclidean geometry established by the medium may scatter light, even in the case of impedance matching. Impedance matching (Jackson, 1998) may signiﬁcantly reduce scattering, but not completely, unless the Ricci tensor (141) vanishes. In this case, the Riemann tensor vanishes as well in three-dimensional space [see exercise 1 in Section 92 of Landau and Lifshitz (1995)]. The Riemann tensor quantiﬁes the measure of curvature, irrespective of the coordinates. If the Riemann tensor vanishes, the geometry is ﬂat—the apparent curvature is an illusion where curved coordinates disguise a straight system: the material is a transformation medium. So only transformation media cause absolutely no local scattering.5 Only they guide electromagnetic waves without disturbing them; they merely transform ﬁelds from electromagnetic to real space. However, some transformation media still cause scattering in situations where the topologies of the two spaces differ from each other. In Section 5 we discuss two examples of topological scattering (though for space–time transformations), the optical Aharonov–Bohm effect and analogs of the event horizon. Let us return to Fermat’s principle. In the regime of geometrical optics (Born and Wolf, 1999), the phases of electromagnetic waves advance more rapidly than the variations of the dielectric properties of the material. The medium determines the wavelength λ, and so λ varies with the variations of the material properties. For geometrical optics to be a valid approximation, the gradient of the wavelength should be small, |∇λ| 1 (Landau and Lifshitz, 1977). We represent the electric-ﬁeld components as

Ei = Ei eiϕ ,

(142)

where Ei is a slowly varying envelope and ϕ the rapidly advancing phase. The gradient of the phase ∇ϕ describes the wave vector k, and the

5 In one dimension, all impendance-matched non-moving dielectrics are transformation media, so

impedance-matched waveguides are reﬂectionless (Jackson, 1998), but some non-impedance-matched materials (with soliton index proﬁles) are reﬂectionless too (Landau and Lifshitz, 1977; Gupta and Agarwal, 2007).

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Transformation Optics and the Geometry of Light

wavelength is given by λ = 2π/|k|. We substitute the ansatz (142) into the wave equation (140) and take only the dominant terms into account, terms that contain products of the ﬁrst derivatives of the phase or ω2 , ignoring all other terms, including the curvature contribution. The amplitude Ei is common to all remaining terms, and so the wave equation reduces to the dispersion relation

gij ϕ,i ϕ,j =

ω2 . c2

(143)

The dispersion relation contains the speed of light in the medium, c . For example, for an isotropic medium with refractive index n = ε = μ, we have the inverse metric tensor gij = n−2 1 and c = c/n. For anisotropic media, the eigenvalues of the matrix gij characterize c for light rays propagating in the directions of its eigenvectors; c /c is the square root of the corresponding eigenvalue of gij . To obtain Fermat’s principle, we take advantage of the connection between optics and classical mechanics (the connection that inspired Schrödinger’s quantum mechanics in analogy to wave optics). We read the dispersion relation (143) as the Hamilton–Jacobi equation of a ﬁctitious point particle that draws the spatial trajectory of the light ray (Landau and Lifshitz, 1976). For this, we need to identify the wave vector ϕ,i as the canonical momentum i.e., as the derivative of some Lagrangian L with respect to the velocity x˙ i . We might be tempted to employ the Lagrangian (113) of the geodesic equation, but it is wise to use

1 L2 = gij x˙ i x˙ j . 2 2

(144)

∂L = gij x˙ j ∂x˙ i

(145)

∂L i 1 1 x˙ − L = gij x˙ i x˙ j = gij ϕ,i ϕ,j . i ˙ 2 2 ∂x

(146)

L = In this case we obtain

ϕ,i = and the Hamiltonian

H =

Since the metric does not depend on time, the Hamiltonian is conserved (Landau and Lifshitz, 1976). The conserved quantity is the “energy” of the ﬁctitious particle. Our choice (144) of the Lagrangian is consistent with the

Maxwell’s Equations

115

dispersion relation (143) if we put

H =

ω2 . 2c2

(147)

This proves that L is a suitable Lagrangian for light rays. The resulting Euler–Lagrange equations give the geodesic equation (116). So light rays follow a geodesic of the metric gij ; they take an extremal optical path given by the properties of the medium (133): light follows Fermat’s principle.

4.4. Space–Time Geometry So far we considered spatial geometries or coordinate transformations in space, but there are also important examples of transformation media that mix space and time (Leonhardt and Philbin, 2006). In Section 5 we discuss two cases in detail, the optical Aharonov–Bohm effect of a vortex and optical analogs of the event horizon. Here, we write down the foundations for the theory of space–time transformation media, Maxwell’s equations in a space–time geometry. First, let us introduce space–time coordinates. The role of the Cartesian coordinates {x, y, z} is played by the Galilean system {ct, x, y, z}, where t denotes time. In Section 3 we developed differential geometry in three-dimensional space, but it is easy to extend the treatment to four dimensions—the indices just take one more value and the Levi–Civita tensor has one more index. In this manner, one can treat arbitrary coordinates in space–time. The appropriate distance in Galilean coordinates {ct, x, y, z} in ﬂat space–time is, however, not given by the four-dimensional Euclidean metric, but rather by the Minkowski metric (Landau and Lifshitz, 1995):

ds2 = c2 dt2 − dx2 − dy2 − dz2 .

(148)

Here, we use the Landau–Lifshitz convention for the metric where spatial distances are counted as negative contributions (one can also use the opposite metric −ds2 where space counts as positive). The metric (148) rather describes a measure of time. It is customary to denote the time coordinate by x0 and the three spatial coordinates by xi where the Latin indices run over {1, 2, 3}. The metric in a general space–time coordinate system (or in curved space–time) is

ds2 = gαβ dxα dxβ ,

(149)

where the Greek indices run over the four values {0, 1, 2, 3}. The component g00 is usually positive throughout space–time, as in the Minkowski metric (148), and the determinant g is always negative.

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Transformation Optics and the Geometry of Light

It turns out that the theory of transformation media and materials that mimic curvature also works in four-dimensional space–time. This is proved in the Appendix, which provides a more challenging example of tensor algebra. There we show that the free-space Maxwell equations in arbitrary right-handed space–time coordinates can be written as the macroscopic Maxwell equations in right-handed Cartesian coordinates {ct, x, y, z} with Plebanski’s constitutive equations (Plebanski, 1960) 1 Di = ε0 εij Ej + [ijk]wj H k , c

1 Bi = μ0 μij H j − [ijk]wj Ek , c

√ −g ij ε =μ =− g , g00 ij

ij

wi =

g0i . g00

(150) (151)

Space–time geometries appear as media. The constitutive equations (151) turn out to reveal an important hidden property of electromagnetism: electromagnetic ﬁelds are conformally invariant in space–time. Suppose we compare two geometries, one with the metric (149) and the other with a metric where we re-scale equally space and time at each point, but the scaling factor may vary over space–time,

gαβ → (xν ) gαβ .

(152)

This is not a coordinate transformation in general: we compare two different geometries. They measure space–time distances differently, but the angles between world lines are the same. As a result of the conformal scaling (152), the inverse metric tensor gαβ scales with −1 and the determinant g with 4 . So εij , μij , and wi do not change, but these are the only quantities that depend on the geometry. Consequently, the electromagnetic ﬁeld does not notice a conformal space–time transformation; electromagnetism is invariant if space and time are re-scaled equally. However, if we only altered the measure of space by a spatially dependent factor n, the new geometry would behave like a dielectric medium with refractive index proﬁle n. Conformal invariance is the basis of Penrose diagrams (Wald, 1984) where the entire causal structure of inﬁnitely extended space–time is condensed, by a conformal factor, into a ﬁnite map one can draw and discuss. In Section 5.4, we apply the conformal invariance of electromagnetism to discuss the space–time geometry generated by moving media (Gordon, 1923; Leonhardt, 2000). For transformation media, we can generalize the constitutive equations (150) and (151) to allow for a handedness change in the spatial part of the space–time coordinate transformation and also to allow for a curvilinear spatial coordinate system in electromagnetic space. Our previous result

Transformation Media

117

(136) shows how to incorporate these possibilities in the permittivity and permeability in the constitutive equations (151). In addition, we can express the constitutive equations (150) and (151) in index-free form if we denote the permittivity and permeability matrices by ε and μ, respectively, and understand εE, etc., as a matrix product. Our ﬁnal constitutive relations are then (Leonhardt and Philbin, 2006) w w × H, B = μ0 μH − × E, c c √ −g ij g0i ij ij g , wi = . ε = μ = ∓√ γ g00 g00

D = ε0 εE +

(153) (154)

In addition to the familiar impedance-matched electric permittivity ε and magnetic permeability μ, a transformation that mixes space and time mixes electric and magnetic ﬁelds. A space–time geometry appears as a magneto-electric medium, also called a bi-anisotropic medium (Serdyukov et al., 2001; Sihvola et al., 1994). The mixing of electric and magnetic ﬁelds is brought about by the bi-anisotropy vector w that has the physical dimension of a velocity. In Section 5.4 we show that w is closely related to the velocity of the medium (for slow media, w is proportional to the velocity). Moving media are naturally magneto-electric (Landau and Lifshitz, 1993)—a moving dielectric responds to the electromagnetic ﬁeld in its local frame, but this frame is moving and motion mixes electric and magnetic ﬁelds by Lorentz transformations (Jackson, 1998). Such phenomena have been observed before special relativity was discovered, for example, in the Röntgen effect (Leonhardt and Piwnicki, 1999a; Röntgen, 1888) or, indirectly, in Fizeau’s demonstration (Fizeau, 1851) of the Fresnel drag (Fresnel, 1818). More recently, moving optical media have been shown to generate analogs of the event horizon (Philbin et al., 2008).

5. TRANSFORMATION MEDIA The concept of transformation media has been the key idea for the design of invisibility devices (Greenleaf, Lassas, and Uhlmann, 2003a,b; Leonhardt, 2006b; Pendry, Schurig, and Smith, 2006), an idea that was put into practice using electromagnetic metamaterials (Schurig et al., 2006). Moreover, the idea that inspired the surge of interest in metamaterials in the ﬁrst place, the perfect lens (Pendry, 2000), turned out to represent an example of transformation optics as well (Leonhardt and Philbin, 2006). Furthermore, the optical Aharonov–Bohm effect (Cook, Fearn, and Milonni, 1995; Hannay, 1976; Leonhardt and Piwnicki, 1999b, 2000) and optical analogs of

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Transformation Optics and the Geometry of Light

the event horizon (Leonhardt, 2003a; Philbin et al., 2008) can be understood as cases of media that perform space–time transformations (Leonhardt and Philbin, 2006). In this section we discuss these examples in some detail.

5.1. Spatial Transformation Media Let us ﬁrst focus on some general properties of spatial transformation media. These are media that perform purely spatial coordinate transformations of electromagnetic ﬁelds. We develop an economic form of the theory that allows quick calculations and rapid judgment on the physical properties of spatial transformation media. For this, we distinguish two different coordinate systems and three matrices of metric tensors: the coordinates xi of electromagnetic space with metric tensor gi j , which in physical coordinates xi appears as the tensor components gij . The physical coordinates xi are not necessarily Cartesian, but characterized by the metric γij of physical space, and neither are the electromagnetic coordinates required to be Cartesian, because it is often wise in physics to adopt the coordinates to the particular physical problem under investigation. Transformation media are made of anisotropic materials, in general, that are characterized by ε and μ tensors. In curvilinear coordinates, the components of the dielectric tensors are given by Eqs. (55) and (136) as

√ g j ε = μ = ± √ gi j i i j γ ij

ij

(155)

where ± indicates the handedness of the physical coordinates with respect to the electromagnetic coordinates, + for right-handed and − for lefthanded coordinate systems; g and γ denote the determinants of gij and γij . We obtain from the transformation (36) of the metric tensor

g = det(gi j ) det 2 (i i ) =

det(gi j ) det2 (i i )

=

g det2 (i i )

.

(156)

We employ a convenient matrix form for the dielectric tensors (155) by deﬁning the matrices

G ≡ (gi j ),

≡ (γij ),

i

≡

i

=

∂xi . ∂xi

(157)

Transformation Media

119

The ﬁrst index and the second index label the rows and the columns of the matrices, respectively. In terms of the matrix representation (157), we obtain for the dielectric tensors (155)

√ det G G −1 T . ε=μ= √ det det

(158)

The equality of the electric permitivity ε and the magnetic permeability μ means that transformation media are impedance-matched (Jackson, 1998) to the vacuum, but in practice impedance-matching can be relaxed at the expense of introducing some slight additional scattering (Cai et al., 2007a,b; Schurig et al., 2006). For an anisotropic medium, the eigenvalues of the dielectric tensors εc and μc in Cartesian coordinates describe the dielectric responses along the three axes of the medium at each point in space; the axes are given by the eigenvectors. For transformation media, the tensor εc = μc is symmetric, because it is constructed from the symmetric inverse metric tensor gi j in Eq. (155). Consequently, the matrix εc has three real eigenvalues a with the orthogonal eigenvectors a, the three orthogonal axes of dielectric response. Here we show how to calculate the eigenvalues a from the dielectric tensor (158) in curvilinear coordinates. Consider the mixed tensor ε = εil γlj . Suppose we transform this tensor from curvilinear to Cartesian coordinates by the transformation matrix c . According to the rules of tensor transformations conveniently indicated by the index positions in εi j , the curvilinear ε turns into the Cartesian εc = c ε −1 c ,

−1 and vice versa ε = −1 c εc c . Consequently, c a is an eigenvector of ε with eigenvalue a ; the eigenvalues of ε are those of the dielectric tensor in Cartesian coordinates. Note that the eigenvectors of ε are not orthogonal in general (and εi j is not symmetric). Nevertheless, the eigenvalues of ε directly give the dielectric functions of the transformation medium. To illustrate why this procedure offers considerable economy in calculations, we discuss the example of a cylindrical transformation. Suppose that the medium transforms the radius r in cylindrical coordinates, but preserves the angle φ and the vertical coordinate z,

r = r(r ), r =

x2 + y2 ,

r y r x , y= , z = z . x= 2 2 2 2 x +y x +y

(159)

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Transformation Optics and the Geometry of Light

We calculate the transformation matrix from {x, y, z} to {x , y , z } and express the result as

⎛

R cos2 φ +

r sin2 φ r

r cos φ sin φ r

R−

⎜ ⎜ ⎜ = ⎜ R − r cos φ sin φ ⎜ r ⎝ 0

r cos2 φ + R sin2 φ r 0

⎞ 0

⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎠ 1

(160)

with

R=

dr , dr

det =

r R. r

(161)

We obtain the dielectric tensor in Cartesian coordinates

⎛ ⎜ ⎜ ⎜ r ⎜ ⎜ εc = rR ⎜ ⎜ ⎜ ⎝

r2 R2 cos2 φ + 2 sin2 φ r r2 R2 − 2 cos φ sin φ r

2 r R2 − 2 cos φ sin φ r

r2 cos2 φ + R2 sin2 φ r2

0

0

⎞ 0 ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ 0 ⎟ ⎟ ⎠ 1 (162)

calculate the eigenvalues, and obtain after some algebra

a = R

r r r , , . r R r rR

(163)

Alternatively, we calculate ε in the coordinate system that is best adapted to cylindrical transformations: both in electromagnetic and in physical space we use cylindrical coordinates. From the line element in cylindrical coordinates ds2 = dr2 + r2 dφ2 + dz2 , we read off the metric tensors = diag(1, r2 , 1) and G = diag(1, r2 , 1). For transformations of the radius = diag(R, 1, 1). Consequently,

r r r ; ε = diag R , , r R r rR

(164)

this elementary calculation reveals the dielectric properties of the cylindrical transformation medium. If the transformation opens a hole in physical

121

Transformation Media

(a)

(b)

2

2

1

1

y9

y

21

21

22

22

22

21

x9

1

2

3

22

21

x

1

2

3

FIGURE 11 Invisibility for waves. The device transforms waves emitted in physical space as if they propagate through empty space in transformed coordinates. (a) Transformed space. The figure shows the propagation of a spherical wave emitted at the point ( −1.5, 0, 0). (b) Physical space. The invisibility device turns the wave of (a) into physical space by the coordinate transformation illustrated in Figure 10. The cloaking layer with radii R1 = 0.5 and R2 = 1.0 deforms electromagnetic waves such that they propagate around the invisible region and leave without carrying any trace of the interior of the device.

space, anything inside this hole is decoupled from the electromagnetic ﬁeld: it has become invisible, as Figure 11 shows.6

5.2. Perfect Invisibility Devices Transformation media that have holes in their coordinate grids in physical space act as perfect invisibility devices. Consider the following simple example in spherical coordinates (Pendry, Schurig, and Smith, 2006). Suppose the device transforms the radius r in electromagnetic space, but does not affect the spherical angles,

r = r(r ),

θ = θ,

φ = φ .

(165)

We obtain from Eq. (42) the matrices = diag(1, r2 , r2 sin2 θ) and G = diag(1, r2 , r2 sin2 θ), and, as in the previous subsection, the transformation matrix of the devices is = diag(R, 1, 1) with R = dr/dr . Consequently,

2 r2 r2 r2 1 1 2 r ε = 2 diag R , 2 , 2 = diag R 2 , , . (r R) r r r R R

(166)

6 The corresponding ﬁgure in Leonhardt and Philbin (2007b) is not correct. We thank T. Tyc for pointing

this out.

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Transformation Optics and the Geometry of Light

r

R1

0

R2

r9

FIGURE 12 Cloaking transformation. Radial transformation performed by a spherical cloaking device. The radius r of spherical coordinates in electromagnetic space is transformed to the radius r in physical space, and vice versa. R1 denotes the inner and R2 the outer radius of the cloaking shell: physical regions with radius r < R1 are not reached by electromagnetic waves and for r > R2 the electromagnetic coordinates agree with the coordinates of physical space.

Figure 12 shows an example of a radial transformation when r(r ) reaches a ﬁnite value R1 at the origin in electromagnetic space where r = 0. This means that in physical space the entire spherical shell of radius R1 corresponds to a single point in electromagnetic space. Anything lurking inside that sphere has become excluded from the electromagnetic ﬁeld: it has become invisible. Figure 12 also shows that beyond the radius R2 , electromagnetic and physical coordinates agree—R2 describes the outer radius of the cloaking device, whereas R1 is the inner radius. If electromagnetic and physical coordinates agree beyond the outer layer of the cloak, then outside of the device electromagnetic waves are indistinguishable from waves propagating through empty space. The device guides electromagnetic waves around the enclosed hidden object without causing disturbances. So the object has not only disappeared from sight, but the act of hiding has become undetectable; the scenery behind the cloaking device would show no sign of the object or of the device. In short, this transformation makes a perfect cloaking device. This behavior has been veriﬁed using scattering theory (Ruan et al., 2007) and the most appropriate boundary conditions for the inner layer of the cloak have been identiﬁed (Greenleaf et al., 2007a,b). Fortunately perhaps, perfect invisibility (Pendry, Schurig, and Smith, 2006) is severely limited. Figure 13 shows the trajectories of light rays passing through the cloaking shell. The rays make a detour around the hidden

Transformation Media

123

(b)

(a)

R1

R2

FIGURE 13 Spherical cloaking device. The figure shows the trajectories of light rays. (a) Two-dimensional cross section of rays striking the device, diverted within the annulus of cloaking material contained within R1 < r < R2 to emerge on the far side undeviated from their original course. (b) Three-dimensional view of the same process. (From Pendry, Schurig, and Smith, 2006. Reprinted with permission from AAAS.)

core of the device, but they must arrive at the same time as if they were propagating through empty space. So there is only one option: inside the cloaking device the phase velocity of light must exceed c. This is possible in principle (Boyd and Gauthier, 2002; Milonni, 2004), but only in regions of the spectrum with narrow bandwidth that correspond to resonances in the material. But, to make matters worse, light rays straddling the inner lining of the cloak must go around the inner core in precisely the same time it would take for them to traverse a single point in electromagnetic space, in zero time. Light must propagate at inﬁnite phase velocity! One can put these thoughts into precise mathematical terms (Leonhardt and Philbin, 2006) by considering the product ε1 ε2 ε3 of the three dielectric functions of the transformation medium. This product is the product of the eigenvalues of ε , the determinant of ε . We obtain from formula (158)

√ det G 1 ε1 ε2 ε3 = det(ε ) = √ . det det

(167)

√ We showed in Section 3.2 that det G describes the volume element, here the volume element in electromagnetic space. The volume of a single point is zero; hence at least one of the eigenvalues of the dielectric tensor is zero—the speed of light reaches inﬁnity at the inner lining of the cloak. This inevitable feature restricts the performance of perfect invisibility devices to a single-frequency set by the particular metamaterial used. Perfect invisibility devices are impractical, but imperfect devices inspired by these ideas may be possible (Leonhardt and Tyc, 2009).

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5.3. Perfect Lenses The perfect lens, made by negative refraction (Pendry, 2000; Veselago, 1968), turns out to be an example of a transformation medium as well (Leonhardt and Philbin, 2006). As in perfect invisibility devices, the unusual properties of the lens are based on an unusual topology of the transformation. For achieving invisibility, electromagnetic space does not cover the entire physical space, whereas for perfect lenses electromagnetic space turns out to be multi-valued: single points in electromagnetic space are mapped to multiple points in physical space. Consider in Cartesian coordinates the transformation x(x ) illustrated in Figure 14, whereas all other coordinates are not changed. In the fold of the function x(x ), each point x in electromagnetic space has three faithful images in physical space. Obviously, electromagnetic ﬁelds at one of those

x9

x

FIGURE 14 Perfect lens. Negatively refracting perfect lenses employ transformation media. The top figure shows a suitable coordinate transformation from the physical x axis to the electromagnetic x ; the lower figure illustrates the corresponding device. The inverse transformation from x to x is either triple or single-valued. The triple-valued segment on the physical x axis corresponds to the focal region of the lens: any source point has two images, one inside the lens and one on the other side. Since the device facilitates an exact coordinate transformation, the images are perfect with a resolution below the normal diffraction limit (Born and Wolf, 1999): the lens is perfect (Pendry, 2000). In the device, the transformation changes right-handed into left-handed coordinates. Consequently, the medium employed here is left-handed, with negative refraction (Veselago et al., 2006).

125

Transformation Media

(a)

(b)

2

2

1

1

y

y

21

21

22

22

22

21

x

1

2

3

22

21

x

1

2

3

FIGURE 15 Propagation of electromagnetic waves in a perfect lens. The lens facilitates the coordinate transformation shown in Figure 14. Spherical waves in electromagnetic space are transformed into physical space. (a) The wave of Figure 11(a) is emitted outside the imaging range of the lens. The wave is transformed by the lens, but the device is not sufficiently thick to form an image. (b) A wave is emitted inside the imaging range, creating two images of the emission point, one inside the device and one outside, corresponding to the image points of Figure 14.

points are perfectly imaged onto the others: the device is a perfect lens (see Figure 15). Perfect lensing was ﬁrst analyzed (Pendry, 2000) as the imaging of evanescent waves in a slab of negatively refracting material, waves that may carry images ﬁner than the optical resolution limit (Born and Wolf, 1999). Various aspects of this idea have been subject to a considerable theoretical debate (Minkel, 2002; Nieto-Vesperinas and Garcia, 2003; Pendry, 2003; Pendry and Smith, 2003; Valanju, Walser, and Valanju, 2003), but recent experiments (Fang et al., 2005; Grbic and Eleftheriades, 2004; Melville and Blaikie, 2005) conﬁrmed sub-resolution imaging. Our pictorial argument leads to a simple intuitive explanation of why such lenses are indeed perfect. We also immediately obtain from the theory of transformation media the reason why perfect lensing requires left-handed media with negative ε and μ (Marqués, Martin, and Sorolla, 2008; Milonni, 2004; Sarychev and Shalaev, 2007; Shelby, Smith, and Schultz, 2001; Veselago et al., 2006): inside the device, i.e., inside of the x fold, the derivative of x(x ) becomes negative and the coordinate system changes handedness. The electromagnetic left-handedness of negative-index materials appears in a left-handed coordinate system. We obtain from our recipe (158) the compact result

dx dx dx , . ε = μ = diag , dx dx dx

(168)

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At the place where the transition between right-handed and left-handed coordinates occurs, the derivative of x(x ) is undeﬁned and so are ε and μ, but this has no adverse effect in practice (Leonhardt, 2003b). It simply describes the discontinuity at the boundary of the medium where our transformation x (x) automatically gives the correct boundary conditions (Jackson, 1998), because Maxwell’s equations are satisﬁed arbitrarily close to the boundary. Equation (168) shows that when dx /dx is −1 inside the device and +1 outside, we obtain the standard perfect lens based on an isotropic left-handed material with ε = μ = −1 inside (Pendry, 2000; Veselago, 1968). Furthermore, we see that lenses with ε = μ = −1 are not the most general choice. One could use an anisotropic medium to magnify perfect images by embedding the source or the image in transformation media with |dx /dx| = 1. Note that the perfect lens made by negative refraction is not the only case of perfect imaging (Born and Wolf, 1999). Maxwell (1854) proposed a curious device called the ﬁsh-eye. In Maxwell’s ﬁsh-eye, light goes around in circles in such a way that any point in the device is imaged to a partner point with inﬁnite precision. It turns out (Luneburg, 1964) that Maxwell’s ﬁsh-eye also is an example of a transformation medium, but one with non-Euclidean geometry: here physical space is not mapped to ﬂat space, but to our favorite example of a curved space, the surface of a sphere. In the following we use this visualization of the ﬁsh-eye to explain how it works. Consider the stereographic projection from the surface of a sphere with radius r0 to the x, y plane as shown in Figure 16. In formulae,

x=

x , 1 − z /r0

y=

y 1 − z /r0

with

x2 + y2 + z2 = r02

(169)

and the inverse transformation

x =

2x 2y (r/r0 )2 − 1 , r 2 = x2 + y 2 . , y = , z = r 0 1 + (r/r0 )2 1 + (r/r0 )2 (r/r0 )2 + 1 (170)

Imagine light propagating on the surface of the ﬁctitious sphere with uniform refractive index n0 . In this case, the line element of electromagnetic space is

ds2 = n20 dx2 + dy2 + dz2 .

(171)

Transformation Media

127

z9 x9

x

FIGURE 16 Maxwell’s fish-eye. The left figure illustrates the stereographic projection described in Eq. (169). A line is drawn from the north pole through a point on the sphere. The image of this point is the intersection of this line with the plane through the equator. The top right figure shows light rays on the surface of the sphere and projected on to the equatorial plane; the bottom right figure shows the rays in the plane where the dotted circle indicates the equator. Light rays originating from one point meet again at the antipodal point: Maxwell’s fish-eye makes a perfect lens.

Similar to Eq. (3) we express the differentials dx , dy , and dz in terms of the differentials in physical space and obtain

ds2 = n2 dx2 + dy2 ,

n=

2n0 r02 x2 + y2 + r02

.

(172)

By the way, this result also proves that the stereographic projection is a conformal map of the surface of the sphere on the two-dimensional plane.7 We can easily extend this procedure from two-dimensional surfaces to threedimensional spaces by considering the stereographic projection from the hyper-surface of the four-dimensional sphere to three-dimensional space or simply by rotating the x, y plane around one axis, say the y axis, while performing two-dimensional projections. Because of rotational symmetry, we obtain in 3D

ds2 = n2 dx2 + dy2 + dz2 ,

n=

2n0 r02 x2 + y2 + z2 + r02

.

(173)

7 The stereographic projection, invented by Ptolemy, is an ingredient of the Mercator projection used in

cartography; on the complex plane, the Mercator projection is the logarithm of the stereographic projection (Nehari, 1952).

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Transformation Optics and the Geometry of Light

Let us return to 2D where we can draw pictures, as shown in Figure 16. On the surface of the sphere, light rays propagate along geodesics, the great circles. It is one of the remarkable properties of the stereographic projection (Nehari, 1952) that circles on the sphere are transformed into circles on the plane. So, in physical space, light goes around in circles as well. The great circles originating from one source point on the sphere meet again at the antipodal point. In the stereographic projection, the image of the antipodal point is the reﬂection of the source on a circle, the circle with the radius r0 of the sphere. Here the source is perfectly imaged: Maxwell’s ﬁsh-eye makes a perfect lens (Born and Wolf, 1999). However, this is quite an unusual instrument: both the source and the image are embedded in the non-uniform refractive index proﬁle (173) of the ﬁsh-eye. In contrast, for the ﬂat lens made by negative refraction (Pendry, 2000), source and image are outside the device; the lens acts across some distance. The imaging range is the distance from the lens where multiple images are formed, where x(x ) is multi-valued. As Figure 14 shows, the imaging range is equal to the thickness of the lens for the standard case where dx /dx = −1 and, accordingly, = μ = −1. Because of losses in the material, the imaging range has been very small in practice though (Fang et al., 2005; Grbic and Eleftheriades, 2004; Melville and Blaikie, 2005). In another twist of the story, perfect lenses are predicted to cause unusual quantum effects, they lead to repulsive Casimir forces (Leonhardt and Philbin, 2007a). The Casimir force is a force caused by, literally, “nothing”—by the zero-point energy of the quantum vacuum (Lamoreaux, 1999; Mandel and Wolf, 1995; Milonni, 1994). What is zero-point energy? Imagine the electromagnetic ﬁeld as a superposition of inﬁnitely many stationary modes. Some of the mode coefﬁcients may be zero, but according to quantum physics being zero is a positive statement: the modes are in vacuum states. Each mode oscillates like a harmonic oscillator with its eigenfrequency ων . The vacuum state is the ground state of the harmonic oscillator, but this state carries the energy ων /2 (Landau and Lifshitz, 1977). The sum of the ground-state energies of all the modes is called zeropoint energy. So even in the absence of any electromagnetic ﬁelds, in the complete quantum vacuum state, the zero-point energy remains. Moreover, this energy is inﬁnite for inﬁnitely many modes. On the other hand, the mode frequencies ων depend on the boundary conditions. Consider, as Casimir (1948) did, quantum electromagnetism inside two parallel mirrors with distance a. Although the total zero-point energy is inﬁnite, the difference U between the zero-point energies per area at a and at ∞ turns out to be ﬁnite and negative (Casimir, 1948; Milonni, 1994), U(a) = −

cπ2 , 720 a3

(174)

Transformation Media

129

a mesmerizing combination of natural constants. Read U(a) as the potential for the force between the mirrors with distance a. Since the potential falls with decreasing distance, the mirrors are attracted to each other. This attraction caused by the quantum vacuum has been observed (Lamoreaux, 1997) and good quantitative agreement with theoretical predictions was found (Chan et al., 2001). Normally, the Casimir force is weak, but it may become important for small distances a, distances on the scale of nanomachines (Ball, 2007). Imagine that we place a perfect lens between the mirrors. The lens should be sufﬁciently thick such that the mirrors lie within the imaging range b of the lens. The lens maps physical space to electromagnetic space where the mirror distance is reduced to

a = 2b − a ,

a < 2b.

(175)

Electromagnetic space is empty, apart from the mirrors; there they are attracted to each other with potential U(a ), but in physical space this attraction turns into repulsion, because when a decreases, a must grow. We obtain for the force per area

f =−

dU(a ) da cπ2 dU =− = . da da da 240 a4

(176)

This is not the only example of repulsive Casimir forces, but so far it is the only one where repulsion is explained by a simple visual argument. Another example is a miniature sandwich of different dielectric media, silicon, bromobenzene, and gold (Munday, Capasso, and Parsegian, 2009). Also in this case the Casimir force is repulsive for reasons that are far less intuitive (Dzyaloshinskii, Lifshitz, and Pitaevskii, 1961). It turns out that the perfect lens needs to have gain for sustaining a repulsive Casimir force (Leonhardt and Philbin, 2007a), but, in principle, it represents a simple case where two mirrors repel each other across some distance, without any adjacent mediator (such as the bromobenzene in a layer of silicon, bromobenzene, and gold). The Casimir repulsion could be sufﬁciently strong such that one of the mirrors would overcome gravity, levitating on zero-point energy.

5.4. Moving Media Perfect invisibility devices and perfect lenses are made of spatial transformation media—metamaterials that facilitate transformations in space, but that do not alter the measure of time. Here we develop a physical interpretation of electromagnetism under space–time transformations. We

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Transformation Optics and the Geometry of Light

show that they correspond to moving media (Leonhardt, 2000; Leonhardt and Piwnicki, 1999b) [or media with moving dielectric properties (Philbin et al., 2008)]. For this we translate Fermat’s principle (1) into a space– time geometry for moving media with isotropic and equal ε = μ = n. We describe the electrodynamics of moving media in an inertial frame in Cartesian coordinates, the laboratory frame, but as an intermediate step we take the liberty of performing transformations to other inertial frames of reference. Imagine the medium consists of inﬁnitesimally small cells moving with velocities u. The velocity proﬁle may vary in space and time, but for a given space–time point we can always construct a locally co-moving inertial frame. In such a frame, the medium is not moving, at least at the speciﬁc point we are considering, and so we can apply here the standard theory of electromagnetism in media at rest. In particular, Fermat’s principle (1) suggests that the electromagnetic ﬁeld perceives here a spatial geometry with the line element dl2 = n2 (dx2 + dy2 + dz2 ). We translate the spatial geometry into a space–time geometry with the space–time line element ds2 = c2 dt2 − dl2 , and hence the metric tensor gαβ = diag(1, −n2 , −n2 , −n2 ) with the determinant −n6 and the inverse matrix gαβ = diag(1, −n−2 , −n−2 , −n−2 ). The reader easily veriﬁes from Eq. (151) that this geometry indeed describes a medium at rest with ε = μ = n. The local geometry of the medium is still purely spatial; it inﬂuences the spatial part of the metric, but not the measure of time. We can change this by using the conformal invariance of Maxwell’s equations (in the absence of external charges and currents). We may multiply the space–time metric by an arbitrary factor, called conformal factor, and still obtain exactly the same constitutive equations (150) as before. Multiplying diag(1, −n2 , −n2 , −n2 ) by n−2 we obtain the new metric tensor gαβ = diag(n−2 , −1, −1, −1) with determinant −n−2 and inverse matrix gαβ = diag(n2 , −1, −1, −1). This geometry inﬂuences the measure of time (in the metric element g00 ) but not the spatial part of the metric. We write the metric tensor as

gαβ = ηαβ + (n−2 − 1) uα uβ ,

gαβ = ηαβ + (n2 − 1) uα uβ ,

ηαβ = diag(1, −1, −1, −1) = ηαβ ,

(177) (178)

and uα = (1, 0, 0, 0) = uα . Then we use a key insight from special relativity (Landau and Lifshitz, 1995): ηαβ describes the metric tensor (148) of ﬂat space–time in an inertial frame (the Minkowski metric). Since all inertial frames are equivalent, ηαβ does not change in transformations from one inertial frame to another, for example, in the transformation from the

Transformation Media

131

laboratory frame to the locally co-moving frame and vice versa. Using this insight from relativity, we also develop a physical interpretation for uα and uα that allows us to translate the gαβ and gαβ back to the laboratory frame: we regard uα as the local four-velocity of the medium,

uα = ηαβ uβ =

dxα . ds

(179)

Here ds is the line element (148) in ﬂat space–time

u2 ds = ηαβ dx dx = c dt − dx − dy − dz = 1 − 2 dt2 c 2

α

β

2

2

2

2

2

(180)

expressed in terms of the three-dimensional velocity vector of the medium

u=

dx . dt

(181)

For zero velocity, i.e., in the locally co-moving frame, we get uα = uα = (1, 0, 0, 0). Hence we can indeed interpret the uα and uα in the metric (177) as the local four-velocities. The important point is that uα transforms like the four-dimensional coordinate vector xα , as the position of the index in uα indicates, because ds is invariant. Consequently, the expressions (177) are not only true in locally co-moving frames, but in the laboratory frame as well. All we have to do is to calculate uα and uα for non-zero velocities (the local velocities of the moving medium). We obtain from Eqs. (179)–(181) the expressions

(1, u/c) uα = , 1 − u2 /c2

(1, −u/c) uα = . 1 − u2 /c2

(182)

In this way, we have established the space–time geometry of moving media discovered in 1923 by Walter Gordon [and independently rediscovered several times (Leonhardt and Piwnicki, 1999b; Quan, 1956, 1957/58)]. In matrix form, the metric tensors appear as

⎛

⎞ n2 − 1 c2 n2 − u2 cu ⎜ 2 ⎟ c 2 − u2 ⎜ c − u2 ⎟ αβ g =⎜ ⎟, 2 2 ⎝n −1 ⎠ n −1 c u −1 + 2 u⊗u 2 2 2 c −u c −u g = det gαβ = −n−2 ,

(183)

(184)

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Transformation Optics and the Geometry of Light

⎛

gαβ

c2 n−2 − u2 ⎜ c 2 − u2 ⎜ =⎜ ⎝ 1 − n−2 cu c 2 − u2

⎞

1 − n−2 cu c 2 − u2 n−2 − 1 −1 + 2 u⊗u c − u2

⎟ ⎟ ⎟. ⎠

(185)

We obtain from Eq. (154) the dielectric tensors of moving media (that are locally isotropic and impedance matched)

n ε=μ= 1 − u2 n2 /c2

u2 1− 2 c

1 + 1 − n−2

u⊗u c2

≈ n 1 (186)

in the limit of low velocity, u/c 1. Furthermore, since for moving media gαβ contains mixed space–time elements g0i , we also obtain the vector

w=

n2 − 1 u ≈ (n2 − 1) u. 1 − u2 n2 /c2

(187)

In this way, we have found a possible physical interpretation for the bi-anisotropy vector w of electromagnetism in space–time geometries: w may appear as the velocity of a moving medium.

5.5. Optical Aharonov–Bohm Effect Perfect invisibility devices and perfect lenses exploit transformations to non-trivial topologies in space—excluded regions in physical space or folds in electromagnetic space. Here we study a simple example, the propagation of light through a vortex, that turns out to represent a transformation medium with multi-valued space–time geometry. Consider a vortex in a ﬂuid, say in water ﬂowing down a plug hole. A vortex concentrates the vorticity ∇ × u of the swirling ﬂow u in one line, the vortex core. In Cartesian coordinates, the velocity proﬁle of a straight vortex is given by the expression

⎛

⎞ −y ⎝ x ⎠, u= 2 x + y2 0 W

(188)

because for this proﬁle ∇ × u vanishes, except at the z axis where u diverges, but the circulation u · dr gives the ﬁnite value 2πW. All vorticity is concentrated along the vortex line.

Transformation Media

133

It might be a useful exercise for the reader to express the velocity proﬁle (188) of the vortex in cylindrical coordinates (the most natural coordinates for this situation). According to Eqs. (46) and (26) the coordinate-basis vectors of the cylindrical coordinates are

er = cos φ i + sin φ j,

eθ = −r sin φ i + r cos φ j,

ez = k.

(189)

The metric tensor is the familiar γij = diag(1, r2 , 1). One obtains for the velocity proﬁle of the vortex

⎛ ⎞ W 0 u i = 2 ⎝ 1 ⎠, r 0

ui = (0, W, 0).

(190)

We immediately see from the expressions (104) of the curl in arbitrary coordinates that ∇ × u vanishes, because ui is constant. The vorticity is concentrated in the z axis that is excluded from the cylindrical coordinates. Imagine the vortex is illuminated from the side. Any moving medium drags light, as Fresnel (1818) discovered (deducing the correct result without knowing special relativity) and as Fizeau (1851) observed for water in uniform motion (in interferometric measurements without lasers). The basics of Fresnel’s drag are very simple: light propagating with the ﬂow is advanced, whereas light propagating against the current lags behind. (The degree of dragging, Fresnel’s dragging coefﬁcient, depends on special relativity though.) Dragged by the moving medium, light rays experience phase shifts and, if the phase fronts are deformed, light rays are deﬂected. For ﬂow speeds u much smaller than the speed of light in the medium, c/n, we expect that the phase shift is proportional to the integral of u along the propagation of light. For regions with vanishing vorticity, we can deform the contours of the phase integrals u · dr and so the phase difference between light waves that have passed the vortex on different sides is proportional to the circulation u · dr, a constant. Hence we expect that the water vortex does not deﬂect light, but imprints a characteristic phase slip, in analogy to the Aharonov–Bohm effect (Aharonov and Bohm, 1959; Peshkin and Tonomura, 1989; Tonomura, 1998). In the Aharonov–Bohm effect, charged matter waves, electrons for example, are not deﬂected by a vortex in the magnetic vector potential, but experience a characteristic phase shift. The optical analog of the Aharonov–Bohm effect was ﬁrst described by Jon Hannay in his PhD thesis (Hannay, 1976) and has also been independently rediscovered (Cook, Fearn, and Milonni, 1995). The effect could be used to detect quantum vortices with slow light (Leonhardt and Piwnicki, 2000) and it is related to the Aharonov–Bohm effect with

134

Transformation Optics and the Geometry of Light

surface waves (Berry et al., 1980; Roux et al., 1997; Vivanco et al., 1999) and to the gravitational Aharonov–Bohm effect (Stachel, 1982). Assuming that in the optical Aharonov–Bohm effect the phase modu lation is proportional to the integral of u · dr, we guess a space–time coordinate transformation that should describe the wave propagation. Then we use our formalism to verify that this guess is correct. The integral of the velocity proﬁle (188) gives

u · dr = W

−y dx + x dy =W x2 + y 2

y = Wφ. d arctan x

(191)

This phase should modulate the time evolution of the wave. The water has a uniform refractive index n that simply rescales the wavelength. Therefore, we assume in physical space, using cylindrical coordinates,

ct = ct − aφ ,

r=

r , n

φ = φ ,

z=

z , n

(192)

with a constant a to be determined later. In electromagnetic space, we obtain

ct = ct + aφ,

r = nr,

φ = φ,

z = nz.

(193)

Electromagnetic space is ﬂat and empty with the Minkowski line element (180) in primed coordinates expressed in physical coordinates,

ds2 = (cdt + adφ)2 − n2 (dr2 + r2 dφ2 ) − n2 dz2 .

(194)

So the metric tensors are

⎛

gαβ

1 0 ⎜0 −n2 =⎜ ⎝a 0 0 0

a 0 a2 − n2 r2 0

⎞ 0 0 ⎟ ⎟, 0 ⎠ −n2

g = det(gαβ ) = −r2 n6 , ⎛ a a2 0 ⎜1 − n 2 r 2 2 r2 n ⎜ 1 ⎜ 0 0 − 2 ⎜ αβ n g =⎜ ⎜ 1 a ⎜ 0 − 2 2 ⎜ n2 r2 n r ⎝ 0 0 0

(195)

⎞ 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟. ⎟ 0 ⎟ ⎟ 1⎠ − 2 n

(196)

(197)

Transformation Media

135

We recall that in cylindrical coordinates the spatial metric is described by the matrix = diag (1, r2 , 1) with determinant γ = r2 and obtain from the constitutive equations (154) the dielectric tensors

ε = μ = n diag (1, r−2 , 1),

(198)

ε = μ = n 1;

(199)

and hence

the medium is isotropic with refractive index n. We also get from the constitutive equations (154) the bi-anisotropy vector in cylindrical coordinates

wi = (0, a, 0)

(200)

that corresponds to the velocity proﬁle (190) in the low-velocity limit (187) for

a = (n2 − 1) W.

(201)

These results prove that the space–time transformation (193) generates from electromagnetic waves in empty-space solutions of Maxwell’s equations for the vortex (188) in the limit of low velocities. However, although this solution describes the dominant features of the wave propagation through the vortex, it does not have the right topology in general; the solution is not periodic in the angle φ. For example, for a monochromatic wave of frequency ω, the phase grows by 2π a ω after completing one circle, which is not an integer multiple of 2π in general. If we take the coordinate transformation (192) literally, physical space has become multi-valued, with a branch cut in the direction of incidence. This branch cut describes the phase slip of light after passing through the vortex, but it cannot possibly be mathematically rigorous. Figure 17 shows the correct solution due to Aharonov and Bohm (1959) when additional scattering resolves the topological dilemma of the coordinate transformation (192), but this is a subject beyond the scope of this chapter and so is light propagation in rapidly spinning media that describes how light may get sucked into the vortex (Leonhardt and Piwnicki, 1999b).

5.6. Analog of the Event Horizon Transformation media are at the heart of invisibility devices, perfect lenses, and the optical Aharonov–Bohm effect; here we explain that they also describe much of the essential physics at the event horizon of a black hole. In 1972, William Unruh invented a simple analogy of the event horizon

136

Transformation Optics and the Geometry of Light

FIGURE 17 Aharonov–Bohm effect. A fluid vortex generates the optical Aharonov–Bohm effect described by the coordinate transformation (192). Light, incident from the right, is Fresnel-dragged by the moving medium: light propagating with the flow is advanced, whereas light propagating against the current is retarded. The wave should develop the phase slip shown in the left figure. However, although the transformation (192) is exact, physical space–time would be described in multi-valued coordinates here. Instead of the simple phase slip, the light turns out to exhibit the characteristic interference pattern illustrated in the right figure (Aharonov and Bohm, 1959).

as an illustration for a colloquium at Oxford: black holes resemble rivers (Unruh, 2008). Imagine a river is ﬂowing toward a waterfall. Suppose that the river carries waves that propagate with speed c relative to the water, but the water is rapidly moving with velocity u that at some point exceeds c . Beyond this point, waves are swept away toward the waterfall. The point of no return is the horizon of the black hole. Imagine another situation: a river ﬂowing out into the sea, getting slower. Waves coming from the sea are blocked at the line where the ﬂow exceeds the wave speed (Suastika, 2004; Suastika, de Jong, and Battjes, 2000). Here the river establishes the analog of the white hole, an object that nothing can enter. Such analogies occur also in many other situations and they illustrate some essential features of the event horizon (see Figure 18). Horizons are perfect traps: one would not notice anything suspicious while passing the horizon with the ﬂow, but to return is impossible. In the other direction, trying to escape against the current, waves get stuck at the horizon; they freeze with dramatically shrinking wavelengths. These analogs are not mere analogies, they are mathematically equivalent to wave propagation in general relativity [as long as higher-order dispersion is irrelevant (Brout et al., 1995b; Jacobson, 1991; Unruh, 1995)]. For example, sound waves in moving ﬂuids behave like waves in a certain space–time geometry that depends on the ﬂow and the local velocity of sound (Moncrief, 1980; Unruh, 1981; Visser, 1998; White, 1973). Or, as we have already explained, light experiences a

Transformation Media

137

Faster water

Event horizon Slower water

FIGURE 18 Aquatic analog of the event horizon. Current can stop fish moving upstream and mimic an event horizon. A moving optical medium captures the physics at the event horizon, too. A pulse in an optical fiber creates such a moving medium (Philbin et al., 2008). (From Cho, 2008. Reprinted with permission from Peter Hoey.)

moving medium as the effective space–time geometry (177). At the horizon, where the ﬂow u reaches the speed of light c/n, the measure of time g00 in Gordon’s metric (183) vanishes. Time comes to a standstill. Close to the horizon the wavelength is dramatically reduced (until due to optical dispersion the refractive index changes, tuning the light out of the grip of the horizon). For short wavelengths, the lateral dimensions of the horizon are irrelevant. Therefore, the optics at the horizon is essentially one-dimensional. Consider a one-dimensional model: both medium and light move along the z axis; the electromagnetic ﬁeld vectors are pointing orthogonal to z in the x, y plane. In the following we show that this situation corresponds to a space–time transformation medium. We write down a coordinate transformation that maps one-dimensional wave propagation in moving media to propagation in empty electromagnetic space. How to ﬁnd this transformation? In an empty one-dimensional space, light would freely propagate with the speed c as a superposition of wavepackets that move in either positive or negative direction while maintaining their shapes. So these two wavepackets depend only on

t± = t ∓

z . c

(202)

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Transformation Optics and the Geometry of Light

In physical space, the wavepackets do not propagate with the speed of light in vacuum, but with the velocity v± that is given by the relativistic addition (Landau and Lifshitz, 1995) of the ﬂow speed u and the speed of light ±c/n in positive or negative direction,

t± = t −

c n v± = u . 1± cn u±

dz , v±

(203)

So we expect that the one-dimensional wave propagation in the moving medium corresponds to waves in empty space by the transformation

c t =

c (t− + t+ ) , 2

x = x ,

y = y ,

z =

c (t− − t+ ). 2

(204)

To prove this assertion, we apply the theory of transformation media. For u = (0, 0, u), the effective space–time geometry of the moving medium is described by the metric (183) with

⎛

gαβ

c 2 n2 − u 2 c 2 − u2 0 0

⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ 2 ⎝ (n − 1) cu c 2 − u2

0

⎞ (n2 − 1) cu c 2 − u2 ⎟ ⎟ ⎟ ⎟ 0 ⎟. ⎟ 0 ⎟ ⎟ n2 u2 − c2 ⎠ c 2 − u2

0

−1 0 0 −1 0

0

(205)

We transform gαβ to electromagnetic space

β

gα β = αα gαβ β

(206)

with the matrix αα that we obtain by differentiating the new coordinates xα with respect to the old coordinates xα of physical space

⎛ ⎜1 0 0 ⎜ ⎜ ⎜0 1 0 α α = ⎜ ⎜ ⎜0 0 1 ⎜ ⎝ 0 0 0

⎞ (n2 − 1) cu ⎟ c 2 − n 2 u2 ⎟ ⎟ ⎟ 0 ⎟. ⎟ 0 ⎟ ⎟ 2 2 n(c − u ) ⎠ c 2 − n 2 u2

(207)

Transformation Media

139

The result is the diagonal matrix

g

α β

n2 (c2 − u2 ) n2 (c2 − u2 ) = diag ,− 2 , −1 , −1 c 2 − n 2 u2 c − n2 u2

(208)

with the inverse

gα β

c 2 − n 2 u2 c2 − n2 u2 , − , −1 , −1 = diag 2 2 n (c − u2 ) n2 (c2 − u2 )

(209)

and the determinant of the metric

g = −

(c2 − n2 u2 )2 . n4 (c2 − u2 )2

(210)

The matrix gα β describes the geometry in electromagnetic space. To ﬁnd out how this geometry appears as a medium, we use the constitutive equations (150) in electromagnetic space, with primed metric tensors instead of unprimed metric tensors. Since gα β is diagonal, the bi-anisotropy vector w vanishes: in electromagnetic space, the medium is at rest. We also obtain

εx = μx = εy = μy = 1.

(211)

Since electromagnetic waves propagating in the z direction are polarized in the x, y plane, their electromagnetic ﬁelds only experience the x and y components of the dielectric tensors. Consequently, for one-dimensional wave propagation, electromagnetic space is empty, waves are free here, whereas in physical space they appear as modulated wavepackets according to the transformations (203) and (204). We mapped one-dimensional electromagnetic waves in moving media onto waves in empty space, but what happens at the horizon? Why are horizons special? Suppose, without loss of generality, that the medium moves in the negative z direction and develops a black-hole horizon at z = 0: for positive z, the ﬂow speed |u| lies below c/n, and for negative z, the ﬂow exceeds c/n. For the waves that propagate against the current in the positive z direction, the transformation (203) develops a pole at the horizon. We linearize u(z) here and obtain

ln |z| , t+ = t − α

1 du

α= . 1 − n−2 dz 0

(212)

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Transformation Optics and the Geometry of Light

For any given time t, the entire range of t+ is ﬁlled for z > 0 or z < 0, for either side of the horizon. So electromagnetic space has become multivalued and physical space is cut into two distinct regions, because waves conﬁned to either one of the two sides would never interact with each other. But this is not entirely true (Hawking, 1974). Picture a wavepacket that, after having barely escaped from the horizon, propagates in a region where the medium moves at uniform speed. The wavepacket is purely forward-propagating, so it consists of a superposition of plane waves with purely positive wavenumbers k and corresponding positive frequencies ω. On the other hand, recall the result from complex analysis (Ablowitz and Fokas, 1997) that Fourier transforms with positive spectra are analytic on the upper half plane [because integrals over k containing the analytic function exp(ikz) converge and are continuous for positive k and positive Im z]. Consequently, if we analytically continue the wave packet on the complex z plane, it should be analytic for Im z > 0, even if we trace its evolution back to the horizon. Analytic functions are continuous, so the escaping wavepacket must have partially originated from beyond the horizon— some part of the wave must have tunneled through. To calculate the tunnel amplitude, we use the following method (Damour and Rufﬁni, 1976): we combine two waves exp(−iωt+ ) localized on either side of the horizon such that they seamlessly form an analytic function on the upper half z plane. From

ω ω exp −π + i ln(−z) = exp i ln(z) , α α

(213)

it follows that we should give the waves on the left side of the horizon the prefactor exp(−πω/α). Then the two waves combined make an analytic function. On the left side, the medium moves faster than the speed of light in the medium. So here the phase of the wave moves backwards. In locally co-moving frames, the wave would oscillate with negative frequencies. Such negative frequencies have recently been observed in water waves (Rousseaux et al., 2008). The negative-frequency wave makes a negative contribution to the energy, so we ought to subtract the modulus square of its amplitude in the total intensity of the wave [a technically more involved argument (Leonhardt, 2003a) conﬁrms this reasoning]. Therefore, the ratio N of the tunnel intensity to the total spectral intensity is

ω !−1 exp −2π 2πω α ω = exp −1 N= . α 1 − exp −2π α

(214)

Transformation Media

141

According to the quantum ﬁeld theory at horizons (Birrell and Davies, 1984; Brout et al., 1995a) positive- and negative-frequency photon pairs are spontaneously created from the quantum vacuum, because they do not cost any energy. They are emitted at the tunneling rate (214) that one can also interpret as the Planck spectrum (Mandel and Wolf, 1995)

ω N = exp kB T

−1

!−1 (215)

with temperature (kB denotes Boltzmann’s constant)

kB T =

α . 2π

(216)

Black holes are not black after all. The horizon emits light with the Planck spectrum (215) and temperature (216). The light consists of entangled photon pairs where each pair contains one photon with positive frequency and the other with negative frequency, one emitted outside and the other inside the horizon. Seen from one side of the horizon, the black hole is a blackbody radiator. This ﬂash of insight, due to Stephen Hawking (1974), has been one of the most inﬂuential predictions of modern theoretical physics, because it illuminated a vastly unexplored intellectual landscape: Hawking’s theory supports Bekenstein’s idea (Bekenstein, 1973) that the horizon carries entropy that is proportional to its area and it is a quantum phenomenon of a space–time geometry. Hawking’s effect thus connects three vastly different areas of physics—thermodynamics, quantum mechanics, and general relativity. In modern attempts for ﬁnding a quantum theory of gravity, such as loop quantum gravity (Rovelli, 1998) and superstring theory (Green, Schwarz, and Witten, 1987), the correct prediction of the Bekenstein–Hawking entropy has been used as a benchmark. However, the Hawking temperature of solar-mass black holes lies about eight orders of magnitude below the temperature of the cosmic microwave background, so most probably, there is no chance of directly observing Hawking radiation in astrophysics. The benchmark of some of the most advanced theories of physics seems destined to remain as theory. On the other hand, laboratory analogs of the event horizon may demonstrate the physics behind Hawking radiation (Novello, Visser, and Volovik, 2002; Philbin et al., 2008; Unruh and Schützhold, 2007; Volovik, 2003). For example, one could perhaps generate a detectable amount of Hawking radiation using few-cycle light pulses (Brabec and Krausz, 2000; Kärtner, 2004) in photonic-crystal ﬁbers (Russell, 2003). According to non-linear ﬁber optics (Agrawal, 2001), the pulses behave, for all practical purposes, like one-dimensional moving media (Leonhardt and König, 2005; Philbin

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Transformation Optics and the Geometry of Light

et al., 2008): due to the optical Kerr effect, they create additional contributions to the refractive index that move with the light pulses—media that move at the speed of light. Apart from optical dispersion (Born and Wolf, 1999), such moving media resemble the case considered in this chapter. How Hawking radiation emerges from elementary processes in gravity has remained largely a mystery, but laboratory analogs have a chance to, quite literally, shed light on one of the most fascinating pieces of physics, the creation of light at horizons.

6. CONCLUSIONS Optical media appear as geometries and geometries appear as media. Or, to be more precise, dielectric media appear to light as if they were changing the geometry of space (or of space–time if the media are moving or are bi-anisotropic). The modiﬁed and sometimes distorted geometry perceived by light is the cause of many optical illusions, including the ultimate illusion: invisibility (Leonhardt, 2006b; Pendry, Schurig, and Smith, 2006). To deliberately create such an illusion, one may design the desired geometry ﬁrst and then construct a dielectric material for implementing it in practice. The simplest geometries are created by coordinate transformations where light propagation in physical space appears to be curved, but where, in a transformed space, light propagates along straight lines. Media that facilitate coordinate transformations are known as transformation media. Connections between geometry and optics have a long and distinguished history: Fermat’s principle has inspired the principle of least action in classical mechanics, the connection between geometrical optics and wave optics motivated quantum mechanics and path integrals in quantum ﬁeld theory and statistical mechanics, and the principle of least action inspired the idea of inertial motion along geodesics in general relativity, but transformation optics is probably the ﬁrst case where geometrical ideas that typically belong to the lofty sphere of general relativity have become practically useful in rather down-to-earth engineering applications. Both sides beneﬁt from this connection: engineers get motivated to learn and appreciate the tools of geometry, because applications are a great incentive to learn, and applied mathematicians or relativists are confronted with and often delighted by the way in which these ideas appear in the real world, both in technical applications and also in some natural phenomena on Earth. Connections between theory and experiment and between applied and fundamental science have been the greatest strength of science. The subject of this chapter, transformation optics, is an example of such connections. We hope to have written it in a form such that both practical-minded

Conclusions

143

engineers and theoretical-minded physicists and mathematicians ﬁnd some treasure (and pleasure) in it. This chapter is a primer, an introduction to the geometry of light and the concepts of transformation optics. We focused on the principal ideas and introduced them without assuming much prior knowledge, only standard mathematics, some basic physics and sufﬁcient stamina of the reader. Being a primer, the chapter necessarily omits many aspects of this ﬁeld. We analyzed only four examples of transformation media—cloaking devices, perfect lenses, vortices, and horizons—but these four cases illustrate characteristic non-trivial topologies, each one with different physics, and they have been experimentally veriﬁed at least to some extent. We did not discuss transformation media that re-scale space without changing the topology (see, for example, Kildishev and Shalaev, 2008; Schurig, Pendry, and Smith, 2007), nor did we describe intriguing ideas that have not been implemented yet, such as electromagnetic wormholes (Greenleaf et al., 2007a,b). We omitted the analysis of active cloaking devices (Miller, 2006) or plasmonic coverings (Alu and Engheta, 2005; Milton and Nicorovici, 2006), because they are not directly related to transformation media (but perhaps the latter may appear as transformation media in disguise). We also did not discuss cloaking devices that combine coordinate transformations with refractive-index proﬁles that cannot be transformed away, ﬁrst ideas of non-Euclidean cloaking (Hendi, Henn, and Leonhardt, 2006; Leonhardt, 2006c; Leonhardt and Tyc, 2009; Ochiai, Leonhardt, and Nacher, 2008). Furthermore, we did not explain in detail how transformation media are implemented in practice, where the properties of metamaterials come from and how they are made, because this is a subject that ﬁlls entire books (Marqués, Martin, and Sorolla, 2008; Milton, 2002; Sarychev and Shalaev, 2007). We focused on the main ideas and some connections between optics, in particular, transformation optics, and other areas of physics and mathematics. It is quite remarkable how many different ideas optics combines and connects, but also here we were forced to make omissions. For example, we did not discuss applications of transformation media in acoustics (see, for instance, Chen and Chan, 2007; Cummer et al., 2008; Fang et al., 2006; Milton, Briane, and Willis, 2006) and in quantum mechanics (Leonhardt, 2006b). We also focused primarily on the classical optics of transformation media and only mentioned their quantum optics without going into detail. Last, and least, we did not even attempt to represent the recent literature related to transformation optics, because this literature is too recent and rapidly growing—had we included it this chapter would be outdated in a very short time. We hope to have compensated for these shortcomings and omissions by being clear and pedagogical in the main ideas and by focusing on the “new things in old things” and explaining “old things in new things,” by telling the aspects of the story that we believe are already guaranteed to last and to remain inspiring for a long time.

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Transformation Optics and the Geometry of Light

ACKNOWLEDGMENTS We are privileged for having beneﬁted from many inspiring conversations about “geometry, light and a wee bit of magic.” In particular, we would like to thank John Allen, Sir Michael Berry, Leda Boussiakou, Luciana Davila-Romero, Mark Dennis, Malcolm Dunn, Ildar Gabitov, Greg Gbur, Andrew Green, Awatif Hendi, Julian Henn, Chris Hooley, Sir Peter Knight, Natalia Korolkova, Irina Leonhardt, Renaud Parentani, Harry Paul, Paul Piwnicki, Sir John Pendry, Wolfgang Schleich, David Smith, Stig Stenholm, Tomáš Tyc, and Grigori Volovik. Our work was supported by the Leverhulme Trust, the Engineering and Physical Sciences Research Council, the Max Planck Society, and a Royal Society Wolfson Research Merit Award.

Appendix A

145

APPENDIX A The physical principle underlying general relativity is that all the laws of physics should be expressible as tensor equations in four-dimensional (possibly curved) space–time (Misner, Thorne, and Wheeler, 1973). As a consequence of the transformation properties of tensors, this means that the laws of physics will have the same form in any space–time coordinate system. This property of tensor equations is called general covariance. In this Appendix, we show that the free-space Maxwell equations in generally covariant form are equivalent to Maxwell’s equations in a material medium with constitutive equations (150) and (151). The space–time form of Maxwell’s equations is written in terms of the electromagnetic ﬁeld tensor Fμν , constructed from the the E and B ﬁelds; in a right-handed coordinate system

⎛

Fμν

0 ⎜E1 =⎝ E2 E3

−E1 0 −cB3 cB2

−E2 cB3 0 −cB1

⎞ −E3 −cB2 ⎟ . cB1 ⎠ 0

(A1)

The generally covariant Maxwell equations are (Misner, Thorne, and Wheeler, 1973)

F[μν;λ] = F[μν,λ] = 0,

ε0 ε0 Fμν ;ν = √ −gFμν ,ν = jμ , −g

(A2)

where jμ = (ρ, ji /c) is the four-current, and the square brackets denote antisymmetrization. This last operation produces a completely antisymmetric tensor with three indices. In Eq. (A2), F[μν,λ] is explicitly

F[μν,λ] = Fμν,λ + Fνλ,μ + Fλμ,ν ,

(A3)

since the electromagnetic ﬁeld tensor already is antisymmetric: Fμν = −Fνμ . The forms of (A2) containing partial rather than covariant derivatives are obtained by using the antisymmetry of Fμν and Fμν . In a left-handed coordinate system, the relation between Fμν and the magnetic ﬁeld differs by a sign from that in Eq. (A1). In this Appendix, we will conﬁne ourselves to right-handed systems; this issue of handedness is dealt with in Section 4. We deﬁne a quantity H μν by H μν = ε0 −gFμν = ε0 −ggμλ gνρ Fλρ =⇒

Fμν =

1 √ gμλ gνρ H λρ ε0 −g

(A4) (A5)

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Transformation Optics and the Geometry of Light

and regard H μν as being constructed from D and H ﬁelds as follows:

⎛

D1

D2

0

H 3/c

−H 3/c

0

H 2/c

−H 1/c

0

⎜−D1 ⎜ H μν = ⎜ ⎝−D2 −D3

Then, by introducing a new four-current J μ = equations (A2) can be written as

D3

⎞

−H 2/c⎟ ⎟ ⎟. H 1/c ⎠ 0

√ −gjμ , the free-space

H μν,ν = J μ ,

F[μν,λ] = 0,

(A6)

(A7)

which are the macroscopic Maxwell equations (130) in right-handed Cartesian coordinates. The constitutive equations are given by Eqs. (A1) and (A4)–(A6). To obtain relations between the vector ﬁelds D, H and E, B, consider ﬁrst the components F0i ; from Eqs. (A1), (A5), and (A6) one obtains

Ei =

1 1 gi0 gj0 − gij g00 Dj − √ √ [jkl]g0j gik H l . ε0 −g ε0 c −g

(A8)

We simplify this result as follows. The identity

gμλ gλν = δνμ

(A9)

gives giλ gλ0 = 0 =⇒ gi0 = −

1 gij gj0, g00

(A10)

g0λ gλi = 0 =⇒ gi0 = −

1 ij g gj0 , g00

(A11)

gjλ gλi = gj0 g0i + gjk gki = δij .

(A12)

Use of Eqs. (A10) or (A11) in Eq. (A12) produces the following two relations

1 i0 k0 g − 00 g g gkj = δij , g ij

g

ik

1 gkj − gk0 gj0 g00

= δij ,

(A13)

Appendix A

147

which reveal inverse-related 3 × 3 matrices. In view of Eqs. (A12) and (A13), multiplying Eq. (A8) by gli and contracting on the index i yields

√ ε0 −g ij j 1 Di = − g E + [ijk]gj0 H k , g00 cg00

(A14)

the ﬁrst of the constitutive equations (150) with (151). To obtain the second constitutive relation, we employ the tensors dual to Fμν and H μν . This requires use of the 4D Levi–Civita tensor, which is given in a right-handed system by

μνλρ =

−g [μνλρ],

1 μνλρ = − √ [μνλρ], −g

[0123] = +1. (A15)

The dual tensors ∗Fμν and ∗Hμν are deﬁned by (Misner, Thorne, and Wheeler, 1973) ∗ μν

F

∗

1 μνλρ 1 Fλρ =⇒ Fμν = μνλρ ∗Fλρ , 2 2

(A16)

1 1 μνλρ H λρ =⇒ H μν = μνλρ ∗H λρ ; 2 2

(A17)

=

Hμν =

so they have components ⎛

∗ μν

F

0 −cB 1 ⎜ 1 ⎜ =√ ⎜ −g ⎝−cB2 −cB3 ⎛

∗

Hμν

cB1 0 E3 −E2

−H 1 /c 0 D3 −D2

0 1 ⎜ H ⎜ /c = −g ⎜ 2 ⎝H /c H 3 /c

cB2 −E3 0 E1

−H 2 /c −D3 0 D1

⎞ cB3 E2 ⎟ ⎟ ⎟, −E1 ⎠ 0 ⎞ −H 3 /c D2 ⎟ ⎟ ⎟. −D1 ⎠ 0

(A18)

(A19)

Re-expressed in terms of the dual tensors, the constitutive equations (A4) and (A5) read ∗ Hμν = ε0 −ggμλ gνρ ∗Fλρ , (A20) ∗ μν

F

=

1 √ gμλ gνρ ∗Hλρ , ε0 −g

(A21)

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Transformation Optics and the Geometry of Light

where we applied the four-dimensional version of the double vector product (70)

ν μ ν μνλρ λρστ = −2 δμ σ δτ − δτ δσ .

(A22)

Writing out ∗H0i using Eqs. (A18)–(A20), one ﬁnds

ε0 c2 ε0 c g00 gij − gi0 gj0 Bj + √ [jkl]gj0 gik El . Hi = − √ −g −g

(A23)

Comparison of this with Eqs. (A8) and (A14) shows that

√

Bi = −

−g ij j 1 g H − [ijk]gj0 Ek , cg00 0 c2 g00

(A24)

which is the second of the constitutive equations (150) with (151). In this way we derived Plebanski’s constitutive equations (150) and (151) (Plebanski, 1960). Note that several other relations between D, H, E, and B are contained in Eqs. (A4), (A5), (A20) and (A21). For example, to express D and H in terms of E and B, we need to take only the time–space components of (A4), obtaining

Di = ε0 −g gi0 gj0 − gij g00 Ej − ε0 c −g[jkl]gk0 gij Bl ,

(A25)

and the required formulae are Eqs. (A23) and (A25).

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CHAPTER

3 Photorefractive Solitons and Their Underlying Nonlocal Physics Eugenio DelRe*, Bruno Crosignani† and Paolo Di Porto†

Contents

1 Introduction 2 Observing a Photorefractive Soliton 2.1 The Basic Apparatus 2.2 Nonlocal Phenomenology 3 Charge Migration and Intrinsic Spatial and Temporal Scales 3.1 Time Nonlocality 3.2 Spatial Nonlocality 3.3 Electro-Optic Response 4 Nonlinear Beam Propagation 5 Intrinsic Scales and a Condition for a Local Kerr-Like Soliton 5.1 Saturation Length and a Spatially Local Response 5.2 A Spatially Local Kerr-Like Effect 5.3 The Dielectric Response Time 5.4 A Time-Averaging Effect 5.5 Cumulative Response and Imprinting 5.6 Quasi-Steady-State Solitons: A Signature of Time Nonlocality 6 Observable Effects of Spatial Nonlocality in 1D Solitons 6.1 Physical Origin 6.2 Self-Bending 6.3 Nonperturbative Effects 6.4 Harnessing 1D Spatial Nonlocality 7 Nonlocality and Anisotropy in 2D Solitons 7.1 An Intrinsic Nonlocality 7.2 Optical Propagation

154 156 156 159 161 162 163 164 165 167 167 168 171 173 173 174 177 177 178 182 185 187 187 189

* Dipartimento di Ingegneria Elettrica e dell’Informazione, Università dell’Aquila, 67100 L’Aquila, Italy † Dipartimento di Fisica, Università dell’Aquila, 67100 L’Aquila, Italy Progress in Optics, Volume 53, ISSN: 0079-6638, DOI: 10.1016/S0079-6638(08)00203-5. Copyright © 2009 Elsevier B.V. All rights reserved.

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7.3 A “Hidden” Contribution to the Nonlinearity 7.4 Direct Observation of Two-Dimensional Anisotropic Response 7.5 Round or Circular-Symmetric 2D Solitons 8 Conclusions and Perspective References

190 191 193 196 196

1. INTRODUCTION In classical physics, the distinction between particle and wave behavior holds in a clear-cut manner only when the latter can be correctly described by models leading to linear equations. In the general case, that is, when nonlinearity plays a role, waves can qualitatively behave like particles, the principal example being solitons. These are waves that form in many physical systems and (i) are and remain localized in space or time during their dynamics and (ii) interact with other similar wave-like particles. An important ﬁeld of investigation is that of optical solitons, that is, light beams that do not suffer distortion due to diffraction or pulses that do not spread due to dispersion. When light is made to interact with matter, that is, essentially with electrons bound to nuclei, its description as a propagating ﬁeld distinct from matter produces a generally nonlinear dynamic, unless the interaction is small enough that its effect on the stable electron states is well described by harmonic-like displacements. Typically, when a pulse of light travels down an optical ﬁber under conditions in which its associated oscillating electric ﬁeld is comparable with the static ﬁeld holding the outermost electrons, propagation is, to a good approximation, described by the so-called nonlinear Schroedinger equation, which contains a term cubic in the optical electric ﬁeld (Sulem and Sulem, 1999). The result is that there exist pulses that form solitons, which in this case are termed temporal solitons since they are localized in the propagation direction (Hasegawa and Tappert, 1973; Mollenauer, Stolen, and Gordon, 1980). These “Kerr solitons” essentially propagate without suffering the distortions due to material dispersion and can bounce and collide with other similar pulses. The origin of Kerr nonlinearity is local in space at the atomic scale and in time for times longer than the characteristic dynamic scales of bound electrons (of the order of 10−16 s, i.e., shorter than the period of the optical ﬁeld). This locality gives rise to a highly tractable model that is to be considered among the basic paradigmatic equations governing solitons (Degasperis, 1998). On the other hand, this very locality means that the effect is intrinsically a product of the instantaneous interaction with each single atom, and, for a given atomic system, the pulse must have an appropriately elevated intensity. In general, the wave-like nature affects ﬁeld dynamics and hence localization in the full space–time domain,

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and hence nonlinearity can form optical solitons of various kinds. For example, spatial solitons are those optical waves in which the nonlinearity balances spreading due to diffraction, thus conﬁning them in one or two directions orthogonal to the propagation axis (transverse plane) (Segev, 1998; Segev and Stegeman, 1998; Stegeman and Segev, 1999; Trillo and Torruellas, 2001). For a local nonlinearity, the formation of a spatial soliton would require an enormous amount of energy, since as opposed to a pulse conﬁned in a ﬁber, forming a temporal soliton, here the intensity should be elevated throughout a ﬁnite spatially extended portion of the material. It is a fact that spatial solitons readily form only in materials in which this locality, characteristic of the Kerr effect, is replaced by a more elaborate spatio-temporal nonlinear response, which means that it depends on the optical intensity at previous instants and at different surrounding positions. Clearly, this poses, in general, a difﬁcult task in modeling theoretically and interpreting spatial soliton phenomenology. Among the various nonlocal mechanisms that provide an accessible basis for optical spatial solitons, which range from thermal and reorientational effects to photosensitive materials, our aim here is to discuss the physics underlying photorefractive spatial solitons. The discovery of photorefractive solitons starts with the prediction of (Segev et al., 1992) and is described in detail in the original set of seminal papers on the subject (Castillo et al., 1994; Christodoulides and Carvalho, 1994, 1995; Crosignani et al., 1993; Duree et al., 1993, 1994; Segev et al., 1994a,b; Singh and Christodoulides, 1995). Photorefraction occurs in speciﬁc materials where the combined effect of two physical mechanisms intervenes (Günter and Huignard, 2006; Solymar, Webb, and Grunnet-Jepsen, 1996; Yeh, 1993): the ﬁrst is the formation of a light-induced inhomogeneous charge distribution in the crystal due to the interplay of in-band deep acceptor and photosensitive donor impurities; the second is the simultaneous change in the local index of refraction produced by the charge distribution. The arising optical nonlinearity stems from the fact that the propagating ﬁeld distribution produces an index of refraction change that modiﬁes its own propagation dynamics. While in general this produces a complicated dynamical evolution of the light beam, the soliton forms when a speciﬁc optical wave distribution produces an index of refraction pattern capable of exactly balancing the natural beam diffraction. In this photorefractive process, temporal and spatial nonlocalities are, respectively, associated with the ﬁnite time and space scales of the charge migration process. To understand the optical behavior, the basic strategy is to consider the paraxial propagation equation for inhomogeneous media for the slowly varying amplitude of the optical ﬁeld coupled, through the material-speciﬁc electro-optic response, to the model describing the light-induced formation of the space–charge ﬁeld, the band-transport or Kukhtarev model. Our aim is to provide an account of the recent progress in the physical understanding

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of the phenomenon, including the origin of quasi-steady-state solitons and round two-dimensional (2D) solitons, which is something beyond to providing an extensive review of photorefractive soliton phenomenology, as can be found in a number of review articles (Crosignani et al., 1998; DelRe, Crosignani, and Di Porto, 2001; DelRe et al., 2006; Krolikowski, Luther-Davies, and Denz, 2003).

2. OBSERVING A PHOTOREFRACTIVE SOLITON 2.1. The Basic Apparatus In a typical apparatus (Segev, Shih, and Valley, 1996), a continuous-wave visible laser beam is made to propagate inside a photorefractive crystal, and the beam intensity distribution is detected and analyzed. The principal detection scheme is to image the input and output transverse beam intensity distribution I(x, y, z, t) onto a charge-coupled-device (CCD) camera and through this analysis to infer underlying nonlinear beam evolution. A less precise direct detection is to image the intensity distribution inside the material itself, for example, collecting spuriously scattered light in a direction orthogonal to the propagation axis (say the z-axis). Given the intrinsically nonlocal nature of the phenomenon, observations do not in themselves allow the full analysis of the nonlinearity. Time nonlocality can be analyzed by comparing the intensity distribution at different times, considering the general dynamic process I(x, y, z, t) (Duree et al., 1993), whereas for spatial nonlocality a dedicated method able to detect the underlying space–charge distribution (and hence the “hidden” anisotropy and nonlocality) has been developed in a sub-class of photorefractive samples: soliton electro-holography (DelRe, Tamburrini, and Agranat, 2000). This technique is based on the nonlinear combination, afforded by the quadratic electro-optic effect, of an externally applied bias ﬁeld and the soliton-associated space–charge ﬁeld. A second, less versatile, technique to analyze spatial nonlocality and anisotropy is not to limit the detection to the dynamics of single spatial solitons but to carry out a form of nonlinear spectroscopy by investigating soliton–soliton interaction (Krolikowski et al., 1998). As a basis for our present discussion, we should begin by describing the schematic of the experimental apparatus and the principal phenomenology. The light beam that is to form the soliton is obtained in the following manner: given that the photorefractive nonlinearity builds up in time, the most accessible manner is to use a continuous-wave laser source that has an emission within the absorption spectrum of the impurities hosted within the photorefractive crystal, but clearly in the transparent spectrum

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of the undoped dielectric. For example, a He–Ne laser oscillating at 545 nm wavelength will generally lead to charge dislocation in an iron-doped sample of strontium–barium–niobate (SBN), potassium–niobate (KNbO3 ), and most samples of lithium–niobate (LiNbO3 ). Because the doping structure generally leads to a whole set of different impurity levels, the actual wavelength is not an issue and so neither is the spectral quality. A different story holds for the spatial mode, which for standard bright soliton experiments is preferably a TEM00 Gaussian mode, as would emerge from any standard gas laser. Beam polarization is important since the electro-optic response is anisotropic. In the basic scheme, the launch beam should be polarized linearly in the direction of the optical axis, for ferroelectric samples, and in the direction of the applied external ﬁeld in cases in which no spontaneous polarization exists. The linear polarization avoids both polarization evolution and tumbling during the propagation, which would lead to a propagation-dependent optical response, and allows the use of the generally most effective electro-optic coefﬁcient, although different schemes have certainly been looked into (Fazio et al. (2003)). Beam power is not a main issue in the apparatus: the lower the power implemented, the slower the dynamics, all other issues, such as ambient noise and detector noise, considered. The actual response time is sample-dependent. For example, in a typical experiment in SBN, a TEM00 beam focused down to a 10 μm spot will form a soliton in terms of tens of seconds for a microwatt power level. Evidently, in the charge separation, which originates from an absorption, what matters is the local intensity I(x, y, z, t), and under conditions in which much larger beams must be used, such as for the formation of one-dimensional (1D) solitons (that is solitons that are so extended in one transverse direction that they only diffract and suffer nonlinear effects in the other) (Kos et al., 1996), power levels even three-orders of magnitude higher must be envisaged. The actual size and shape of the beam at the input facet of the sample is also an important factor. There are two basic things that must be considered: (i) crystal length (the length of the linear/nonlinear propagation) Lz and (ii) the intrinsic spatial scales that are involved in the charge migration, the source of which we term spatial nonlocality. As for the ﬁrst consideration, the observation of a spatial soliton must be under conditions in which the linear paraxial propagation Lz must give rise to substantial spreading and distortion due to diffraction. This can be evaluated for a launch Gaussian beam by the comparison of Lz to the diffraction length Ld (Yariv, 1989). For a typical ferroelectric crystal with index of refraction n0 = 2.5 and a wavelength λ = 0.5 μm, a beam of width w0 = 10 μm will have Ld 103 μm, a reasonable situation to investigate solitons will be in crystals with Lz greater than several millimeters. For shorter samples, a smaller w0 can be used, but then the second consideration must be taken into account. As discussed in what follows, smaller

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spatial scales activate spatial-nonlocal processes associated to charge diffusion and charge saturation so that simply reducing w0 will not generally bring to the beneﬁt of a simple soliton observation. For example, in a typical scheme, the use of w0 of the order of 1 or 2 μm will even hamper soliton formation (DelRe, Ciattoni, and Palange, 2006). The shape of the beam also affects the nature of the soliton, whether it be 1D or 2D (2D—that is, a beam that linearly diffracts in both transverse dimensions). A beam that is circular with a w0x w0y can lead to a 2D soliton, which is a beam for which diffraction does not produce a spreading of the beam in both the directions transverse to the propagation direction, whereas a beam that has w0x w0y can be used to investigate 1D solitons since diffraction in this case would distort and spread the beam only in the tighter x-direction. In both cases, to observe the most basic phenomena, the beam is focused onto the input facet of the sample, either in the direction of the facet itself or at a small angle, to observe angled soliton formation or soliton collisions. To achieve a 10 μm wide beam from a typical laser source, the beam from the laser is ﬁrst expanded and collimated by using two confocal lenses, to a spot size of several millimeters, and then focused down to the desired size by a third focusing lens, typically with a focal length of the order of several centimeters. The crystal itself is chosen and conﬁgured so as to lead to a predominantly drift-based nonlinearity. It is thus biased along the direction of the optical polarization and principal crystal axis (for example, the xdirection). This can be achieved either through a pair of planar electrodes applied to the two opposite x-facets, a geometry that typically requires the use of voltages on the order of the kilovolt (for millimeter-sized samples), or through more elaborate systems, depending on the crystal used. For example, in using paraelectric KLTN (potassium–lithium–tantalate– niobate), the sample can be biased through top- or one-sided electrodes, and this permits the observation of solitons with much lower voltage levels (down to tens of volts) (D’Ercole et al., 2004). In some cases, the crystal must also be temperature-controlled, and the crystal holder has a feedbackcontrolled Peltier junction. Furthermore, in many cases, the crystal thermal (or dark) conductivity must be artiﬁcially enhanced and controlled: here, the sample itself is illuminated by a constant intensity distribution, the background illumination. This illumination can be achieved by extracting part of the basic launch beam through a beam-splitter and expanding it into a plane wave and then recombining it with the launch beam through a second beam splitter. In order for the beam itself to not suffer the effects of the index pattern leading to the soliton, it can be sent through a λ/2 waveplate so as to have a polarization orthogonal to the soliton polarization, along the y-direction, for which the electro-optic effect is normally inefﬁcient. In other schemes, the background illumination can be delivered from the side of the crystal and can originate from a different laser source or even a lamp. An important parameter in the set-up is the ratio between the input peak intensity of the diffracting beam Ip and that of the

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plane wave (Ib ). Typically, this ratio u20 = Ip /Ib will not be less than several units and determines the saturation of the nonlinearity (Segev, Shih, and Valley, 1996). Under conditions in which the background illumination is absent, the nonlinearity builds up in time and provides transient soliton phenomena, since in the end, saturation sets in and attenuates greatly the nonlinearity (Duree et al., 1993). Finally, the detection scheme is given as follows: detection is normally achieved by collecting the transmitted light beam along the propagation z axis through a single collecting lens, with a focal length of several centimeters, placed immediately after the sample. This lens images a speciﬁc plane in the crystal onto the photosensitive region of a CCD camera that then collects the data on the intensity distribution I(x, y, z, t). To the effect that no pattern is present in the sample, that is, before any nonlinear effects have set in, the system can be used to detect the evolution of the beam from the input facet to the output facet. As the pattern is formed in the volume of the sample, this simple volume microscope-like technique is distorted; so, the only detection that can be easily carried out is at the output of the sample, for z = Lz . In some cases, the propagation dynamics can be observed through a lateral facet of the sample. This, however, is only feasible when strong scattering is involved, something that is only reasonable when the beam is close to the crystal surface. In fact, only 1D solitons have ever been observed in this manner, since here the beam actually hits the top surface of the sample.

2.2. Nonlocal Phenomenology Of the various experiments normally carried out, we are here interested in those that highlight the nonlocal features of the effect, which are listed below. The techniques to study these effects are principally the following: (i) time dynamics, by which the intensity distribution is analyzed for different instants of time from the moment in which the beam is actually launched in the sample (Denz et al., 1999); (ii) position dynamics in the transverse plane, by which the actual position of the soliton is detected in comparison to the normal center of the diffracting beam (Pismennaya et al., 2008); (iii) soliton electro-activation, by which the underlying space–charge ﬁeld pattern is detected by analyzing the propagation of beams that do not alter the soliton space–charge distribution under conditions in which the applied voltage is changed (DelRe, Ciattoni, and Agranat, 2001); (iv) soliton collisions and interactions (Shih and Segev, 1996); (v) soliton ellipticity studies, by which the details of the shape of the trapped soliton beam are analyzed to detect anisotropy in the response (Zozulya and Anderson, 1995; Zozulya et al., 1996). The main nonlocal features of photorefractive solitons are listed below. As will be discussed, these are associated with the underlying charge migration process and its intrinsic spatio-temporal scales (or, equivalently,

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the “nonlocality” of the response) and with the geometrical nature of the electro-static problem: 1. The value of the beam intensity plays no fundamental role in a wide range of values (Segev et al., 1994b). This is the most evident departure from a local nonlinear response, for which beam intensity is one of the parameters driving the process. It is also the reason for which spatial solitons can be observed with extremely low optical power beams. The origin of this intensity-independence is associated to the charge migration process, by which mobile electrons hop from donor impurity to donor impurity through drift and diffusion: while the rate at which the hopping proceeds depends on the rate of photoexcitation and hence on the intensity, the mechanism itself only depends on a sequence of independent single-photon absorptions. 2. The photorefractive response has memory in that the space–charge pattern underlying the soliton phenomenon persists for a ﬁnite time after the optical excitation has been removed (Morin et al., 1995). In fact, since the response is associated to a charge hopping process, the absence of light simply allows no further evolution or change to intervene in the space–charge distribution (apart from slow thermal relaxation). 3. The soliton process has a time scale in that, for a given intensity, the soliton wave forms after a ﬁnite transient and, in some conditions, the soliton state itself only persists for a temporal window (quasi-steadystate solitons) (Duree et al., 1993). As above, this is a direct consequence of the underlying charge migration process and its intrinsic time scales. 4. Since the soliton process has ﬁnite time scales, these allow the observation of so-called incoherent solitons (Mitchell et al., 1996; Mitchell and Segev, 1997), that is, beams that have a ﬂuctuating intensity distribution at small time scales but a smooth envelope for longer time scales. Once again, the mechanism allowing soliton formation is the fact that the charge migration dynamics average out the optical excitation at time scales much shorter than their own. 5. Photorefractive solitons can also form as circularly symmetric 2D waves, even though the underlying system is highly anisotropic (DelRe et al., 1998c; Shih et al., 1995, 1996). This is associated with a speciﬁc interplay between the anisotropy and the transverse spatial nonlocality of the charge separation process (Crosignani et al., 1997; DelRe et al., 2004; Gatz and Herrmann, 1998; Krolikowski et al., 1998). 6. Some kinds of photorefractive spatial solitons undergo self-bending, and this depends on the size of the beam itself (Christodoulides and Carvalho, 1994; Pismennaya et al., 2008). This indicates the presence of an intrinsic transverse spatial scale to the point that, for small enough beams (but far above the nonparaxial limit), no soliton can actually form (DelRe, Ciattoni, and Palange, 2006).

Charge Migration and Intrinsic Spatial and Temporal Scales

161

7. The observation of diffusion-driven effects, that is, solitons that emerge because of charge diffusion, is one of the basic spatially nonlocal processes involved in charge migration (Crosignani et al., 1999). These clearly cannot exist in local nonlinearity, where diffusion is intrinsically absent.

3. CHARGE MIGRATION AND INTRINSIC SPATIAL AND TEMPORAL SCALES The nonlocality that underlies photorefractive solitons originates from photorefraction, the mechanism by which light produces, in appropriate materials, an observable index of refraction change associated to a reversible dislocation of charge from photosensitive impurities. Photorefraction can be thought of as a consequence of different simultaneous processes: (1) a continuous absorption of a small fraction of the propagating light beam on behalf of donor impurities (or, analogously of acceptor impurities); (2) the migration of the photoinduced charges; (3) the change of the local index of refraction of the material produced by the presence of a quasi-static electric ﬁeld (Solymar, Webb, and Grunnet-Jepsen, 1996). The three processes are circularly linked, since, for example, Process (1) produces the free charges that undergo Process (2), while it is Process (2) that gives rise to a change in the local electric ﬁeld that intervenes in Process (3). Now Process (3) inﬂuences beam propagation, i.e., the driving force of Process (1). Instead of reducing the entire system to a sequence of single photon propagation, absorption, electron excitation, migration, and recombination events (the microscopic description), the physical variable we are most interested in is the macroscopic light beam intensity distribution in space and time I(x, y, z, t), averaged over the aforementioned elementary events. Nonlocality stems from the basic fact that no absorbed photon leading to Process (1) will ever suffer the effects of its absorption through the ensuing Processes (2) and (3); yet considering the light intensity as a single entity I, we are immediately cast in the condition in which the value of I(r , t ) inﬂuences I(r, t), where t > t and r = r . This basic fact embodies the time and space nonlocality. The three processes imply one speciﬁc temporal and one spatial scale. To identify these, we brieﬂy review the photorefractive model. Since we are considering the averaged behavior over the microscopic events, Process (1), i.e., the absorption and recombination of the photoinduced charges, can be modeled through an effective rate equation,

∂ + N = (β + sI)(Nd − Nd+ ) − γNNd+ . ∂t d

(1)

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Photorefractive Solitons and Their Underlying Nonlocal Physics

Here, the rate of change in the concentration of photosensitive ionized donors Nd+ is proportional to the probability per unit time of the ionization of the donor site. This can be described by a term proportional, through the constant s (associated with the absorption cross-section), to the product of the concentration of nonionized donors (Nd − Nd+ ) with the local intensity of the light beam I and an independent thermal excitation event, described by a term proportional to the sole concentration of nonionized donors, the constant β being thus the thermal excitation rate. Equally independent are the processes of recombination, and this gives rise to a decrease in Nd+ that is approximately proportional to the probability per unit time that an electron in the conduction band interact with an ionized donor site, i.e., a term proportional to the product between the concentration of free electrons N and Nd+ through the recombination constant γ.

3.1. Time Nonlocality In most experimental situations, the charge migration [Process (2)] will occur through a multi-stage sequence of elementary processes, by which each single electron is photoexcited, migrates, and recombines. In our macroscopic view, we consider the average motion of the charges on spatiotemporal scales larger than the microscopic migration steps (i.e., electron mean free path and corresponding time scale). At these scales, electrons can be thought of as moving in the crystal under the inﬂuence of drift and diffusion (neglecting photovoltaic effects such as those investigated in Fazio et al. (2004)), giving rise to a current density J described by

J = qμNE + kb Tμ∇N,

(2)

where −q is the electron charge, μ is the charge mobility, kb is the Boltzmann constant, T is the crystal temperature, and E is the local quasi-static electric ﬁeld. The local averaged charge density is ρ = q(Nd+ − Na − N), where we have considered the contribution of the ionized donors, the free electrons, and a small concentration of acceptor impurities Na (Nd /Na = α 1) that at thermal equilibrium are all practically ionized. In turn, charge dynamics obey the laws,

∂ ρ+∇ ·J=0 ∂t ∇ · E = ρ, where is the crystal dielectric tensor.

(3) (4)

Charge Migration and Intrinsic Spatial and Temporal Scales

163

In general, two time scales come into play: the charge recombination time (or charge lifetime) τr = 1/(Nγ) [see Eq. (1)] and the dielectric relaxation time τd = /(μNq) [see Eqs. (2–4)]. The two are considerably different (here is the order of magnitude of the dielectric components). In fact, τr involves local excitation/recombination events, whereas τd is related to an average spatial displacement through the dielectric, which is generally slow. Thus, for most conﬁgurations, τr τd . Considering the macroscopic dynamics at time scales of τd , we can assume that the free charge generation and recombination processes are at every instant at equilibrium [∂Nd+ /∂t = 0 in Eq. (1)], so that (β + sI)(Nd − Nd+ ) γNNd+ . Time dynamics involves hence a single time scale τd , and we shall later identify the resulting time process with a slow cumulative build-up of charge separation, the basis for time nonlocality.

3.2. Spatial Nonlocality This equilibrium relationship can be further simpliﬁed considering that Nd+ = ρ/q + Na + N, and, for a typical achievable illumination N Na so that (β + sI)(Nd − ρ/q) γN(ρ/q + Na ), where we have furthermore approximated Nd − Na Nd (since α 1). At this point, we can substitute

N=

(β + sI)(Nd − ρ/q) γ(ρ/q + Na )

(5)

in Eq. (3) to achieve

⎡

1− γ ∂((0)E) + E(β/s + I) ∇ ·⎣ qμsα ∂t 1+ ⎛ +

1−

kB T ∇ · ⎝(β/s + I) q 1+

∇·((0)E) αNa q ∇·((0)E) Na q

∇·((0)E) αNa q ∇·((0)E) Na q

⎞⎤ ⎠⎦ = 0,

(6)

where (0) is to recall the fact that for the considered time scales, the dielectric response is quasi-static. Inspecting Eq. (6), we ﬁnd that having considered the average dynamics of the photoinduced charge system brings with it two distinct spatial scales: one associated with the diffusion process of Eq. (2), the other with the combination of Eqs. (1) and (4). Of these two scales that intervene in Eq. (6), the saturation scale constitutes what can be termed the spatial nonlocal trait of the phenomenon and will be later discussed in paragraph 5.1.

164

Photorefractive Solitons and Their Underlying Nonlocal Physics

3.3. Electro-Optic Response In considering Process (3), that is, the inﬂuence of E on the propagation of light, we must address the mechanism by which a low-frequency component (the quasi-static E) couples to a high-frequency electromagnetic component Eopt characterizing the light ﬁeld, a coupling that must be intrinsically nonlinear. This coupling can be usefully modeled by assuming that E alters the material dielectric properties, and these produce an optical response through the standard effect by which the optical ﬁeld gives rise to a high-frequency polarization in the medium that reacts on the optical ﬁeld itself: the electro-optic effect. In the basic linear limit, this dielectric response can be described, for each monochromatic component of Eopt of angular frequency ω, through the introduction of the electric displacement vector D, tied to Eopt by the high-frequency material dielectric tensor D = Eopt , where is the frequency-dependent two-index tensor with components ij so as to produce a vector through the contraction of the material properties with the optical ﬁeld vector. In describing light propagation, it is useful to introduce the index of refraction tensor n2 = /0 or its equivalent inverse form (1/n2 ). The changes in this tensor produced by the low-frequency E arise as a consequence of a nonlinear component of the susceptibility of the material, i.e., of its polarization and electric displacement vector. If compared to the spatial and temporal nonlocality of the charge migration, the electro-optic effect can be considered local. While E will undergo the complicated and nonlocal process of Eq. (6), the alteration of dielectric and optical response can be described by changes in the (1/n2 ) = (1/n2 )|E=0 + (1/n2 ) tensor. These depend adiabatically (or parametrically) on the value of E at the given position and instant of time, through a phenomenological (local) relationship

1 2 n

= rijk Ek + sijkl Ek El ,

(7)

ij

where rijk and sijkl are, respectively, the linear and quadratic electro-optic tensors, E = (Ex , Ey , Ez ), and summing on repeated indices is assumed. The two-index-tensorial nature derives once again from the standard representation of the dielectric response that ties D to Eopt , and the expression on the right-hand side is the most general that allows the combination of the low-frequency vector with the material properties, truncated at second-order terms in E itself, which gives a two-index tensor. The considerable difference in the temporal scales further ensures that the highfrequency spectrum of the optical ﬁeld remains essentially unaltered, and the electro-optic tensors depend on the well-deﬁned high-frequency ω of each component.

Nonlinear Beam Propagation

165

4. NONLINEAR BEAM PROPAGATION To address soliton formation, we must also address the mechanism by which the changes introduced by Process (3) on the dielectric, and hence optical, response [Eq. (7)] induce changes in the actual propagation of light. This closes the loop into a nonlinear process, since the partial absorption of the optical ﬁeld produces the slow changes in the quasi-static electric ﬁeld, which in turn reacts at fast polarization scales back onto the optical ﬁeld. We consider for the moment a monochromatic optical ﬁeld, a situation that reproduces the greater majority of experiments that use continuouswave laser sources. In this case, the optical signal can be approximated by a ﬁeld of angular frequency ω = 2πc/λ, where λ is the vacuum wavelength, so that Eopt = Eω exp(−iωt). The macroscopic wave equation for Eω is

∇ 2 Eω − ∇(∇ · Eω ) = −ω2 μ0 Eω ,

(8)

where μ0 is the magnetic permeability in vacuum. For most conditions, the second term in Eq. (8) turns out to be negligible. To understand this, consider the Maxwell equation

∇ · Dω = ρω /0 ,

(9)

where ρω is the ω frequency component of the charge density ρ. Now, consider that the dielectric only hosts free-charges that are produced in the photorefractive process. These separate in a time scale τd , which is such that τd 1/ω. Hence, the only contributions to ρω are polarization charges. This implies that

0 = ∇ · Dω = ∇(ω : Eω ) = ∂i (ij E(ω)j ),

(10)

where Dω = ω Eω , and the subscript ω is suppressed for the components of . Using the principal axes (i.e., ij = δij i ), and assuming that these axes are slowly dependent on position (see below), Eq. (10) becomes

E(ω)i ∂i i + i ∂i E(ω)i = 0.

(11)

From Eq. (11), we see that the order of magnitude of ∇ · Eω ≡ ∂i Ei must be ∼Eω /, where is the spatial scale of variation of ω . This variation, due to the inhomogeneity, is in turn related, in our description, to the low-frequency electric ﬁeld E, and hence with the transverse scale of the optical intensity distribution I (assuming that is larger than

166

Photorefractive Solitons and Their Underlying Nonlocal Physics

the intrinsic spatial scale of the charge migration process), and thus comparing the ﬁrst two terms of Eq. (8), we note that the ﬁrst scales with ∼(1/λ)2 and the second with ∼(1/λ)(1/). Insomuch that λ ( is typically the beam width), Eq. (8) is well approximated by the Helmholtz equation

∇ 2 Eω + k02 n2 Eω = 0,

(12)

where k0 = ω/c, and n2 is a tensor that, because of the electro-optic effect, is also point-dependent, i.e., n2 = n2 (r, t). It is now clear that, for Eq. (12) to be valid, the principal axes of (and hence of n2 ) must not appreciably change on scales of the order of . We now cast the Helmholtz equation [Eq. (12)] in a more tractable form by writing Eω = Ax (x, y, z) exp(ik1 z)uˆ x + Ay (x, y, z) exp(ik2 z)uˆ y , where k1 = k0 n1 and k2 = k0 n2 , with n2i = i /0 (i = 1, 2), i.e., the principal dielectric eigenvalues. We obtain the set of equations 2 ∇⊥ Ax + 2ik1

xy ∂Ax xx = −k02 Ax − k02 Ay ei(k2 −k1 )z , ∂z 0 0

(13)

2 Ay + 2ik2 ∇⊥

∂Ay yx yy = −k02 Ax ei(k1 −k2 )z − k02 Ay , ∂z 0 0

(14)

where ∇⊥ = (∂xx + ∂yy ). These are equivalent to Eq. (12) under conditions in which the terms ∂zz Ax and ∂zz Ay have been neglected, which is justiﬁed when the scale of variation of Ax and Ay in the z-direction is much larger than λ (slowly varying approximation), as occurs when the beam is conﬁned in the transverse x, y plane to scales λ (paraxial propagation). The terms ij are the components of the electro-optic perturbation to , i.e., = E=0 + , with respect to the principal axes of the unperturbed E=0 . In terms of the commonly used phenomenological relationships of Eq. (7), we note that 0 = −(1/n2 ), which with respect to the principal axes is such that

ij =

−0 n2i δip

1 n2

pq

δqj n2j

=

−0 n2i n2j

1 n2

.

(15)

ij

The nonlinear propagation can then be described by (i) determining the low-frequency electric ﬁeld E produced by the optical intensity I = |A|2 = |Ax |2 + |Ay |2 [Eq. (6)], (ii) determining the resulting local index of refraction modulation using Eq. (7), and (iii) inserting this modulation through Eq. (15) into the parabolic equations Eqs. (13,14).

Intrinsic Scales and a Condition for a Local Kerr-Like Soliton

167

5. INTRINSIC SCALES AND A CONDITION FOR A LOCAL KERR-LIKE SOLITON 5.1. Saturation Length and a Spatially Local Response To begin with, we consider those conditions in which, in the 1D case, the space–time nonlocality effects can be brought to play a somewhat lesser role, to the extent that the possibility of enucleating an approximate local (Kerr-like) model can be pursued [for example, the cases analyzed in Segev, Shih, and Valley (1996)]. When the background illumination Ib co-propagating with the soliton is of an intensity comparable to its peak intensity Ip , for an appropriate choice of soliton width x [soliton fullwidth-at-half-maximum (FWHM)] and applied external bias ﬁeld, beam dynamics is found to reach a soliton-like steady-state conﬁguration after an initial transient (Segev et al., 1994b). In this case, we self-consistently consider the steady-state 1D version of Eq. (6)

⎡ d⎢ ⎣E(I + Ib ) dx

1− 1+

d dx ((0)1 E) αNa q d ((0) 1 E) dx

Na q

⎛ +

kB T d ⎜ ⎝(I + Ib ) q dx

1− 1+

d dx ((0)1 E) αNa q d ((0) 1 E) dx

⎞⎤ ⎟⎥ ⎠⎦ = 0,

Na q

(16) where x is the single transverse direction on which the optical intensity I depends, parallel to the external applied ﬁeld direction and to the optical polarization. In this expression, we have considered the fact that to self-consistently describe soliton propagation, we expect to have a limited dependence of I on the propagation variable z, an approximate symmetry with respect to translation along z that implies that E can only have one transverse component along the x axis, i.e., E = (E(x), 0, 0). Furthermore, we indicate with I = I(x) the intensity of the soliton beam, whereas Ib is the sum of β/s and the background intensity. As a consequence, the vector (0)E gives ((0)1 E, 0, 0), where (0)1 is the principal component along the optical axis 1 = x. Equation (16) is a simpliﬁed form of Eq. (6), but it still involves spatial derivatives. For example, the term proportional to kB T/q is the result of charge diffusion, as described in Eq. (2), which, by its very nature, is a spatially nonlocal process. Evidently, insomuch that this contribution plays a predominant role, no local relationship between the nonlinear n(E) and I(x) can arise (such as the basic Kerr relationship n = nb + n2 I 2 , where nb is the unperturbed index of refraction and n2 is a material-dependent constant), since the relationship between E(x) and I(x) is nonlocal. The same can be said of the terms (d/dx)((0)1 E)/αNa q and (d/dx)((0)1 E)/Na q. These last contributions are sometimes referred to as due to charge displacement and saturation. In fact, in the photorefractive process, charges can be made to optically migrate from an illuminated region to a dark one

168

Photorefractive Solitons and Their Underlying Nonlocal Physics

and remain displaced in the measure in which the migrated electrons can recombine with the thermally ionized donor sites Nd+ . The concentration of these is Na , since in the absence of light all the acceptors have stripped their neighboring donors. Consequently, the concentration of stably displaced charges ρ = (d/dx)((0)1 E) in a given portion of the crystal is limited by the quantity qNa . In cases in which the photogenerated charge diplaced in the formation of a soliton is not sufﬁcient to reach this saturation limit, that is, the optical intensity distribution I produces a concentration of photoexcited electrons N Na , the term (d/dx)((0)1 E)/Na q will amount to a small correction to effects, and (d/dx)((0)1 E)/αNa q, being that α 1, can be safely neglected. Thus, Eq. (16) becomes

E(Ib + I)

1 1+

(0)1 dE Na q dx

⎛ ⎞ kB T d ⎝ 1 ⎠ = g, (Ib + I) + q dx 1 + (0)1 dE

(17)

Na q dx

where g is a constant proportional to the steady-state current density and related to the boundary conditions, which can be treated perturbatively. The ﬁrst step is to compare the single terms to introduce normalized variables for I, E, and x. A natural choice for I is to introduce Q = (Ib + I)/Ib . Since solitons typically emerge when the crystal is biased, a suitable normalization for E is to consider Y = E/E0 , where E0 is the amplitude of the bias electric ﬁeld in the region in which the soliton is forming. For example, if a pair of plane electrodes in the x-direction L apart deliver the voltage V L to the crystal, 0 Edx = V. If L , that is, if the transverse region in which the light-induced charges that render E = E0 is small, E0 V/L. As regards to the spatial scale for x, it could be normalized to , but since this scale is in fact not predetermined by the experimental conditions but an effect of the nonlinear process, a different scale is desired. Having introduced Y = E/E0 , the terms (1 + (d/dx)((0)1 E)/Na q) suggest the introduction of ξ = x/xq = x/[(0)1 E0 /(Na q)]. This scale xq is therefore a natural scale of the migration process: the saturation length. In these terms Eq. (17) becomes

Q Q YQ +a − Y = G, 1 + Y 1 + Y (1 + Y )2

(18)

with a = Na kB T/(0)1 E20 and G = g/E0 Ib . The prime stands for (d/dξ).

5.2. A Spatially Local Kerr-Like Effect As can be understood, this scale implies a negligible inﬂuence of spatial nonlocality when the characteristic beam scale is xq . In this condition,

Intrinsic Scales and a Condition for a Local Kerr-Like Soliton

169

we can follow a perturbative approach (DelRe et al., 1998a): Eq. (18) can be formally rendered explicit

Y=

Q GY Y G −a + +a . Q Q Q 1 + Y

(19)

Now the local (ﬁrst term of RHS) and nonlocal contributions are distinct. A local Kerr-like nonlinearity will emerge when these nonlocal terms are comparatively smaller than the ﬁrst term. To establish when this is possible, consider that spatial derivatives scale with the parameter η = xq /, and this is the sole governing parameter since a ∼ 1, as can be readily estimated for a typical experimental apparatus using SBN, with (0)1 ∼ 103 0 , Na ∼ 1 × 1022 m3 , and reasonably accessible values of E0 ∼ 1 kV/cm (Kos et al., 1996). With these values, xq ∼ 0.1 μm, and if we plan to trap into a soliton a beam with an intensity FWHM ∼ x ∼ 10 μm, η = xq /x ∼ 0.01 represents a smallness parameter so that

Y (0) =

G + o(η). Q

(20)

This ﬁrst dominant term is a local term, in the sense that the ﬁeld at a given location Y(ξ) depends on the optical intensity only at that (same) location Q(ξ). When the bias ﬁeld is delivered so that L , G 1 (Segev, Shih, and Valley, 1996), and hence Y (0) = 1/Q, or E(x) E0 / (1 + I(x)/Ib ). To formulate the corresponding nonlinearity, we must insert this ﬁeld in the electro-optic response of Eq. (7). The beam is propagating along a principal axis of |E=0 , with its polarization parallel to the crystal optical axis. Effects are described by the Eqs. (13) and (14). Specializing to the case of ferroelectric SBN, xy = −0 n21 n22 rxyx E(x) = 0 on consequence of its 4 mm tetragonal symmetry that implies that rxyx = 0. The two propagation equations are uncoupled. If the input beam has Ay (z = 0) = 0, the process can be described by the single equation, Eq. (13), with xx = −0 n4x (1/n2 )xx and (1/n2 )xx = rxxx E(x) = r33 E(x), in terms of the contracted indices used in literature [for example, in Yariv (1989)]. The quadratic electro-optic effect described by the tensor sijkl has been neglected, as is wholly justiﬁed in the ferroelectric phase (4 mm) [see Yariv and Yeh (2003) for a detailed description of the electro-optic effect]. Propagation is then described by (setting Ax = A)

∂xx A + 2ik1

∂A xx = −k02 A ∂z 0

(21)

170

Photorefractive Solitons and Their Underlying Nonlocal Physics

that is written in the standard form with n(I) as

ik1 ∂A(x, z) i ∂2 A(x, z) = n(I)A(x, z), − ∂z 2k1 ∂x2 n1

(22)

1 E0 n(I) = − n31 r33 . 2 1 + I(x)

(23)

where

Ib

In cases in which I(x) Ib , n(I) −(1/2)n31 r33 E0 + (1/2)(n31 r33 E0 /Ib )I so that the effect is, apart from a constant shift, a Kerr self-focusing effect that supports a wide variety of soliton effects and phenomenology. Analogously, we can formulate the nonlinearity for crystals that have a quadratic electro-optic response, such as paraelectrics (for example, KLTN) (DelRe et al., 1998b; Segev and Agranat, 1997). In these, no axis of spontaneous polarization exists, and the linear electro-optic tensor is zero (rijk = 0). Again, for a beam that is propagating along a principal axis of |E=0 , with its polarization parallel to one of the three crystal axes, only one of the two parabolic equations, Eqs. (13) and (14), intervenes, and xx = −0 n4x (1/n2 )xx , where nx = ny because of the absence of birefringence without an electric bias ﬁeld. Hence, (1/n2 )xx = sxxx (E(x))2 = s11 (E(x))2 . Appreciable quadratic electro-optic effects are normally observed only slightly above the Curie temperature, and in these conditions, the lowfrequency susceptibility is also temperature-dependent. For this reason, it is more convenient to make use of the phenomenological expansion of Eq. (7) in terms of the low-frequency polarization vector P as opposed to the static electric ﬁeld E, i.e.,

1 2 n

= gijkl Pk Pl ,

(24)

ij

where gijkl are the phenomenological constants of the quadratic electrooptic tensor in terms of the components of P. In distinction to the sijkl , these are now approximately temperature-independent, except for temperatures in the critical region. For noncritical conditions, since the unbiased sample is centrosymmetric and dielectrically linear, P = 0 ((0)r − 1)E. The resulting saturated Kerr-like nonlinearity is therefore

E20

1 n(I) = − n31 g11 20 (0)2r 2

1+

I(x) Ib

2 ,

where g11 = gxxxx , n1 = nxx = nyy = nzz , and thus k1 = k2 = k3 .

(25)

Intrinsic Scales and a Condition for a Local Kerr-Like Soliton

171

5.3. The Dielectric Response Time Analyzing the nonlinearity in Eq. (23), we note that the strength of the response depends only on the ratio I(x)/Ib , where Ib is not a materialdependent quantity, but can be artiﬁcially tailored through the background illumination. This appears unusual, unless we recall that Eq. (23) is only valid at steady-state. To appreciate this, consider that the nonlinear propagation described by Eqs. (22) and (23) is invariant for A → fA and Ib → f 2 Ib (being that I = |A|2 ) so that if both soliton beam and background illumination are obtained from the same laser source through a beam splitter (as is commonly done), and its intensity (but not mode structure) varies in time, it will not in anyway affect the soliton. As for the understanding of any steady-state and its underlying dynamical equilibrium, we must consider the nature of the transient leading to it (Fressengeas, Maufoy, and Kugel, 1996; Fressengeas et al., 1997; Maufoy et al., 1999; Wolfersberger et al., 2003). To this end, we reconsider the full version of the space–charge formation model [Eq. (6)]. For the understanding of the spatial local process of Eq. (23), we can proceed along the lines leading to Eqs. (19) and (20). Evidently, as for the natural scales of x, E, and I in Eq. (16), Eq. (6) has a natural time scale τd = (0)γ/(qμsαIb ), a scale that is sometimes referred to as the dielectric time constant. Hence, normalizing the time parameter τ = t/τd , the time-depedent version of Eq. (20) (truncated at zero order in η) reads (DelRe and Palange, 2006)

∂Y (0) (ξ, τ) + Q(ξ, τ)Y (0) (ξ, τ) = G, ∂τ

(26)

and no time dependence in the boundary conditions is considered (G = 1) [such as in (Tosi-Belefﬁ et al., 2000)]. It is understood that we have in this manner assumed that the conditions in which spatial nonlocal terms can be neglected are not appreciably affected and altered by the time dynamics. The corresponding time dependence is not at all trivial. In particular, even though the natural time scale τd (associated with charge mobility) is identiﬁed, the process does not generally manifest a single overall time constant (Dari-Salisburgo, DelRe, and Palange, 2003) (see Figure 1). This would not even be true under conditions in which Q is approximately independent on τ, since it depends on ξ. One simpliﬁed scale that has some use is, from Eq. (26), τs = 1/Qmax , i.e., ts τd Ib /Ip , which represents the shortest time scale available. As it turns out, this is the scale on which typically the strongest self-focusing is observed (Dari-Salisburgo, DelRe, and Palange, 2003). In turn, for the formation of solitons, the spatial and time dependence of Q is an essential feature, and this implies that the accumulation process has a spectrum of different time scales that changes with Q as the beam settles into its steady-state conﬁguration. This complicated situation

172

Photorefractive Solitons and Their Underlying Nonlocal Physics

Normalized FWHM Dxout /Dxin

1.4

(a)

1.2 1.0

4.0

(b)

3.0 2.0 1.0 0.0

0

1

2

3

4

5

Normalized time t /␶s

FIGURE 1 Soliton formation for weak (a) and strong (b) z dynamics in the collapsing stage. The continuous lines are the exponential (a) and stretched exponential fit β

e−(τ/τs ) , with β = 0.89 (b) [see Dari-Salisburgo, DelRe, and Palange (2003)].

is what is generally termed “time-nonlocality” as is best appreciated in the formally equivalent integral version of Eq. (26) τ − Qdτ

Y (0) = Ge

0

⎛ ⎜ ⎝1 +

τ

τ

dτ e0

⎞ Qdτ

⎟ ⎠.

(27)

0

Here, the presence of time integrals indicates the dependence of Y at a given time τ on previous values of Q at τ < τ, and hence, through the propagation coupling of Y and Q afforded by Eqs. (13) and (14), to previous values of Y itself. The full complexity of this behavior emerges during transients, i.e., when Q changes in an appreciable manner with time, for example, during the ﬁrst stages of evolution, during which the beam passes from a diffracting distribution to a self-guided soliton. It is now evident how the apparently peculiar independence of phenomenology on the single intensity levels is an artifact of the steady-state, since during the transients [Eq. (27)], the electric ﬁeld Y depends not only on Q but also on τ, which contains through τd an explicit dependence on Ib (and not on I/Ib ). This is associated with the fact that, as can be appreciated directly from the form of Eq. (6) [the time term is not proportional to (I + Ib )], the rate of change of the charge density depends on the local absolute value of optical intensity, and is ultimately associated with an intensity independent characteristic of the system, the electron mobility μ.

Intrinsic Scales and a Condition for a Local Kerr-Like Soliton

173

Moreover, this dependence on the actual value of the optical intensity is lost when τ no longer intervenes in determining Y. For example, at steadystate ∂τ Y (0) = 0 so that Eq. (20) holds, where only the relative intensity I/Ib intervenes. Clearly, this peculiar dependence on only the relative intensity I/Ib is broken when the system is perturbed, since a change for example in Q will rearrange charge and hence introduce effects associated to τ and μ.

5.4. A Time-Averaging Effect The nonlocal time-nature of the process suggests that the nonlinearity described by Eqs. (22) and (23) could be extended to the average values of Q (that is, Q = 1 + I/Ib ) for optical beams that have a random or deterministic dependence on time with a short enough characteristic scale (coherence time or period) τc , insomuch that Q is independent on τtime. This seems compatible with the presence of the cumulative terms 0 Qdτ in Eq. (27), but the fact that exponential functions are involved requires a slightly more detailed analysis. We can rewrite Q = Q + Q so that Eq. (27) becomes −Qτ−

Y (0) = Ge

τ

⎛ Qdτ

0

⎜ ⎝1 +

τ

dτ e

Qτ +

τ 0

0 τ

= Ge

−Qτ− Qdτ 0

τ +G

⎞ Qdτ

⎛ ⎝eQ[τ −τ] e

⎟ ⎠

⎞ τ − Qdτ τ

⎠dτ .

(28)

0

If Q ﬂuctuates around zero with an amplitude ∼Q and has a time ﬂuctuation scale τc 1/Qmax , we have

Y

(0)

−Qτ

= Ge

+G

τ 0

G eQ[τ −τ] dτ = Ge−Qτ + 1 − e−Qτ Q

(29)

so that, for τ 1/Q, Y 0 = G/Q is time-independent, and hence Q is also time-independent which validates our initial hypothesis of a timeindependent Q. Speciﬁc examples of this response are the so-called incoherent solitons, the emblematic case being the photorefractive self-trapping of light from an incandescent bulb (Mitchell and Segev, 1997).

5.5. Cumulative Response and Imprinting In considering the transient beam effects, we note that the nonlinear propagation described by Eq. (22) is parametrically dependent on time, since the

174

Photorefractive Solitons and Their Underlying Nonlocal Physics

scales of variation of the space–charge ﬁeld E (that is, of Y) are much longer than the time of ﬂight within the sample. So, as time passes, accordingly Eq. (27) changes, and along with it the nonlinearity too changes with the light obeying the time-dependent nonlinear 1D version of Eq. (22),

i ∂2 A(x, z, t) ik1 ∂A(x, z, t) − = n(I(x, z, t))A(x, z, t). 2 ∂z 2k1 n1 ∂x

(30)

A tacit but extremely important consequence of this cumulation is that, if the optical intensities I and Ib are switched off at a given instant, say t∗ , the build-up process is halted, insomuch that the natural thermal conductivity represented by the dark illumination β/s is negligible. This means that in many crystals photorefraction can be considered an “imprinting” technique, and the crystal maintains a permanent memory of previous nonlinear processes through Eq. (27) (D’Ercole et al., 2004). More generally, if the beams leading to a given accumulation of response are attenuated by a factor f , all the time scales are increased to τs = f τs so that for a given time interval there exists a large enough value of f for which the system effectively supports propagation without any further appreciable change in the index pattern. This implies an approximately linear guiding effect in the pattern generated by previous nonlinear propagation. This signature of time-nonlocality is even more evident when we consider that photorefractive time dynamics depends on the absorption cross-section, associated to the parameter s, since τd = (0)γ/(qμsαIb ). Now absorption is wavelength-dependent so that in a typical doped ferroelectric crystal, a visible λ1 will have s(λ1 ) s(λ2 ) if λ2 is in the near infrared region (>800 nm). In this case, even having the system support propagation of intense optical beams, the low value of s makes the corresponding τd so large that no dynamics is actually observed. Insomuch that the electrooptic effect is only weakly dependent on λ, the photorefractive system acts like a passive optical component (Lan, Shih, and Segev, 1997).

5.6. Quasi-Steady-State Solitons: A Signature of Time Nonlocality The cumulative nature itself forwards speciﬁc nonlinear phenomena, the principal of these being the so-called quasi-steady-state solitons that occur for intervals of time τ ∼ 1/Qmax , and under conditions of Q 1 for all values ξ except for the very distant tails (Ip Ib ) (Fressengeas et al., 1998). In this case, the second term in the parenthesis of Eq. (27) is negligible, 1/Q since it amounts to 0 dτ eQτ ∼ (1/Q) 1. For these times, the material τ response is Y (0) Ge− 0 Qdτ , which, in a manner different from the Kerrlike response G/Q of Eq. (20), is, at each given instant, also self-focusing (DelRe and Palange, 2006). We thus expect transient self-trapping and solitons that are not observable in time-local systems, and these are, as

Intrinsic Scales and a Condition for a Local Kerr-Like Soliton

175

mentioned, the quasi-steady-state solitons. Far from being a peculiarity, they are a well-known manifestation of photorefraction. Even though in this present discussion we concentrate on 1D solitons, they are observed also as 2D nonlinear beams, in various crystal types and structural phases, as bright and dark, single component and incoherent solitons, and form the building block of many applications. One reason for their usefulness is the condition Ip Ib , which basically means that no artiﬁcial background illumination is used at all. This at once simpliﬁes the experimental apparatus, and also permits the sequential imprinting of different soliton patterns in different parts of the sample, since the use of an artiﬁcial Ib illuminates all regions and hence erases the previously written ones, unless all guiding patterns are “written” simultaneously [as in Lan et al. (1999)]. To reach an understanding of quasi-steady-state solitons, we begin by assuming self-consistently that for a given interval τc < τ < τp the normalized beam intensity Q becomes approximately time-independent and factors out of the integrals of Eq. (27) (typically τc ∼ τs ). This is suggested by experiments, which consistently lead to a “soliton time plateau” during which optical beam evolution is approximately absent, and the duration of this plateau is τp − τc τc so that the complicated contribution to the integrals associated with the initial transient phase (where Q is inherently time-dependent) can be neglected. The result is that Eq. (27) can be approximated by

Y (0) e−Qτ + 1/Q − (1/Q)e−Qτ → Y (0) e−Qτ ,

(31)

the second form being valid because of the condition Q 1, true except for the very tails of the beam shape and for time scales involved τ such that τ ln Q/Q (DelRe and Palange, 2006). To formulate the basic nonlinear propagation equation, we note that the slowly varying part of the optical ﬁeld A [i.e., I(x, z, t) = |A|2 ] obeys the parabolic wave equation, Eq. (30). For the example recalled above of a ferroelectric sample of SBN, n = −(1/2)n31 r33 E, and deﬁning n0 = −(1/2)n31 r33 E0 , Eq. (31) implies that n = n0 Y (0) = n0 e−Qτ . For paraelectrics, such as room temperature KLTN, the electro-optic response is quadratic in the electric ﬁeld, n = −(1/2)n31 g11 20 ((0)r − 1)2 E2 , where g11 is the effective quadratic electro-optic effect, and Eq. (31) implies that n = n0 (Y)2 = n0 e−2Qτ , having deﬁned n0 = −(1/2)n31 g11 20 ((0)r − 1)2 E20 . For both cases, the single form

n = n0 e−mQτ

(32)

holds, with m = 1 (2) for ferroelectrics (paraelectrics). The meaning of this time-changing nonlinearity is not straightforward. To grasp some of its fundamental features, we can proceed in the following

176

Photorefractive Solitons and Their Underlying Nonlocal Physics

manner: we take a given instant of time τ for which the assumptions leading to Eq. (32) hold. We then analyze the corresponding soliton by considering solutions of the form A(x, z) = u(x)eiz (Ib )1/2 , where is a constant (the soliton propagation constant). These are those sub-classes of solutions of the nonlinear wave equation of Eq. (30) that have a z-invariant intensity distribution I = |A|2 , and since a ﬁxed τ is chosen, it is intended that all the variables also depend parametrically on τ itself. Imposing this condition, inserting Eq. (32) into Eq. (30), we obtain the nonlinear equation

[u + (1/2k1 )∂xx u] = −(k1 /n1 )n0 e−mu τ u, 2

(33)

where the approximation Q I/Ib is implemented (Q 1) . We then generalize the normally used self-consistent approach (Segev, Shih, and Valley, ˜ 1996) by changing wave variable in Eq. (33) from u(x) to w(ξ) = (mτ)1/2 u(ξ), where ξ = x/d is the transverse coordinate normalized to the nonlinear length d = (−2k1 b)−1/2 , and b = (k1 /n1 )n0 . The result is

2 d2 w(ξ)/dξ 2 = − γ − e−w w(ξ),

(34)

where γ = n1 /(n0 k1 ) is the normalized propagation constant. To relate γ to the relevant experimental parameters, we note that Eq. (34) can be integrated once through quadrature leading to the algebraic relationship

p2 − p20 = −γw2 + γw02 − e−w + e−w0 , 2

2

(35)

1/2 where p = dw/dξ, p0 = (dw/dξ)ξ=0 , w0 = w(ξ = 0) = u(ξ = 0)(mτ) = 1/2 u0 (mτ) (u0 = Ip /Ib ). To predict bright solitons, that is optical beams that have a bell-shaped amplitude and intensity distribution in the transverse ˜ ˜ →0 plane, we must impose that (du/dξ) ξ=0 = 0, i.e., p0 = 0, and that u(ξ) ˜ and du/dξ → 0 for ξ → ∞ , which implies that w(ξ) → 0 and dw/dξ → 0 for ξ → ∞ for any ﬁnite value of τ (and even for τ → ∞ for a polynomial 2 or exponential tail shape). This leads to γ = (1 − e−w0 )w0−2 and the soliton proﬁle equation

2 2 d2 w(ξ)/dξ 2 = − (1 − e−w0 )w0−2 − e−w w(ξ),

(36)

characterized by a spatially local exponential nonlinearity. This nonlinearity is intrinsically a result of time nonlocality. It only emerges during the transient. The calculation leads to a family of soliton solutions that are characterized by their so-called “existence conditions” (Segev, Shih, and Valley, 1996), i.e., that subset of observable features that distinguish

Observable Effects of Spatial Nonlocality in 1D Solitons

177

them from other (nonsolitonic) solutions. Typically, the parameters are the normalized peak intensity u(0) = u0 and the normalized intensity FWHM ξ, and the resulting single-valued curve ξ = ξ(u20 ) forms the “existence curve.” In the present case, this approach can be generalized in terms of the parameters w0 = u(ξ = 0)(mτ)1/2 = u0 (mτ)1/2 and ξ (i.e., the normalized soliton intensity FWHM ξ = x/d), which contains time as an implicit parameter. In this manner, Eq. (30) with Eq. (32) is said to give rise to a generalized set of nonlinear Schroedinger equations. The relevance of this solitonic picture to quasi-steady-state solitons is once again not straightforward. Evidently, the solution of Eq. (36) is different for different instants of time. Furthermore, the experimental situation is such that the input launch beam and hence waveform u(ξ) is ﬁxed. In turn, the analysis holds self-consistently under conditions in which the waveform u determined by the soliton equation, Eq. (36), turns out to be on its own account approximately independent of time. This occurs only during the aforementioned plateau. One parameter that can be explicitly evaluated, because it occurs speciﬁcally during the plateau, is the maximum value of nonlinear self-action and, correspondingly, the minimum value of xmin , which turns out to be

xmin =

ξmin λ 2πn21 am

−m/2

E0

,

(37)

where ξmin 3.07, a1 = (r33 )1/2 , and a2 = 0 (0)r (g11 )1/2 . Comparison of this model to experiments in photorefractive KLTN indicates an overall agreement with Eq. (37) (DelRe and Palange, 2006). Equation (37) does not contain u0 , Ib , or Ip . More radically, Eq. (36), which dictates the salient physical features of the self-trappping process, is invariant for transformations of the type Tf (t, I) → ( ft, f −1 I), I being the intensity of the sole soliton beam (this behavior is not true for screening solitons [see Eq. (23)]. Furthermore, the analysis indicates that what we term a transient or quasi-steady-state soliton is driven by underlying sequence in time of different soliton-supporting mechanisms. These features, absent in any time-local mechanism, once again testify to the fundamentally peculiar nature of the photorefractive nonlinearity.

6. OBSERVABLE EFFECTS OF SPATIAL NONLOCALITY IN 1D SOLITONS 6.1. Physical Origin It is charge migration that transfers to the photorefractive nonlinearity both time and spatial nonlocality. In particular, electron charges will

178

Photorefractive Solitons and Their Underlying Nonlocal Physics

migrate not only as a consequence of being locally excited and feeling the external bias ﬁeld but also in response to the concentration of electrons in nearby regions, whether because these repel or attract them, or simply because of statistical diffusion, which accompanies any process that involves inhomogeneity and thermalization [Eq. (2)]. The best way to approach this nonlocality is to follow the steps leading to Eq. (20): consider ﬁrst the 1D process at steady-state, i.e., when the charge distribution does not further evolve in time. Here, spatial nonlocality can essentially be identiﬁed with the appearance of contributions to the nonlinear response, i.e., to the light-induced ﬁeld Y, that depend not only on Q but also on its derivatives Q , Q , and so on. The starting point is Eq. (19), where, however, we must take into account the corrections to the local solution, Eq. (20), that scale in the smallness parameter η. A ﬁrst correction is obtained by iterating the zero-order solution Y (0) = G/Q into Eq. (19), and the resulting expression for Y is

Y (1) =

Q Q G 2 G −a − + o(η2 ). Q Q Q Q

(38)

The second term, of ﬁrst order in η, involves a spatial derivative and is the principal contribution of the charge diffusion terms proportional to the temperature coefﬁcient a in Eq. (18). To this order in η, it can therefore be identiﬁed with the diffusion ﬁeld, i.e., that ﬁeld produced by the thermalization of the mobile charges generated by the spatially resolved optical ﬁeld. The third term of Eq. (38) is also of ﬁrst order in η, since each derivative scales with η itself. From a physical point of view, the origin of this nonlocal contribution is not diffusion, since, for example, temperature does not intervene (through the coefﬁcient a). Its origin can best be appreciated considering Eq. (16), where it is clearly associated with the actual density of the displaced charge dE/(0 (0)r dx), which justiﬁes its identiﬁcation with a “charge-displacement” contribution. The issue now is how these spatially nonlocal perturbations to the Kerrlike nonlinearity of Eq. (23) affect optical propagation and hence soliton formation.

6.2. Self-Bending The ﬁrst effect is that even though the ﬁrst order in η corrections are generally small in so much that η itself is small, they have one major implication: they alter the symmetry properties of the nonlinear response. For example, an optical beam with the property I(x) = I(−x), where x = 0, coincides with the propagation axis, in a Kerr-like nonlinearity where n(x) ∝ I(x), n(x) = n(−x). So, a Gaussian beam will undergo a symmetric focusing and not be deﬂected in any particular direction. This order of things clearly

Observable Effects of Spatial Nonlocality in 1D Solitons

179

changes when the terms of Eq. (38) are considered. The absence of symmetry can appear to forbid the formation of solitons. For example, an even intensity distribution at input associated with a fundamental Gaussian beam symmetrically spreads because of diffraction, a spreading that can be balanced by the symmetric contribution to the nonlinear index of refraction. Seen differently, the self-focusing due to photorefraction is balanced by natural diffraction. Now, what diffractive effect present in the initial Gaussian beam can possibly counter an asymmetric self-focusing? Evidently, the distortion, although perturbative, will generally accumulate in a long propagation and considerably alter the Kerr-saturated picture (DelRe, D’Ercole, and Palange, 2005). In many cases, it leads to the process termed self-bending (Carvalho, Singh, and Christodoulides, 1995; Christodoulides and Carvalho, 1994). To address the issue, we consider the electro-optic response of a ferroelectric crystal in the conditions leading to Eqs. (22) and (23), i.e., for a linear electro-optic effect with, instead of the E resulting from Y (0) [i.e., Eq. (20)], the approximate solution Y (1) as given in Eq. (38). The resulting nonlinear propagation equation describing paraxial optical 1D phenomenology is then (DelRe, D’Ercole, and Palange, 2005)

i

∂w ∂2 w w |w|∂|w|/∂ξ |w|∂|w|/∂ξ + 2 − +α w+β w = 0, 2 2 ∂ζ ∂ξ 1 + |w| 1 + |w| (1 + |w|2 )3

(39)

√ the wave being expressed in terms of w(ξ, ζ), with A = w Ib , ξ = x/x0 , and ζ = z/z0 . Here x0 = 1/ 2k1 /z0 , z0 = n1 /(k1 n0 ), k1 = 2πn1 /λ, and n0 = (1/2)n31 r33 E0 , with λ being the optical wavelength. The nonlocal parameters are α = aβ = 2kb T/(qE0 x0 ) and β = 2xq /x0 (note that here α and β are not those used elsewhere). It is not generally possible to analyze Eq. (39) analytically, but a numerical investigation reveals (DelRe, D’Ercole, and Palange, 2005) that for conditions in which the asymmetric contributions are able to accumulate to observable effects, for low values of u0 , where u20 = Ip /Ib , solitons will simply not form for any value of external bias E0 (and hence n0 ), and considerable nonsoliton effects predicted typical of Airy waves appear (Christodoulides and Coskun, 1996). These effects are characterized, as expected, by an asymmetric scattering of light in the transverse plane, a scattering that depends on the actual values of α and β. In this case, the nonlocality cannot be reconciled with soliton formation. However, as saturation sets in for larger values of u0 (u0 ≥1), a regime that best reﬂects the greater part of experiments, self-trapping is predicted for speciﬁc values of external bias, as for standard 1D solitons in which the spatial nonlocality is made to play a negligible role (short propagation conditions). Here, even though an asymmetric contribution is playing its part, yet the localized wave forms without dissolving through radiation.

180

Photorefractive Solitons and Their Underlying Nonlocal Physics

The signature of the asymmetric response is not in a distortion of the soliton, but in the fact that instead of propagating along the z-direction, it now follows a parabolic like trajectory, the phenomenon known as soliton self-bending (Carvalho, Singh, and Christodoulides, 1995). The intriguing circumstance associated with this perturbative effect of nonlocality appears when we ﬁnd the existence conditions of the self-trapped bending beams, as previously discussed in the case of quasisteady-state solitons. In saturated conditions, the corresponding existence curve in the (u0 ,ξ) plane hardly varies changing the propagation length Lz and is almost unvaried with respect to the existence curve of the local treatment for which α and β are artiﬁcially set to zero [Eq. (23)]. The point is that this does not occur because the nonlocal components play no role, since these make the beam follow a parabolic trajectory of curvature B = xs /L2z , with xs being the beam lateral shift at the output of the beam. To understand this behavior, we follow the established solitonic analysis by self-consistently imposing a symmetry to the system on consequence of propagation invariance, reducing the nonlinear Eq. (39) to an ordinary differential equation. The system has a built-in symmetry that allows the reduction in an accelerated frame: searching for soli transverse tons of the type w(ξ, ζ) = u() exp −iζ (b2 /6)ζ 2 − (b/2)ξ + q , where u is real, = ξ − (b/2)ζ 2 , and q and b are matching parameters (which have nothing to do with the symbols used elsewhere), we ﬁnd the ordinary nonlinear equation

u, + qu − bu −

u u2 u u2 u + α + β = 0, 1 + u2 1 + u2 (1 + u2 )3

(40)

where subscripts identify derivatives. Aﬁrst straightforward observation is that the transformation introduces a characteristic linear transverse phase chirp −bu. Considering, for the moment, only charge diffusion (i.e., β = 0), the nonlocal term, for the saturated (i.e., u0 1) regime, can be approximated to αu2 u /(1 + u2 ) αu . Apart from beam tails, the soliton bell-shaped structure is a quasi-Gaussian and u u0 exp(−2 /20 ) (0 deriving from x), with u −(2/20 )u. Approximations are warranted by the fact that the term is an o(η). Thus, Eq. (40) manifests a property which results particularly enlightening. The nonlocal part gives approximately itself a linear phase chirp: exactly the result of the symmetry transformation. Thus, if we choose to describe the beam in the system in which b = 4α/20 , the nonlinear propagation equation reduces to

u, + qu −

u = 0, 1 + u2

(41)

181

Observable Effects of Spatial Nonlocality in 1D Solitons

which is none other than the screening equation obtained by fully neglecting the nonlocal components, as in Eqs. (20–23). This at once forwards an explanation to the existence-condition invariance and suggests that the soliton should follow that exact trajectory with b = 4α/20 , as has been found true (DelRe, D’Ercole, and Palange, 2005). The leading term in the new description of the self-bent beam supports a soliton, with its propagation invariance and its particle-like manifestations. Moreover, it is the very same soliton that would appear in the absence of nonlocal processes, with the very same existence conditions, albeit along a parabolic trajectory. Because of the nonlocal perturbation, solitons emerge under conditions in which the full nonlinearity is actually local, and this occurs on the trajectory that allows for this circumstance, i.e., that for which the system of reference is transformed so as to allow for a saturated Kerr manifestation. In this, we identify the intrinsically “local” nature of photorefractive solitons under conditions in which the spatial nonlocality is simply a perturbation. Considering the general case of β = 0 the existence conditions remain those of the local Kerr-saturated case and the trajectory of the soliton is again parabolic [see Figure 2]. In distinction to the previous contribution to spatial nonlocality, however, the charge displacement term βu2 u / (1 + u2 )3 does not directly share the same scaling as the diffusive one, and (b)

(a)

8

B(m⫺1)

⌬␰

12

4

0

1.2

0.8

0.4

0.0 0

2

4

6

8

10

0

2

u0 (d)

(c)

4

6

8

10

u0 u0 ⫽ 1.4

(e)

u0 ⫽ 3

(f)

u0 ⫽ 10

z

FIGURE 2 Numerical simulations of 1D soliton self-bending under conditions in which charge displacement plays a role (i.e., both α = 0 and β = 0). (a) The predicted existence conditions that remain on the existence curve of the local model with α = β = 0; (b) the parabolic curvature parameter B; evidence of self-trapping to a 7-μm soliton along the Lz = 9.6 −mm propagation of the originally diffracting beam (c), for various values of u0 (d–f) [see (DelRe, D’Ercole, and Palange, 2005).]

182

Photorefractive Solitons and Their Underlying Nonlocal Physics

although it once again introduces a chirp, it is distorted by the 1/Q2 factor, a fact that does not allow a straightforward analytical interpretation.

6.3. Nonperturbative Effects Nonlocality can, in turn, radically change the soliton regime. A direct inspection of Eq. (18) indicates that a nonperturbative regime exists when the charge saturation term Y −1, this corresponding in general to the divergence of terms in Eq. (6) when ∇ · ((0)E)/Na q −1. In these conditions, we expect the whole picture of charge-separation at the basis of the Kerr-like screening model of Eq. (22) to radically change. The magnitude of Y scales self-consistently with the transverse scale , and these nonpertubative effects come into play when attempts are made to form miniaturized solitons, i.e., with widths of the order of only a few microns (DelRe, Ciattoni, and Palange, 2006). This regime, which turns out to be different from the standard soliton regime, is especially appealing since the guided mode would have an optimal overlap with standard modes delivered by optical ﬁbers, which are typically of several micrometers in width. More fundamentally, nonperturbative effects help to identify the ultimate limits to the transverse miniaturization. To analyze the situation, let us reconsider the full model, Eq. (6), but without making the substitution of Eq. (5) so that

kB T ˜ ∂E ˜ + EN + ∇ N = 0, ∇ · τd ∂t q

(42)

with

˜ =Q N

1− 1+

(0)∇·E αNa q (0)∇·E Na q

≡

γ N. Ib α

(43)

We have singled out Eq. (43) for the normalized conduction band electron density from Eq. (42), because the nonperturbative effects involve speciﬁcally the expression in Eq. (43). Equation (43) [and consequently Eq. (42)] becomes singular and unphysical when ∇ · E −qNa /(0). For a beam, this occurs when its transverse spatial scale is comparable with lim determined by the condition that the optically induced electric ﬁeld reaches, at some instant and in a given region, what is generally termed the saturation ﬁeld, E ∼ (Na q/(0))lim ≡ Esat . This typically implies, for a given steady-state soliton, an lim of the order of a few microns (note that when E ∼ E0 , lim = xq ). For example, in a ferroelectric SBN with (0) = 103 0 , Na = 2 × 1022 m−3 , n1 = 2.5, and a linear electro-optic

Observable Effects of Spatial Nonlocality in 1D Solitons

183

coefﬁcient r33 = 200 pm/V, lim ∼ 3 μm for a λ = 0.5 μm and an intensity ratio u20 = Ip /Id = (4)2 . Similarly, for a sample of KLTN with (0) = 104 0 and a quadratic electro-optic coefﬁcient g11 = 0.12 m2 C−4 , the limiting beam width in the same conditions for steady-state screening solitons is lim ∼ 6 μm. Experiments in this regime of ∼ lim indicate that, although in the ﬁrst stage of evolution the beam appears to undergo progressive self-focusing accompanied by self-bending, a phenomenology that is qualitatively analogous to transient self-trapping (DelRe, Ciattoni, and Palange, 2006), as conﬁnement sets in, the beam acquires a characteristic asymmetric feature that ultimately evolves into a lateral component that saps the principal beam wave and hampers the formation of a single bell-shaped, self-focused, and self-trapped propagation [see Figure 3]. To describe the effect, we proceeded to modify the approximate description by including those physical mechanisms absent in Eqs. (42) and (43) but present in the full nonlinear band-transport model of Eqs. (1–4), which prevent its breakdown for ∼ lim . Direct inspection of Eq. (43) indicates that the origin of the singularity is in having neglected a kind of “electron self-action.” More precisely, due to the large concentration of acceptors, one is brought to simplify the approach to the description of soliton response by neglecting the free-electron concentration N in the Gauss law Nd+ = N + Na 1 + ((0)1 /qNa )∇ · E and combining it with the photoionization equation to explicitly obtain N. This amounts to neglecting the effect of electron density on electron photoionization (self-action). Where ∇ · E −qNa /(0)1 , the acceptor impurity charge density is the principal contribution to the space charge density, since the ionized donor density Nd+ locally equals the free electron density N (and the two charge densities cancel out), a condition of charge saturation where electron density can evidently not be neglected. Taking into account this electron self-action ˜ yields an equation, which is structurally the same as Eq. (42), but with N replaced by

E 1 ˜ −Q − χ 1 + ∇ · N= 2α qNa

2

E E 1 Q+χ 1+∇ · , + 4χQ α − ∇ · + 2α qNa qNa (44) where χ = γNa /(sIb ), and is physically associated with the ratio between the dark electron density and the acceptor density. On this basis, i.e., on the modiﬁed approach of Eqs. (42) and (44), numerical results are able to describe beam evolution as observed in experiments.

184

Photorefractive Solitons and Their Underlying Nonlocal Physics

(b)

(a)

0

50 100 150 200 250 300

0

50 100 150 200 250 300

x (mm)

50 100 150 200 250 300

0

x (mm)

0

0

50 100 150 200 250 300

0

(h)

50 100 150 200 250 300

0

x (mm)

50 100 150 200 250 300

x (mm)

x (mm)

(k)

50 100 150 200 250 300

50 100 150 200 250 300

x (mm)

(g)

(j)

50 100 150 200 250 300

50 100 150 200 250 300

x (mm)

(i)

(d)

x (mm)

(f)

x (mm)

0

0

x (mm)

(e)

0

(c)

0

(l)

50 100 150 200 250 300

x (mm)

0

50 100 150 200 250 300

x (mm)

FIGURE 3 Observation of beam dynamics in the micron-sized regime where evidence of a more complex dynamical behavior is observed. Transverse intensity distribution (top) and profile (bottom) in normalized units of the (a) input 4.5 μm 1D Gaussian beam; (b) output after a 6 mm propagation along the z axis at t = 0; (c–l) after t = 10 min intervals [see (DelRe, Ciattoni, and Palange, 2006)].

One feature that is found is the absence of a distinct cutoff separating standard soliton phenomenology from nonsolitonic dynamics. The point is that the picture based on a critical lim is a clear-cut condition for the steady-state regime, but things get more involved during beam

Observable Effects of Spatial Nonlocality in 1D Solitons

185

transients. In effect, the regions where partial, substantial, and even total charge saturation takes place give rise to complex transient dynamics, which include elaborate bending and breathing, and can locally induce transient scales ∼ lim .

6.4. Harnessing 1D Spatial Nonlocality Spatial nonlocality is therefore an intrinsic property of the charge migration process that mediates optical nonlinearity in photorefractives. It alters and generally distorts spatial soliton phenomenology, to the point that it would appear that microscopic photorefractive solitons are impossible to achieve, and hence that photorefraction itself is not a viable avenue to such pioneering ﬁelds as nonlinear nonparaxial optics. This is true for a standard spatial soliton, but more involved schemes have been recently suggested and are being experimentally investigated (Ciattoni et al., 2008b). For example, the photorefractive material can be biased instead of with a single homogeneous ﬁeld, with one that alternates in space along the propagation direction z. If the alternating period is small with respect to the diffraction length of the beam, the result is an effective nonlinearity for which the nonlocal terms give rise to a symmetric soliton-supporting response. This is a speciﬁc example of how a sequencing of nonlinear response on a scale that is smaller than the characteristic beam propagation scale produces an optical propagation driven by an effective nonlinearity that, although a product of the underlying combination, can have different quantitative and qualitative features, i.e., a natural hybridization produced by the system wave dynamics. Consider therefore the case of the scalar propagation of a paraxial beam through a medium that has the nonlinear response ∞

n(I, Ix , z) = c0 (I, Ix ) + m=1

2πmz , cm (I, Ix ) cos Lv

(45)

where z is the propagation direction, Ix = ∂I/∂x takes into account the spatial nonlocality [the terms of order η in Eq. (38)], and Lv the longitudinal scale of variation. Assume now that Lv is such that Lv Ld , where Ld is the diffration length of the beam. Setting A(x, z) = A0 (x, z) + δA(x, z), where A0 is that part of the ﬁeld that has a longitudinal scale of variation Ld , and δA is (i) longitudinally rapidly varying, at a scale Lv , and (ii) it is uniformly in the condition |δA| |A0 |; the resulting nonlinear evolution equation for A0 is

!

1 ∂2 ∂ i + ∂z 2k1 ∂x2

"

A0 = −

k1 neff A0 , n1

(46)

186

Photorefractive Solitons and Their Underlying Nonlocal Physics

where

#

neff

1 = c0 + 2

Lv λ

2

∞ m=1

$

"%& ! 2 2 ∂c0 cm cm ∂c0 ∂ I + I ∂I m2 ∂Ix ∂x m2 I = I0

. (47)

Ix = ∂I0 /∂x

Therefore, by appropriately choosing the sequence of longitudinal known nonlinearities, we are in the position to generate effective nonlinearities, which are qualitatively different from their underlying constituents. Consider now a centrosymmetric photorefractive sample where the external bias potential is delivered on the facets x = L/2 and x = −L/2 by means of two standard plate electrodes, compared to a system of alternating electrodes along the propagation z-axis. The nonlinear refractive index change can be shown to lead to (Ciattoni et al., 2008b)

ψ χ ∂I + n = α cos(κv z) I + Ib ∂x I + Ib

2 ,

(48)

where ψ = VIb /((L/2) cosh(κv (L/2))), Lv = 2π/κv , so that light dynamics is here governed by a nonlinear response belonging to the general class of Eq. (45). According to Eqs. (46) and (47), the optical ﬁeld A0 experiences the effective nonlinear refractive index change

neff =

α 2 2 2 + 2χ (I ) ψ , x 2(I + Ib )2

(49)

where we have approximated neff = c0 (neglecting higher order contributions). Compared to the standard nonlinearity (i.e., for κv = 0), the effective case has acquired a reﬂection symmetry (x → −x) for an even intensity proﬁle I(x) = I(−x), a symmetry that is intrinsically unavailable for the standard photorefractive response, as discussed in Eq. (38). Moreover, those very processes of charge saturation that generate the nonperturbative effects discussed before can be decreased and ultimately suppressed. In this manner, it is reasonable to conclude that micron-sized photorefractive solitons will propagate in a straight line [see Figure 4]. Recently, the same approach has been numerically demonstrated for 2D solitons (Ciattoni et al., 2008a).

Nonlocality and Anisotropy in 2D Solitons

187

(1b)

x (mm)

0 210 220

(2b)

10

x (mm)

230 0

0

200

400

600

800

1000

200

400

600

800

1000

200

400

600

800

1000

210

(3b)

10

x (mm)

0

0

210 0

z (mm)

FIGURE 4 Devising a bending-free self-trapping nonlinearity in photorefractives. (1b), (2b), and (3b) : Steady-state optical intensity profiles under conditions in which Lv Ld , Lv Ld , and Lv < Ld , respectively. Note the transition into the effective nonlinear regime in case (3b) [see Ciattoni et al. (2008b)].

7. NONLOCALITY AND ANISOTROPY IN 2D SOLITONS 7.1. An Intrinsic Nonlocality While spatial nonlocality can, in certain conditions, be a perturbation for 1D effects, it plays a fundamental role when the optical beam is conﬁned in both transverse directions, that is, in those phenomena that are termed 2D (Belic et al., 2002). In particular, it is the basis for the formation of round 2D photorefractive solitons. Here, the spatial nonlocality is not something directly connected to xq , but to the geometric set-up.

188

Photorefractive Solitons and Their Underlying Nonlocal Physics

To appreciate this, consider the formation of the space–charge ﬁeld, as described in Eq. (6), valid under conditions in which charge saturation does not lead to nonperturbative effects, as described previously. As done in arriving at the approximate Eq. (20), we now consider the steady-state case (Crosignani et al., 1997)

⎡ ∇ · ⎣E(β/s + I)

1− 1+

∇·((0)E) αNa q ∇·((0)E) Na q

⎛ 1− kb T ∇ · ⎝(β/s + I) + q 1+

∇·((0)E) αNa q ∇·((0)E) Na q

⎞⎤ ⎠⎦ = 0, (50)

which becomes, to zero order in η = xq / (where is now the characteristic scale of the transverse proﬁle in both transverse directions),

∇ · Y(0) Q = 0

(51)

and the irrotational condition

∇ × Y(0) = 0.

(52)

Here we have to introduce, alongside the ξ = x/xq , also the η˜ = y/xq (˜η helps to distinguish the normalized spatial variable from the smallness parameter η = xq /). In the 1D case, having neglected all orders in η naturally gives rise to a spatially local response, that is, the simple Kerr-like effect of Eq. (23). Now, the fact that the light distribution Q(ξ, η˜ ) is spatially resolved implies, through Eq. (51), that Y is also spatially resolved both in the ξ and η˜ axes so that the conditions of Eq. (52) introduce a nontrivial behavior, which turns out to be, even without diffusion or charge-saturation, intrinsically nonlocal. To see this, note that the most general solution of Eqs. (51) and (52) can be written in the form

Y(0) (ξ, η˜ ) =

1 1 ex − ∇ × v, Q Q

(53)

where ex is a unit vector in the x-direction and v(ξ, η˜ , z) is an arbitrary vector ﬁeld determined by imposing the condition of Eq. (52). For zindependent intensity proﬁles, which are self-consistently associated with soliton-like propagation, we can assume that v = (0, 0, f (ξ, η˜ )), where f obeys (Crosignani et al., 1997) 2 ∇⊥ f − ∇⊥ f · ∇⊥ ln (1 + I/Ib ) =

∂ (1 + I/Ib ) ∂η˜

(54)

Nonlocality and Anisotropy in 2D Solitons

with ∇⊥ =

(0)

∂ ∂ ∂ξ , ∂η˜

189

. It is clear from Eqs. (53) and (54) that, in general, the

value of Y at a given point depends on the value of Q and on its spatial derivatives. Hence, we can conclude that, although we have neglected all components of the response that scale with η, yet the very electro-static nature of the charge-migration process, a product of the 2D structure of the beam in the transverse plane and the planar electrode bias boundary conditions, gives rise to a spatial nonlocality. It is this simple observation that ultimately provides the most complete understanding of photorefractive solitons.

7.2. Optical Propagation To ﬁnalize our analysis of 2D propagation, the response of Eq. (53) must be inserted into the electro-optic response of Eq. (15). The ﬁrst issue is that, in general, Y(0) is not simply parallel to the x-axis, but changes direction and amplitude in a complicated fashion, especially at the beam boundary (Zozulya and Anderson, 1995). This would imply that the beam polarization changes and evolves across the beam proﬁle (because of the ﬁeld-induced birefringence), effectively depolarizing the ﬁeld and introducing a propagation-dependent response effect that excludes the possibility of forming spatial solitons. The fact is that materials that support optical self-trapping have a weak electro-optic response when the beam is polarized along the x-direction and the electric ﬁeld is along the y-direction so that, even though the resulting ﬁeld E has a ﬁnite spatially resolved component Ey , the resulting nonlinear response is effective only for the Ex component, and coupling from the x to the y optical mode is weak. To better grasp this, we can start from Eq. (13) 2 ∇⊥ Ax + 2ik1

xy ∂Ax xx = −k02 Ax − k02 Ay ei(k2 −k1 )z . ∂z 0 0

(55)

Conversion from the initial x polarization to the y mode is driven by the electro-optic element xy that reads

xy = −0 n21 n22

1 n2

.

(56)

xy

For a ferroelectric sample, from Eq. (7)

1 2 n

= rxyk Ek = rxyx Ex + rxyy Ey ,

(57)

xy

where rxyx = r43 and rxyy = r42 [see, for reference, Yariv (1989)]. In crystals that readily support 2D solitons, such as ferroelectric SBN, these

190

Photorefractive Solitons and Their Underlying Nonlocal Physics

coefﬁcients are small so as to produce, for the typical propagation distances, negligible effects (note that in the present nomenclature x is along the crystal axis, i.e., in the z-direction of most crystal literature). Propagation is therefore approximately described by (setting Ax = A) 2 ∇⊥ A + 2ik1

∂A = k02 n41 r33 Ex A, ∂z

(58)

where Ex = E0 Y(0) · ex , as per Eqs. (53) and (54).

7.3. A “Hidden” Contribution to the Nonlinearity The result of nonlocality in the nonlinear propagation described by Eq. (58) due to Ex , the x-component of the ﬁeld solution of Eqs. (53) and (54) is not directly observable through standard soliton phenomenology. This, a sort of “hidden” trait, has ignited much of the original debates and discussions on the nature of 2D photorefractive solitons (Shih et al., 1995, 1996; Zozulya and Anderson, 1995; Zozulya et al., 1996). In fact, it turns out that for most conditions of interest, the nonlocality determined by this increased dimensionality in the transverse beam distribution gives rise to an anisotropic nonlinear response, i.e., of Ex in Eq. (58), characterized by two lateral lobes in the x-direction (the direction of the external bias), lateral in the sense that they form in regions not illuminated by the beam itself (the trait of nonlocality) (Gatz and Herrmann, 1998; Zozulya and Anderson, 1995). The lateral lobes are antiguiding, whereas the nature of the response where the beam actually propagates is a guiding one [see Figure 5]. In this manner, (a)

2

(c) 2

1

1

y x

KLTN z 180 mm

21

m)

50

22

2150

100 290

0 x (mm)

90

y (m

⫹

200 mm

0

0

Dn3104

⫺

250 mm

(b)

150 150

FIGURE 5 Top-sided electrode geometry and lateral anisotropic modes: (a) schematic; (b) detail of soliton-supporting region (c) numerical prediction of trapping pattern n with lobes [see D’Ercole et al. (2004)].

Nonlocality and Anisotropy in 2D Solitons

191

the lobes affect propagation by introducing an anisotropy in the two transverse directions x and y. This anisotropy excludes the circular symmetry of the self-trapped mode A(x, y) and hence of the solitons, without being directly observable in the soliton beam. Some experiments on soliton beam ellipticity agree with this picture: the nonlinearity acts differently along the x and y axes (Calvo et al., 2003; Zozulya et al., 1996). Many experiments provide direct and conclusive evidence of round solitons (DelRe et al., 1998c; Shih et al., 1995, 1996). A second method to gain information on these lobes is to study soliton dynamics that involve the immediate surroundings of the soliton beam itself. This can be done by investigating soliton–soliton scattering (Krolikowski et al., 1998).

7.4. Direct Observation of Two-Dimensional Anisotropic Response Direct observation of the index pattern underlying a 2D soliton is not a generally trivial task. In fact, one could imagine creating a 2D interferometric scan of the sample, in the propagation z-direction. Unfortunately, the soliton is self-bending so that it exists in the full three-dimensional (3D) environment, and a 3D interferometric scan of a macroscopic volume is not readily accessible. There is, however, a class of materials in which a direct observation can be carried out. This occurs when the material supporting the 2D soliton is in its paraelectric phase, as occurs for room-temperature KLTN, nano-poled SBN, and photorefractive organic polymers (Chauvet et al., 2006; Chen et al., 2003; Shih and Sheu, 1999). In this case, the electrooptic response of Eq. (7) is quadratic (i.e., rijk = 0), and a technique, termed soliton electro-activation or soliton electro-holography, can be implemented that allows the viable and accessible study of the response pattern and, speciﬁcally, of the lateral lobes (DelRe, Tamburrini, and Agranat, 2000). It is so that soliton electro-activation has become, on the applicative side, an instrument for fast electro-optic beam manipulation, and on the fundamental side, the preferential instrument to assess and detect anisotropy and spatial nonlocality (DelRe, Ciattoni, and Agranat, 2001; DelRe et al., 2005) [see Figure 6]. The principle at the basis of soliton electro-ativation can be understood as follows: consider for simplicity the 1D case, such as that leading to Eq. (22). Under conditions in which the soliton has formed, inside the crystal we ﬁnd the electric ﬁeld E(x). This ﬁeld can be ascribed to the combined effect of the photogenerated charge distribution ρ(x) and the external bias ﬁeld E0 (produced by the lateral facet electrodes). We can thus write

E(x) = E0 + Esc (x),

(59)

Photorefractive Solitons and Their Underlying Nonlocal Physics

Output V 5 0

192

Input

50 mm

5 mm

0 mm

25 mm

FIGURE 6 Electro-optic readout at E0 = 0 (zero-field soliton electro-activation) after the formation and interaction of a 1D and a 2D soliton for different distances between the beams. Input distributions are shown for reference in the bottom row. The role of the anisotropic and asymmetric lobes underlying the 2D soliton in the interaction is explicit [see DelRe et al. (2005)].

where we have identiﬁed the space–charge ﬁeld Esc (x) and (d/dx)Esc = ρ/0 . Once the charge migration has occurred, we can read out the index pattern (attenuating the beam intensity or changing the beam wavelength) for different values of external bias ﬁeld E0 = E0 , under conditions in which ρ and hence Esc remain unaltered. If we consider the case of a ferroelectric, the index pattern supporting the soliton n(x) = −b1 E(x), with b1 = (1/2)n31 r33 , is simply shifted

n(x) = −b1 (E(x) − E0 + E0 ) = −b1 E(x) − b1 (E0 − E0 ).

(60)

In other words, the index pattern is the same as that forming the soliton, n(x) = −b1 E(x), but for an additive constant. While this changes the phase of the propagating light, it has no appreciable effect on the diffraction and propagation dynamics. In a paraelectric, we have in turn that the soliton-supporting pattern n(x) = −b2 E(x)2 , where b2 = (1/2)n31 20 ((0)r − 1)2 g11 , becomes

n(x) = −b2 E(x)2 − 2b2 E(x)(−E0 + E0 ) − b2 (−E0 + E0 )2 .

(61)

The ﬁrst term on the RHS is simply the pattern originally supporting the soliton, the third is an additive constant, but the second is a nontrivial double-product term that strongly modiﬁes propagation of the read beam, since it depends on x, and it can be controlled through E0 . For example, for E0 = 0, we have

n(x) = −b2 (Esc )2 .

(62)

Nonlocality and Anisotropy in 2D Solitons

193

Now, Esc = 0 in the illuminated region so that the index of refraction becomes lower in this region (b2 > 0), and the originally guiding pattern becomes antiguiding. To a certain degree, it is thus possible to render the parts of the pattern that are guiding, antiguiding, and those that are antiguiding, guiding. When applied to the lateral lobular structure arising through spatial nonlocality in the 2D, the zero-ﬁeld read-out (i.e., with E0 = 0) makes the lateral lobes, which are antiguiding when the soliton is activated, guiding (DelRe, Ciattoni, and Agranat, 2001). This, combined with the fact that the central pattern becomes antiguiding, allows the investigation of the lobes as guiding index pattern structures. In this sense, even though the soliton bends and generally is a 3D manifestation in the volume, yet the propagation and guiding of light beams for E0 = 0 becomes their direct detection. In this 2D case, we have directly from Eq. (53),

E(0) (x, y) =

E0 E0 ex − ∇ × v(x, y), Q(x, y) Q(x, y)

(63)

and the index pattern of Eq. (58) becomes, under conditions in which (0) (0) g12 /g11 1, nxx = −b2 (Ex ), where Ex = E(0) · ex . Rewriting E(0) = E0 + Esc , we once again have, for example, for E0 = 0

n(x)xx = −b2 (Esc · ex )2 .

(64)

In this situation, regions that have (Esc )x = 0 become guiding. This occurs exactly in the lobe-like structure (Gatz and Herrmann, 1998), and hence the zero-ﬁeld read-out detection of the nonlocality is possible.

7.5. Round or Circular-Symmetric 2D Solitons Taken that a 2D intensity distribution Q(x, y) generates an anisotropic nonlocal response even at the lowest order in η, the question arises as to how a round or circular symmetric intensity distribution can self-trap into solitons, as observed in many cases and discussed previously. The fact is that 2D self-trapping is a remarkable manifestation of a speciﬁc combination between anisotropy and nonlocality associated to higher order corrections in η [starting from the full model of Eq. (6)]. A model without these corrections cannot simply predict and hence describe conditions in which a circular-symmetric soliton forms (Crosignani et al., 1997; Gatz and Herrmann, 1998). Identifying and demonstrating this condition can be achieved through a numerical and experimental comparative study, and this further allows the formulation of a qualitative picture as regards to the speciﬁc physical

194

Photorefractive Solitons and Their Underlying Nonlocal Physics

mechanism that leads to the circular-symmetric 2D photorefractive solitons (DelRe et al., 2004). As regards to the experimental study, the rationale is to ﬁnd and investigate two conditions that would be physically identical in a “local” model of Eq. (63), but that lead to qualitatively distinct self-trapping phenomena: round solitons in one, whereas in the other, no accessible system parameters allow for the symmetric manifestation. Starting from Eq. (50), the key is to play with the diffusion and saturation components that scale with η = xq /, which depends on the transverse spatial scale, by using different input beam sizes 0 = x0 = y0 , analyzing the relationship between shape and size in the steady-state regime. What is observed is that as the transverse scale becomes smaller (and hence η increases), round solitons can form. The smaller 0 activates a relevant nonlocal (asymmetric) component due to charge diffusion and saturation, which at once tilts the index pattern and suppresses one of the two lateral lobes. The process is negligible for larger values of 0 , since this leads to smaller nonlocal components. It thus turns out that the lensing in the x-direction, generally stronger than in the ydirection, is weakened by the suppression of one of the lobes. The resulting asymmetry due to the enhancement of the remaining lobe causes soliton bending, without appreciably affecting the transverse shape [see Figure 7]. (a)

(b) 3.5 3.0 2.5 220

20 0

x (mm)

20

(c)

220

0

y (mm)

E (kV/cm)

E (kV/cm)

4.0 1.5 1.4 1.3 1.2 1.1 250

0

x (mm)

50

250

y (mm)

(e)

(d)

50 mm

50 0

25 mm

12.5 mm

FIGURE 7 Nonlocal components to the response cause one lobe to be suppressed and support round solitons. Numerical calculation of the x-component of the space–charge field neglecting propagation effects, for (a) 0 = 6.5 μm and (b) 0 = 14 μm. Observed output intensity distribution with a zero-field readout (E0 = 0), for (c) 0 = 6.5 μm, (d) 11 μm, and (e) 14 μm, under conditions in which the soliton FWHM x = 0 [see DelRe et al. (2004)].

Nonlocality and Anisotropy in 2D Solitons

,0 5 9 mm /,5 9 mm Δys 5 25 mm

,0 512 mm /,5 9 mm Δys 5 250 mm

,0 515 mm /,515 mm Δys 5 15 mm

Input

,0 510 mm /,5 10 mm Δys 5 11 mm

195

EO read-out

Soliton

Diffraction

50 mm

(a)

(b)

(c)

(d)

FIGURE 8 Anisotropy versus nonlocality when pinning a soliton to a striation. (a) In a first experiment, a beam that at input has 0 = x0 y0 10 μm is launched into a slab inhomogeneity, which supports a ys 11 μm mode in the y-direction. The beam forms a round soliton of 10 μm ( 0 ) after t 200 s; the underlying index of refraction structure indicates a distinct lateral lobular structure, whose analysis reveals an approximate symmetry along the x-direction (i.e., the two lobes have equal shape and peak intensity), in contrast to what occurs in the bulk, where circular symmetry is achieved only at the expense of a considerably weakening of one lobe through charge-displacement and charge-diffusion components. (b) Repeating the experiment in a region with slab-like patterns supporting wider guided modes, with ys > , so as to weaken the pinning of the soliton to the inhomogeneity, round soliton formation at = x y 9 μm leads to a very different picture, in which, as for homogeneous media, the lobes are strongly asymmetric. (c) When the same experiment is repeated with an 0 =12 μm, round solitons are not observed for any value of external field, but rather self-trapping occurs for a smaller (ultra-focused) 9 μm < 0 , and the lobe structure is again highly asymmetric. (d) A soliton of 0 15 μm in a ys 15 μm slab-inhomogeneity is shown, and the round soliton forms with a underlying lobular index pattern that is approximately symmetric (along the x-direction), a result that is impossible without the striation [see Pierangelo et al. (2006)].

This picture implies two independent conditions (i) that the resulting slanted index pattern support the soliton, and (ii) that one of the lobes be suppressed. Condition (i) is equivalent to verifying that the parameters (E0 , u0 , ) giving rise to the soliton-supporting refractive index pattern n compensate diffraction. This condition can be estimated neglecting all nonlocal and anisotropic effects. Condition (ii) implies that, in the region manifesting the lobes, the conﬁguration leading to round solitons must

196

Photorefractive Solitons and Their Underlying Nonlocal Physics

be associated with a given precise relationship between the nonlocal xcomponent of the space–charge ﬁeld response, En = Ed + Es (due to the diffusion component Ed and charge saturation one Es ) and the corresponding anisotropic part Ea (the lobes). This relationship must be preserved for all conditions leading to round solitons, for different parameters (E0 , u0 , ), implying a condition on their scaling that can be experimentally tested. An estimate of the diffusion and saturation ﬁeld indicates, respectively, that Ed ∝ (kB T/q)u20 (1 + u20 )−1 −1 and Es ∝ 0 (0)r E20 u20 (1 + u20 )−3 −1 (Na q)−1 . In turn, the numerical analysis of the electric ﬁeld problem, for a given circular symmetric intensity proﬁle I with a transverse size , indicates that Ea ∝ E0 . A further conﬁrmation of this picture is afforded by experiments when 2D solitons are pinned to crystal striations. Here, the presence of a guiding slab-like structure breaks the symmetry in the y-direction, and thus the combination of the lobes and this permanent pattern can lead to round solitons that do not bend, i.e., for which no spatial nonlocality, associated to terms in which η intervenes, plays a role (Pierangelo et al., 2006) [see Figure 8].

8. CONCLUSIONS AND PERSPECTIVE Photorefractive solitons are, and have been for more than a decade, the arena for a vast and innovative effort in nonlinear optics and nonlinear science as a whole. In this paper, we have focused on those aspects of the photorefractive soliton-supporting nonlinearity that are associated to the underlying physical mechanism, a light-driven charge migration in a full three-dimensional time-dependent setting. This migration process introduces nonlocality in both space and time, a feature that has a prevailing importance in many observed phenomena, starting from the observation of round 2D solitons. This nonlocality is discussed as the basis for the understanding of many of the peculiarities of the observed self-trapping and to render explicit the great potential that the direct coupling of an optical wave to an intrinsically nonlocal charge migration mechanism affords. Ultimately, even though the charge migration process is not directly observed in experiments that detect the optical beam intensity, yet this “hidden” nonlocality is at the very heart of photorefractive self-trapping.

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DelRe, E., D’Ercole, A., and Palange, E. (2005). Mechanisms supporting long propagation regimes of photorefractive solitons. Phys. Rev. E 71, 036610. DelRe, E., D’Ercole, A., Palange, E., and Agranat, A. J. (2005). Observation of soliton ridge states for the self-imprinting of ﬁber-slab couplers. Appl. Phys. Lett. 19, 191110. DelRe, E., De Masi, G., Ciattoni, A., and Palange, E. (2004). Pairing space-charge ﬁeld conditions with self-guiding for the attainment of circular symmetry in photorefractive solitons. Appl. Phys. Lett. 85, 5499–5501. DelRe, E., and Palange, E. (2006). Optical nonlinearity and existence conditions for quasi-steady-state photorefractive solitons. J. Opt. Soc. Am. B 23, 2323. DelRe, E., Segev, M., Christodoulides, D., Crosignani, B., and Salamo, G. (2006). Photorefractive solitons. Chap. 11 In “Photorefractive Materials and Their Applications I” (Gunter, P., Huignard, J. P., eds.), Springer-Verlag, New York. Springer Series in Optical Sciences, Volume 113/2006. DelRe, E., Tamburrini, M., and Agranat, A. J. (2000). Soliton electro-optic effects in paraelectrics. Opt. Lett. 25, 963–965. DelRe, E., Tamburrini, M., Segev, M., Refaeli, E., and Agranat, A. J. (1998c). Two-dimensional photorefractive spatial solitons in centrosymmetric paraelectric potassium-lithiumtantalate-niobate. Appl. Phys. Lett. 73, 16–18. Denz, C., Krolikowski, W., Petter, J., Weilnau, C., Tschudi, T., Belic, M. R., Kaiser, F., and Stepken, A. (1999). Dynamics of formation and interaction of photorefractive screening solitons. Phys. Rev. 60, 6222–6225. Duree, G. C., Shultz, J. L., Salamo, G. J., Segev, M., Yariv, A., Crosignani, B., Di Porto, P., Sharp, E. J., and Neurgaonkar, R. R. (1993). Observation of self-trapping of an optical beam due to the photorefractive effect. Phys. Rev. Lett. 71, 533–536. Duree, G., Salamo, G., Segev, M., Yariv, A., Crosignani, B., Di Porto, P., and Sharp, E. (1994). Dimensionality and size of photorefractive spatial solitons, Opt. Lett. 19, 1195–1197. Fazio E., Ramadan W., Bertolotti M., Petris A., Vlad V. (2003). Complete characterization of (2+1)D soliton formation in photorefractive crystals with strong optical activity. J. Opt. A: Pure Appl. Opt. 5, S119–S123. Fazio E., Renzi F., Rinaldi R., Bertolotti M., Chauvet M., Ramadan W., Petris A., Vlad V. (2004). Screening-photovoltaic bright solitons in lithium niobate and associated singlemode waveguides. Appl. Phys. Lett. 85, 2193–2195 Fressengeas, N., Maufoy, J., and Kugel, G. (1996). Temporal behavior of bidimensional photorefractive bright spatial solitons. Phys. Rev. E 54, 6866–6875. Fressengeas, N., Maufoy, J., Wolfersberger, D., and Kugel, G. (1997). Experimental transient self-focusing in Bi12TiO20 crystal. Ferroelectri. 202, 193–202. Fressengeas, N., Wolfersberger, D., Maufoy, J., and Kugel, G. (1998). Build up mechanisms of (1 + 1)-dimensional photorefractive bright spatial quasi-steady-state and screening solitons. Opt. Commun. 145, 393–400. Gatz, S., and Herrmann, J. (1998). Anisotropy, nonlocality, and space-charge ﬁeld displacement in (2 + 1)-dimensional self-trapping in biased photorefractive crystals. Opt. Lett. 23, 1176–1178. Günter, P., and Huignard, J. P. (eds.) “Photorefractive Materials and Their Applications 1: Basic Effects,” Springer-Verlag, New York. Springer Series in Optical Sciences, Volume 113/2006. Hasegawa, A., and Tappert, F. (1973). Transmission of stationary nonlinear optical pulses in dispersive dielectric ﬁbers. Appl. Phys. Lett. 23, 142–145. Kos, K., Meng, H. X., Salamo, G., Shih, M. F., Segev, M., and Valley, G. C. (1996). Onedimensional steady-state photorefractive screening solitons. Phys. Rev. E 53, R4330–R4333. Krolikowski, W., Luther-Davies, B., and Denz, C. (2003). Photorefractive solitons. IEEE J. Quant. Electron. 39, 3–12.

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Sulem, C., and Sulem, P. (1999). The nonlinear Schrödinger equation self-focusing and wave collapse. Series: Applied Mathematical Sciences, Vol. 139. Springer, New York. Tosi-Belefﬁ, G. M., Presi, M., DelRe, E., Boschi, D., Palma, C., and Agranat, A. J. (2000). Stable oscillating nonlinear beams in square-wave-biased photorefractives. Opt. Lett. 25, 1538–1540. Trillo, S. and Torruellas, W. (eds.), 2001. “Spatial Solitons.” Springer-Verlag, Berlin Heidelberg. Wolfersberger, D., Lhomme, F., Fressengeas, N., and Kugel, G. (2003). Simulation of the temporal behavior of one single laser pulse in a photorefractive medium. Opt. Commun. 222, 383–391. Yariv, A. (1989). “Quantum Electronics. third edition.” John Wiley and Sons, New York. Yariv, A., and Yeh, P. (2003). “Optical Waves in Crystals: Propagation and Control of Laser Radiation.” Wiley Series in Pure and Applied Optics, John Wiley and Sons, New Jersey. Yeh, P. (1993). “Introduction to Photorefractive Nonlinear Optics.” Wiley, New York. Zozulya, A. A., and Anderson, D. Z. (1995). Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electricﬁeld. Phys. Rev. A 51, 1520–1531. Zozulya, A. A., Anderson, D. Z., Mamaev, A. V., and Saffman, M. (1996). Self-focusing and soliton formation in media with anisotropic nonlocal material response. Europhys. Lett. 36, 419–424.

CHAPTER

4 Stimulated Scattering Effects of Intense Coherent Light Guang S. He*

Contents

1 Historical Remark of Stimulated Scattering Studies 2 Scope and Organization of This Review 3 Stimulated Raman Scattering 3.1 Brief Theoretical Description of SRS 3.2 Raman-Active Materials 3.3 Experimental Features of SRS 3.4 Stimulated Rotational Raman Scattering 3.5 Stimulated Electronic Raman Scattering 3.6 Stimulated Hyper-Raman Scattering 3.7 Stimulated Spin-Flip Raman Scattering 4 Stimulated Brillouin Scattering 4.1 Physical Mechanism of SBS 4.2 Theoretical Description of SBS 4.3 Materials for SBS Generation 4.4 Experimental Features of SBS 5 Stimulated Rayleigh-Wing Scattering and Stimulated Thermal Rayleigh Scattering 5.1 Stimulated Raleigh-Wing Scattering 5.2 Stimulated Thermal Rayleigh Scattering 6 Stimulated Kerr Scattering 6.1 Observations of Super-Broadband Stimulated Scattering in Kerr Liquid-Core Fiber Systems 6.2 Physical Models of Stimulated Rayleigh–Kerr Scattering and Stimulated Raman–Kerr Scattering 6.3 Brief Theoretical Description of SKS 6.4 Experimental Studies of SKS 7 Stimulated Rayleigh–Bragg Scattering 7.1 Discovery of Frequency-Unshifted Stimulated Scattering in a Two-Photon-Absorbing Medium 7.2 Physical Model of SRBS 7.3 Pump Threshold Requirement of SRBS

202 203 204 204 209 214 222 223 225 226 228 228 231 234 236 243 243 246 249 249 251 253 258 263 263 264 266

* Institute for Lasers, Photonics and Biophotonics, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA Progress in Optics, Volume 53, ISSN: 0079-6638, DOI: 10.1016/S0079-6638(08)00204-7. Copyright © 2009 Elsevier B.V. All rights reserved.

201

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Stimulated Scattering Effects of Intense Coherent Light

7.4 Experimental Results of SRBS in a Multi-Photon-Absorbing Medium 8 Phase-Conjugation Property of Backward-Stimulated Scattering 8.1 Definition and Unique Feature of Optical Phase-Conjugate Wave 8.2 The Mechanism of Generating PC Wave by Backward–Stimulated Scattering 8.3 Phase-Conjugation Experimental Studies on BSS Acknowledgments References

267 269 269 271 273 276 276

1. HISTORICAL REMARK OF STIMULATED SCATTERING STUDIES In the second year (1961) after the invention of the ﬁrst laser device (Maiman, 1960), when researchers investigated the spectral structure of a ruby laser using a nitrobenzene liquid cell as the Q-switched element, they observed an unexpected ∼767-nm lasing line added in the previously known ∼694.3-nm lasing line of ruby (Woodbury and Ng, 1962). This newly observed stimulated emission line could not be explained based on the ﬂuorescence property of the ruby crystal itself. Shortly after that observation, the researchers found that the frequency difference between these two stimulated radiation lines was just equal to the frequency of the strong Raman-vibration-mode of the nitrobenzene liquid. The researchers then realized that they actually observed the stimulated Raman scattering (SRS) from the nitrobenzene cell excited by the intense 694.3-nm ruby lasing line (Eckhardt et al., 1962). In 1964, the stimulated Brillouin scattering (SBS) was ﬁrst demonstrated by Chiao, Townes, and Stoicheff (1964) in crystal samples of quartz and sapphire excited by ruby laser pulses. This type of stimulated scattering is based on the interaction of an intense coherent light ﬁeld with the induced intense coherent supersonic acoustic ﬁeld, through the mechanism of electrostriction effect. Mash et al. (1965) reported their ﬁrst observation on stimulated Rayleigh-wing scattering (SRWS) in Kerr liquids, excited by ruby laser pulses. This effect was characterized by the red-shifted broad band (10–15 cm−1 ) stimulated scattering due to the reorientation of anisotropic liquid molecules. Herman and Gray (1967) published their theoretical paper of stimulated thermal Rayleigh scattering (STRS) in a linearly absorbing medium and predicted an anti-Stokes shift in an extent of the half of the pump spectral linewidth. Almost at the same time, Rank et al. (1967) reported their experimental observations on this effect in several iodine-doped liquid samples.

Scope and Organization of This Review

203

To explain the observed super-broadband (≥300–600 cm−1 ) stimulated scattering from Kerr liquids (CS2 and benzene) ﬁlled hollow ﬁber samples (He et al., 1985), He and his colleagues proposed their theoretical models and experimental results on stimulated Rayleigh–Kerr scattering (SRKS; He and Prasad, 1990b) and stimulated Raman–Kerr scattering (SRmKS; He, Burzynski, and Prasad, 1990), respectively. The key explanations for these two effects are based on the assumption that the optical reorientational Kerr effect and Raman-induced Kerr effect are essentially light scattering processes, accompanied with the energy transfer from the incident optical ﬁeld to the scattering anisotropic liquid molecules. The scattering molecules will further convert the energy they received from the optical ﬁeld into the reorientation work to overcome the viscosity of the liquid. Recently, another stimulated scattering effect, namely stimulated Rayleigh–Bragg scattering (SRBS) effect, is observed in a two-photonabsorbing dye-solution medium (He, Lin, and Prasad, 2004). This effect is characterized by no frequency shift, while the positive feedback is provided by the reﬂection from an induced stationary standing-wave Bragg grating (He et al., 2005). It is also indicated that the same effect can be generated in any multi-photon-absorbing medium. In this case, the key role is the multi-photon excitation-induced refractive-index change of the scattering medium.

2. SCOPE AND ORGANIZATION OF THIS REVIEW Stimulated scattering of intense coherent light is one of the most important parts of nonlinear optics (Boyd, 2002; He and Liu, 2000; Kaiser and Maier, 1972; Shen, 1984). The studies of various stimulated scattering effects greatly enrich our understanding and knowledge of the interaction between intense coherent light and matter. The stimulated scattering is one of the most useful approaches for frequency conversion and ampliﬁcation of coherent emission. In addition, stimulated scattering is also a very efﬁcient technique to generate phase-conjugate (PC) waves that exhibit the capability to remove aberration inﬂuences from a gain medium and/or a propagation medium. The purpose of this chapter is to provide an overview of the development and progress of stimulated scattering studies over the past four to ﬁve decades. The key issues are focused on the principles (mechanisms or models), basic experimental features, and technical applications of various stimulated scattering effects and related processes. Stimulated scattering is a very broad research subject. Since 1960s, a huge number (more than 5000) of journal papers have been published in this speciﬁc area. Due to length limitation of this chapter, the number

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Stimulated Scattering Effects of Intense Coherent Light

of reference citations have to be strongly restricted. For this reason, only about 370 papers are cited in this chapter, based on the novelty, signiﬁcance, innovation, technical merit, or sequence of their publication dates. As the paper search is based on the SciFinder system by using appropriate key words, some publications of importance may still be inadvertently missed. For the same reason, some other stimulated scattering-related issues [such as impulsive stimulated scattering, stimulated scattering in plasma, and coherent anti-Stokes Raman spectroscopy (CARS) effect] are not included in this chapter. The key issues of the so-called impulsive stimulated (Brillouin, Raman, and thermal) scattering studies are focused on the dynamic responses of short (or ultrashort) laser pulse-induced phonon ﬁelds, not on the generation of stimulated scattering light with a new frequency. In a similar manner, studies of stimulated scattering in plasmas are mainly aimed at exploring opto-physical properties of the plasma medium. Finally, the so-called CARS effect is a Raman-enhanced fourwave frequency-mixing (FWFM) process, which requires phase-matching condition and can only occur in the forward direction.

3. STIMULATED RAMAN SCATTERING 3.1. Brief Theoretical Description of SRS In general, Raman scattering is a molecular scattering process accompanied by an electronic, vibrational, vibration-rotational, or a pure rotational transition (Raman and Krishnan, 1928). A clear and rigorous description of this scattering process can be given only by quantum theory of radiation, in which both the medium and the optical ﬁeld have to be treated as a combined quantized system [see Heitler, (1954)]. The Raman scattering process can occur in two possible ways as schematically shown in Figure 1a and b. Figure 1a shows the elementary process for Raman scattering with a Stokes frequency shift, while Figure 1b shows that for Raman scattering with an anti-Stokes shift. To describe these two elementary processes, a key concept of intermediate states must be introduced. For clarity of explanation, each elementary Raman scattering process can be visualized as a “two-step” event. For the case shown in Figure 1a, in the ﬁrst step there is the annihilation of an incident photon of frequency ν0 with the simultaneous excitation of a molecule from the ground state a to the intermediate state; in the second step, this molecule returns to Raman excited state c with the simultaneous creation of a scattered photon at the frequency ν = ν0 − νr . Here, one can see that the connection between these two steps is the intermediate state that means an uncertain status for the scattering molecule. In other words, at

Stimulated Raman Scattering

205

Intermediate state Intermediate state

ν0

ν0

ν0 ⫺ ⌬νr

c

c ⌬νr

a

ν0 ⫺ ⌬νr

ν0 ⫹ ⌬νr

ν0 (a)

a

ν

ν0

ν0 ⫹ ⌬νr

ν

(b)

FIGURE 1 Quantum-transition diagrams of molecular scattering: (a) the Stokes-shifted Raman scattering, and (b) the anti-Stokes-shifted Raman scattering. The corresponding spectra are shown in the bottom.

the considering moment, this excited molecule is not situated in its any speciﬁc excited state, instead it is situated in its all allowed eigen states with a certain probability distribution. Since the energy spread range is so broad for all eigen allowed states, according to uncertainty principle, the staying time of molecule in this intermediate state should be inﬁnitely short. For this reason, the two steps mentioned above actually occur in the same time, and the Stokes-shifted elementary Raman scattering process must be recognized as a single and instantaneous process. The same physical explanation can be also applied to the anti-Stokes Raman scattering process shown in Figure 1b: in the ﬁrst step, there is an annihilation of an incident photon of frequency ν0 with the simultaneous excitation of a molecule from Raman excited state c to the intermediate state; in the second step, this molecule returns to ground state a with a simultaneous creation of a scattered photon of frequency ν = ν0 + νr . Once again, the whole elementary process should be recognized as a single and instantaneous process.

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Stimulated Scattering Effects of Intense Coherent Light

3.1.1. Raman Scattering Cross Section If the molecule number in the eigenstate a within a unit volume is denoted by Na and the interaction volume between the incident light ﬁeld and the scattering medium is V, the overall probability for Na V molecules to scatter one photon into an arbitrary scattering mode over 4π solid angle per second can be written as (He and Liu, 2000)

W = 8πNa cσ · [n(ν0 ) + n(ν0 )n(ν )].

(1)

Here, n(ν0 ) and n(ν ) are the photon degeneracy of the incident light and Raman scattering light, respectively; c is the light velocity in free space, and σ is the differential Raman scattering cross section that is deﬁned by (in SI units)

σ=

2π c

4

ν0 ν3 (4πε0 )2 h2

(p0 )ab (p)bc b

νba − ν0

(p)ab (p0 )bc + νba + ν

2 .

(2)

Here, (p0 )ab and (p)bc are the matrix elements of the components of the molecular electric-dipole moment operator in the polarization direction of the incident light ﬁeld and the scattered light ﬁeld, respectively; νba is the molecular transition frequency from the eigenstate a to eigenstate b, ε0 is the permittivity of free space, and h is Planck’s constant. For a given scattering medium, the value of σ can be theoretically calculated if the related molecular eigenfunctions are known. However, it can be much easily determined by experimental measurements. In Eq. (1), the ﬁrst term represents the spontaneous Raman scattering and the second term represents the SRS (Hellwarth, 1963). If the incident light is a high-intensity laser beam, its photon degeneracy can be n(ν0 ) ≥ 1010 , which makes it possible that n(ν ) >> 1. In this case, the second term in Eq. (1) may become much greater than the ﬁrst term; thus, we can rewrite Eq. (1) as

W(t) = 2Na cσ · n(ν0 ) · n(ν ).

(3)

This corresponds to the situation of SRS, where is the solid angle occupied by the stimulated scattering light.

3.1.2. Gain Coefficient and Threshold Condition of SRS The quantity W determined by Eq. (3) represents the probability of Na V molecules to scatter one photon into the solid angle per second, upon the

Stimulated Raman Scattering

207

action of n(ν0 ) photons in a given mode of the incident light. On the other hand, within a unit time interval there are many pump photon modes passing through the medium, but the total photon numbers per second are simply determined by P0 /(hν0 ), where P0 is the power of the incident pump beam. After taking account of that, overall increase of stimulated scattering photons per second will be

P0 d P = 2Na cσ · · n(ν ). dt hν hν0

(4)

Here, P is the power of the stimulated scattering beam. It is further assumed that the beam sections for the incident light and stimulated scattering light are nearly the same; then Eq. (4) becomes

dI = 2Na cσ · I0 · n(ν ) · (ν/ν0 ), dt

(5)

where I and I0 are the intensities of the stimulated scattering beam and the incident beam. Moreover, the photon degeneracy of the scattering beam is related to the intensity by

n(ν ) =

I , 2hν ν /λ2

(6)

where and ν are the solid angle and spectral linewidth of the SRS beam. Substituting Eq. (6) to Eq. (5) and assuming ≈ lead to

λ2 dI = Na cσ I0 · I . dt hν0 ν

(7)

For simplicity, here we consider the steady-state stimulated scattering process. In that case, all related physical quantities are not explicit functions of time t, and in Eq. (7), we can let dt = dz/c and obtain

λ2 dI (z) = Na σ I0 · I (z). dz hν0 ν

(8)

Here, z is the spatial variable along the propagation direction of the stimulated scattering beam. The solution of the above equation is

I (z) = I (0) · eGz ,

(9)

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Stimulated Scattering Effects of Intense Coherent Light

where G is the exponential gain factor determined by

G = Na σ

λ2 I0 = gs I0 , hν0 ν

(10)

and gs is the steady-state exponential gain coefﬁcient,

gs = Na σ

λ2 . hν0 ν

(11)

Obviously, gs is a material parameter of the Raman medium. To obtain Eq. (10), we have employed the small-signal approximation, i.e., the depletion of the input pump beam within the medium is negligible, and I0 can be approximately recognized as a constant. From Eq. (11), one can see that the exponential gain coefﬁcient gs is proportional to the molecule density Na at the initial level, the differential Raman cross-section σ, and is inversely proportional to the spectral width ν of the SRS. Usually, ν can be assumed nearly equal to the spontaneous Raman scattering linewidth. For any type of stimulated scattering, there is a threshold requirement, i.e., the gain over unit propagation length should be larger or much larger than the losses due to various attenuation mechanisms. When an optical resonant cavity is employed, the threshold condition of SRS can be written as

e(G−α)L · R ≥ 1,

(12)

where α is the overall exponential attenuation coefﬁcient due to various loss mechanisms, L is the effect gain length of the Raman medium, and R is the reﬂectivity of the cavity mirrors. Without using an optical cavity, the threshold requirement for generating single-pass SRS is

e(G−α)L >> 1.

(13)

Any type of stimulated scattering is initiated from very weak spontaneous scattering signals; the photon degeneracy for the latter is ≤10−2 –10−4 and ≥107 –109 for the former. Therefore, the net stimulated scattering ampliﬁcation value should be around 109 –1013 . For this reason, one may employ the following expression

(G − α)L ≥ 20−30

(14)

as a rough threshold criterion for stimulated scattering generation (e.g., Bruesselbach et al., 1995; Hagenlocker and Rado, 1965).

Stimulated Raman Scattering

209

3.2. Raman-Active Materials Up to date, a greater number of Raman-active materials have been studied for SRS generation. Depending on pump laser wavelengths and Raman transition frequencies of the materials, the spectral range of SRS can extend from vacuum ultraviolet to far-infrared. The energy conversion efﬁciency from the pump laser frequency to SRS output frequencies can be readily ≥30−50%. The materials suitable for efﬁcient SRS generation should fulﬁll three basic requirements: (i) high transparency at both the pump and SRS wavelengths, (ii) a larger differential Raman cross-section, and (iii) a good resistance to breakdown and damage induced by high pump energy or intensity. The frequency-shift values (in units of cm−1 ) of SRS are determined by the strongest Raman transition modes of the chosen scattering media; usually, these values for vibrational SRS range from several hundreds to several thousands of cm−1 . In this subsection, we brieﬂy summarize some typical materials for SRS studies and applications based on their vibrational Raman transitions.

3.2.1. Liquids Most fundamental studies of SRS have been done in a larger number of transparent organic solvents, such as benzene (C6 H6 ), carbon disulﬁde (CS2 ), nitrobenzene, carbon tetrachloride (CCl4 ), acetone (Colles, 1969; Eckhardt et al., 1962; Geller, Bortfeld, and Sooy, 1963; Ghaziaskar, Mullett, and Lai, 1993; von der Linde, Maier, and Kaiser, 1969). Among all these liquids, CS2 is one of the most commonly used liquids for fundamental studies of SRS and self-focusing because of its very strong Raman mode (∼665 cm−1 ) and the reorientational Kerr effect. On the other hand, dimethyl sulfoxide (DMSO) is one of the most efﬁcient Raman medium with a larger frequency shift (∼2916 cm−1 ) (Decker, 1978). In general, SRS always occurs at the strong peak positions of the corresponding spontaneous Raman spectra (Ghaziaskar, Mullett, and Lai, 1993). Water and heavy water are the two interesting liquids for SRS study as they have very broad Raman spectral width (more than several hundreds of cm−1 ) (Dheer, Madhavan, and Rao, 1975; Tcherniega et al., 2000; Yui et al., 2002). The SRS in liquid gases (such as N2 and O2 ) (Brueck and Kildal, 1982; De Martino, Frey, and Pradere, 1980; Frey et al., 1977) and in liquid Br2 (Bridges et al., 1982) have been reported. All above-mentioned liquids generally exhibit a relatively large Raman cross-section for their vibrational Raman modes and can withstand higher pump intensities without breakdown or bubbling.

3.2.2. Gases Highly efﬁcient vibrational (or vibration-rotational) SRS can be achieved in molecular gas systems, including H2 , D2 , and CH4 (Minck, Terhune,

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Stimulated Scattering Effects of Intense Coherent Light

and Rado, 1963), CO2 (Audibert and Joffrin, 1972), O2 , N2 , and CO (Kovacs and Mack, 1972), HCl (Frey, Pradere, and Ducuing, 1977), HF (De Martino, Frey, and Pradere, 1978), CH3 I (Lang et al., 1992), and SF6 (Wittmann, Nazarkin, and Korn, 2000). Among all these gaseous systems, the compressed H2 is the earliest and most extensively investigated medium for generating vibrational (or vibration-rotational) SRS (DeMartini and Ducuing, 1966; Johnson, Duardo, and Clark, 1967; Wilkerson, Sekreta, and Reilly, 1991) because of its high conversion efﬁciency and extremely broad spectral range covered by multi-order anti-Stokes vibrational SRS lines. The frequency shift associated with the strongest vibrational transition [Q(1) mode] is ∼4155 cm−1 . Under conditions at room temperature and at 10–80 atm pressure with a gas cell length of 30–100 cm, the conversion efﬁciency from the pump energy to the ﬁrst-order Stokes energy could be 50%, pumped by 10-ns and 248-nm laser pulses (Loree, Sze, and Barker, 1977), and the energy conversion efﬁciency from the pump laser to all SRS components could be ∼60%, by using 50-ns and 351-nm pump laser pulses (Lou, 1989). The efﬁcient vibrational SRS can also be generated in H2 pumped by picosecond- and femtosecond-laser pulses (Krylov et al., 1996). The energy conversion efﬁciencies reached 23 and 13% in the ﬁrst and second Stokes lines by using 2.9-ps and 392-nm pump laser pulses (Fischer and Schultz, 1997); pumped with 300-fs and 390-nm laser pulses, the energy conversion efﬁciency was measured to be 18% (Krylov et al., 1998). The continuous-wave SRS operation in H2 was achieved by using 532-nm pump radiation from a cw Nd:YAG laser device; the power conversion efﬁciency into the ﬁrst Stokes line was 35% at an input power level of 7.6 mW (Brasseur, Repasky, and Carlsten, 1998). The most impressed feature of vibrational SRS in H2 is the bright visible emission of the multi-order anti-Stokes lines when a near-IR pump laser beam is employed. These multiple anti-Stokes coherent emission lines are mainly due to the Raman-enhanced four-wave frequency mixing (FWFM) (or four-photon parametric interaction) processes, which require phasematching conditions to be satisﬁed. In addition to H2 , methane (CH4 ) is another highly efﬁcient gas system for vibrational SRS generation working in nanosecond-regime (Sentrayan, Michael, and Kushawaha, 1993), picosecond-regime (Vodchits et al., 2008), and femtosecond-regime (Koprinkov et al., 1999).

3.2.3. Crystals Calcite (CaCO3 ) is one of the well-known crystals used for early SRS study (Eckhardt, Bortfeld, and Geller, 1963), because of its high optical quality and a larger Raman cross-section at a vibrational mode of ∼1086 cm−1 .

Stimulated Raman Scattering

211

Since 1980s, a number of nitrate and tungstate artiﬁcial crystals have been developed for SRS generation (Eremenko, Karpukhin, and Stepanov, 1980; Hulliger, Kaminskii, and Eichler, 2001; Zverev, Basiev, and Prokhorov, 1999), including Ba(NO3 )2 , BaWO4 , SrWO4 , KGd(WO4 )2 , and PbWO4 . For Ba(NO3 )2 crystal, the frequency shift of the strongest Raman mode is ∼1048 cm−1 and the energy conversion efﬁciency into the ﬁrst Stokes line could be 48% by using ∼1.3-μm and ∼10-ns pump pulses with cavity enhancement (Murray et al., 1995) and 25% by using 532-nm and 22-ps pump pulses with single-pass conﬁguration (Zverev et al., 1993). For BaWO4 crystals, the frequency shift of the strongest Raman mode is ∼925 cm−1 and the conversion efﬁciency could be ∼50% by using 1064-nm and 3.5-ns pump pulses with cavity enhancement (Cerny et al., 2002) and ∼15% by using 355-nm and 20-ps pump pulses (Cerny et al., 2002). For SrWO4 crystals, the frequency shift of the strongest Raman mode is ∼921 cm−1 and the total conversion efﬁciency into the ﬁrst and second Stokes lines could be up to ∼70% by using 1064-nm and 12-ns pump pulses with cavity enhancement (Ding et al., 2006), and the overall conversion efﬁciency into all Stokes and anti-Stokes lines was 62% by using 532-nm and 30-ps pump pulses (Hu et al., 2006). For KGd(WO4 )2 crystal, the Raman shift is ∼901 cm−1 and the conversion efﬁciency into the ﬁrst Stokes line could be 18% by using 532-nm and 28-ps pump pulses (Cerny et al., 2000). Finally for PbWO4 crystal, the Raman shift is ∼901 cm−1 and the conversion efﬁciency into the ﬁrst Stokes line could be up to ∼50% by using 532-nm and 80-ps pump pulses (Gad, Eichler, and Kaminskii, 2003). In addition to the ionic crystals mentioned above, some ferroelectric single crystals have also been studied for SRS generation, such as LiNbO3 (Amzallag et al., 1971; Kurtz and Giordmaine, 1969), LiIO3 (Ammann and Decker, 1977; Holz et al., 1980), and KTiOPO4 (Chen, 2005; Pasiskevicius, Canalias, and Laurell, 2006; Pasiskevicius et al., 2003). For this type of Raman crystals, the special feature is that some of their vibrational modes are both Raman and infrared active; therefore, the SRS with phonon–polariton coupling can be observed. Another type of Raman active solids is some covalent crystals such as natural and manmade diamonds with the strongest Raman mode of ∼1332 cm−1 (Eckhardt, Bortfeld, and Geller, 1963; Kaminskii, Ralchenko, and Konov, 2004), quartz crystal (Tannenwald and Weinberg, 1967), and silicon crystal (Ralston and Chang, 1970). Silicon is one of the most abundant material and highly useful for the massive integrated circuit industry. For SRS generation, the frequency shift of the strongest vibrational mode is ∼521 cm−1 ; the conversion efﬁciency is measured to be ∼20% by using 1064-nm and 10-ns pump pulses (Grassl and Maier, 1979). Very recently, cw SRS operation in a silicon waveguide conﬁguration with cavity enhancement has been successfully achieved (Rong et al., 2005, 2007).

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3.2.4. Lasing-Raman Crystals A number of rare-earth metal-doped ionic crystals have been developed for both lasing and SRS purposes. This new type of crystals include Nd3+ : KGd(WO4 )2 and Nd3+ :KY(WO4 )2 (Andryunas et al., 1985), Nd3+ :PbWO4 and Nd3+ :NaY(WO4 )2 (Kaminskii et al., 1999), Nd3+ :Bi12 SiO20 (Kaminskii et al., 1999), Nd3+ :SrWO4 (Ivlena et al., 2003), Yb3+ :KGd(WO4 )2 (Lagatsky, Abdolvand, and Kuleshov, 2000), and Yb3+ :KLu(WO4 )2 (Liu et al., 2005). In experiments, one of these single crystal samples is placed in a cavity consisting of two mirrors that can provide suitable optical feedback for both the lasing wavelength generated by the doped metal ions and the SRS wavelengths generated from the crystal matrix; therefore, one can observe the output containing both lasing line and SRS lines. In this case, the whole device can be very compact by using a semiconductor diode-laser as a pump source. These materials can be working in continuous wave (cw) mode (Demidovich et al., 2005), quasicw mode (Findeisen, Eichler, and Peuser, 2000) and in Q-switched mode (Sulc et al., 2007).

3.2.5. Optical Solid-Core Fibers Optical ﬁbers are among the most efﬁcient candidates for generating multi-order SRS emission because of their high optical quality, long interaction length, and the capability of retaining high local light intensity. The early observation of SRS in a single-mode silica glass ﬁber of short length (9 m) was reported by Stolen, Ippen, and Tynes (1972); by using nanosecond- and 532-nm pump pulses, the observed SRS emission wavelength was 545 nm, corresponding to a peak Raman mode of ∼460 cm−1 with a spectral width broader than 100 cm−1 . Later, when a ﬁber sample of much longer length (175 m) was pumped by 1064-nm laser pulses, more than four orders of Stokes SRS lines were generated (Lin et al., 1977). The multi-order Stokes SRS components could be efﬁciently generated in single (or multi-mode) fused silica ﬁbers of ten to hundred meters long, pumped by laser pulse in nanosecond-regime or picosecond-regime (Lin and French, 1979; Pini et al., 1983; Rothschild and Abad, 1983). In these cases, the major mechanism for generating the multi-order Stokes emission is the cascaded stimulated scattering of the strongest fundamental vibration mode(s), ensured by the high local optical intensity and very long gain length. By using single-mode ﬁber of a much longer length (e.g., more than 1 km), the SRS can be generated by using a cw pump beam (Sasaki et al., 1981); when the input cw power of a 1.064-μm pump beam was ∼5.7 W, the ﬁrst-order Stokes output (1.12 μm) power was ∼5 W; therefore the conversion efﬁciency was ≥80% (Irrera, Mattiuzzo, and Pozza, 1988).

Stimulated Raman Scattering

213

3.2.6. Liquid- and Gas-Filled Hollow Fiber Systems The advantages of high local optical intensity and long gain length can also be retained by utilizing hollow optical ﬁber systems ﬁlled with Raman active liquids or gases. The earliest experiment of SRS generation in a 1-m long and CS2 -ﬁlled hollow glass ﬁber of 12-μm core diameter, pumped by ∼1-μs and 514.5-nm laser pulses with cavity enhancement, was reported by Ippen (1970). In this case, the refractive-index value (∼1.6) of CS2 is larger than the silica-glass wall (∼1.46); therefore, total internal reﬂection on the interface between the liquid core and the glass wall can be ensured. Later on, the SRS generation in other liquid-core ﬁber systems were also reported, such as C6 H6 -ﬁlled ﬁber sample (Schaefer and Chabay, 1979; Zolin and Samokhina, 1977), CCl4 - and CBrCl3 -ﬁlled ﬁber samples (Chraplyvy and Bridges, 1981), and liquid bromine-ﬁlled ﬁber system (Bridges et al., 1982). The high-pressure gas-ﬁlled hollow glass ﬁber (capillary) systems have also been used for SRS generation. In this case, the total internal reﬂection cannot be ensured, as the refractive-index of the gas core medium is lower than the glass wall; however, for the near-forward propagating pump beam and SRS beam, the leaking-off loss can still be considerably small within a length of 50–100 cm due to the high reﬂection at grazing angles. The H2 - and CH4 -ﬁlled hollow ﬁber samples of 0.2-mm core diameter and 1-m length were employed to generate SRS by using 1064-nm and 35-ns pump pulses (Berry and Hanna, 1983) and by using 532-nm and 70-ps pump pulses (Hanna, Pointer, and Pratt, 1986). The advantage of this type of conﬁguration is a much lower threshold requirement than ordinary gas cell devices. Even pumped with femtosecond laser pulses, the energy conversion efﬁciency still could be ≥20% (May and Sibbett, 1983; Sali et al., 2005). More recently, the SRS generation in a H2 -ﬁlled photonic crystal hollow-core ﬁber system is reported by Benabid et al. (2002) and Couny, Benabid, and Carraz (2007).

3.2.7. Microdroplets The observation of SRS in liquid droplets of H2 O, D2 O, and ethanol was ﬁrst reported by Snow, Qian, and Chang (1985). These droplets with 20–40-μm sizes were excited by 532-nm and 10-ns laser pulses, and regularly spaced multiple stimulated scattering spectral peaks were observed within the spontaneous Raman spectral proﬁle. As indicated by the authors, these discrete narrow spectral peaks correspond to morphology-dependent resonances of the droplet that can be recognized as a micro-cavity. After that, similar studies were conducted in CCl4 droplet (Qian and Chang, 1986), in C6 H6 and toluene droplets (Lin and Campillo, 1997; Lin, Eversole, and Campillo, 1992), and in liquid-hydrogen droplet (Uetake, Sihombing, and Hakuta, 2002). The experimental features of using liquid droplets to

214

Stimulated Scattering Effects of Intense Coherent Light

observe SRS are the low-excitation threshold and high gain provided by the micro-cavity enhancement. The measurements of temporal behavior of the SRS emission from the droplets show that the output stimulated scattering signals can last a much longer time period than the input pump pulse duration (Pinnick et al., 1988; Qian et al., 1990; Uetake, Sihombing, and Hakuta, 2002); the decay of the output SRS emission from the droplet is mainly determined by the cavity lifetime of the oscillating stimulated scattering photons rather than the pump pulse duration.

3.3. Experimental Features of SRS 3.3.1. Effective Gain Length and Asymmetry Between Forward and Backward SRS When the pump laser beam is a quasi-parallel beam, the gain length for the forward SRS is determined by the path length of the Raman medium. However, if the pump beam is a focused laser beam, the geometrical gain length (lg ) is mainly determined by the focal depth (or the so-called Raleigh region) within which a higher local pump intensity is maintained. In the latter case, by properly choosing the beam diameter and focusing length, the geometrical gain length should be comparable with the optical path length (l0 ) of the Raman medium. In practice, the value of lg can be drastically increased when the Raman medium is in a conﬁguration of channel waveguide or optical ﬁber. In that case, the high local intensity of a focused pump laser beam can be basically maintained within the whole pass length of the waveguide material (i.e., lg ≈ l0 ). Therefore, a higher net gain and lower pump threshold requirement can be obtained. For single-pass SRS devices, the stimulated scattering could be observed in one of the following three possible manners: (i) nearly symmetric forward and backward SRS output, (ii) a stronger forward output and a weaker backward output, and (iii) only a forward SRS output. The ratio between the forward stimulated scattering and backward stimulated scattering (BSS) is essentially determined by the ratio between the geometrical gain length (lg ) and the longitudinal coherent length (lcoh ) of the input pump ﬁeld. According to the uncertainty principle, lcoh ≈ c · tcoh ≈ 1/(δν), where c is the light speed; tcoh and δν are the coherent time and spectral linewidth (in units of cm−1 ) of the pump beam, respectively. If the pump source is a cw or pulsed laser beam with a longitudinal coherent length (lcoh ) longer than the geometrical gain length (lg ) for a given conﬁguration, the forward and backward SRS output should be identical, because the interaction length between the coherent pump ﬁeld and stimulated scattering ﬁeld in both directions is the same and is equal to lg . If lcoh is shorter than lg , one expects a stronger forward output and a weaker backward output; because in this case, the effect gain length for the

Stimulated Raman Scattering

215

forward is still determined by lg , but for the backward it is determined by lcoh . Furthermore, if lcoh is signiﬁcantly shorter than lg , no backward SRS can be observed. In practice, if the pump source for SRS is a cw laser or a pulsed laser with pulse duration in the nanosecond-regime or ≥50 ps, the spectral linewidth is usually narrower than 0.1–1 cm−1 ; therefore we have lcoh ≥ 1−10 cm. For most SRS experiments using liquid cells or solid rods, the geometrical gain lengths are usually in the same (1–10 cm) range so that nearly the same SRS signals should be observed in both forward and backward directions (e.g., Bryant and Golombok, 1991). On the contrary, if the pump input is a pulsed laser beam with pulse duration ≤10 ps or in femtosecond-regime, the coherent length will be nearly the same as the pulse’s geometrical length; for example, if the pulse duration is ∼10 ps, the pulse length will be ∼3 mm. In this case, the probability for generating forward stimulating scattering will be much greater than that for generating BSS. As one of the early experimental examples, Carman et al. (1969) conducted a SRS study in a 50-cm long CCl4 cell pumped by 694.3-nm and 5-ps laser pulses; strong SRS signals were observed in the forward direction, but no backward SRS signal was observed. When picoseconds- or femtoseconds-laser pulses were used to pump a Raman medium, even for the forward SRS, the effective gain length undergoes an additional restriction from the group-velocity dispersion effect, which is due to the spatial separation between the pump pulse and the frequency-shifted SRS pulses after certain propagation length. That is one of the reasons to explain when the pump pulse duration changed from nanosecond-regime to picosecond-regime and then to femtosecondregime, the SRS threshold was getting higher and higher (Gazengel, Phu Xuan, and Rivoire, 1979; Stolen and Johnson, 1986).

3.3.2. Self-Focusing Effect in SRS Processes One of the early ﬁndings in SRS studies is that the self-focusing effect may take place with the SRS process together. Many experiments showed that for a given Kerr liquid sample pumped by 10–20-ns laser pulses, the measured threshold energy (or power) for stimulated scattering generation was also the threshold value for observing a self-focusing effect (Maier and Kaiser, 1966; Shen and Shaham, 1967; Wang, 1966). Under these circumstances, the measured threshold values for SRS depend not only on the Raman parameters of the gain media but also on their optical Kerr constants that inﬂuence the self-focusing behavior in these liquids. From viewpoint of semiclassical theory, this fact is easy to understand, because both effects can be phenomenologically described by the complex third-order nonlin(3) ear susceptibility χR , of which the imaginary part determines the Raman gain behavior and the real part determine the Raman-resonance-enhanced

216

Stimulated Scattering Effects of Intense Coherent Light

refractive-index change (Maker and Terhune, 1965; Shen and Bloembergen, 1965). Once the input pump laser beam has been self-focused or selftrapped, the local pump intensity drastically increased and thus leaded to the start of SRS generation. The self-focusing effect involves very complex processes (Akhmanov, Khokhlov, and Sukhorukov, 1972; Chiao, Garmire, and Townes, 1964; Shen, 1975); it can be caused by different mechanisms and depends on many different factors, such as pulse duration, beam size and transverse intensity proﬁle, focusing optics, and material’s properties (Gazengel et al., 1976; Maier, Wendl, and Kaiser, 1970; Shapiro, Giordmaine, and Wecht, 1967). For these reasons, the apparent SRS threshold values measured under different experimental conditions cannot be simply used to determine the exponential gain coefﬁcients of different types of materials. The self-focusing may not only lead to the apparent decrease of the pump threshold but also inﬂuence the further formation of cascaded multiorder Stokes generation. Once the input pump intensity is high enough to generate the ﬁrst-order Stokes SRS, there are two existing collinear coherent beams of ω0 and ωs1 ; the frequency difference between them is just nearly equal to the peak Raman transition frequency (ωr ) of the medium. For this resonant interaction, the induced refractive-index changes at these two frequencies can be written as (He, 1987)

ωr − (ω0 − ωs1 ) · |E0 (ωs1 )|2 [ωr − (ω0 − ωs1 )]2 + 2 ωr − (ω0 − ωs1 ) n(ωs1 ) ∝ · |E0 (ω0 )|2 [ωr − (ω0 − ωs1 )]2 + 2 n(ω0 ) ∝

⎫ ⎪ ⎪ ⎬ .

⎪ ⎪ ⎭

(15)

Here is the half spectral linewidth, |E0 (ω0 )|2 and |E0 (ωs1 )|2 are the local intensities of the pump beam and the ﬁrst-order Stokes beam. When (ω0 − ωs1 ) approaches ωr from the low-frequency side, n is positive and reaches its maximum value at the position of (ω0 − ωs1 ) = ωr − . In this situation, the self-focusing takes place easily for both beams; as a result, their local intensities can be signiﬁcantly increased. Such a selffocusing effect will be favorable for the generation of cascaded high-order SRS and may also promote the generation of so-called Raman-enhanced FWFM that will be described later separately. During the experiment, the two-beam coupled and Raman-enhanced self-focusing was observed, when the frequency difference between the two beams was tuned to approach the Raman mode frequency (∼1086 cm−1 ) (He et al., 1984). There were also some experimental results that directly demonstrated the self-focusing-promoted generation of multi-order Stokes components in optical silica-glass ﬁbers (Baldeck, Ho, and Alfano, 1987; He, 1987).

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217

(a)

␭0

␭s 1

␭s 2

␭s 3

␭s 4

␭s 5

␭s 6

␭s 7

␭s 8

(b)

␭

FIGURE 2 Beam sizes and spectral structures of various SRS components generated in (a) a GeO2 -doped multi-mode silica fiber and (b) a P2 O5 -doped multi-mode silica fiber pumped with 532-nm and 10-ns laser pulses. (After He, Liu, and Liu, 1989.)

As an example, in Figure 2, the photographs of relative transverse spot size of the transmitted pump beam and various-order Stokes SRS beams from two multi-mode silica ﬁber samples of 100-m length and 50-μm core diameter (He, Liu, and Liu, 1989) are shown. From Figure 2, one can clearly see that the beam size for a high-order Stokes component is remarkably smaller than that for the transmitted pump beam or a low-order Stokes component. In particular, for the GeO2 -doped ﬁber sample, the spot size of the eighth-order Stokes component is only about 1/10 of that of the transmitted pump beam.

3.3.3. Raman-Enhanced Four-Wave Frequency Mixing (FWFM) in SRS Processes From the early studies of SRS in bulk liquids or crystals, it is known that the ﬁrst-order and the cascaded high-order Stokes SRS components are always collinear with the pump beam along the axial direction. In addition to this type of pure SRS generation, which is characterized by the collinear interaction without phase-matching requirement, some ringlike coherent emission components with different Stokes- or anti-Stokes shifts can also be observed in forward direction along different angles apart from the pump beam direction (Chiao and Stoicheff, 1964; Garmire, 1965;

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Stimulated Scattering Effects of Intense Coherent Light

He et al., 1986; Maker and Terhune, 1965; Zeiger et al., 1963). This type of coherent Raman emission originates from Raman-resonance-enhanced FWFM processes, which require phase-matching conditions and occur only in the forward direction. If the Raman medium is a solid or liquid and the Raman shift value is quite large, owing to refractive-index dispersion effect, the phase-matching conditions cannot be satisﬁed in the same (axial) direction. As a result, different frequency components should propagate along different directions. The coherent emission components produced by FWFM processes usually manifest conical-shaped spatial structures and can be observed only in the forward direction. Based on these two features, researchers can easily distinguish them from the real SRS components that are always along the axial direction and can also be observed in the backward direction. There are two major reasons that can explain why SRS processes are often accompanied by FWFM processes. First, a Raman medium efﬁcient for SRS generation is also efﬁcient for the Raman-enhanced FWFM. Second, during SRS processes, a Raman-enhanced refractive-index change may lead to a Raman-enhanced self-focusing as mentioned above, which may signiﬁcantly increase the local intensity of the existing optical waves. As a result of this type of self-focusing, both high-order SRS and high-order FWFM can be observed. In practice, one of the most useful methods to distinguish FWFM from SRS is to measure the spectra of the coherent Raman emissions (including both SRS and FWFM signals) as a function of propagation directions (angles) at different pump intensity levels. Such an optical set-up is schematically shown in Figure 3, in which a focused pump beam passes through a Raman medium. The transmitted pump beam and the forward coherent Raman output are recollimated by a lens. If a screen is placed to observe the near-ﬁeld pattern of the output beams, one will see a strong

Lens 1

Lens 2 FWFM signals SRS signals FWFM signals

Pump beam (To grating)

Raman medium

Screen & Slit

FIGURE 3 Optical setup for simultaneously measuring the spectral and spatial structures of coherent Raman emissions.

Stimulated Raman Scattering

219

central spot that contains the transmitted pump beam and collinear SRS components. When the pump intensity increases, one may see additional coherent Raman emission rings that are generated by FWFM processes. To arrange the overall output beams passing through a vertically placed slit in the central position and then let the selected strip-shaped beams to be reﬂected from a dispersion grating, one can ﬁnally separate different coherent emission components on both the spectral scale and the spatial (angular) scale. By measuring the spatial structures of different spectral components, the sequence of their appearance, and the dependence of their appearance on the pump intensity level, researchers can make a clear distinction between the contributions from SRS and FWFM. When the pump intensity is slightly higher than the threshold for SRS, usually, there is only a ﬁrst-order Stokes (λs1 ) SRS observed in the screen as a central spot. When the pump intensity increases to a moderate level, the second-order Stokes (λs2 ) SRS is observed in the same axial direction due to the cascaded stimulated scattering process; at the same time, a ring-shaped emission at the λs2 wavelength and the ﬁrst-order anti-Stokes wavelength λas1 can be simultaneously observed in the screen corresponding to different conic angles. The generation of these two rings is due to FWFM process, possibly following the phase-matching shown in Figure 3a:

k0 (axial) + ks1 (axial) = ks2 (ring) + kas1 (ring),

(16)

where k0 and ks1 are the wave vectors of the pump beam and collinear ﬁrst-order Stokes SRS beam, and ks2 and kas1 are the wave vectors of the two conic beams. At a high pump level, one may further see the third-order Stokes (λs3 ) SRS along the axial direction and two more ring emission at the same λs3 wavelength and the second-order anti-Stokes (λas2 ) wavelength. In these cases, the two emission rings may be generated by FWFM processes following the phase-matching indicated by Figure 4b and c:

2ks2 (ring) = ks1 (axial) + ks3 (ring) . 2kas1 (ring) = k0 (axial) + kas2 (ring)

(17)

It should be noted that the practically observed spectral structures might be either simpler or more complicated than what we discussed earlier. As an experimental example, the measured spectral and spatial structure of coherent Raman emission from a 10-cm liquid C6 H6 cell is given in Figure 5. To obtain this result, the optical set-up shown in Figure 3 was employed, the sample was pumped by 532-nm and ∼8-ns laser pulses, and the transverse intensity distributions of different spectral components were recorded by a digital CCD camera.

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Stimulated Scattering Effects of Intense Coherent Light

kas1

ks2

k0

ks1 (a) )

(ring k as1

g)

ks

g)

(rin k as1

3 (r

ing

(rin

)

g)

k as

ks1(axial)

)

ng

(ri

ks

k s2

(rin 2

2

k0 (axial)

(b)

(c)

FIGURE 4 Phase-matching conditions of FWFM for generating (a) second-order Stokes ring and first-order anti-Stokes ring, (b) third-order Stokes ring, and (c) second-order anti-Stokes ring.

3˚

4.

␭as2

␭as1

␭0

␭s1

␭s2

FIGURE 5 Spatial structures of spectra of the forward SRS and FWFM output from a 10-cm benzene liquid cell at a pump level of I0 ≈ 300 MW/cm2 . The SRS frequency shift for benzene is 992 cm−1 . Since various spectral components experienced different attenuation via spectral filters, this figure does not indicate the real intensity ratio among different spectral components.

If the Raman sample is a gaseous medium and the pressure (density) is not very high, the refractive-index dispersion effect in a considered spectral range is relatively small so that the phase-matching conditions for FWFM may be approximately fulﬁlled along the input pump beam direction; then a nearly axially propagating multi-order anti-Stokes coherent emission can be observed in the forward direction through the FWFM processes. However, once the gas pressure is high enough and the dispersion effect cannot be neglected, the FWFM emission will appear as a conic radiation. Shown

221

1 cm-div

Stimulated Raman Scattering

0.15

0.29

0.49 0.69 Hydrogen pressure (MPa)

0.88

FIGURE 6 Variation of the near-field pattern of the sixth-order anti-Stokes pulses with hydrogen pressure. (From Moriwaki et al., 1993; copyright © 1993 American Institute of Physics.)

in Figure 6 are the photographs of near-ﬁeld pattern for the sixth-order anti-Stokes FWFM emission from an 80-cm long H2 -gas cell taken at different pressure values; the pump beam was 299-nm and 8-ns laser pulses with ∼100 mJ (Moriwaki et al., 1993).

3.3.4. Temporal Behavior and Pulse Compression of SRS Owing to the threshold requirement of SRS, the pulse duration of SRS is always shorter than that of the input pump pulse. When the pump level is just slightly higher than the threshold value, the duration of a SRS pulse might be much shorter than that of the pump pulse; however, at high pump levels, the duration for the forward SRS pulse can be quite close to that for the pump pulse. As discussed in the beginning of this subsection, the effective gain length for the backward SRS could be shorter or much shorter than the forward SRS. A shorter effective gain length means a higher threshold requirement for intensity or power density of the pump beam. For this reason, under appropriate experimental conditions, a much shorter backward SRS pulse can be obtained. An early observation of pulse shortening for backward SRS generation in a CS2 liquid sample was reported by Maier, Kaiser, and Giordmaine (1966); the pump was 694.3-nm and 12-ns ruby laser pulses, whereas the shortest pulse duration they measured for the backward SRS was ∼30 ps. In another experiment using a 55-cm long Raman cell ﬁlled with 55-atm CH4 , the pulse duration of input 308-nm pump pulses could vary from 3.3 to 1.5 ns, but for the backward ﬁrst-order Stokes SRS at 338.4 nm, the measured pulse duration was only about 0.17 ns; the pulse compression

222

Stimulated Scattering Effects of Intense Coherent Light

ratio was about 17× (Nassisi and Pecoraro, 1993). By using the same CH4 as the Raman medium pumped by 496-nm and 560-fs laser pulses, the measured backward ﬁrst-order Stokes SRS pulse duration was ∼85 fs; the pulse compression ratio was about 6.5 × (Jordan et al., 1994). There was one more experimental example reported by Kurbasov and Losev (1999), where the Raman medium was a 1.5-cm long KGd(WO4 )2 crystal pumped by 532-nm and 20-ps laser pulses, the measured backward SRS pulse duration was ∼1.7 ps, and the compression ratio was about 12×.

3.4. Stimulated Rotational Raman Scattering In a molecular gaseous medium, the SRS is usually generated through its vibrational or vibration-rotational transitions. However, the purerotational transitions associated with SRS can also be produced under appropriate conditions. In this case, the Raman shift ranges from several tens to several hundreds of reciprocal centimeters; these shift values are much smaller than a vibrational transition-related frequency shift. The earliest observation of stimulated rotational Raman scattering (SRRS) in hydrogen (H2 ) and deuterium (D2 ) gases was reported by Minck, Hagenlocker, and Rado (1966), using 10-ns and 6943-nm ruby laser pulses as a pump source. Later, the SRRS had also observed in other molecular gases, such as CO2 , O2 , and N2 O (Mack et al., 1970), as well as N2 (Averbakh, Makarov, and Talanov, 1978; Henesian, Swift, and Murray, 1985). The gases of H2 and D2 are the two well-studied media for generating SRRS. The partial reason was that, by using a high-power pulsed CO2 -laser of 10.2–10.6-μm wavelength as the pump source, these two gases could provide a tunable coherent radiation over the spectral range of 13–18 μm, which was specially recognized to be useful for isotope separation of UF6 (Byer, 1976). For H2 gas, the strongest SRRS occurs at S0 (1) line with a shift of 587 cm−1 for ortho-hydrogen (spins parallel) or at S0 (0) line with a shift of 354 cm−1 for para-hydrogen (spins opposite); the natural ortho-para ratio is 3:1 in normal room-temperature conditions (Johnson, Duardo, and Clark, 1967). For ortho-D2 gas, the strongest SRRS takes place at S0 (2) line with a shift of 414 cm−1 in the temperature range of (300–200) K or at S0 (0) line with a shift of 170 cm−1 in the temperature range lower than 160 K (Minck, Hagenlocker, and Rado, 1966). To create the priority for stimulating rotational Raman transition over vibrational transitions, the pump light should be circularly polarized and the gas pressure should be considerably lower than that suitable for vibrational SRS operation. In experiments, the energy conversion efﬁciency from the input pump pulse energy to the output SRRS energy could be 70% (in H2 ) pumped by a 1064-nm and 3-ns laser beam (Basov et al., 1979), 60–80% (in H2 , D2 , and HD) pumped by a 532-nm and 10-ns laser beam

Stimulated Raman Scattering

223

(Hanson and Poirier, 1993), and 32% (in H2 ) pumped by a 308-nm and 23-ns laser beam (Perrone, De Nunzio, and Panzera, 1998). Promoted by the potential application for laser isotope separation, the most SRRS studies were focused on the generation of ∼16-μm coherent and tunable emission in para-H2 via S0 (0) transition of 354 cm−1 frequency shift pumped by ∼10-μm CO2 laser pulses. Although in this case there is no competition from the vibrational transition, the gain in this working wavelength range is remarkably low, since the exponential gain coefﬁcient of stimulated scattering is inversely proportional to the scattering wavelength. To signiﬁcantly improve the effective gain factor, two major measures can be adopted. One is to increase the interaction length inside a gas cell and the other is to reduce the sample temperature to enhance the molecular population in the low rotational level. For these purposes, a multi-pass cell of 2–4-m length was used in conjunction with two spherical mirrors providing 20–30 passes, and the H2 gas boiled from liquid hydrogen was kept at a liquid N2 temperature (Robinowitz et al., 1978) or at room temperature (Byer and Trutna, 1978). The pump source usually was a transversely excited atmospheric CO2 laser tunable among its different lines, with the pulse duration of 70–150 ns and pulse energy of 2–5 J. The pump beam should be circularly polarized, the forward Stokes stimulated scattering is circularly polarized with opposite sense, while the stimulated backscattering has the same sense of circular polarization as the pump beam. For H2 working at 77–100 K and 440–530 Torr, the energy conversion efﬁciency for SRRS could be ≥30–50% and the quantum efﬁciency could be ≥50–70% (Midorikawa et al., 1985; Robinowitz et al., 1979). For H2 working at 300 K and 760 Torr with a seeded signal, the energy efﬁciency was ∼30% and the quantum efﬁciency was ∼50% (Carlsten and Wenzel, 1983). For D2 gas working at 100 K and ∼400 Torr, the energy conversion efﬁciency reached to 17–32% (Suda et al., 1997; Tashiro et al., 1986). More recently, by using a H2 gas-ﬁlled hollow photonic crystal ﬁber of 7.2-μm core diameter and several meters length, pumped by 1064-nm and 0.8-ns laser pulses, the SRRS energy efﬁciency is increased to 86% and the quantum efﬁciency is increased to 92% (Benabid et al., 2004).

3.5. Stimulated Electronic Raman Scattering For a gaseous medium consisting of atoms, such as a metal vapor or inert gas, there is no vibrational or rotational energy state structure, the Raman transition can only occur among their different electronic states, whose energy spacing can be much greater than that of different vibrational states of molecular systems. In practice, the stimulated electronic Raman scattering (SERS) has been observed in a number of metal atomic vapor systems. Metal vapors are easy to prepare for SRS studies. They exhibit relatively simple and well-known energy-level structures and most

224

Stimulated Scattering Effects of Intense Coherent Light

importantly a quite large number of energy levels can be chosen for resonant enhancement purposes. One can see from the expression of differential cross-section of Raman scattering [Eq. (2)] that, when the pump photon energy (ν0 ) is quite close to one of the spacing (νba ) between the initial state and any excited state, the denominator of one summed term becomes very small, and thus the contribution to the cross-section from this speciﬁc term will become very large and dominate others. This is the so-called one-photon resonanceenhanced Raman process, and it is extensively used to achieve a higher SERS gain. The highly efﬁcient electronic-transition SRS studies have been accomplished in a series of metal vapors including potassium (K) (Bernage, Niay, and Houdart, 1981; Cotter et al., 1975; Kung and Itzkan, 1976; Ohde et al., 1996; Rokni and Yatsiv, 1967; Sorokin et al., 1967; Wynne and Sorokin, 1975), thallium (Tl) (Weingarten et al., 1972), cesium (Cs) (Cotter, Hanna, and Wyatt, 1976; Harris et al., 1984; Sorokin and Lankard, 1973), barium (Ba) (Carlsten and Dunn, 1975; Cotter and Zapka, 1978; Djeu and Burnham, 1977; Manners, 1983; Sapondzhyan and Sarkisyan, 1983; Verma, Jaywant, and Iqbal, 1985), lead (Pb) (Akiyama et al., 1993; Burnham and Djeu, 1978; Lou, Guo, and Huo, 1989; Marshall and Piper, 1990), bismuth (Bi) (Burnham and Djeu, 1978), rubidium (Rb) (Niay, Bernage, and Bocquet, 1979), indium (In) (Chilukuri, 1996; Takubo, Tsuchiya, and Shimazu, 1981), and thulium (Tm) (Verkhovskii et al., 1982). In most cases, the one-photon resonant enhancement has been applied to increase the conversion efﬁciency. The temperature range of those metal vapors could be from 300 to 1200◦ C, the length of the heat-pipe oven is about 30–100 cm, and the input pump radiation can be high-power laser pulses with the duration in the order of nanoseconds or picoseconds. The frequency shift of the output SERS signal can be as large as from ∼8000 to ∼25 000 cm−1 . When the input pump wavelengths are in the UV or visible range, the output Stokesshifted SERS wavelengths can be in visible or near-IR range accordingly. Under the optimum conditions, the quantum conversion efﬁciency can be more than 30–60% and the energy conversion efﬁciency can be more than 20–40%. Cs and Ba vapors are the two most interesting and efﬁcient media for SERS studies. For Ba vapor, the reported highest quantum conversion efﬁciency was 80%, pumped with nanosecond laser pulses of ∼351 nm, the output SERS wavelength was ∼585 nm (Djeu and Burnham, 1977). For Cs vapor, the quantum conversion efﬁciency could be 40–50% with nanosecond pump pulses (Cotter et al., 1977) and 20–40% with picosecond pump pulses (Sarkisyan, 1988; Wyatt and Cotter, 1980, 1981). It is interesting that even pumped with the cw dye-laser radiation tunable around 4550– 4590 nm, the SERS signals were also observed in a Cs vapor system with magnetic ﬁeld resonances (Sharma, Happer, and Lu, 1984). Finally, the

225

Stimulated Raman Scattering

continuous SERS operation was also achieved in a neon (Ne) gas system pumped by 588.2-nm cw dye-laser beam, which generated two stimulated scattering lines (659.9 nm and 588.2 nm) (Xia et al., 1993).

3.6. Stimulated Hyper-Raman Scattering So far in this section, all the discussed SRS effects are generated through a one-photon excitation process. Taking aforementioned SERS as an example, as shown in Figure 7a, the single-pump photon energy is chosen close to a given electronic state level to achieve the one-photon resonance enhancement. In the regime of semi-classical theory of nonlinear optics, this type of one-photon-excited SRS processes can be phenomenologically described by third-order nonlinear susceptibility (χ(3) ). The resonant-enhancement principle can also be extended to a two-photon-excited SRS process. In this case, as shown in Figure 7b, the two-photon energy of the pump beam is quite close to the spacing between the initial level 1 and a two-photon absorption (2PA) allowed excited state 3; the stimulated scattering may occur by virtue of the annihilation of two pump photons at frequency ν0 and the creation of one scattered photon at frequency νs . This is the so-called two-photon-excited stimulated hyper-Raman scattering (SHRS) effect; in semi-classical theory this effect can be recognized as a ﬁfth-order (χ(5) ) nonlinear process. Moreover, the similar resonance enhancement can be further applied to a three-photon-excited SRS process. In the latter case, as shown in Figure 7c, the stimulated scattering may occur with the annihilation of three pump photons and the simultaneous creation of one scattered photon, provided that the resonant enhancement can be achieved either by 4 3

ν0 3

3 νs

ν0

νs

νs

ν0 2

2

ν0

2 ν0

ν0 1

1

1 (a)

(b)

(c)

FIGURE 7 Schematic diagrams for (a) one-photon resonance-enhanced SRS process, (b) two-photon–excited SHRS process, and (c) three-photon–excited SHRS process.

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Stimulated Scattering Effects of Intense Coherent Light

choosing the two-photon-pump energy close to a 2PA allowed real level 3 or by choosing the three-photon pump energy close a three-photon absorption allowed real level 4. This is the so-called three-photon-excited SHRS effect, which can be recognized as a seventh-order (χ(7) ) nonlinear process. In general, two-photon-excited SHRS processes are more difﬁcult to be generated than the regular one-photon-excited SRS processes, but are easier than the three-photon-excited SHRS processes. In experiments, the two-photon-excited SHRS has been produced in a number of metal vapors including potassium (K) (Yatsiv, Rokni, and Barak, 1968), cesium (Cs) (Vrehen and Hikspoors, 1977), sodium (Na) (Cotter et al., 1977; Moore, Garrett, and Payne, 1988; Mori et al., 1986), strontium (Sr) (Reif and Walther, 1978), and lithium (Li) (Kroekel, Ludewigt, and Welling, 1986). The output SHRS wavelengths can be shorter or longer than the pump wavelength, depending on the energy state structures of the medium and the pump wavelength or other experimental conditions. The conversion efﬁciency from the input pump energy to the output SHRS energy could be ≥1–2% (Cotter et al., 1977; Kroekel, Ludewigt, and Welling, 1986); the quantum conversion efﬁciency as high as 64% was reported in Na vapor (Mori et al., 1986). In addition to metal vapors, the two-photon-pumped SHRS has also been observed in neutral krypton (Kr) atomic gas (Shahidi, Luk, and Rhodes, 1988) excited by ∼193-nm and ∼5-ps laser pulses; the output stimulated scattering wavelength range was 2.5–2.7 μm. The same effect was also observed in hydrogen (H2 ) molecular gas medium (Czarnetzki, Wojak, and Doebele, 1989) excited by ∼1-ns pump pulses of ∼193-nm wavelength with a tunable range of ∼450 cm−1 ; the output stimulated scattering wavelength range was about 750–840 nm. Particularly, Wang et al. (1992) reported their observation of SHRS signal around 556 nm in an organic crystal (MBA-NP) excited by 1054-nm and 10-ps laser pulses. The Raman shift frequency was ∼990 cm−1 , although it did not appear in the spontaneous Raman spectrum as a strong peak. Finally, the three-photon-excited SHRS was also observed in the vapor of lithium (Li) (Chen, Han, and Wu, 1993); the pump beam was ∼571-nm and ∼7-ns dye laser pulses that produced ∼395-nm output of SHRS with a measured efﬁciency of 2 × 10−6 .

3.7. Stimulated Spin-Flip Raman Scattering This is a special SRS effect that was ﬁrst observed in a semiconductor (InSb) crystal (Patel and Shaw, 1970). The unique feature of this effect is that the Raman medium is a semiconductor sample placed in an intense dc magnetic ﬁeld. The scattering centers are the electrons in the conduction band, and the Raman transition takes place between the Zeeman-splitting sublevels of those electrons. It is more important that the frequency interval

Stimulated Raman Scattering

227

of the Zeeman-splitting energy levels is directly dependent on the applied magnetic ﬁeld. Therefore, with a frequency-ﬁxed pump laser source, one can generate a frequency-tunable SRS output by changing the magnitude of the magnetic ﬁeld. InSb is a typical semiconductor sample for the study of stimulated spinﬂip Raman scattering (SSFRS). In the presence of an applied magnetic ﬁeld, the energy level of the conduction electron will be split into a series of equally spaced sublevels that are referred to the Landau levels labelled with different orbital quantum numbers of n (Smith, Dernis, and Harrison, 1977). Furthermore, if we consider the interaction between the electron spin and the magnetic ﬁeld, each Landau level should involve two sublevels corresponding to two possible spin states: the lower sub-level of spin-up state and the higher sub-level of spin-down state. For each Landau level, the interval between these two sublevels is determined by

E = |g∗ |βB,

(18)

where g∗ is the effective gyromagnetic ratio, β is the Bohr magneton and B is the magnetic induction of the applied ﬁeld. In the normal situation of an InSb sample, the most electrons in conduction band populate in the low sublevel of their n = 0 Landau level. Under the action of an incident optical ﬁeld, a conduction electron can take a Raman transition from its low spin-up state to the high spin-down state within the same n = 0 Landau level and scatter a red-shifted photon simultaneously. The frequency shift between the input photon and scattered photon is determined by

ν = ν0 − νs = E/h = |g∗ |βB/h,

(19)

where ν0 and νs are the frequencies of the input light and the scattered light. This is the basic mechanism of the so-called spin-ﬂip Raman scattering effect. From Eq. (19), we can see that the frequency of the scattering light can be tuned by changing the applied dc magnetic ﬁeld. In experiments, the most commonly used scattering medium is the n-type InSb crystal. The carrier concentration of the sample is ne ≈ 1015 −1016 cm−3 , the working temperature 3–30 K, the refractive index ∼4, and at low magnetic ﬁeld g∗ ≈ −50. The pump laser sources could be a pulsed CO2 -laser with ∼10.6-μm wavelength (Allwood et al., 1970; Patel, Shaw, and Kerl, 1970) or with its frequency-doubled output of ∼5.3-μm wavelength (Haefele, 1974; Irslinger et al., 1971; Mellish, Dennis, and Allwood, 1971), or a cw CO-laser with (5.2–5.3)-μm output wavelengths (Mooradian, Brueck, and Blum, 1970; Pascher, Appold, and Häfele, 1978). At a high magnetic ﬁeld level (∼10 tesla), the Raman shift range can be 150– 250 cm−1 , and the average tuning rate is about 22–15 cm−1 /tesla within the magnetic ﬁeld range of 0–8 tesla. The output spin-ﬂip SRS can be tuned

228

Stimulated Scattering Effects of Intense Coherent Light

over a spectral range of 11–14 μm when pumped with ∼10.6-μm laser beam or over a spectral range of 5.5–6 μm when pumped by ∼5.3-μm laser beam. It is shown that under proper experimental conditions, in addition to the Stokes SRS, the second-order Stokes component and anti-Stokes component could also be generated through the FWFM process (Aggarwal et al., 1971; Shaw and Patel, 1971). When the cw CO-laser source is used as a pump source with 1–3 W output power, a conversion efﬁciency up to 30–50% was achieved (Brueck and Mooradian, 1971; DeSilets and Patel, 1973). In addition to InSb, the spin-ﬂip SRS can also be generated in several other semiconductor crystals including CdS (Scott and Damen, 1972, pumped with 493-nm and 500-ns laser beam), InAs (Eng, Mooradian, and Fetterman, 1974, pumped with a 339-nm and 250-ns laser beam), Hg1−x Cdx Te (Sattler, Weber, and Nemarich, 1974; Norton and Kruse, 1977, pumped with a 10.5-μm pulsed laser beam), Pb1−x Snx Te (Yasuda and Shirafuji, 1979, pumped with a 10.5-μm pulsed laser beam), and Cd1−x Mnx Se (Heiman, 1982, pumped with a visible and 5-ns dye laser beam).

4. STIMULATED BRILLOUIN SCATTERING 4.1. Physical Mechanism of SBS Brillouin scattering is generated through the interaction between an incident monochromatic optical wave ﬁeld and the elastic acoustic-wave ﬁeld of a transparent continuous medium. The feature of this effect is that the frequency shift of the scattering light is dependent on the scattering angle and the acoustic velocity in the medium (Brillouin, 1922). Spontaneous Brillouin scattering is resulted from the interaction between the input light ﬁeld and the thermally ﬂuctuating acoustical ﬁelds, whereas the stimulated Brillouin scattering (SBS) is based on the interaction between an intense coherent optical ﬁeld and the induced coherent electrostriction ﬁelds in the scattering medium (Chiao, Townes, and Stoicheff, 1964). It is different from Raman scattering that for an elementary Brillouin scattering process, there is no energy or momentum exchange between the input optical ﬁeld and each individual molecule of the medium; in other words, the molecular energy and momentum remain unchanged. However, the energy and momentum exchange can take place between the input optical ﬁeld and the induced electrostrictive acoustic ﬁeld. In this sense, the Brillouin scattering essentially is a parametric interaction between these two ﬁelds, and therefore, some phase-matching requirements should be fulﬁlled due to the conservation of energy and momentum. To derive

Stimulated Brillouin Scattering

229

these phase-matching conditions, one may rely on a quantized-ﬁeld model to describe the interaction between the optical ﬁelds and the induced acoustic ﬁeld. In this approach, the elementary Brillouin scattering process can be recognized as a parametric interaction between an input photon, a scattered photon, and an induced phonon inside the medium. The conservation of energy and momentum of the whole system can be fulﬁlled in the following two possible ways.

4.1.1. Way I: Stokes Scattering Generation This process can be described as the annihilation of an incident photon and the simultaneous creation of one scattered photon and one induced phonon. In this case, the conservation of energy and momentum requires that

ν0 = νs + νa , k0 = ks + ka

(20)

where ν0 , νs , and νa are the frequencies of the incident photon, scattered photon, and induced phonon, and k0 , ks , and ka are the wave vectors of these three quanta, respectively. The feature of this scattering process is that the partial energy of the input optical ﬁeld is transferred to the acoustic ﬁeld. The Stokes frequency shift of the scattered light is determined by the phase-matching condition as shown in Figure 8. From Eq. (20), we can see that ν0 ≈ νs and k0 ≈ ks because νa << ν0 , νs . Therefore from Figure 8b one can ﬁnd that

θ 1 ka ≈ k0 sin . 2 2

(21)

Here, ka = 2π/λa = 2πνa /va , k0 = 2πn/λ0 = 2πnν0 /c; λ0 and λa are the wavelengths of the incident light and the phonon; c/n and va are the velocities of the photon and phonon in the scattering medium; n is the refractive index of the medium; θ is the angle between the incident light and the observed scattered light.

ks ␪

ka ka

ks

␪

k0 (a)

k0 (b)

FIGURE 8 Phase-matching condition of Stokes Brillouin process for (a) the forward scattering and (b) the backward scattering.

230

Stimulated Scattering Effects of Intense Coherent Light

Based on Eq. (21) and the ﬁrst equation of Eq. (20), we obtain

ν = ν0 − νs = νa = 2ν0

θ nva sin . c 2

(22)

From the above equation, one can see that when θ = π, for backward scattering light, the frequency shift reaches its maximum, i.e.,

νmax = ν0 − νsmax = 2ν0

nva . c

(23)

In experiment, we can accurately measure the frequency shift by using a Fabry–Perot etalon or an optical heterodyne technique; then the velocity of the induced phonon ﬁeld in a medium can be determined based on Eq. (22) or Eq. (23). In addition, from Figure 8, we can see that for the backward scattering light with θ = π, the phase-matching is collinear, i.e., the induced phonon that is in the supersonic frequency range will propagate along the same direction as the incident light.

4.1.2. Way II: Anti-Stokes Scattering Generation This process can be described as the annihilation of one incident photon, one existing phonon, and the simultaneous creation of one scattered photon. The conservation of energy and momentum requires that

ν0 + νa = νas , k0 + ka = kas

(24)

where νas and kas are the frequency and wave vector of the anti-Stokes scattering light. The feature of this process is that the partial energy of the existing supersonic acoustic ﬁeld in the medium is transferred to the scattering light ﬁeld. In this case, the phase-matching condition is shown in Figure 9. The frequency shift of the anti-Stokes scattering light is

kas ␪

ka

ka kas

␪

k0 (a)

k0 (b)

FIGURE 9 Phase-matching condition of the anti-Stokes Brillouin process for (a) the forward scattering and (b) the backward scattering.

Stimulated Brillouin Scattering

231

determined by

ν = νas − ν0 = νa = 2ν0

θ nva sin . c 2

(25)

When θ = π, the backward scattering light has the maximum frequency shift. In this case, however, the phonon ﬁeld should propagate along the opposite direction of the incident light beam. In practice, one may arrange the transmitted pump beam to be reﬂected from a mirror and backwardpassed through the scattering medium. In this particular case, the backward pump beam can interact with the existing forward supersonic ﬁeld to generate anti-Stokes-shifted SBS component.

4.2. Theoretical Description of SBS The elementary processes of SBS are related to the interaction between the intense coherent optical waves and the induced electrostrictive acoustic waves in an optical medium. To have a quantitative description of these processes, one needs ﬁrst to consider the acoustic ﬁeld equation of the medium under the action of an intense optical ﬁeld, then to give Maxwell’s equations taking account of the opto-elastic effect and ﬁnally to solve the following coupled ﬁeld equations (in SI units):

∇ 2E −

1 γ ∂2 εr ∂ 2 E = (ρ E) c2 ∂t2 c2 ρ0 ∂t2

⎫ ⎪ ⎪ ⎪ ⎬ .

⎪ αa ∂ρ ε0 γ 1 ∂2 ρ ⎪ ⎭ − 2 2 = 2 ∇ 2 (E2 )⎪ ∇ 2ρ − va ∂t va ∂t va 2

(26)

Here, E is the optical ﬁeld, ρ is the variable density function of the scattering medium, ρ0 is the average density value; εr is the average relative dielectric constant without applying an intense optical ﬁeld, γ is the electrostrictive coefﬁcient, va and αa are the velocity and attenuation coefﬁcient of the acoustic wave in the medium. In Eq. (26), the ﬁrst equation describes the spatio-temporal variation of the optical ﬁeld, while the second equation represents the variation of the induced electrostrictive acoustic ﬁeld. For simplicity, we assume that the optical ﬁeld is composed of two monochromatic components. One is the pump wave of frequency ω0 and another is the backward Stokes SBS wave of frequency ωs . The frequency of the induced electrostrictive acoustic wave is ωa = ω0 − ωs . The optical ﬁeld and the induced acoustic ﬁeld can be accordingly written as

E = E0 (r)e−iω0 t + Es (r)e−iωs t , ρ = ρa (r)e−ωa t

(27)

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Stimulated Scattering Effects of Intense Coherent Light

where E0 , Es , and ρa are the amplitude functions of these three components. Substituting the above expressions into Eq. (26), we obtain the following amplitude wave equations for the three waves:

[∇ 2

+ k02 ]E0 (r)

ω2 γ = − 20 [Es (r)ρa (r)] c ρ0

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

ωs2 γ , [E0 (r)ρa∗ (r)] ⎪ ⎪ c2 ρ0 ⎪ ⎪ ⎪ ε0 γ ⎪ ⎭ [∇ 2 + ka2 + iαa ka ]ρa (r) = 2 ∇ 2 [E0 (r) · E∗s (r)]⎪ va 2 [∇ 2 + ks2 ]Es (r) = −

(28)

where k0 , ks , and ka = ωa /va are the magnitudes of wave vectors of the three components. To solve the above equations, we assume that these three components are plane waves. Among them the incident pump light wave and the induced acoustic wave are propagating along the same z-axis direction, while the backward SBS wave is propagating along −z direction; both optical waves are linearly polarized in the same direction. Based on these assumptions, the amplitude functions of these three waves can be simply expressed as

⎫ E0 (z) = e0 A0 (z)eik0 z ⎪ ⎪ ⎬ −ik z s . Es (z) = e0 As (z)e ⎪ ⎪ ⎭ ρa = ρa0 (z)eika z

(29)

Here, e0 is the unit vector along the polarization of the light ﬁeld; A0 (z), As (z), and ρa0 (z) are the amplitude of the three waves. Substituting the above expressions into Eq. (28) and applying the slowly varying amplitude approximation to the three waves, the analytical solutions of Eq. (28) can be ﬁnally obtained (Tang, 1966). By omitting the details of such a procedure, the steady-state solutions for the intensity variation of the forward pump beam and the backward SBS beam can be expressed as

⎫ ∂I0 (z) ⎪ = −gs Is (z)I0 (z)⎪ ⎬ ∂z , ⎪ ∂Is (z) ⎭ = −gs I0 (z)Is (z) ⎪ ∂z

(30)

where gs is a coefﬁcient determined by

gs =

ω02 γ 2 (αa va /2) , 3 n0 c ρ0 va (ωs − ωs )2 + (αa va /2)2

(31)

Stimulated Brillouin Scattering

233

where ωs is the central frequency of the Stokes SBS determined by Eq. (23), n0 is the average refractive index of the medium. From Eq. (30), we see that the intensity of the pump beam decreases along its propagation (z-axis) direction, while that of the backward SBS beam increases along the −z direction. If the depletion of the pump beam over a short distance can be nearly neglected, the pump intensity I0 (z) in the second equation of Eq. (30) can be replaced by its initial value, i.e.,

dIs (z) = −gs I0 (0)Is (z). dz

(32)

The solution of this equation is

Is (0) = Is (z)e gs I0 (0)z = Is (z)eGs z .

(33)

Here Is (0) and Is (z) are the intensities of the backward SBS beam in the incident plane and exit plane, respectively; I0 (0) is the intensity of the pump beam in the incident plane. The exponential gain factor of the BSS is a function of the frequency ωs , i.e.,

Gs (ωs ) = gs (ωs )I0 (0) =

ω02 γ 2 (αa va /2) I0 (0), 3 n0 c ρ0 va (ωs − ωs )2 + (αa va /2)2

(34)

where gs (ωs ) is the exponential gain coefﬁcient determined by Eq. (30). At the central frequency position of ωs = ωs , the exponential gain coefﬁcient reaches its maximum value

gsmax =

ω02 γ 2 ω02 γ 2 2 2 = , n0 c3ρ0 va αa va n0 c3ρ0 va δνB

(35)

where δνB is the spontaneous Brillouin scattering linewidth. According to the deﬁnition of the electrostrictive coefﬁcient of a medium,

γ = ρ0

∂εr ∂ρ

,

(36)

T

where γ is proportional to the average density ρ0 ; therefore, gsmax is proportional to ρ0 . So far we have assumed that the pump light is a monochromatic wave and its spectral linewidth is much smaller than δνB so that the gain linewidth of SBS is mainly determined by δνB . However, if the spectral linewidth of the pump laser beam is much greater than δνB , the spectral width of the gain curve will be mainly

234

Stimulated Scattering Effects of Intense Coherent Light

determined by the spectral linewidth of the pump beam. In particular, if the pump light exhibits a Lorentzian spectral proﬁle, Eq. (34) can be phenomenologically expressed as

Gs (ωs ) = gs (ωs )I0 (0) =

ω02 γ 2 (ω0 /2) I0 (0), n0 c3 ρ0 va (ωs − ωs )2 + (ω0 /2)2

(37)

where ω0 is the full spectral linewidth of the pump laser beam. In this case, the maximum exponential gain coefﬁcient will be

gsmax =

ω02 γ 2 2 . 3 n0 c ρ0 va ω0

(38)

Comparing Eq. (38) with Eq. (35) indicates that, when the pump linewidth is much greater than the spontaneous Brillouin scattering linewidth, the maximum exponential gain coefﬁcient will be getting smaller. Resembling with the case of SRS, the threshold condition for SBS process without using an optical cavity also can be expressed as

e(G−α)L >> 1,

(39)

where α is the overall exponential attenuation coefﬁcient and L is the effective gain length of the medium. Furthermore, according to Eq. (14), at the threshold level of the SBS generation, the net gain factor should be (G − α)L ≥ 20−30. So far, we have only considered the generation of Stokes SBS generation. The same derivation procedure can also apply to the generation of anti-Stokes SBS, provided that there is an existing intense acoustic wave propagating along the opposite direction with respect to the pump laser beam. It can be achieved experimentally by allowing the transmitted pump beam reﬂected from an external mirror and re-enter the same medium. In such a case, the backward propagating pump beam will interact with the existing forward acoustic wave and produce a forward antiStokes SBS.

4.3. Materials for SBS Generation In principle, any types of optical media transparent for a given pump laser wavelength can be used for generating SBS. In practice, good materials chosen for SBS generation should meet three basic requirements, which are (i) low losses for both the optical pump ﬁeld and the generated supersonic ﬁelds, (ii) capable of withstanding high-pump light

Stimulated Brillouin Scattering

235

intensity without any damage, breakdown, or boiling, and (iii) reliable opto-physical and opto-chemical stability. Here, we summarize some typical materials that have been commonly employed for SBS-related fundamental studies and applications.

4.3.1. Liquids Most common colorless organic solvents can be used for SBS generation. There were some earliest SBS studies based on organic liquids, such as CS2 , CCl4 , H2 O, acetone, methanol, benzene, toluene, n-hexane, methanol, ethanol, chloroform, and so on (Brewer and Rieckhoff, 1964; Burleﬁnger and Puell, 1965; Emmett and Schawlow, 1968; Garmire and Townes, 1964; Maier, Rother, and Kaiser, 1967; Pine, 1966; Walder and Tang, 1967a). In the 1980s, some of these organic solvents, which have a superior UV transmission property, were employed for SBS generation by using UV laser pulses from excimer lasers (Armandillo and Proch, 1983; Bourne and Alcock, 1984; McIntyre, Boyer, and Rhodes, 1987; Slatkine et al., 1982). Since the 1990s, the need of high brightness Nd:YAG laser devices have promoted the exploration of new liquid materials, which can provide a high conversion efﬁciency and withstand the pump action of 1064-nm and ns-laser pulses of high energy and/or high repetition rate. These new materials include a number of tetrachlorides of Group IV elements (SiCl4 , TiCl4 , GeCl4 , SnCl4 , and SiCl4 ) (Shilov et al., 2001; Volynkin et al., 1985); Freon-113 (Offerhaus, Godfried, and Witteman, 1996); a series of heavy ﬂuorocarbon liquids (FC72, -75, -77, -84, -3277, -3293) and perﬂuoropolyether (HT-70, -110, -135, -200, -230, -270) (Park et al., 2006; Yoshida et al., 1997). SBS were also observed in liquid crystals in isotropic phase (Rao, 1970; Roy, Rao, and Bronk, 1981) and in liquid helium (Winterling, Walda, and Heinicke, 1968). For most experimental studies in these liquid materials, the pump source is a single axial-mode solid laser device and the measured frequency-shift values of the backward SBS are in the range of 0.15–0.3 cm−1 .

4.3.2. Gases From the explanation of Eq. (35), it is known that the exponential gain coefﬁcient of SBS is proportional to the density of the materials so that highpressure gases transparent in the working spectral range can be employed for generating SBS. The gaseous systems utilized for SBS generation include H2 , N2 , CH4 , CO2 (Hagenlocker and Rado, 1965; Hagenlocker, Minck, and Rado, 1966; Rank et al., 1966), several heavy gases such as SF6 (Saito et al., 1970), Xe (Duignan, Feldman, and Whitney, 1987; Kovalev et al., 1972), CCl2 F2 , CClF3 , C2 F6 (Damzen, Hutchinson, and Schroeder, 1987; Hovis and Kelley, 1989; Tomov, Fedosejevs, and McKen, 1985). The most SBS measurements in gaseous systems have been performed by using Q-switched and single axial-mode solid-pulsed lasers. At the

236

Stimulated Scattering Effects of Intense Coherent Light

pressure range of 10–100 atm, the measured frequency-shift values are around 0.01–0.03 cm−1 .

4.3.3. Solids Various optical solid materials, including crystals, glasses, plastics, and optical ﬁbers, can be used for generating SBS. In history of nonlinear optics, the ﬁrst SBS effect was demonstrated in crystals of quartz and sapphire (Chiao, Townes, and Stoicheff, 1964). The SBS in fused quartz and crown glass (K-8) was reported by Mash et al. (1965). Later on, Pohl and Kaiser (1970) investigated SBS in several types of optical glasses (FK3, F2, SF6, BK7) and compared them to the typical liquid media; Dietz and Wiggins (1972) studied SBS in plastic samples (PMMA and PEMA) and measured the optical damage thresholds. On the other hand, Ippen and Stolen (1972) accomplished the generation of SBS through a 20-m long single-mode optical glass ﬁber and demonstrated the low-pump threshold due to the high net gain in comparison with the bulk solid or liquid samples. Furthermore, the cw-laser pumped SBS generation had been achieved in several kilometer-long single-mode or multi-mode ﬁbers with the power conversion efﬁciency ≥50–60% (Cotter, 1982; Harrison et al., 1999; Uesugi and Ikeda, 1981). Actually, the ﬁber length for SBS generation can be further shortened to 400 m for two cascaded ﬁbers (de Oliveira et al., 1993), to 10 m for a multi-mode ﬁber (Eichler et al., 2002), to 5 m for a single-mode As2 Se3 -glass core ﬁber (Abedin, 2005), to 2 m for a single-mode tellurite glass ﬁber (Abedin, 2006) and for a photonic crystal ﬁber (Dainese et al., 2006), and even to 1 m for an internally tapered ﬁber sample (Heuer and Menzel, 1998). In addition to the solid materials mentioned above, the SBS generation in a number of inorganic crystals has been reported by several research groups including the following: Faris, Jusinski, and Hickman (1993) (in KD*P, CaF2 , etc.), Dubinskii and Merkle (2004) (in TeO2 single crystal), and Sonehara et al. (2007) (in PbMoO4 , LiNbO3 , etc.). Finally, the SBS can also be generated in organic crystals. For example, Yoshimura et al. (1998) reported the efﬁcient SBS generation in organic crystals of LAP and d-LAP pumped by 1064-nm and nanosecond-laser pulses. The frequency-shift values for backward SBS in solid materials are around 0.3–1.0 cm−1 .

4.4. Experimental Features of SBS The basic optical layouts for SBS experiments are schematically shown in Figure 10. Figure 10a shows the most common optical set-up for SBS observation, where the pump source is a pulsed or cw laser device working in a single axial mode or otherwise with a linewidth narrower than 10−1 –10−2 cm−1 ; the combination of a polarizing prism and a quarter-wave

Stimulated Brillouin Scattering

(a)

Pump laser

Polarizing prism

␭/4 plate

237

F SBS medium

SBS output

(b)

Pump laser

SBS output

Cylindrical lens (c)

Pump laser

SBS output

(d)

Pump laser

Laser amplifier

SBS output

FIGURE 10 Optical configurations for SBS experiments: (a) single-pass set-up for backward output, (b) cavity-enhanced backward or large-angle SBS output, (c) cavity-enhanced 90◦ SBS generation, and (d) backward SBS passing through a laser amplifier.

plate plays the role of output coupler for the backward SBS signals and prevents the latter from re-entering the pump laser cavity; the focusing element (F) usually is a single lens or a lens combination (inverted telescope) to focus the pump beam into the scattering medium to ensure a higher local intensity and a longer effective gain length. The medium chosen for highly efﬁcient SBS generation should be of high optical quality and transparency at the working wavelengths. For this purpose, sometimes the liquid materials should be puriﬁed by vacuum distillations (Eichler et al., 1992) or ultraﬁltered through a membrane of pore size

238

Stimulated Scattering Effects of Intense Coherent Light

≤20–100 nm to remove the suspended particulate impurities (Dane et al., 1995; Kmetic et al., 1998). Shown in Figure 10b is the set-up for generating cavity-enhanced backward (θ = 180◦ ) or large-angle (θ < 180◦ ) SBS. For the latter case, the frequency shift of the stimulated scattering can be smoothly and precisely tuned by changing the scattering angle θ value; however, the interaction length between the pump beam and stimulated scattering beam will be limited by their crossed angle. Figure 10c shows the special set-up for θ = 90◦ SBS generation, where the pump beam is focused by a positive cylindrical lens to form a high-intensity focal line along the 90◦ direction inside the scattering medium (Emmett and Schawlow, 1968; Pine, 1966); if necessary an external cavity can be adopted to enhance the stimulated scattering. Finally, the set-up shown in Figure 10d is most useful for a practical laser oscillator–ampliﬁer system. In an ordinary laser oscillator–ampliﬁer system, the optical quality of the laser beam from the oscillator could become degraded after passing through an ampliﬁer due to the aberration inﬂuence from the latter. If one utilizes a SBS medium as a phase-conjugation mirror and let the SBS beam backward passing through the ampliﬁer, this beam will get further ampliﬁed because the frequency shift is usually smaller than the gain width of the ampliﬁer. Most importantly, the aberration inﬂuence from the ampliﬁer on this ampliﬁed beam can be ﬁnally removed due to the optical phase-conjugation property of backward SBS (see Section 8).

4.4.1. Pump Threshold and Nonlinear Reflectivity of SBS In experimental studies of SBS, there are two important parameters that should be measured: one is the threshold requirement of pump intensity (in w/cm2 ) and the other is the nonlinear reﬂectivity deﬁned as the ratio between the backward SBS power (or energy) and the input pump power (or energy). For most commonly used liquid, gaseous, and solid materials and under narrow spectral line pump conditions, the measured nonlinear reﬂectivity can be ≥30–50% at moderate pump levels. In some optimized experimental conditions, the nonlinear reﬂectivity can be up to ≥80–90% and the pump levels can be 20–30 times higher than the threshold value. There were some experimental studies on SBS generation by using broad linewidth pump pulses (Filippo and Perrone, 1992, 1993; Perrone and Yao, 1994). In particular, O’Key and Osborne (1992), and Cook and Ridley (1996) had investigated the pump bandwidth-dependent characteristics of SBS over a large variation range (0.005–27 cm−1 ). All these studies showed that the SBS can also be quite efﬁciently generated by using broad spectral linewidth; the nonlinear reﬂectivity can be ≥30–50% at high pump levels. The limitation for high power and high repetition-rate operation of SBS systems is the breakdown occurring in liquid or gaseous materials or the

Stimulated Brillouin Scattering

239

permanent damage in solid materials. These breakdown and damage are caused by extremely high local optical and/or acoustic ﬁeld and more likely occur in those spots where impurities or defects are located.

4.4.2. Cavity Enhanced and Cascaded SBS Effects In the very early stage of SBS studies, some researchers did try to put the Brillouin medium inside the pump laser cavity (not shown in Figure 10), and the SBS signals were easily observed due to high intracavity pump intensity and the lasing gain effect (Alcock and C. DeMichelis, 1967; Burleﬁnger and Puell, 1965). In this case, the pump beam and the originally generated backward SBS beam repeatedly passed through the scattering medium and changed their directions after each reﬂection from the cavity mirror. As a result of cascaded SBS processes, multi-order Stokes and antiStokes components could be generated (see the explanation of Figure 9). Similarly, if a scattering medium is excited by two counter-propagating laser beams, the multiple orders of Stokes and anti-Stokes components can also be created due to the same mechanisms (Ye et al., 2007). When a scattering medium is placed outside the pump laser cavity, as shown in Figure 10, an optical isolator is often adopted to avoid the backward SBS beam to re-enter the pump laser cavity. Otherwise, it may disturb the lasing behavior and make the output spectral structures and temporal behavior more complicated (Wiggins et al., 1967). Several experimental demonstrations for external cavity enhancement were reported by Heupel et al. (1997) and Buiko et al. (1999). Some other optical feedback conﬁgurations for SBS enhancement were proposed by Wong and Damzen (1990). Finally, it is shown that a SBS cell could be employed as a cavity mirror of a laser oscillator, and in the meantime it can also play the role of a passive Q-switching element (Ostermeyer and Menzel, 1999; Su et al., 2004).

4.4.3. Spectral and Hypersonic Measurements For fundamental studies of SBS, there are three parameters essentially important to explore the properties of pump-laser-induced acoustic ﬁeld in hypersonic frequency range: (i) the frequency shift of SBS, (ii) the speed of hypersonic wave, and (iii) the decay- or life-time of the induced hypersonic phonons. There are two methods to measure the frequency shift of SBS: the ﬁrst is the use of a Fabry–Perot (F-P) interferometer; the second is the use of optical heterodyne detection. For the ﬁrst method, the spectral resolution of a F-P device is determined by the spacing and reﬂectivity of the two mirrors of which the interferometer is composed, and the resolving power can be roughly ≥1−3 × 106 in visible and near-IR ranges. On the other hand, an optical heterodyne detection is usually composed of a high-speed

240

Stimulated Scattering Effects of Intense Coherent Light

photo-detector and an electrical spectrum analyzer, and the working principle is based on detecting the temporal beating between the pump beam and the SBS beam (Brewer, 1966; Garmire and Townes, 1964). The spectral resolution for this technique can be higher in orders of magnitude than ordinary F-P devices. Recently, there are more reported experiments of using this technique for SBS study (e.g., Abedin, 2005; McElhenny et al., 2008). Knowing the backward SBS frequency-shift value for a given scattering medium, one can directly use Eq. (23) to determine the speed of the induced hypersonic acoustic wave along the forward direction. Based on this method, researchers can investigate various acoustic properties in hypersonic frequency range, such as acoustic speed dispersion effect (by changing the scattering angle), the temperature or pressure effect, the propagation direction dependence in an anisotropic medium, and the estimations of photo-elastic parameters. Some early examples were reported by Goldblatt and Litovitz (1967); Hsu and Kavage (1965); Korpel, Adler, and Alpiner (1964); Madigosky, Monkewicz, and Litovitz (1967); Walder and Tang, 1967b; and Meixner et al. (1972). The lifetime or decay behavior of SBS-induced hypersonic phonons can be directly measured by using two-pulse technique. The ﬁrst strong pump pulse is used to excite the induced hypersonic ﬁeld via SBS process, and the second collinearly propagating but optically delayed weak probe pulse is employed to detect the reﬂection from the existing hypersonic ﬁeld induced travelling-wave grating. Measuring the reﬂected signal of the probe pulse as a function of time delay, the decay constant or lifetime of the hypersonic phonon can be accordingly determined (e.g., Damzen, Hutchinson, and Schroeder, 1987; Heinicke, Winterling, and Dransfeld, 1971; Leiderer, Berberich, and Hunklinger, 1973). It has been recently shown (Zhu, Gauthier, and Boyd, 2007) that the similar principle can be utilized to reconstruct the stored light information after certain delay in an optical ﬁber system through SBS process. Alternatively, by using counterpropagating strong pump beam and weak tunable probe beam, one can measure the SBS gain curve from the spectral width of which the lifetime of the hypersonic phonon can be indirectly estimated (Amimoto et al., 1991; Faris, Jusinski, and Hickman, 1993; Kalogerakis et al., 2007). Finally, Table I gives the rough ranges of some key parameters related to SBS processes in common Brillouin media.

4.4.4. Delayed Start of SBS: Hypersonic Field Accumulation Effect Most experimental SBS studies have been performed by using pump laser pulses of 10–20-ns duration. When the pump levels are not too high compared with the threshold, researchers can ﬁnd an obvious delay between the start points of the pump pulse and the SBS pulse. This delay cannot be

241

Stimulated Brillouin Scattering

TABLE I The rough range of key parameters for SBS in common scattering media Scattering media

Brillouin shift (in cm−1 )

Brillouin shift (in GHz)

Hypersonic speed (m/s)

Hypersonic phonon lifetime (ns)

Gases Liquids Solids

0.01–0.05 0.15–0.3 0.3–1

0.3–1.5 4.5–9 9–30

150–500 1000–1700 4000–6000

1–30 0.2–5 1–10

(a) Power (a.u.)

4 Pump pulse Stokes pulse

3 2 1 0 0

10

20 Time (ns)

30

40

SBS intensity

Pump intensity

(b)

0

10 20 30 40 50 60 t (ns)

70

0

10

20

30 40 t (ns)

50

60

70

FIGURE 11 (a) Temporal profiles of the 1064-nm pump pulse (dashed line) and the SBS pulse (solid line) generated in fluorocarbon (FC-75) liquid. (After Kmetic et al., 1998.) (b) Intensity distribution of the pump pulse train (left) and SBS pulse train (right) generated in SF6 gas cell. (After Mullen, 1990; copyright © 1990 IEEE.)

simply explained by the pump intensity threshold requirement. As one of many similar experimental examples, Figure 11a shows the oscilloscopemeasured temporal proﬁles of the 1064-nm pump pulse and the SBS pulse generated in a ﬂuorocarbon (FC-75) liquid sample (Kmetic et al., 1998). From this ﬁgure, one can see that the SBS pulse started near to the peak position of the pump pulse and lasted over the second half of the pump pulse. Considering that the pump pulse has nearly a symmetric pulse shape, for each two equal intensity points, the SBS only occurs in the second half section of the pump pulse. This is not a result of the intensity

242

Stimulated Scattering Effects of Intense Coherent Light

threshold requirement; instead, it is due to the accumulation requirement for the induced hypersonic ﬁeld. The similar result obtained under different conditions is shown in Figure 11b, where the pump source was a pulse train from a mode-locked and Q-switched Nd:YAG laser, the duration of each single pulse was ∼0.2 ns, the pulse spacing was ∼7.5 ns, the width of the whole pulse train was ∼45 ns, and the SF gas pressure was ∼22 atm with an estimated phonon lifetime of ∼18 ns (Mullen, 1990). From the latter ﬁgure, one can see there is an obvious delay before the burst of the ﬁrst SBS pulse. The above-mentioned intrinsic delay of SBS generation essentially reﬂects the non-instantaneity of the interaction between the input laser ﬁeld and the induced hypersonic ﬁeld through the electrostrictive mechanism. The induced acoustic ﬁeld is related to a collective macroscopic movement of great number of particles (molecules, ions, or atoms) of which the medium is composed; therefore, the growth of the induced acoustic wave ﬁeld is slower than the intensity change of the input pump laser ﬁeld. On the other hand, the induced hypersonic ﬁeld must be strong enough to provide a high enough nonlinear reﬂectivity for backward Stokes-shifted optical signals. For these two reasons, there should be a certain accumulation time requirement for the growth of the hypersonic ﬁeld and for the burst of the SBS output signals. Once the acoustical ﬁeld’s accumulation process is accomplished and the SBS is started, the remaining part of the pump pulse proﬁle is easy to reproduce by the SBS through reﬂection from the existing hypersonic grating. This may explain the tail behavior of SBS generation shown in Figure 11. It is understood that when the pump level is getting higher, this accumulation time period will be getting short, which is proved by most experimental observations. The upper limit of this accumulation time should be around the hypersonic phonon lifetime, while the lower limit should be much longer than the period of the supersonic waves. The acoustic ﬁeld accumulation-related decay behavior under various conditions has been reported by many other research groups including the following: Greiner-Mothes and Witte (1986); Wirth et al. (1992); Osborne and O’Key (1992); Nizienko et al. (1994); and Yoshida et al. (1999). Based on all these types of results, one may conclude that it is difﬁcult to generate efﬁcient SBS by using a single pump pulse of duration shorter than 10−9 s.

4.4.5. SBS Pulse Compression Effect Backward SBS is a very straightforward and efﬁcient method to compress the laser pulses of nanosecond duration. The compressed pulse usually exhibits a fast-rising edge and a slowly falling tail. The length of the tail part depends on the pump pulse shape and the pump energy level. When the pump level is not too high comparing to the threshold level, it is

Stimulated Rayleigh-Wing Scattering and Stimulated Thermal Rayleigh Scattering

243

easy to produce a much narrower SBS pulse. In general experiments, the compression ratio between the pump pulse duration and the SBS pulse width can be around 10 (Damzen and Hutchinson, 1983; Hon, 1980; Kuwahara et al., 2000; Marcus, Pearl, and Pasmanik, 2008; Shilov et al., 2001; Tomov, Fedosejevs, and McKen, 1984; Wiggins, Wick, and Rank, 1966). In some other experiments, researchers could even get a much higher pulse compression ratio, i.e., ≥20–50 (Bourne and Alcock, 1984; McIntyre, Boyer, and Rhodes, 1987; Neshev et al., 1999; Schiemann, Ubachs, and Hogervorst, 1997).

5. STIMULATED RAYLEIGH-WING SCATTERING AND STIMULATED THERMAL RAYLEIGH SCATTERING 5.1. Stimulated Raleigh-Wing Scattering The stimulated Rayleigh-wing scattering (SRWS) is reported by Mash et al. (1965) when they observed a broad (10−15 cm−1 ) stimulated scattering spectrum on the Stokes side of the pump line in CS2 (and other Kerr-type liquids), excited by 694.3-nm and ∼13-ns ruby laser pulses. Figure 12 shows the photograph of the F-P interferograms obtained at two different pump levels. According to the authors of this report as well as several other papers (Bloembergen and Lallemand, 1966; Starunov, 1968; Zaitsev et al., 1967a) this stimulated scattering effect could be described by a phenomenological expression of the exponential gain coefﬁcient

g(ν) ∝ ␭0

(2πντ) . 1 + (2πντ)2 50 cm⫺1

(40) ␭0

W

W

(a)

(b)

␭0

␭0

FIGURE 12 Interferograms of SRWS from a CS2 liquid cell at the pump levels of (a) ∼100 MW and (b) 85–90 MW. Free dispersion region of the F-P interferometer is 50 cm−1 . The label W indicates the main spectral range predicted by the SRWS theory. (After Mash et al., 1965.)

244

Stimulated Scattering Effects of Intense Coherent Light

Here, ν = ν0 − ν is the frequency shift of the stimulated scattering, and τ is the molecular reorientation relaxation (or Debye) time of a given Kerr liquid. This expression predicts (i) a negative gain on the anti-Stokes side of the pump line, (ii) a zero gain at the pump line position, and (iii) a positive gain on the Stokes side with the maximum located at the frequency-shift position of νmax = (2πτ)−1 . For CS2 liquid, which manifests the strongest reorientational Kerr effect, τ = 1.5−2 ps, one would expect that the frequency shift of the spectral maximum will be νmax = 2.7−3.5 cm−1 on the Stokes side, which should be easily detected by a F-P interferometer or even a high-resolution grating spectrograph. However, in Figure 12, no spectral maximum on the red side of the pump line could be identiﬁed. Instead, we see a continuous spectral-broadening starting from the pump line position and monotonously decaying toward the long-wavelength direction. Based on the aforementioned comparisons, it is realized that even in the beginning of SRWS studies, there were some inconsistencies between the experimental observations and the theoretical predictions. As another effort to verify the validity of Eq. (40), Denariez and Bret (1968) did a direct gain-curve measurement by using pump-probe approach based on a ruby laser working with a single-axial mode and nanosecond duration. The measured gain data as a function of the frequency shift is shown in Figure 13. To obtain these results, an external signal beam of frequency ν was backward-passed through the nitrobenzene cell pumped with a forward laser beam of ﬁxed frequency ν0 . The input probe (signal) beam was actually a backward SBS beam generated in another liquid cell; by changing different liquids ﬁlled in the second cell, the frequency ν of the signal beam could be varied. In Figure 13, the ﬁtting curves in solid-lines are based on Eq. (40) and the spectral peak on the top curve corresponds to the additional SBS ampliﬁcation in nitrobenzene. Unfortunately, in the spectral region of ν = 0−0.15 cm−1 that should be most essential for comparison, there was no experimental data that could be used for comparison. Therefore, these results cannot be used as strong evidence to support the theoretical prediction given by Eq. (40). From the viewpoint of spectral resolution, it is better to investigate the SRWS effect by using the pump pulses of ns-duration with a very narrow spectral linewidth, i.e., ≤10−2 cm−1 for single-axial mode operation or ≤10−1 cm−1 for using a saturable absorber as the passive Q-switching element. Moreover, if the scattering cell is not too long and the pump level is not too high, it may be better to detect the stimulated backscattering signals to avoid the strong inﬂuence from the pump line. Bol´shov et al. (1970) reported the spectral-broadening properties of the pump line and the SRS line in both forward and backward directions in a CS2 cell of 10–30-cm length, pumped by 532-nm and ∼10-ns laser pulses from a frequencydoubled Nd:glass laser working with a single-axial mode. They observed

Stimulated Rayleigh-Wing Scattering and Stimulated Thermal Rayleigh Scattering

245

g (cm MW⫺1)

10⫺2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

ν0 ⫺ν⬘(cm⫺1)

FIGURE 13 Measured exponential gain coefficient of SRWS amplification in nitrobenzene as a function of frequency shift ν for parallel (top curve) and perpendicular (bottom curve) polarization of the pump beam and the signal beam; the solid lines are the theoretical curves. (After Denariez and Bret, 1968; copyright © 1968 American Physical Society.)

very broad (≥150 cm−1 ) and monotonously decaying spectral-broadening on the Stokes side of the pump line; at the same time, they also observed a noticeable broadening on the anti-Stokes side, which was contrary to the prediction given by Eq. (40). SRWS studies on CS2 liquid had also been conducted by several research groups using 15–30-ps laser pulses as the pump source. In those cases, the pump spectral linewidth was relatively broad (1−2 cm−1 ) due to the transform limit; therefore, only semi-quantitative information about the possible frequency shift and spectral-broadening structure could be extracted from the experiments. Among these studies, Sacchi, Svelto, and Zaraga (1972) reported their observation of a red-shifted sharp line (comparing to the pump linewidth) at the position of ∼2.8 cm−1 from the 694.3-nm pump line; when the input pulse was shorter than ∼5 ps, they further observed several other stimulated scattering lines on the Stokes side at positions of 19.3, 23.8, 27.8, and 33.2 cm−1 . However, these results had not been reproduced by other later experimental studies under similar conditions. For example, Maillotte, Monneret, and Froehly (1990) reported

246

Stimulated Scattering Effects of Intense Coherent Light

on the observation of super-broadband (≥100−400 cm−1 ) stimulated scattering on the Stokes side of the pump line in a 3-cm long liquid cell pumped by 432-nm and 35-ps laser pulses. Miller and Boyd (1992) investigated the stimulated backscattering in a 10-cm long CS2 liquid cell pumped by 532-nm and 25-ps laser pulses of linear or circular polarization. They found that at a high pump level, there was no frequency shift and instead there was a small but measurable broadening to the anti-Stokes side. Wang and Rivoire (1993) pursued a further study on SWRS; they examined the spectral-broadening behavior in both forward and backward directions and measured the temperature effects of the CS2 liquid sample. Their experimental results showed that when the temperature changed from 189 to 279 K, the spectral peak position of the backward SWRS scattering did not change, although based on Eq. (40) this peak position should be a function of Debye time (τ), which would be changed over a very broad temperature range. In the same work, it was also indicated that there was a slight broadening on the anti-Stokes side. It should be pointed that the reported spectral measurements results, obtained near to or slightly higher than threshold pump level, were not regular and reproducible due to the intrinsic instability and competition with other nonlinear processes. Also when the self-focusing-produced ﬁlamentation takes place in a Kerr liquid cell, the stimulated scattering spectrum may be deeply reformed due to the inﬂuence from self-phase modulation. The self-phase modulation often leads to a quasi-modulated spectral structure on both sides of the pump line, as demonstrated by Dorkenoo, von Wonderen, and Rivoire (1998). In conclusion, there are several major features of experimental results that cannot be well explained either quantitatively or qualitatively by the original SRWS theoretical consideration and the representative equation [Eq. (40)]. One may ask what is wrong with this equation. One insight is the role of the key parameter τ in Eq. (40), which is the reorientational relaxation time of the anisotropic liquid molecules. To date, researchers are well familiarized with that a characteristic relaxation time usually determines the spectral linewidth of the scattering process (such as spontaneous Raman and Brillouin scattering) and not the frequency shift. Another problem with the original theory of SRWS is the lack of a clear physical model to explain the speciﬁc mechanism of the SRWS generation.

5.2. Stimulated Thermal Rayleigh Scattering The early theoretical analysis of stimulated thermal Rayleigh scattering (STRS) was given by Herman and Gray (1967). Based on the consideration of intense light ﬁeld-induced density and temperature ﬂuctuations in a linearly absorbing medium, they derived an expression of exponential gain

Stimulated Rayleigh-Wing Scattering and Stimulated Thermal Rayleigh Scattering

247

coefﬁcient for backward STRS generation,

g(ν = ν0 − ν ) = gmax ·

−ν(δν0 + δνR ) . (ν)2 + (δν0 + δνR )2 /4

(41)

Here, ν0 is the pump frequency, ν is the simulated scattering frequency, δν0 is the pump spectral linewidth, and δνR is the spontaneous Rayleigh scattering linewidth. From this equation, one can see that the gain can be obtained only on the anti-Stokes side of the pump line, and at the position of

ν = −(δν0 + δνR )/2,

(42)

the gain reaches its maximum value, which is determined by the material coefﬁcients and is proportional to the linear absorption coefﬁcient of the medium. In liquids, usually δνR is much narrower than δν0 ; hence one would expect an anti-Stokes shift with the value equal to the half of the pump laser linewidth. Following this theoretical prediction, almost at the same time, Rank et al. (1967) reported their experimental results of STRS in transparent liquids (CCl4 and CS2 ) with added iodine as the absorber. The pump source was the 694.3-nm and 10-ns laser pulses with a linewidth varied from 0.019 to 0.025 cm−1 . They measured the spectral shift by using a F-P interferometer and conﬁrmed an anti-Stokes shift of value (∼0.01 cm−1 ) that was near to the value of half of the pump linewidth, as predicted by Eq. (42). The typical results are shown in Figure 14. To obtain these results, the pump beam (its polarization was rotated by 90◦ ) and the BSS beam were simultaneously passed through the F-P interferometer and a specially arranged four-quadrant analyzing sectors. In such a way, as shown in Figure 14a, the pictures in quadrants I and III are the interferograms formed by the STRS beam, while quadrants II and IV are formed by the pump beam. Under this special arrangement, the experimental uncertainty of small spectral-shift measurement might be essentially limited by the system error of the combined four-quadrant analyzer. For example, taking a very careful look on Figure 14a, one can ﬁnd that there is an obvious anti-Stokes shift between quadrants I and IV, whereas there is no evident frequency shift between quadrants I and II or between quadrants II and III. Furthermore, Figure 14b shows that at a low linear absorption coefﬁcient value (α = 0.05 cm−1 ), there is only the backward SBS that can be seen with an obvious Stokes shift. At a slightly higher value of α = 0.10 cm−1 , as shown in Figure 14c, the newly generated STRS line can be seen with the SBS line together. However, in the latter case, apparently the STRS line is also red-shifted, which is in contradiction with the result shown in Figure 14a. For this

248

Stimulated Scattering Effects of Intense Coherent Light

Pump

STRS

(II)

(I)

SBS

STRS SBS

(III)

(IV)

Pump

Pump

Pump

STRS (a)

(b)

(c)

FIGURE 14 F-P interferograms of the 694.3-nm pump line and backward STRS line from an I2 -doped CCl4 solution, measured through a four-quadrant combined analyzer. (a) Linear absorption coefficient α = 0.27 cm−1 and F-P spacing d = 70.4 mm; (b) α = 0.05 cm−1 and d = 6.8 mm; (c) α = 0.10 cm−1 and d = 6.8 mm. (Reproduced from Rank et al., 1967; copyright © 1967 American Physical Society.)

reason, the similar results obtained by using the same four-quadrant analyzer could not be conclusively served as evidence supporting Eq. (42) (Cho et al., 1968; Wiggins et al., 1968). In fact, Bespalov, Kubarev, and Pasmanik (1970) repeated the measurement in the same I2 -added CCl4 (and other transparent solvents) samples pumped by 694.3-nm laser beam of 0.02 cm−1 linewidth, and no frequency shift was observed with a spectral resolution of 0.005 cm−1 , although Eq. (41) predicted a shift value of 0.01 cm−1 . Particularly, Darée and Kaiser (1971) reconsidered the theoretical description of STRS under transient condition and gave a gain coefﬁcient expression with no frequency shift, which was in striking contrast to the original steady-state theory mentioned above. In addition, they also re-measured the spectral behavior of STRS in the same I2 +CCl4 sample and showed no frequency shift within the accuracy of approximately 10−2 cm−1 . During 1967–1970, some Russian researchers had also studied the issue of stimulated scattering around the pump wavelength position in a transparent or linear absorbing medium. Zaitsev et al. (1967b) proposed the concept of “stimulated temperature (entropy) scattering” (STS) in a transparent liquid medium, predicting a Stokes-side shift at the position of the half of the spontaneous Rayleigh scattering line (without considering the inﬂuence of the pump linewidth). They did try to observe this predicted effect in a ruby laser-pumped benzene cell along the side (90◦ ) direction. With this unusual arrangement (i.e., observing the linear scattered signals of the BSS beam), they observed the multi-order SBS lines and an unshifted line (within the spectral resolution) in the F-P interferograms and assumed

Stimulated Kerr Scattering

249

that the latter was the STS line. As they did not observe the backscattering directly, the possibility cannot be excluded that the observed unshifted line was due to the linear scattering of the strong pump beam. Three years later, Kyzylasov, Starunov, and Fabelinskii (1970) extended their original theory of STS in a non-absorbing medium to a linearly absorbing medium and renamed the stimulated scattering in the former case as STS-I type effect and in the latter case as STS-II type effect. They related the ﬁrst type effect to the electrocaloric process and the second type effect to the direct linear absorption at the pump wavelength. According to their modiﬁed theory, for liquids the STS-I process should produce a Stokes-shift, while the STS-II effect should bring an anti-Stokes shift. In both cases, the absolute value of the frequency shift would be |ν| = (δν0 + δνR )/2. However, they experimentally found that in an absorber (I2 )-added liquid (benzene or ethyl alcohol) sample, with large light absorption coefﬁcient, the measured STS-II line shift could exceed the half of the pump linewidth by several times. Besides this, other theoretical analyses further showed that even for absorbing liquids, the STS-II effect can occur either on the Stokes side or anti-Stokes side, depending on the thermal parameter property of the medium (Starunov, 1970; Zel’dovich and Sobel’man (1970)). Looking back the history of early studies of stimulated thermal scattering in absorbing media, one may realize that there are two major problems or uncertainties. First, in spite of discrepancies among these different theoretical predictions, no clear physical model (mechanism) was available to support the proposed theoretical conclusion. Second, all early experiments were based on a single axial-mode laser source with a narrow spectral linewidth (usually ≤0.02 cm−1 ), hence most of the frequencyshift measurements of STRS or STS were actually limited by the spectral resolution of the system used. It is quite obvious that a much conclusive result concerning the possible line shift will be easily obtained by using a pump laser with a broader linewidth (e.g., δν0 ≥ 0.1–1 cm−1 ). Some recently reported results using the pump beam of a broader spectral linewidth will be described in Section 7.

6. STIMULATED KERR SCATTERING 6.1. Observations of Super-Broadband Stimulated Scattering in Kerr Liquid-Core Fiber Systems In 1985, a super-broadband forward stimulated scattering was observed either on the Stokes-side of the pump line position from a CS2 -core ﬁber sample or on the Stokes-side of the SRS line position from a benzene-core ﬁber sample (He et al., 1985). These observations can be repeated with different ﬁber lengths (2 or 7 m) and with different pump wavelengths

250

Stimulated Scattering Effects of Intense Coherent Light

(a)

␭s3

␭s2

␭ s1

␭0

(b)

␭s3

␭s2

␭s1

␭0

(c)

␭s6

␭s5

␭s4

␭s3

␭s2

␭s1

␭0

FIGURE 15 Spectral photographs of the forward stimulated scattering from a 7-m-long liquid-core fiber under various pump intensity levels. (a) CS2 sample, λ0 = 563 nm, Raman shift 656 cm−1 ; (b) C6 H6 sample, λ0 = 532 nm, Raman shift 992 cm−1 ; (c) CCl4 sample, λ0 = 532 nm, Raman shift 460 cm−1 . (After He and Prasad, 1990a.)

(532 nm or ∼560 nm). Later on, more complete experimental results and the physical explanations were presented by He and Prasad (1990a). Figure 15 shows the photographs of the spectral structures of the forward stimulated scattering from a liquid-core hollow ﬁber ﬁlled with CS2 , benzene, and CCl4 , respectively. The ﬁber was 7-m long with a 100-μm core diameter; pump wavelengths were 532 nm (from a Q-switched Nd:YAG laser) or 563 nm (from a pulsed dye laser); pump linewidth was ∼0.05 cm−1 and pulse duration was ∼10 ns. These spectra were measured by a grating spectrograph with a spectral resolution better than 0.03 nm; the pump intensity in the input end of the ﬁber sample could be varied in the range of 102 –103 MW/cm2 . For the CS2 -ﬁlled ﬁber sample, as shown in Figure 15a, more than three orders of Stokes SRS are observed, which can be easily explained by cascaded SRS processes due to high local pump intensity and long gain length. In addition, a super-broadband-stimulated scattering contribution is added on the red side of the transmitted pump line and each SRS line. The broadening range of this additional stimulated scattering contribution can reach 500–700 cm−1 . For the benzene-ﬁlled ﬁber sample, as shown in Figure 15b, we can see a super-broadband-stimulated scattering contribution added to three

Stimulated Kerr Scattering

251

cascaded Stokes SRS lines. However, in this case no super-broadening was added on the pump line position. Finally, for the CCl4 -ﬁlled ﬁber sample, as shown in Figure 15c, we can see more than six orders of Stokes SRS lines. However, no spectralbroadening was observed on the pump and SRS lines. It is well known that CS2 and benzene are typical Kerr liquids consisting of anisotropic molecules; but CCl4 is a non-Kerr liquid consisting of isotropic molecules. Based on these facts, we can reasonably assume that the observed super-broadband-stimulated scattering is inherently related to the reorientational optical Kerr effect. Furthermore, the striking difference of the experimental appearances between Figure 15a and b can be well explained by the theoretical analyses described in the following subsections.

6.2. Physical Models of Stimulated Rayleigh–Kerr Scattering and Stimulated Raman–Kerr Scattering In a liquid consisting of anisotropic molecules, the molecules obey a random orientational distribution if there is no applied external electric ﬁeld. In the presence of an intense optical ﬁeld, the anisotropic molecules may take a ﬁeld-induced reorientational motion along the polarization direction of the optical ﬁeld and the liquid becomes partially anisotropic. This phenomenon is the well-known reorientational optical Kerr effect. Unlike the case of a gaseous medium, the anisotropic molecules in a liquid phase cannot re-orientate freely owing to the rotational viscosity of the liquid. This means that they have to get additional energy to overcome the viscosity and to do the reorientation work. If we assume the liquid is transparent to the incident light beam, i.e., there is no resonant absorption occurring at the wavelength of the applied optical ﬁeld, then a red-shifted photon scattering process is the only way to transfer energy from the optical ﬁeld to the scattering molecule. In this case, a molecule making an light induced reorientation movement can be recognized as a scattering center, and the optical Kerr effect can be recognized as a red-shifted photon scattering process. A microscopic picture of the elementary scattering process of the Kerr effect can be described as follows. A molecule (scattering center) receives additional energy through the annihilation of an incident photon and the simultaneous creation of a red-shifted photon, while the molecule exhausts this additional energy to overcome the rotational viscosity and to do the reorientation work within the liquid (He and Prasad, 1989). Depending on the initial and ﬁnal location of the scattering molecule in its eigen energy levels, the elementary scattering can occur in two possible ways, as shown schematically in Figure 16.

252

Stimulated Scattering Effects of Intense Coherent Light

Rayleigh–Kerr scattering

Raman–Kerr scattering Dν

Dν

ν0

ν9

ν0

ν9

␪9 5 0 ␪

␪ ␪0 5 ␪

␪9 5 0

␪0 5 ␪

E

ν9

ν0 (a)

Dνr

E

ν9

νr

ν0 (b)

FIGURE 16 Schematic diagrams of the elementary Kerr scattering process and induced molecular reorientation in a Kerr liquid: (a) Rayleigh–Kerr scattering and (b) Raman–Kerr scattering. (After He and Prasad, 1990a.)

If the scattering molecule stays in the same ground level at the beginning and at the end of an elementary process, as shown in Figure 16a, a red-shifted photon will be observed on the Stokes side of the pump line. This is the Rayleigh–Kerr scattering, which corresponds to the ordinary optical Kerr effect. It can be assumed that at the beginning of this elementary process, the scattering molecule has an arbitrary initial angle (θ0 = θ) between the molecular axis and the polarization direction of the applied optical ﬁeld; at the end of this scattering process, the molecule has a ﬁnal angle of θ = 0. Here, the molecular axis means the direction along which the molecule possesses a maximum polarizability. In the other case, if the induced reorientation is accompanied by a transition of the scattering molecule from the initial ground level to a higher excited level, as shown in Figure 16b, a red-shifted photon can be observed on the Stokes side of a Raman line. This is the Raman–Kerr scattering model proposed by He, Burzynski, and Prasad (1990). According to conservation of energy, for Rayleigh–Kerr scattering the energy loss of the incident photon equals the reorientation work, and for

Stimulated Kerr Scattering

253

Raman–Kerr scattering the energy loss of the incident photon equals the sum of the reorientation work and the Raman excitation energy. Therefore, for these two cases, the reorientation work W can be expressed as

W(θ) = h(ν0 − ν ) (Rayleigh–Kerr) W(θ) = h(νr − ν ) (Raman–Kerr)

,

(43)

where ν0 is the frequency of the incident ﬁeld, ν is the frequency of the Kerr scattering, and νr is the Raman scattering frequency. Obviously, this reorientation work is an increasing function of the orientation-angle change; in other words, a larger θ means a greater reorientation work W(θ) or a larger frequency shift. Since different molecules have different initial orientation angle θ0 with respect to the applied optical ﬁeld, as a result of contributions from a great number of scattering molecules, we will see a continuous broadband spectral distribution on the red side of the frequency ν0 or the frequency νr . It is quite reasonable to expect that once the product of input pump intensity and the effective gain length is large enough, both Rayleigh–Kerr scattering and Raman–Kerr scattering can become stimulated scattering, generally named as stimulated Kerr scattering (SKS).

6.3. Brief Theoretical Description of SKS 6.3.1. Cross-Section of Kerr Scattering In Section 3, we have given the general expression [Eq. (2)] of the differential cross-section for Raman scattering. Now if the Raman medium is a Kerr liquid, the incident light is linearly polarized, the liquid molecule exhibits a maximum polarizability along the z-axis and a minimum polarizability along the x-axis, then the modiﬁed expression of the differential Raman cross-section can be ﬁnally written as (see He, Burzynski, and Prasad, 1990)

σ=

2π c

4

⎤2

⎡

ac α cos2 θ + αac α sin2 θ ⎥ ν0 νr3 ⎢ zz // xx ⊥ ⎥ . ⎢ α ⎦ ⎣ 2 (4πε0 ) α2⊥ + α2// − α2⊥ cos2 θ

(44)

Here, θ is the angle between the polarization direction and the molecular maximum polarizability (z-axis) direction; the parameters

α// = αzz α⊥ = αxx

(45)

254

Stimulated Scattering Effects of Intense Coherent Light

are the maximum and minimum polarizabilities of a ground-state molecule; and

αac zz αac xx

⎫ 1 (pz )ab (pz )bc (pz )ab (pz )bc ⎪ ⎪ = + ⎪ ⎪ h νba − ν0 νba + νr ⎬ b 1 (px )ab (px )bc (px )ab (px )bc ⎪ ⎪ ⎪ = + ⎪ ⎭ h νba − ν0 νba + νr

(46)

b

are the molecular polarizabilities associated with a Raman transition from the state a to state c. If we assume that of molecu the anisotropy 2 2 2 lar polarizability is not very large so that α// − α⊥ α⊥ , then Eq. (44) becomes

σ=

2π c

4

2 ν0 νr3 ac ac ac 2 cos α + α − α θ . zz xx (4πε0 )2 xx

(47)

Moreover, if the second term in the square brackets is much less than the ﬁrst term, Eq. (47) can be ﬁnally simpliﬁed as

σ=

2π c

4

ν0 νr3 r 2 r r r 2 + 2α − α θ , α α cos // ⊥ ⊥ ⊥ (4πε0 )2

(48)

r ac where αr// = αac zz and α⊥ = αxx are the anisotropic molecular Raman polarizabilities related to the transition from the state a to state c. The ﬁrst term in the square brackets of Eq. (48) is independent of θ and will not vanish even when the molecules are Raman-isotropic, i.e., αr// = αr⊥ . This term represents the ordinary Raman scattering of isotropic molecules in liquid. The second term in the square brackets of Eq. (48) is dependent on θ as well as the Raman anisotropy of the molecule. It, therefore, represents the contribution to the Raman–Kerr scattering. The cross-section of this scattering process is

σ(θ) =

2π c

4

ν0 νr3 r r r 2 α 2α − α ⊥ cos θ. (4πε0 )2 ⊥ //

(Raman–Kerr) (49)

Here, θ = θ0 − θ = θ is the orientation-angle change of the scattering molecule, which is related to the frequency shift of the scattered photon through Eq. (43),

ν = νr − ν = h−1 W(θ).

(50)

Stimulated Kerr Scattering

255

Without any speciﬁc knowledge of the relation between θ and ν, one can generally write

θ = f (ν),

(51)

where f (ν) is an increasing function of ν and can be experimentally determined for a given Kerr liquid. Using the above equation, one can rewrite Eq. (49) as

σ(ν) =

2π c

4

ν0 νr3 r r r 2 f (ν) . (Raman–Kerr) α 2α − α // ⊥ ⊥ cos 2 (4πε0 ) (52)

For a Rayleigh scattering process, a = c, νr = ν0 , and we have

αaa zz

= αzz = α//

⎫ 1 2νba (pz )ab (pz )ba ⎪ = 2 ⎪ ⎪ ⎪ h ⎬ νba − ν02 b

αaa xx = αxx

. 1 2νba (px )ab (px )ba ⎪ ⎪ ⎪ = α⊥ = 2 ⎪ ⎭ h ν − ν2 b

ba

(53)

0

Therefore, from Eq. (44), we can obtain the expression of the cross-section for the generalized Rayleigh scattering

σ=

2π c

4

ν04 2 2 2 2 α cos + α − α θ . // ⊥ (4πε0 )2 ⊥

(54)

Here, the ﬁrst term in the square brackets is independent of θ and will not vanish even when the molecules are isotropic, i.e., α// = α⊥ ; so this term represents the ordinary Rayleigh scattering without any frequency shift. The second term in the square brackets of Eq. (54) is dependent on both the molecular anisotropy and the angle change of θ; this term represents the Rayleigh–Kerr scattering. The cross-section of this scattering process is

σ(ν) =

2π c

4

ν04 (4πε0 )2

α2// − α2⊥ cos2 θ.

(Rayleigh–Kerr) (55)

This expression can also be rewritten in a form similar to Eq. (52), i.e.,

σ(ν) ≈

2π c

4

ν04 2α⊥ (α// − α⊥ ) cos2 f (ν) . (Rayleigh–Kerr) 2 (4πε0 ) (56)

256

Stimulated Scattering Effects of Intense Coherent Light

Knowing the differential cross-section of Kerr scattering, one can further consider the exponential gain property of the SKS process.

6.3.2. Exponential Gain of SKS Since the expressions of the cross-section for Raman–Kerr and Rayleigh– Kerr scattering are known, following the same procedures described in Section 3.1, we can give the following formula for SKS (comparing Eq. (9)):

I(ν , L) = I(ν , 0)e gK I0 L ,

(57)

where I(ν , 0) is the initial intensity of the Kerr scattering signal, I0 is the pump intensity, L is the gain length, and gK is the steady-state exponential gain coefﬁcient by (compare Eq. (11))

gK (θ) = Na F(θ)σ(ν)

λ2 . hν δν

(58)

Here, Na is the molecular density in the ground state, F(θ) is the molecular orientation distribution function, σ(ν) is the cross-section of Kerr scattering determined by either Eq. (52) or Eq. (56), and δν is the spectral width of an elementary Kerr scattering process. Since this elementary process involves an optical ﬁeld-induced change of the molecular orientational states, the average time interval of a molecule staying in a certain orientation state is limited by thermal collisions and is characterized by Debye time τ. In a simple case, this molecular orientation-stay time will be the major factor determining the spectral proﬁle of an elementary Kerr scattering process, which can be written as a Lorentzian function

S(δν) =

[1/(4πτ)]2 . (δν)2 + [1/(4πτ)]2

(59)

From this function, we know that the spectral width of the elementary scattering process is

δν =

1 . 2πτ

(60)

For simplicity, we can approximately replace F(θ) by 1/(4π); then the gain coefﬁcient for red-shifted stimulated Raman–Kerr scattering (SRmKS) can be expressed as

gK (ν = νr − ν ≥ 0) Na 2π 4 ν0 νr2 λ2 r r r 2 · α − α α = // ⊥ ⊥ cos [ f (ν)]. 2π c (4πε0 )2 hδν

(61)

Stimulated Kerr Scattering

257

Similarly, for the red-shifted stimulated Rayleigh-Kerr scattering (SRKS), we have

gK (ν = ν0 − ν ≥ 0) ν03 λ2 Na 2π 4 · α⊥ α// − α⊥ cos2 [ f (ν)]. = 2 2π c (4πε0 ) hδν

(62)

The physical meaning of Eqs. (61) and (62) is that while the molecules complete their induced reorientation along the polarization direction of the applied optical ﬁeld, they will stimulate red-shifted scattered photons that form a continuous broadband spectrum on the red side of ν0 position or νr position. Since different red-shifted spectral components correspond to different orientation-angle changes, the overall red-shifted stimulated scattering spectrum shows the nature of inhomogeneous broadening. On the other hand, at the frequency positions of ν0 and νr , the stimulated scattering contributions mainly come from those molecules that have an initial angle of θ0 ≈ 0. The frequency shift of stimulated scattering from these molecules can be nearly neglected. However, the SKS from these molecules will provide a nonzero contribution on the anti-Stokes side of the ν0 line or the νr line due to the intrinsic spectral proﬁle of the elementary Kerr scattering process so that based on Eq. (59), the spectral gain distribution on the anti-Stokes side can be simply written as

gK (ν ≤ 0) = gKmax ·

[1/(4πτ)]2 . (ν)2 + [1/(4πτ)]2

(63)

Here, gKmax is the maximum exponential gain coefﬁcient in the frequency position of ν0 or νr . The physical meaning of this formula is that the spectral distribution of SKS on the anti-Stokes side shows a feature of homogeneous broadening. Similar to Eq. (13), the threshold condition for observing SKS can be expressed by

e[gk (ν)I0 −β]L >> 1.

(64)

Here, β is the exponential attenuation coefﬁcient of the scattering medium. Since the gain coefﬁcient gK is a function of the frequency shift, the threshold requirements for different spectral components are different; therefore, under different pump levels, the observed spectral width of SKS might be apparently different. Nevertheless, once the pump intensity is high enough, all spectral components of the Kerr scattering can be simultaneously stimulated, and then we will see a complete spectral distribution of the SKS.

258

Stimulated Scattering Effects of Intense Coherent Light

Finally, based on the SKS theory, one can expect the following three features. First, the spectral maximum should be located in the pump line position when SRKS is the dominant mechanism. Second, the spectral maximum will be located in the SRS line position when SRmKS is the dominant mechanism. Third, there should be a measurable broadening on the anti-Stokes side of the pump line or the SRS line.

6.4. Experimental Studies of SKS In this subsection, we shall ﬁrst consider some typical experimental results of forward SRKS in a CS2 -liquid-ﬁlled hollow ﬁber system, pumped by 532-nm and 10-ns laser pulses of 0.85 cm−1 linewidth (He and Prasad, 1990b). Figure 17 shows the normalized spectral intensity distributions of the transmitted 532-nm pump line and the forward SRKS contributions from the CS2 -ﬁlled ﬁber system, measured by a grating spectrometer (with the spectral resolution of 0.4 cm−1 around 532 nm) at different input pump intensity levels. In this ﬁgure, it is shown that the spectral-broadening became more obvious with increasing pump intensity, and there is no spectral maximum at the position (indicated here by an arrow) predicted by the early SRWS theory. According to Eq. (62), the normalized Stokes gain function of SRKS can be simpliﬁed as

gK (ν)/gK (0) = cos2 [ f (ν)],

Normalized intensity

1.0

(65)

Resolution: 0.4 cm⫺1

0.8 0.6 (f) (e)

0.4

(d) (c) (b) (a)

0.2 0.0 18,784

18,788

18,792 Wave

18,796

18,800

18,804

number(cm⫺1)

FIGURE 17 Normalized intensity distributions of the transmitted pump line (λ0 = 532 nm) and the added SRKS contributions from a 2.5-m-long hollow fiber filled with CS2 liquid. The arrow indicates the maximum position predicted by SRWS theory. The pump intensities were (a) 1.3 MW/cm2 , (b) 3.9 MW/cm2 , (c) 13 MW/cm2 , (d) 39 MW/cm2 , (e) 130 MW/cm2 , and (f) 390 MW/cm2 . (After He and Prasad, 1990b; copyright © 1990 American Physical Society.)

259

Stimulated Kerr Scattering

where f (ν) is an unknown increasing function of ν. As a trial function, it can be arbitrarily assumed that

f (ν) = (Aν)B ,

(66)

where A and B are two ﬁtting parameters. For comparison, Figure 18 shows the measured exponential gain curve (thin line) at a pump level of 270 MW/cm2 as well as the ﬁtting curves given by two different theoretical models. The dashed-line curve is given by SRWS theory with a ﬁtting parameter of τ = 2 ps, whereas the thick-line curve is given by SRKS theory using the following best-ﬁtting parameters:

A = 7.5 × 109 , B = 0.148.

(67)

Here, ν is in units of cm−1 and f (ν) is in units of degrees. It is clear that the experimental result can be well ﬁtted by using SRKS theory. On the contrary, the experimental result shown in Figure 18 is totally different from that predicted by the early SRWS theory. On the other hand, the similar broadband-stimulated scattering is also observed in the various-order Stokes SRS lines, which is due to SRmKS in the same CS2 -ﬁlled hollow ﬁber system. This fact plus the results shown in Figure 15a imply that the liquid CS2 exhibits a larger value for both (α// − α⊥ ) and (αr// − αr⊥ ). 1.0

Experimental curve Fitting curve by SRKS theory

Normalized gain

0.8

Fitting curve by SRWS theory

0.6

0.4

0.2

0.0 0

20

40

60

80

100

120

Frequency shift (cm21)

FIGURE 18 Normalized exponential gain distribution of the forward SRKS in a CS2 -filled hollow fiber at a pump level of I0 = 270 MW/cm2 . (After He and Prasad, 1990b; copyright © 1990 American Physical Society.)

260

Stimulated Scattering Effects of Intense Coherent Light

However, under the same conditions, the experimental results obtained from a benzene-ﬁlled hollow ﬁber are not the same as that from a CS2 -ﬁlled hollow ﬁber, although both liquids are typical Kerr media (He, Burzynski, and Prasad, 1990). The main difference is indicated by the broadening structures around the transmitted pump line from these two samples. Figure 19 (left) shows the normalized spectral distributions around the transmitted 532-nm pump line from the benzene-ﬁlled hollow ﬁber at different pump intensity levels. It is shown that even at a high pump level (300 MW/cm2 ), the spectral-broadening still is nearly negligible. This fact implies that the anisotropy of polarizability for benzene molecules in ground state is much smaller than that for CS2 molecules in ground state. However, at same pump levels, a much evident spectral-broadening can be easily observed on the Stokes side of the ﬁrst SRS lines, as shown in Figure 19 (right). This is the typical experimental evidence of SRmKS effect, which can also explain the experimental features shown in Figure 15b. Although the aforementioned results were obtained from the forward stimulated scattering in Kerr-liquid core hollow ﬁber systems, the same effects were also observed in bulk Kerr liquids in both forward and backward directions. As an example, Figure 20 shows the spectral photographs of stimulated backscattering from a 10-cm long CS2 liquid cell, pumped by 532-nm and 10-ns pulses with different spectral linewidth (∼0.1 and l0

ls1

Normalized intensity

(benzene)

18770

18790

18810

(e)

(e)

(d)

(d)

(c)

(c)

(b)

(b)

(a) 18830 17770 Wave number (cm21)

(a) 17790

17810

17830

FIGURE 19 Normalized spectral distributions around the transmitted pump line (λ0 = 532 nm) (left) and the first-order SRS line (λs1 = 562 nm) (right) from a 2.5-m-long C6 H6 -filled hollow fiber at various pump intensity levels: (a) 3 MW/cm2 , (b) 9 MW/cm2 , (c) 30 MW/cm2 , (d) 90 MW/cm2 , and (e) 300 MW/cm2 . (After He, Burzynski, and Prasad, 1990; copyright © 1990 American Institute of Physics.)

Stimulated Kerr Scattering

261

(a)

l0 5 532 nm (b)

ls1 5 551 nm

l0 5 532 nm

FIGURE 20 Spectral photographs of the backward stimulated scattering from a 10-cm-long CS2 -liquid cell pumped by 532-nm laser beam with a spectral linewidth of (a) δν = 1 cm−1 and (b) δν = 0.1 cm−1 . The pump intensity was 150 MW/cm2 , and the spectral resolution was ∼9 cm−1 . The second photograph was over-exposed to show the slight broadening. (After He, Cui, and Prasad, 1997.)

1 cm−1 ) (He, Cui, and Prasad, 1997). When the pump linewidth was ∼1 cm−1 , as shown in Figure 20a, there is a smoothly decreasing superbroadband-stimulated scattering on the red side of the pump wavelength; it resembles the characteristic spectral distribution of the forward SRKS from a hollow ﬁber system. In contrast, pumped by the same wavelength but with a much narrower linewidth (∼0.1 cm−1 ), as shown in Figure 20b, the spectral-broadening on the Stokes side of the pump line position is not so obvious, but there is a strong and slightly broadened ﬁrst-order Stokes SRS line. To explain the difference between these two photographs, we must consider the competition effect among several different stimulated scattering processes. It is known that CS2 is a good medium not only for SKS but also for SRS and SBS. Among them, the threshold requirements for SBS and SRS are more sensitively dependent on the pump linewidth than SRSK. Under the broader (∼1 cm−1 ) linewidth pump condition, the SRSK process dominates over other two processes; whereas at a narrower (∼0.1 cm−1 ) pump linewidth, the SBS and SRS processes become the dominant. The observation with a F-P interferometer veriﬁed that the strong sharp line shown in Figure 20b was indeed due to the backward SBS in CS2 liquid.

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Stimulated Scattering Effects of Intense Coherent Light

1.0 0.8

Resolution

0.6

Normalized gain

0.4 0.2 0.0 20.2 20.4 20.6 20.8 21.0 220

0

20 40 Frequency shift (cm21)

60

FIGURE 21 Normalized exponential gain curves for backward stimulated scattering based on the measured data (solid-line) and fitting (dashed-line) by SRWS theory with an assumed value of τ = 1.5 ps. The pump intensity was I0 ≈ 500 MW/cm2 , linewidth 1 cm−1 , and the spectral resolution ∼0.48 cm−1 . (After He, Cui, and Prasad, 1997.)

For comparison, Figure 21 shows the measured normalized exponential gain curve (solid line) of the backward SRKS from the CS2 liquid cell, pumped with the 532-nm laser of ∼1-cm−1 linewidth at an intensity level of 500 MW/ cm2 . In the same ﬁgure, the dashed-line curve is predicted by the SRWS theory. Once again, there is a severe discrepancy between the experimental results and the SRWS theory. So far, we have only considered the results from either a Kerr-liquid core ﬁber sample or a short liquid cell sample, both pumped by ∼10-ns laser pulses. However, using the same sample but pumped by laser pulses with drastically different duration (10−8 –10−13 s), the basic picture of the spectral-broadening due to SKS remains unchanged. For example, by using ∼600-nm and 0.5-ps laser pulses as the pump source, the forward SKS effects can be efﬁciently generated in a 10-cm Kerr liquid cell (He et al., 1993), as well as in a Kerr-liquid-core ﬁber system (He et al., 1991). When the pump pulses are in picosecond or subpicosecond regime and the Kerr liquid is ﬁlled in a long hollow ﬁber or waveguide device, the SBS is not allowed while the forward SKS gain can be extremely high so that the cascaded SKS processes become more important. In this case, the depletion of intensities around the original pump wavelength and the neighboring spectral region becomes more severe, while the accumulation of the cascaded processes may form a new spectral peak at a longer wavelength

Stimulated Rayleigh–Bragg Scattering

263

position. The location of this new spectral peak will depend on the pump level and the interaction length in the Kerr medium (He and Xu, 1992). The similar intensity-dependent spectral peak shift was also observed in a 1-m long CS2 -ﬁlled glass capillary of ∼0.5-mm diameter, pumped by 532-nm and 25–40-ps laser pulses (Wang et al., 1995; Zhou, Wang, and Yu, 1990). Regarding SRmKS, there are other experimental evidence demonstrated in a benzyl alcohol-ﬁlled hollow ﬁber (Qiu, Lu, and Lu, 1991) and phenylethanol-ﬁlled hollow ﬁber (Lu et al., 1992), pumped by 532-nm and 10-ns laser pulses of 0.82 cm−1 linewidth. Finally, it should be noted that the mechanisms of SKS can be used to realize a new type of optical ampliﬁer, which can efﬁciently amplify a super-broadband optical signal (He and Xu, 1992; He et al., 1989).

7. STIMULATED RAYLEIGH–BRAGG SCATTERING 7.1. Discovery of Frequency-Unshifted Stimulated Scattering in a Two-Photon-Absorbing Medium It is mentioned in Section 5 that the early theories of STS in a linearly absorbing medium predicted a frequency shift determined by the half of the pump linewidth. During the same time period, most of the early experiments were performed by using a very narrow pump linewidth (≤0.02 cm−1 ) and the predicted frequency-shift values were within the experimental uncertainty. Therefore, the experimental veriﬁcation of various proposed theoretical predictions had not been completed. In the other respect, the possible 2PA contribution to the stimulated thermal scattering in a pure organic solvent (such as benzene) was ﬁrst proposed by Boissel et al. (1978), although there was a lack of speciﬁc identiﬁcation of the observed stimulated backscattering in benzene pumped by 347-nm and 8-ns laser pulses. The same possibility was also mentioned by Karpov, Korobkin, and Dolgolenko (1991) to explain their observation in a pure liquid hexane sample, pumped by 308-nm and 8-ns laser pulses. Recently they observed the BSS, of which the predicted frequency shift was smaller than the experimental uncertainty, while the linear absorption value was much lower than that required by stimulated thermal scattering theories. Based on this fact, they assumed that the observed stimulated scattering might be attributed to 2PA-induced heating effect at 308-nm in hexane liquid sample (Karpov and Korobkin, 2005). In 2004, He, Lin, and Prasad (2004) reported the ﬁrst unambiguous observation of BSS in a two-photon-absorbing dye-solution sample, pumped by 532-nm and 10-ns laser pulses. The ﬁrst feature of this observation is that there is no frequency shift within the spectral resolution that is much narrower than the half of the pump linewidth. The second feature

264

Stimulated Scattering Effects of Intense Coherent Light

is that without two-photon-absorbing dye, only SBS can be observed in the pure solvent, which implies that the newly observed frequency-unshifted BSS is related with a two-photon excitation process. The third feature is that under same experimental conditions, the pump threshold requirement for observing frequency-unshifted scattering in a two-photon-absorbing dye solution is much lower than that for observing SBS in a pure solvent. To explain this new observation, a straightforward physical model is proposed. According to this model, the feedback mechanism of the frequency-unshifted BSS is the reﬂection from an induced standing-wave Bragg grating formed by the strong pump wave and very weak backward Rayleigh scattering beam. Based on this physical explanation, this type of stimulated scattering is termed as stimulated Rayleigh–Bragg scattering (SRBS).

7.2. Physical Model of SRBS Here, let us consider a linearly transparent but two-photon (or multiphoton)-absorbing medium pumped by a strong laser beam. In our case, the backward-propagating spontaneous Rayleigh scattering can be thought of as an original seed signal, which may interfere with the forward-propagating pump wave to form a standing wave with a spatially modulated intensity distribution. This intensity modulation may further induce an intensity-dependent refractive-index change and create a stationary Bragg grating. A Bragg grating formed in such a way will offer a nonzero reﬂectivity for both the strong forward pump beam and the very weak backward scattering beam. However, the absolute value of the energy reﬂected from the pump beam to the scattering beam will be much greater than that from backward-scattering beam to the pump beam. As a net result, the backward-scattering seed beam becomes stronger, as schematically shown in Figure 22. Moreover, a slightly stronger backward-scattering seed beam will enhance the modulation depth of the induced Bragg grating and consequently increase the reﬂectivity of this grating, which means more energy will transfer from the pump beam to the backward-scattering beam. These processes may ﬁnally make the backward-scattering signals to be stimulated. One can see that there is a typical positive feedback mechanism provided by a stationary Bragg grating formed by two counterpropagating beams. Upon this assumption, 2PA may play a particularly important role in the following two senses. First, accompanying with 2PA, there is a resonance-enhanced refractive-index change that is necessary for forming an effective Bragg grating. Second, 2PA leads to an attenuation inﬂuence mainly on the strong pump beam not on the much weaker backward-scattering beam. In contrast, for a linear absorbing medium, the linear attenuation ratio is the same for both strong pump beam and weak scattering beam, which may prohibit the latter from being ﬁnally

Stimulated Rayleigh–Bragg Scattering

265

Multi-photon absorbing medium

Grating,s reflection Reflected pump beam

Gained (stimulated) backward scattering beam

Forward pump beam Backward scattering beam Induced Bragg grating

FIGURE 22 Schematic illustration for the formation of induced stationary Bragg grating inside the scattering medium, as well as the energy transfer from the strong pump beam to the weak backward scattering beam through grating’s reflection.

stimulated. This can explain why the effect of frequency-unshifted SRBS is much easier to be observed in a two-photon-absorbing medium than in a linear absorbing medium. Comparing SRBS with SBS, one may ﬁnd one thing in common: in both cases, the reﬂection of the pump beam from the induced grating is the only mechanism providing a positive feedback for the initial weak seed signals. The difference is that for SBS, the induced grating is a traveling wave grating and the reﬂected beam experiences a frequency shift due to Doppler effect, whereas for SRBS the induced is a stationary (standingwave) grating that brings reﬂection without any frequency shift. According to the model described above, the key factor for SRBS generation is the induced refractive-index change instead of the pump ﬁeld-induced heating effect. This is because the reﬂectivity of the induced Bragg grating is essentially determined by the modulation depth of induced refractive-index changes (n) in a nonlinearly absorbing medium. There are various mechanisms that can possibly contribute to n, including population change of the absorbing molecules among their different energy levels, nonlinear refractive-index dispersion effect related with nonlinearly absorbing molecules, stationary electrostriction effect and/or possible local heating effect. Different mechanisms have different temporal response features; therefore, through specially designed experiments, it can be determined which mechanism is dominant under given conditions. Based on the same model, one can expect that when the intensity levels of two beams (forward pump and BSS) approach nearly equal to each other, no energy transfer between these two beams will go further; hence, the maximum steady-state nonlinear reﬂectivity should be around 50%.

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7.3. Pump Threshold Requirement of SRBS To give a mathematical description for the proposed Bragg grating model, we can write the intensity of the overall optical ﬁeld inside the scattering medium as

I(z) = (IL + IS ) + 2 IL IS cos(4πn0 z/λ0 ).

(68)

Here IL is the intensity of the forward pump beam, IS the intensity of backward Rayleigh scattering beam, and n0 is the linear refractive index of the scattering medium at the pump wavelength λ0 . This periodic intensity modiﬁcation will produce a refractive-index change with the same spatial period (λ0 /2n0 ) due to third-order nonlinear polarization effect. The spatial modulation of the intensity-dependent refractive-index change can be expressed as

n(z) = n2 I(z) = 2n2 IL IS cos(4πn0 z/λ0 ) = δn cos(4πn0 z/λ0 ). (69) Here, n2 is the nonlinear refractive-index coefﬁcient, the value of which for a given medium is dependent on the speciﬁc mechanism of induced refractive-index change (as we mentioned before); δn is the amplitude of the spatial refractive-index modulation. Light-induced periodic refractive-index changes inside a nonlinear medium may create an induced Bragg phase grating, which in turn provides an effective reﬂection for both beams by the same reﬂectivity R. In an experiment using bulk liquid or solid sample, the laser-induced Bragg grating is essentially a thick hologram grating with a cosinoidal spatial modulation, and its reﬂectivity is simply given by well-known coupled wave theory of thick hologram gratings (Kogelnik, 1969):

R = th2 (πδnL/λ0 ).

(70)

Here L is the thickness of the grating or the effective gain length inside the scattering medium. From Eq. (69), we have

δn = 2n2 IL Is . Then Eq. (70) becomes

R = th2 2πn2 IL Is · L/λ0 .

(71)

The threshold condition for stimulating the backward Rayleigh scattering can be expressed as

RIL >> IS {1 − exp[−α(λ0 )L]},

(72)

Stimulated Rayleigh–Bragg Scattering

267

where α(λ0 ) is the residual linear attenuation coefﬁcient at λ0 . Under threshold condition, we can assume R << 1 and α(λ0 )L << 1, the hyperbolic tangent function in Eq. (71) can be replaced by its arguments, then Eq. (72) can be ﬁnally simpliﬁed as

(2πn2 /λ0 )2 LIL2 >> α(λ0 ).

(73)

The physical meaning of the above condition is that for a given pump level of IL , the backward SRBS is easier to be observed in a two-photon-absorbing medium possessing a larger n2 value, a longer gain length L, and a smaller optical attenuation coefﬁcient α(λ0 ) at the pump wavelength.

7.4. Experimental Results of SRBS in a Multi-Photon-Absorbing Medium The nonlinear medium for the ﬁrst observation of SRBS was a two-photonabsorbing dye (PRL802) solution in tetrahydrofuran (THF of spectroscopic grade); this dye solution is highly transparent at 532-nm wavelength but shows an efﬁcient 2PA at the same wavelength. A 1-cm long quartz cuvette ﬁlled with dye solution of 0.01 M concentration was pumped by 532-nm and 10-ns pulses with two signiﬁcantly different spectral linewidths, i.e., ∼0.08 cm−1 and ∼0.8 cm−1 . In both cases, once the input pump intensity level was higher than a certain threshold level, which was lower than the threshold level for generating SBS or SRS in the pure solvent, a stimulated backscattering beam with no frequency shift could be observed. When pumped with the narrow spectral line, a F-P interferometer was used to measure the spectral structure of the stimulated scattering with a spectral resolution of 0.025 cm−1 ; when pumped with the broad spectral line, the stimulated scattering spectrum was measured by the grating spectrometer with a spectral resolution of 0.11 cm−1. Figure 23 shows the interferograms of (a) the half of the pump beam, (b) the backward SRBS from the dye solution cell, and (c) both beams together. All these photographs were taken by a single pulse shot at an intensity level ﬁve times higher than the threshold value of ∼40 MW/cm2 . From Figure 23a and b, one can see that both the BSS beam and the input pump beam have nearly the same spectral width of ≈ 0.08 cm−1 . Furthermore, from Figure 23c, one can see that there is no frequency shift between these two beams within the spectral resolution of 0.025 cm−1 . This is in contrast to an anti-Stokes shift of /2 ≈ 0.04 cm−1 predicted by the early theory of STRS in a linearly absorbing medium. Finally, shown in Figure 23d is the interferogram of the half of the pump beam and the SBS beam from a 1-cm pure solvent (THF) sample, at a pump level two times higher than the SBS threshold of ∼100 MW/cm2 .

268

Stimulated Scattering Effects of Intense Coherent Light

(a)

(b)

(c)

(d)

FIGURE 23 F-P interferograms of (a) the half of the 532-nm input pump, (b) the backward SRBS beam from a 1-cm PRL802/THF solution, (c) both beams together, and (d) the half of the pump beam and the backward SBS beam from a 1-cm pure solvent (THF) together. Pump linewidth was ∼0.08 cm−1 and the free spectral range of the F-P interferometer was 0.5 cm−1 . (After He, Lin, and Prasad, 2004.)

Figure 24 shows the spectral photographs for the input pump line of ∼0.8 cm−1 linewidth, the output backward SRBS line, and the both lines together (He et al., 2005). Once again, there is no measured frequency shift within a spectral resolution of 0.11 cm−1 , which was much smaller than half of the pump linewidth. The same frequency-unshifted SRBS effect is also observed in a threephoton-absorbing dye solution (PRL-OT04 in chloroform) pumped by 1064-nm and 10–20-ns laser pulses (He, Zheng, and Prasad, 2007), as well as in a two-photon-absorbing semiconductor quantum rod solution sample pumped by the same laser pulses (He et al., 2008b). In conclusion, the preliminary results show that SRBS effect can be efﬁciently generated in any type of multi-photon-absorbing media (He et al., 2008a). Comparing with other known stimulated scattering effects (such as SBS and SRS), SRBS exhibits two favorable features for applications, i.e., the low-pump threshold and no-frequency shift. On the other hand, there are two limitations related to this new effect. First, the nonlinear reﬂectivity of this backward SRBS process should be ≤50% due to the feedback mechanism discussed in the preceding subsection. Second, similar to the case of SBS, so far the SRBS has been generated only by using the laser pulses in the nanosecond-regime and with a spectral linewidth ≤1 cm−1 . Under these conditions, the effective gain length is essentially

Phase-Conjugation Property of Backward-Stimulated Scattering

269

(a)

(b)

(c)

0.63 cm21

FIGURE 24 Spectral photographs of (a) the backward stimulated scattering beam (top), (b) the input 532-nm pump beam (bottom), and (c) both beams together. The spectral resolution is 0.11 cm−1 . (After He et al., 2005; copyright © 2005 American Physical Society.)

determined by the short one between the longitudinal coherent length and the path-length of the scattering sample. When the pump pulse duration is getting much shorter, the spectral linewidth is getting broader due to uncertainty relationship; the effective thickness of the induced Bragg grating will ultimately limited by the coherent length of the pump beam.

8. PHASE-CONJUGATION PROPERTY OF BACKWARD-STIMULATED SCATTERING 8.1. Definition and Unique Feature of Optical Phase-Conjugate Wave The term optical phase conjugation is usually used to describe the wavefront reversion property of a backward-propagating optical wave with respect to a forward-propagating wave. Suppose there is an input quasi-plane monochromatic wave with arbitrary phase distortion deviated from an

270

Stimulated Scattering Effects of Intense Coherent Light

ideal plane-monochromatic wave, i.e.,

E(z, x, y, ω) = E(z, x, y)e−iωt = A0 (z, x, y)ei[kz+ϕ(z,x,y)] e−iωt .

(74)

Here, z is the longitudinal variable along the propagation direction, x and y are the transverse variables along the beam section, ω is the circular frequency of the optical ﬁeld, k = n0 /(2πλ) is the magnitude of the corresponding wave vector, n0 is linear refractive index of the propagating medium, E(z, x, y) is the complex amplitude function, A0 (z, x, y) is the real amplitude function, and ﬁnally, ϕ(z, x, y) is the phase-distortion function describing the deviation of the real wavefront from an ideal plane wave. If there is a backward-propagating wave, which can be expressed as

E (z, x, y, ω) = a · E∗ (z, x, y)e−iωt = a · A0 (z, x, y)e−i[kz+ϕ(z,x,y)] e−iωt , (75) where a is a real constant, then the ﬁeld E (z, x, y, ω) is deﬁned as backward frequency-degenerate phase-conjugate (PC) wave of the original forward ﬁeld E(z, x, y, ω). After passing through a disturbed medium, an ideal plane monochromatic wave will experience a certain aberration inﬂuence, and its ﬁeld function can be expressed by Eq. (74) by introducing the phase distortion factor ϕ(z, x, y). After being reﬂected from an ordinary plane mirror and backward-passing through the same disturbed medium, the aberration inﬂuence on the beam will be accumulated, as schematically shown in Figure 25a. However, if instead of the plane mirror, there is a special nonlinear medium that can provide a PC wave expressed by Eq. (75), after backward-passing through the same disturbed medium, the aberration inﬂuence can be totally removed, as shown in Figure 25b. This is the very unique feature of a backward PC wave based on which researchers may design special systems to realize aberration-free propagation, ampliﬁcation, and restoration of coherent optical signals. The concept of optical phase conjugation can be extended to the frequency-nondegenerate situation. In this case, we have a backwardpropagating ﬁeld with different frequency, which can be expressed as

E (z, x, y, ω ) = a · E∗ (z, x, y)e−iω t = a · A0 (z, x, y)e−i[k z+ϕ(z,x,y)] e−iω t , (76) where E (z, x, y, ω ) is called the backward frequency-nondegenerate PC wave of an original ﬁeld E(z, x, y, ω) expressed by Eq. (74). When a frequency-nondegenerate PC wave backward-passes through the same disturbed medium, as shown in Figure 25b, the aberration inﬂuence, indicated in Eq. (76) by the function ϕ(z, x, y), can also be removed.

Phase-Conjugation Property of Backward-Stimulated Scattering

(a) Input wave

Transmitted wave

Output wave

Reflected wave

Mirror

Disturbed medium

(b) Input wave

271

Transmitted Reflected wave wave

Output wave

Disturbed medium

Phase conjugate reflector

FIGURE 25 (a) An ordinary reflected wave backward-passes through a disturbed medium; the aberration influence is accumulated; (b) a phase-conjugate (PC) wave backward-passes through the disturbed medium; the aberration influence is removed. (After He, 2002.)

In practice, there are three major approaches to generate PC waves: the ﬁrst is using the degenerate FWM method to generate frequencydegenerate PC wave; the second is using the backward-stimulated scattering method to generate nondegenerate PC wave; the third is using the backward cavityless lasing method to generate nondegenerate PC wave (see the review by He, 2002).

8.2. The Mechanism of Generating PC Wave by Backward–Stimulated Scattering The ﬁrst observation of optical phase-conjugation phenomenon was made in an SBS experiment by Zel’dovich et al. (1972). However, during that time period, this observation could not be well explained by the known theory of SBS. For this reason, the results did not receive much attention in research community until 1977, when the theories of PC wave generation via threewave mixing and FWM were suggested (Hellwarth, 1977; Yariv, 1976, 1977) and experimentally proven (Arizonis et al., 1977; Bloom and Bjorklund, 1977). Since then the studies of PC wave generation via backward SBS, SRS, and other types of stimulated scattering have become more interesting for researchers because of its simplicity and high efﬁciency.

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Stimulated Scattering Effects of Intense Coherent Light

Though various types of backward-stimulated scattering (BSS) can be utilized to efﬁciently generate PC waves, it had taken quite a long time for researchers to understand the nature of these processes and to give a clear physical explanation without invoking any cumbersome mathematical treatments. There is a simple and very basic question to ask: why the mechanisms for generating different types of BSS (such as SBS, SRS, and SRWS) are totally different yet they can be used to generate PC wave? To answer this question, one needs to consider a basic platform on which all types of BSS stand for manifesting a phase-conjugation property. The ﬁrst insight was made in 1978 when it was suggested that the phaseconjugation nature of BSS might be connected with a special holographic process occurring in the scattering medium (Kudryavtseva et al., 1978; Sokolovskaya, Brekhovskikh, and Kudryavtseva, 1978). Later, a quasicollinear FWM model was proposed to explain the phase-conjugation property of the BSS, and this model was supported by a rigorous mathematical formulation in the unfocused-beam approximation (He, Liu, and Liu, 1986). According to that model, the PC wave generation via any type of BSS can be visualized as a nondegenerate FWM process; the input pump beam contains two waves (E1 and E2 ) of frequency ω0 , while the BSS beam contains other two waves (E3 and E4 ) of frequency ω . The total backward BSS ﬁeld (E3 + E4 ) can be phase-conjugative with the total input pump ﬁeld (E1 + E2 ), i.e., (E3 + E4 ) ∝ (E1 + E2 )∗ . To explain this model more clearly, it is better to invoke Gabor’s original idea of holograph. In that case, with a coherent light wave passing through a transparent object (phase object), the object is assumed to be such that a considerable part of the wave penetrates undisturbed through it, and a hologram is formed by the interference of the secondary wave arising from the presence of the object with the strong background wave, as clearly described by Born and Wolf (1983). From this viewpoint, after passing through a phase object, the total optical ﬁeld can be expressed as a superposition of two portions:

U = U (i) + U (s) = A(i) eiϕi + A(s) eiϕs = eiϕi [A(i) + A(s) ei(ϕs −ϕi ) ].

(77)

Here, U (i) is the undisturbed part of the transmitted ﬁeld; U (s) is the disturbed part arising from the presence of the object; A(i) and A(s) are their amplitude functions; ϕi and ϕs are the corresponding phase functions. The Gabor principle is applicable to most phase-conjugation experiments based on BSS. In these cases, as schematically shown in Figure 26, E0 (ω0 ) is a quasi-plane pump wave. After passing through an aberration plate or a phase subject, the pump ﬁeld manifests itself as a superposition of two portions: a stronger undisturbed wave E1 (ω0 ) and a weaker distorted wave E2 (ω0 ). These two waves interfere with each other in a scattering (gain) medium and create an induced volume holographic grating due to intensity-dependent refractive-index changes of the medium. Only the undisturbed pump wave E1 (ω0 ) is strong enough to fulﬁl the threshold

273

Phase-Conjugation Property of Backward-Stimulated Scattering

Aberration plate

Scattering medium E2

E4 E3

E1 E4

(Grating)

E0

E2 (Plane pump wave)

[E3 (␻9) 1 E4 (␻9)] 5 a [E1 (␻0) 1 E2 (␻0)]∗ E1: E2: E3: E4:

Undistorted pump wave; Distorted pump wave; Initial backward stimulated scattering wave (reading beam); Diffracted wave from the induced grating.

FIGURE 26 Schematic illustration of the nondegenerate FWM model for phaseconjugation formation of the backward stimulated scattering.

requirement and to generate an initial BSS wave E3 (ω ) that exhibits a regular wavefront as wave E1 (ω0 ). When wave E3 (ω ) passes back through the induced holographic grating region, a diffracted wave E4 (ω ) can be created. Here, we see a typical holographic wavefront-reconstruction process: the induced grating is formed by the regular E1 (ω0 ) wave (reference beam) and the irregular E2 (ω0 ) wave (signal beam), the initial BSS E3 (ω ) is a reading beam with a regular wavefront, and the diffracted wave E4 (ω ) will be the PC replica of the E2 (ω0 ) wave. Moreover, the wave E4 (ω ) will experience an ampliﬁcation with the wave E3 (ω ) together because both waves have the same scattering frequency. In the case of SBS, ω0 ≈ ω , it is a nearly degenerate quasi-collinear FWM process. In the case of SRS, ω0 > ω , there is a nondegenerate and frequency down-converted FWM process. Based on the explanation described earlier, one can see a common mechanism (pump ﬁeld-induced holographic grating) plays the key role for phase-conjugation formation by using BSS method. This mechanism is applicable to all types of BSS processes. Even in the case of using a focused pump laser beam, the relationship (E3 + E4 ) ∝ (E1 + E2 )∗ still holds, provided that some requirements about the pump level, effective gain length, and aberration extent can be fulﬁlled (Liu and He, 1999).

8.3. Phase-Conjugation Experimental Studies on BSS The majority of experimental studies on phase-conjugation properties have been pursued via SBS, mainly motivated by the need of highbrightness and high-energy laser oscillator/ampliﬁer systems for military

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Stimulated Scattering Effects of Intense Coherent Light

applications and laser fusion purpose. The major advantage of using SBS to generate PC waves is the very small frequency shift and high nonlinear reﬂectivity; the disadvantage is that the pump pulse duration should be in the nanosecond-regime. There are also a number of experimental studies that have been done via backward SRS (e.g., Mays and Lysiak, 1979; Sokolovskaya, Brekhovskikh, and Kudryavtseva, 1977; Tomov et al., 1983; Zel’dovich et al., 1977). The advantage of using SRS to generate PC waves is that the pump pulse duration can be in picosecond-regime or even shorter, but the frequency shift is much larger than other types of stimulated scattering. Moreover, the phase-conjugation property was also observed via SRWS (Ferrier et al., 1982; Miller, Malcuit, and Boyd, 1990). The phase-conjugation studies via stimulated scattering in liquid crystals were reported by Khoo, Li, and Liang (1993), Khoo and Liang (2000). Very recently, superior phase-conjugation performances have been demonstrated via SRBS (He et al., 2008b; He, Zheng, and Prasad, 2007). Figure 27 shows a typical experimental set-up for testing the phaseconjugation property of a BSS beam from a given scattering medium. The input pump beam passes through a diaphragm (or mask) for bearing certain near-ﬁeld information, an aberrator for receiving aberration inﬂuence, and then a lens for being ﬁnally focused on the scattering medium. The complete study of phase-conjugation properties for a given BSS beam involves the near-ﬁeld, far-ﬁeld, and polarization status measurements. If the BSS beam is a perfect PC wave of the input pump beam, the results of these three measurements should be exactly the same for both beams. In practice, different experimental studies might be focused on one of these Aberrator

Scattering medium

Beam splitter Diaphragm (or mask)

(Pump beam) Lens (BSS beam) Lens

(BSS beam)

eld r-fi n Fa ectio t de

Near-field detection

CCD camera

FIGURE 27 Optical setup for phase-conjugation experiments based on backward stimulated scattering.

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Phase-Conjugation Property of Backward-Stimulated Scattering

three measurements. From the viewpoint of generating high-brightness coherent light source, the far-ﬁeld ﬁdelity measurement is most important (e.g., Bruesselbach et al., 1995; Nosach et al., 1972; Schelonka and Clayton, 1988; Wang and Giuliano, 1978; Whitney, Duignan, and Feldman, 1982, 1990). For application of optical information reconstruction, the near-ﬁeld ﬁdelity measurement is more interesting (e.g., Kralikova et al., 2000; Slatkine et al., 1982; Sokolovskaya, Brekhovskikh, and Kudryavtseva, 1978, 1983). (b)

(d)

)

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0.8 0.7 0.6 0 .5 0.4 0.3 0 .2 0.1 X (mm 0.0 0.0 )

)

1.0 0.8 0.6 0.4 Y 0.2

)

1.0 0.8 0.6 0.4 Y 0.2

(m

(a)

0.0

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(m

0.8 0 .7 0.6 0.5 0 .4 0.3 0.2 0 X (mm .1 0.0 0.0 )

m

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1.0 0.8 0.6 0.4 Y 0.2

FIGURE 28 Far-field patterns for (a) the pump beam without inserting an aberrator, (b) the backward SRBS beam without inserting an aberrator, (c) the pump beam passing through the aberrator once, (d) the pump beam passing through the aberrator twice, and (e) the backward SRBS beam after passing through the aberrator. Focal length was 50 cm, and the pump energy levels for (b) and (e) are 9 and 15 mJ, respectively. (After He et al., 2008b; copyright © 2008 IEEE.)

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If the aberration inﬂuence is in the form of a small wavefront distortion, researchers can utilize the interferometer method to precisely estimate the phase-conjugation ﬁdelity of the BSS beam (e.g., Lefebvre, Pfeifer, and Johnson, 1992; Liu and He, 1999; Munch, Wuerker, and LeFebvre, 1989; Schelonka, 1987). Some other experiments were related with the compensation of polarization distortion via BSS (e.g., Carr and Hanna, 1987; Miller, Malcuit, and Boyd, 1990; Seidel and Kugler, 1997). Finally, as a latest example of phase-conjugation measurement performed by a backward SRBS beam without any frequency shift, Figure 28 shows the measured far-ﬁeld patterns of the input pump beam, the transmitted pump beam, and the BSS beam under the condition with and without inserting an aberrator (He et al., 2008b). There are two salient features shown in this ﬁgure. First, without inserting the aberrator, the divergency angle of the SRBS beam is even smaller than the input pump beam; this is due to the combination of threshold requirement of local gain and the nonlinear dependence of local gain on the local pump intensity. Second, even with inserting an aberrator on the path of the pump beam, the inﬂuence of this aberrator on the BSS beam is basically removed, indicating a superior phase-conjugation property.

ACKNOWLEDGMENTS The author is grateful to Dr. P. N. Prasad, Distinguished Professor and Executive Director of Institute for Lasers, Photonics and Biophonics, State University of New York at Buffalo, for his valuable support in conducting research activity. The author is also indebted to Dr. Charles Lee for his long-term support through the research projects granted by the U.S. Air Force Ofﬁce of Scientiﬁc Research. Finally, contributions from his colleagues and collaborators in the research on this broad subject are highly appreciated.

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CHAPTER

5 Singular Optics: Optical Vortices and Polarization Singularities Mark R. Dennis*, Kevin O’Holleran† and Miles J. Padgett†

Contents

1 Introduction 2 Historical Review of Vortices in Optics 2.1 Wolter’s Vortex in the Goos–Hänchen Shift for Refraction 2.2 Vortices in Superposition and Diffraction 2.3 Singularities in the Cusp and Diffraction Catastrophes 2.4 Vortices in Random Optical Fields 2.5 Wave Dislocations 3 Creating an Optical Vortex 3.1 Optical Vortices and Laguerre–Gaussian Beams 3.2 Direct Generation of Vortex Beams from Laser Systems 3.3 Generation of Vortex Beams Using Diffractive Optics 3.4 Generation of Vortex Beams Using Lenses and Spiral Phase Plates 3.5 Generation of Vortex Beams Using Spatial Light Modulators 3.6 Vortex Beams with Low Temporal or Spatial Coherence 4 The Structure of Optical Phase Singularities 4.1 The Singular Skeleton of an Optical Field 4.2 Local Anisotropy of Vortex Points 4.3 Topological Reactions of Vortex Points: Unfolding, Creation, and Annihilation 4.4 Vortex Line Geometry in Three Dimensions 4.5 Vortices in Simple Superpositions of Plane Waves 5 Vortices in Random Wavefields 5.1 Statistical Properties of Vortices in Two- and Three-Dimensional Random Waves

294 297 299 300 302 304 305 307 307 308 309 310 311 314 316 316 318 320 321 324 327 329

* H H Wills Physics Laboratory, University of Bristol, Bristol, UK † Department of Physics and Astronomy, University of Glasgow, Glasgow, UK

Progress in Optics, Volume 53, ISSN: 0079-6638, DOI: 10.1016/S0079-6638(08)00205-9. Copyright © 2009 Elsevier B.V. All rights reserved.

293

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5.2 Three-Dimensional Random Wave Simulations: Vortex Fractality and Random Topology 6 Optical Vortices and Angular Momentum 6.1 Optical Spanners 6.2 Orbital Angular Momentum and Quantum Entanglement 6.3 Rotational Frequency Shifts 6.4 Fourier and Possible Uncertainty Relationships between Angle and Orbital Angular Momentum 7 Polarization Singularities 7.1 Introduction: Polarization Singularities in Daylight 7.2 Parametrizing Polarization Singularities 7.3 Polarization Singularities in Anisotropic Media, and Skylight Revisited 7.4 Polarization Singularities in Nontransverse Fields 7.5 Relativistic Electromagnetic Singularities: Riemann-Silberstein Vortices and Poynting Stagnation Lines 8 Conclusions Acknowledgments References

331 332 335 336 339 339 340 340 343 347 348 349 350 352 352

1. INTRODUCTION Had Thomas Young, in his celebrated demonstration of optical interference (Young, 1804), generalized his two-slit experiment to three or more slits, he would have discovered a qualitatively different type of destructive interference.1 For in general, when three or more waves interfere, light vanishes at points, rather than on fringes, in two dimensions. At these places where the intensity of the wave is zero, the phase is undeﬁned (singular), and in general, all 2π phase values occur around the zero, leading to a circulation of the optical energy. These points, which are extremely general features of optical ﬁelds, are known by various terms encompassing these properties: nodal points, phase singularities, wave dislocations, and optical vortices. The optical ﬁeld behind two and three symmetrically arranged point sources representing slits is illustrated in Figure 1. When we draw two-dimensional planes in optics, we are representing a plane sections of the 3D optical ﬁeld, and nodal points in 2D are slices of nodal lines in three dimensions. The nodal points in the xz plane of Figure 1 are inﬁnite straight lines in the y direction, perpendicular to the propagation direction z. Any measurement plane would have to be carefully positioned in z to see the nodal line in this situation. Optical ﬁelds also occur in which the node is along the axis of a propagating beam [for

1 In fact, Young could not have generalized his experiment as he did not use slits at all but rather a single

line obstruction in a pencil of light.

Introduction

(a)

(b)

295

(c)

Imax

␲

Intensity

Phase

I

␹

0

2␲

(d)

FIGURE 1 Near field for Young’s slits. (a) Intensity for two slits, showing bright and dark fringes. (b) Intensity for three slits, where now zeros of intensity occur at points. (c) Phase for three slits, where intensity zeros are places where all phases meet. (d) Inset, showing two phase singularities, whose phases increase in opposite directions.

example, in Laguerre-Gaussian (LG) “donut” modes], appearing as a nodal point in every transverse cross-section. Phase singularities are a phenomenon in a physical ﬁeld of at least two variables (e.g., positions x and y), where the physical quantity represented by the ﬁeld can naturally be represented in a plane, such as the Argand plane of complex numbers. In optics, this is naturally the complex amplitude of a scalar optical ﬁeld, whose modulus is the real amplitude and argument is the phase. For this ﬁeld, we write

ψ = ξ + iη = ρ exp(iχ),

(1)

where ξ and η are the real and imaginary parts, ρ is the real amplitude (giving the intensity I = ρ2 ), and χ is the phase. Just as the angle of polar coordinates is not deﬁned at the origin, the phase is not deﬁned when ψ = 0, and near such a point, the whole 2π range of phases occurs. This means that there is a net change of phase in a circuit C enclosing the zero point, quantized in units of 2π:

s=

1 2π

dr · ∇χ = C

1 2π

dχ.

(2)

C

The integer s (positive or negative) is called the strength, or topological charge, of the singularity. The sign of s is called the sign of the singularity, positive if the phase increases in a right-handed sense.

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In two dimensions, these phase singularity nodes occur at points. This can be most simply seen in Figure 2a: a zero of the complex ﬁeld must be a place where the real part ξ = 0 (which occurs along lines) and the imaginary part η = 0. These two sets of contour lines intersect at points; the nodal points of the complex ﬁeld. In three dimensions, ξ = 0, η = 0, and the complex zeros are curved lines occurring along the intersection of the real and imaginary zero surfaces (Figure 2b). The sense that phase increases around this line, with respect to the right-hand rule, determines a vector direction along the line. In scalar ﬁelds representing a polarized component of the electric ﬁeld, this current is the Poynting vector j, which takes the form (Jackson, 1998)

j = Im ψ∗ ∇ψ = ξ∇η − η∇ξ = I∇χ.

(3)

Thus vector j points in the direction of phase change ∇χ, and phase singularities are therefore vortices of the optical current ﬂow: optical vortices. Phase singularities were recognized as a general phenomenon of wave physics in a seminal paper by Nye and Berry (1974), although special cases had been described earlier. They have similarities to vortices and phase singularities elsewhere in science, such as in chemical and biological systems, reviewed by Winfree (2001). Optical phase singularities have recently become a fashionable topic in optical physics, partly through their relationship with beams carrying orbital angular momentum (Allen et al., 1992; Allen, Padgett, and Babiker, 1999b), and a range of techniques have been developed to generate optical ﬁelds containing vortices. Vortices also occur naturally in optical ﬁelds; in a random optical wave such as a speckle ﬁeld, there are many vortices interspersed between the bright speckles, and the correlation properties of the vortex points are related to those of the ﬁeld. (a)

(b)

FIGURE 2 Zero contours. The zero contours of the real and imaginary parts of a complex scalar field are shown in (a) 2D, where they appear as lines, and (b) 3D, where they appear as surfaces. Vortices occur where these sets of contours cross, at points in (a), and lines in (b).

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Phase singularities are features of scalar optics and occur naturally only in (complex) scalar components of vectorial ﬁelds. In ﬁelds where the state of polarization varies with position, polarization singularities can occur (Nye, 1983a,b; Nye and Hajnal, 1987); these are loci where some particular descriptor of the polarization, such as the polarization azimuth (angle of major axis of polarization ellipse), is not deﬁned. The study of these phenomena has come to be known as “Singular Optics.” Aspects of this subject have been reviewed previously in “Progress in Optics” in general by Soskin and Vasnetsov (2001b); optical vortices and solitons by Desyatnikov, Kivshar, and Torner (2005); polarization and conical refraction by Berry and Jeffrey (2007) and space-varying polarization ﬁelds and their manipulation by Hasman et al. (2005); catastrophe optics by Berry and Upstill (1980). The topic has also been discussed in several other reviews, books, and conference proceedings (Berry, 1981; Berry, Dennis, and Soskin, 2004a; Nye, 1999; Soskin, 1998; Soskin and Vasnetsov, 2001a; Vasnetsov and Staliunas, 1999). Here, we will emphasize a particular facet of singular optics: the appearance of vortices and polarization singularities in linear optics, especially superpositions of plane waves. This emphasis in fact encompasses a wide range of optical phenomena, including diffraction, scattering from rough surfaces and in sky light, as well as the controlled creation of beams carrying orbital angular momentum. For other aspects of singular optics, such as vortices in nonlinear optics, the reader is referred to other reviews (for instance, Desyatnikov, Kivshar, and Torner, 2005, in the nonlinear case). The arrangement of this review will be as follows. In the following section, we will discuss in more detail the structure and physics of phase singularities, illustrated by historical optical and electromagnetic examples. In Section 3, we will consider the various ways by which optical vortex-carrying beams can be created experimentally. Afterwards, we will discuss the morphology of phase singularity points and lines and their occurrence in simple superpositions of plane waves (Section 4). Descriptions of phase singularities in random waves (Section 5) and the connections between phase singularities and optical orbital angular momentum (Section 6) follow. Finally, we will brieﬂy review the ﬁeld of optical polarization singularities (Section 7).

2. HISTORICAL REVIEW OF VORTICES IN OPTICS Phase singularities, that is, points or lines where a cyclic variable is undeﬁned, occur in a wide range of physical systems. An example of a phase singularity unrelated to waves is the undeﬁned time zone of the north pole, since geographically the north pole has undeﬁned longitude. The

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absolute time of local noon at each point on the earth varies cyclically with longitude round the earth—this can be thought of as the phase of a complex function, whose amplitude is the sun’s height above the horizon at local noon. At the north and south poles, the sun is permanently on the horizon (neglecting the effect of seasons), there is no well-deﬁned noon, and all time zone lines converge. Before the international date line was established, the effect of circling this phase singularity on circumnavigating the globe gave rise to the gain/loss of a day, a potentially confusing phenomenon that was apparently observed on Magellan’s circumnavigation (see Winfree, 2001, pp. 10–11, for further historical discussion) and has provided opportunities for ﬁction (Eco, 1995; Verne, 1992). In wave physics, the ﬁrst discovery of phase singularities was apparently the amphidromic points in the tides (Berry, 2000). The tides may be considered as a complex wavefunction, whose constant phase lines are places of simultaneous high tide (cotidal lines2 ) (Cartwright, 1999). Whewell (1836) postulated the existence of amphidromic points based on tidal observations in the North Sea, between the eastern coast of Britain and the western coast of continental Europe. Amphidromic points have been discussed several times in the context of phase singularities in waves and optics (Berry, 1981, 2001; Nye, 1999; Nye and Berry, 1974; Nye, Hajnal, and Hannay, 1988). On the way to his postulate of magnetic monopoles in quantum theory, Dirac (1931) appreciated the existence of phase singularity lines in threedimensional, general quantum wavefunctions. In his theory, the quantum phase singularity lines in three dimensions end on magnetic monopoles, which is, of course, not possible in optical waves (as the phase must be well-deﬁned everywhere, without quantum gauge freedoms). Vortex lines have been subsequently studied in quantum wavefunctions (Hirschfelder, Goebel, and Bruch, 1974; Riess, 1970) and play an important role in many-particle quantum wavefunctions describing superconductivity and superﬂuidity (Tilley and Tilley, 1990). Proposals exist to couple vortices in atomic Bose–Einstein condensates with those in laser ﬁelds (Ruostekoski and Dutton, 2005). In the rest of this section, we will review the important properties of optical vortices and the circumstances in which they appear. The discussion will be themed around several important papers, mainly from the 1940s and 1950s, in which the principal properties of optical vortices were appreciated. In these cases, vortices were observed in different circumstances apparently independently, by Pearcey (1946), Wolter (1950b), Findlay (1951), and Sommerfeld (1954, although the example from this

2 A name proposed by Thomas Young (Cartwright, 1999); although he failed to appreciate phase

singularities in optics, Young was indirectly responsible for their discovery in oceanography!

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textbook was probably found earlier). Beams with properties we now associate with vortices, such as the exp(iφ) phase-dependence of modes in cylindrical cavities, were also described around this time (Stratton, 1941, for example). Further historical precedents of optical vortices were discussed by Soskin and Vasnetsov (2001b).

2.1. Wolter’s Vortex in the Goos–Hänchen Shift for Refraction It appears to have been Wolter (1950b) who ﬁrst discovered a vortex arising in the solution of an optical problem, and his discussion of this vortex and its properties is surprisingly detailed. This problem was the spatial interference structure arising out of the recently discovered Goos–Hänchen shift (Goos and Hänchen, 1947): the center of a ﬁnite-width optical beam, on total reﬂection at a dielectric interface, undergoes a spatial shift on reﬂection. Wolter had previously studied this effect (Wolter, 1950a), representing the beam as the superposition of two plane waves with propagation directions very close, and close to the critical angle. Simply by considering the Fresnel coefﬁcients for reﬂection and transmission of the two waves, Wolter found theoretically the phase pattern shown in Figure 3 near the interface and between glass and air: this pattern is simply the superposition of four plane waves on the glass side (two incident, two reﬂected) and (a) 2

(b) 0.05

1

0 0 21

22 22

21

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20.05

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FIGURE 3 The phase of Wolter’s vortex pattern in total internal reflection, in two different spatial resolutions. The vertical height z = 0 represents the interface between glass (z > 0, n = 1.52) and air (z < 0, n = 1). Incident is a pair of plane waves with equal amplitude with incident directions just above the critical angle, 40.939 ± 0.086◦ . The vortices all occur on the same horizontal line (chosen as x = 0). The gray lines indicate the flow of the Poynting vector, and the black arrows in (a) indicate the mean incidence and reflection directions. The unconventional sequence of these arrows (reflected on the left, incident on the right), represents the ‘reflection’ of the low-intensity region of the beam, and roughly follows the energy flow. x and z are plotted in units of wavelength. [Figure based on Figures 2 and 3 of Wolter (1950b).]

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two evanescent refracted waves on the air side. As shown in the ﬁgure, close to the intensity minimum of the incident wave superposition, there is a street of vortices, alternating in sign, increasing in height above the interface (Figure 3a), with one extremely close to the interface on the glass side (Figure 3b). The circulation around each vortex is too ﬁne to see in Figure 3a. Wolter was intrigued by what he called the “circulating wave” (zirkierende Welle), where he observed, just inside the glass, all phase lines intersecting in a circulating sense at the intensity node, and, on the air side of the interface, a saddle point of phase and energy ﬂow. As he stated, this pair of features is necessary for the energy ﬂow lines to have the right global topology. Since this four-wave interference pattern is sufﬁciently simple, he was able to ﬁnd a stream function whose contours give the ﬂow lines. He stated The circulating wave is new to optics, however an uncomplicated phenomenon; mathematically it can be realized . . . from only four homogeneous waves. At every centre of [energy] circulation a revolution of the phase contours occurs together with a circulation of one period.3 He concluded by speculating that this circulation phenomenon may be quite general in optical ﬁelds. Surprisingly, this early optical example of phase singularities is not particularly well known, despite being highlighted by Rosu (1997). Boivin, Dow, and Wolf (1967) observed a very similar energy ﬂow pattern to Wolter—several vortex points, where each circulation is accompanied by a saddle point nearby—in the free space scalar diffraction ﬁeld near a focus. A more detailed analysis of the vortices near the focus of the Airy ring system (Karman et al., 1998; Nye, 1998) shows a complicated interplay between vortices and saddle points.

2.2. Vortices in Superposition and Diffraction Braunbek (1951), following from Wolter’s observations, noticed that phase gradient lines and energy ﬂow lines always have the same direction, and three plane waves are sufﬁcient to produce phase singularities/ nodes/energy vortices. Furthermore, he observed that the phenomenon is general: “singular lines exist often in general refracted ﬁelds, [on which] the phase is inﬁnitely undetermined and at the same time the amplitude becomes zero, and around which the energy ﬂows in closed stream lines” (Braunbek, 1951, p. 13). In his celebrated book “Optics,” Sommerfeld

3 Quotation from Wolter (1950b) p. 283, translated by U T Schwarz.

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discusses an example of six plane waves with similar direction and wavenumber (Sommerfeld, 1954, Section 2.6a, Figure 2). He notes that the phase of the resulting ﬁeld has the character of a single plane wave (with slowly varying amplitude), except at the zeros, although “just because the amplitude vanishes there, they do not produce any stronger effects than other points of varying intensity.”4 We will discuss the vortex conﬁgurations, which occur in superpositions of few waves, later in Section 4. The most celebrated study involving vortices from this early period is the direct computation and plots by Braunbek and Laukien (1952) of Sommerfeld’s exact solution of a plane wave diffracted by an inﬁnite halfplane. These diffraction ﬁelds, replotted here in Figure 4, became wellknown due to their inclusion in the textbook by Born and Wolf (1959). The vortices clearly visible in the illuminated region may be interpreted as originating from the interference of three waves: the incident wave, the reﬂected wave, and the edge wave from the diffracting point at the end of the half-plane (Berry, 2002b): without the edge wave, the interference pattern on the illuminated side would be a simple standing wave. As Braunbek and Laukien observe, this pattern of vortices is irregular, unlike the case of vortices in the interference of three plane waves. Berry (2001) observed that Newton possibly came close to ﬁnding optical vortices in his analysis of this kind of diffraction.

(a)

(b)

FIGURE 4 The intensity (a) and phase and energy streamlines (b) of Sommerfeld’s exact solution of the diffraction of a plane wave by a half-plane (with Neumann boundary conditions). The incident wave enters from the left, and the nodal points in intensity are phase singularities and circulations of current, on the illuminated side. Four square wavelengths are plotted.

4 Sommerfeld (1954, p. 10); unfortunately, Sommerfeld apparently does not give enough information about

his superposition for it to be recomputed!

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2.3. Singularities in the Cusp and Diffraction Catastrophes Diffraction is at its most general and striking when it decorates the caustics of geometric ray patterns: so-called diffraction catastrophes, which are most simply seen in natural focusing (Berry, 1981; Berry and Upstill, 1980; Nye, 1999). In ray optics, caustics occur where the number of rays through each point changes (discontinuously), and the generic ways this occurs mathematically has been systematically classiﬁed (Poston and Stewart, 1978). In pure ray optics, these caustics are themselves singularities—divergences of intensity. However, when amplitude and phase are included, diffraction smooths away the inﬁnities to such an extent that nodes appear in the pattern. The diffraction pattern around the simplest canonical caustic, the fold, is the Airy function (Olver, 2008), which depends only on one parameter. The simplest caustic whose diffraction pattern displays nodes that are phase singularities is the next in the hierarchy, namely the cusp, and its diffraction pattern was ﬁrst studied by Pearcey (1946). The diffraction pattern Pearcey computed is now known as the Pearcey function (Berry and Howls, 2008):

∞ exp(i(t4 /4 + xt2 /2 + yt)) dt,

Pe(x, y) =

(4)

−∞

where x and y are scaled dimensionless position variables. The caustic itself occurs where the integral cannot be evaluated by stationary phase methods, that is, where the ﬁrst and second t derivatives of the exponent in the integrand simultaneously vanish, which occurs where 27x2 + 4y3 = 0. The intensity and phase of this pattern are shown in Figure 5, together with the rays and associated caustic. Evaluating the integral (4) was a major computational problem in the 1940s, and even with modern computers, its

6 (a)

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6 (c)

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FIGURE 5 The Pearcey diffraction pattern around a cusp caustic. (a) Geometric rays forming the cusp, (b) the intensity, and (c) the phase pattern of the Pearcey function of Eq. (4).

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computation requires elaborate mathematical techniques (Borghi, 2007). The whole hierarchy of diffraction catastrophes can be viewed mathematically as a class of special functions whose integral representations have coalescing saddle points (Berry and Howls, 2008). Nevertheless, the method of stationary phase reveals that the Pearcey function, off the caustic, can be approximated at each point by the rays going through it, correctly weighted by amplitude and phase; in a sense, the singularities in the Pearcey pattern are formed by three-ray interference. Phase singularity lines occur in 3D diffraction catastrophes, such as that of the elliptic umbilic, which was studied mathematically, numerically, and experimentally by Berry, Nye, and Wright (1979). The 3D diffraction pattern here consists of three ribs (cusp lines) that meet at the elliptic umbilic point; there are many phase singularity lines in this pattern, some of which are inﬁnite, curly lines (“antelope horns”) and some of which are closed loops (as shown from experimental measurements in Figure 6). More recently, phase singularity conﬁgurations have been studied in other diffraction catastrophes, including cusps in nonlinear diffraction (Deykoon, Soskin, and Swartzlander, 1999), and those in the 3D hyperbolic umbilic (Nye, 2006b) and swallowtail catastrophes (Nye, 2007). By deﬁnition, the corresponding caustics have domains in which a certain number of rays through each point and the vortex line behavior in the diffraction catastrophes are roughly consistent with the vortices in superpositions of the corresponding number of plane waves. Diffraction catastrophes occur as the limiting case of inﬁnite aperture for other diffraction problems; in two dimensions, the Fraunhofer pattern can approach the Pearcey pattern as the

(a)

(b)

(c)

FIGURE 6 The elliptic umbilic diffraction catastrophe and its dislocation lines. (a) The ray catastrophe locus, consisting of three rib (cusp) lines with three-fold symmetry meeting at the catastrophe point. (b) Intensity images measured at several planes through the structure. (c) Vortex lines, reconstructed from experimental measurements of the diffraction catastrophe, with loops and helical lines represented in different colors.

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aperture is widened (Nye, 2003b), and the two three-dimensional umbilic diffraction catastrophes occur similarly as inﬁnite-aperture diffraction patterns; the vortex lines undergo a complicated evolution with increasing aperture (Nye, 2003a, 2006a). An interesting philosophical approach to singular optics has been suggested by Berry (1991, for example): the simplest, coarsest theory of optics is ray theory, whose singularities, where the intensity diverges, are the caustics. These inﬁnities vanish on the introduction of a more reﬁned theory, namely that of waves, whose phase pattern is nontrivial upon superposition. Phase has its own singularities, where the intensity completely vanishes. Might a more reﬁned optical theory, involving quantum effects, remove this singularity too? This suggestion has been pursued in two different directions: as quantum features of an optical black hole (Kiss and Leonhardt, 2004; Leonhardt, 2002), and in the effect of the ﬂuctuations in the quantum vacuum at the core of an optical vortex (Barnett, 2008; Berry and Dennis, 2004). Catastrophe optics, both in its ray caustic and diffraction aspects, is often not emphasized as a branch of singular optics. However, we note that much of the terminology often used to describe singular optical phenomena, such as genericity, structural stability, and unfolding, all originate from mathematical catastrophe theory and the related singularity theory (Poston and Stewart, 1978). Optical vortices are structurally stable in that their topology does not change on small perturbation (although their position may change); high strength vortices are unstable and unfold to stable features under perturbation. A central message from singularity theory shines through the study of singular optics: topological singularities in optics occur ubiquitously in the most chaotic and random of ﬁelds, although they themselves display a high local degree of order and structure. The presence of optical vortices in random ﬁelds is the topic of the following discussion.

2.4. Vortices in Random Optical Fields Optical vortices, as a ubiquitous phenomenon in interference, frequently occur in reﬂection from or propagation through random, turbulent, and chaotic media. A simple representation of such a random ﬁeld is shown in Figure 7—there are at least as many completely dark points as there are bright speckles. However, due to their phase singularity nature, the presence of nodes has caused considerable trouble in the measurement and interpretation of such scattered waves. The occurrence of phase singularities in random ﬁelds is often seen as a nuisance, as different continuous paths of phase around a singularity leads to difference of 2π upon unwrapping.

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(a)

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(b)

0

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2␲

FIGURE 7 A simulated random optical field (speckle field), computed as a superposition of 729 plane waves with random phases and amplitudes. (a) Intensity; (b) phase, with inset showing singularities (strength +1 black, −1 white).

This problem goes back at least as far as Findlay’s analysis of radio waves reﬂected from the ionosphere (Findlay, 1951). Several algorithmic approaches have been suggested in dealing with the singularities in general phase unwrapping problems (e.g., Buckland, Huntley, and Turner, 1995; Ghiglia and Pritt, 1998). We will discuss the study of optical vortices in speckle patterns themselves in Section 5. The random ﬁelds of speckle patterns, as evident in Figure 7, have a well-deﬁned complex amplitude at each point—they are fully coherent. Phase singularities of the coherence function occur for the generalization of Young’s slits to partially coherent light (Schouten et al., 2003), and correlation vortices occur in controlled ﬁelds in the temporal (Swartzlander and Schmit, 2004) and spatial (Palacios et al., 2004) correlation functions. These correlation vortices have very similar properties to their coherent counterparts (Wang et al., 2006). Spectra of polychromatic waves near nodes have interesting properties (Gbur, Visser, and Wolf, 2002).

2.5. Wave Dislocations The most inﬂuential paper in the study of singular optics is without doubt “Dislocations in wave trains” by Nye and Berry (1974). This study on wave dislocations was inspired by the topographic mapping of the land under the Antarctic ice sheet by radio pulses (Nye, Berry, and Walford, 1972b; Nye, Kyte, and Threllfall, 1972a); it was found that the reﬂected pulses contained phase singularities. This phenomenon was then studied experimentally in unpublished work by J.F. Nye and M.E.R. Walford in Bristol using ultrasound reﬂected from crumpled aluminum foil, leading to the theoretical understanding of phase singularities as a general and ubiquitous feature of wave interference, as presented by Nye and Berry (1974).

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Optical phase singularities were described in this paper as “wave dislocations,” owing to several similarities observed between phase singularities in scalar waves and dislocations in crystal lattices (Read, 1953). This similarity can be demonstrated with the simple wavefunction

ψdislocation = (x + iy) exp(ik · r),

(5)

which represents a vortex along the line x = y = 0 embedded in a plane wave with wavevector k. The examples of k parallel and perpendicular to z are illustrated in Figure 8; from the pattern of constant phase lines, the case of the vortex line perpendicular to k is related to a crystal edge dislocation, whereas the case of the line parallel to k is related to a screw dislocation. In particular, the screw dislocation has helical wavefronts, whose handedness depends on whether k · zˆ > 0 (left-handed helicoid) or k · zˆ < 0 (right-handed helicoid). Mixed edge-screw dislocations occur when the carrier wave direction is neither perpendicular nor parallel, and here, as in general, the wavefronts around the singularity are helicoids.

(a)

(b)

(c)

(d)

FIGURE 8 Dislocations in waves and crystals. In (a) and (b), the real and imaginary nodal surfaces of the wave of Eq. (5) are plotted: (a) edge dislocation, k = k yˆ ; (b) screw dislocation, k = k zˆ ; (c) 2D crystal edge dislocation; and (d) crystal screw dislocation.

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The wavevector of the carrier wave was thus equated to the Burgers vector of the crystal dislocations. Of course, the optical current always circulates around the singularities, regardless of their edge/screw nature, although the streamlines of current further from the vortex line display interesting spiral features (Berry, 2005). Although, as we have seen, there were several previous studies of vortices in various optical situations, it does not appear that these observations about the phase structure around the singularities were previously described. Nevertheless, there has been some difference of nomenclature in the literature (e.g., Soskin and Vasnetsov, 2001b); in particular, all optical vortices are referred to as “screw dislocations”, with the term “edge dislocation” applied to any nodal line in the transverse plane, particularly those that are 2D sheets in three dimensions (such as those in TEM laser modes and Hermite-Gauss modes). These latter nodal lines and sheets are not structurally stable—in superpositions of several beams, they are replaced by vortex points and lines. Recently, the name “vortex sheets” was suggested for these structures (Wang et al., 2008). The optical vortex conﬁguration in monochromatic ﬁelds is static. However, in pulses and quasimonochromatic waves, they can move. This type of waveﬁeld, originally studied by Nye and Berry, was subsequently considered in more detail by the Bristol group (Nicholls and Nye, 1986, 1987; Nye, 1981; Wright and Nye, 1982). More recently, there has been interest in creating femtosecond vortex beams (Mariyenko, Strohaber, and Uiterwaal, 2005; Moh et al., 2006). For waveﬁelds more general than Eq. (5), such as the random ﬁelds of the previous section, the distinction between edge and screw wave dislocations is less evident. However, further crystal-like geometry in singular waveﬁelds was considered by Nye (1991), and the nature of the Burgers vector as a measure of twisting of the phase helicoids was considered by Dennis (2004). It is the appearance of optical vortices on the axis of laser modes, and their relationship with optical orbital angular momentum, that has led to the explosion of interest in optical vortices in the last two decades, and it is to the experimental generation of such beams that we now turn.

3. CREATING AN OPTICAL VORTEX 3.1. Optical Vortices and Laguerre–Gaussian Beams As discussed in Section 2.5, the ﬁrst serious study experimentally and theoretically of beams containing vortices was the work by Nye, Berry, and Walford in Bristol in the 1970s, in their work on the ultrasound scattered from a rough surface. The ﬁrst modern day recognition that an optical

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vortex could be deliberately produced within optical ﬁelds was possibly by Vaughan and Willetts (1979), who studied the properties of light beams with a helical phase structure and their interference along the beam axis. Coullet, Gil, and Rocca (1989) used the term “optical vortex” to describe a possible laser mode that could occur in cavities with large Fresnel number. The natural laser mode containing an optical vortex is one of the Laguerre-Gauss (LG) family, given by

ψ LG,p (R, φ, z) =

2||+1 p! R|| exp(iφ) zR − iz p π(|| + p)! w||+1 (1 + iz/zR )||+1 zR + iz 0

||

× exp − R2 /w02 (1 + iz/zR ) Lp

2R2 2) w02 (1 + z2 /zR

, (6)

where w0 is the 1/e width of the Gaussian in the waist plane (z = 0), || and || p are the indices of the associated Laguerre polynomial Lp , and zR = kw02 /2, the Rayleigh range of the beam. Of most interest to us is the integer , which gives the number of 2π phase cycles around the optical vortex centered on the beam axis. The set of LG modes is a choice for a complete orthogonal basis set representing arbitrary light beams, contrasting to the more familiar TEM, or Hermite-Gaussian (HG) modes, which can be represented by a Gaussian times a product of Hermite polynomials in x and y separately. The simplest, distinct, LG mode (LG1,0 ) appears as a single annular ring of intensity with a 2π phase singularity along the beam axis.

3.2. Direct Generation of Vortex Beams from Laser Systems Tamm and Weiss (1990) demonstrated that the residual astigmatism of a laser could be controlled such that the TEM1,0 and the TEM0,1 modes were frequency degenerate, coherently interfering to give a TEM∗1,0 . The transverse intensity pattern has the appearance of a ring donut and can be equivalently expressed as an incoherent sum of LG modes LG1,0 and LG−1,0 , which are characterized by having a strength ±1 vortex at their center. Although these two modes have identical intensity distributions, when passed through a pair of cylindrical lenses they are transformed into a TEM1,0 or TEM0,1 mode respectively, thus enabling the sign of the vortex in the original mode to be identiﬁed. Tamm and Weiss used this information to control the coupling of a small fraction of the TEM∗1,0 amplitude back into the laser such that the output could be stabilized or switched between LG1,0 and LG−1,0 modes. This was perhaps the ﬁrst optical system that could generate an optical beam containing a single vortex that could

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be switched from one handedness to the other. Subsequently, several others, including Ishaaya et al. (2003) and Okida and Omatsu (2007), have developed similar laser systems that emit LG beams directly—however, most of these are quite complicated and certainly restricted in terms of the different modes that they can create.

3.3. Generation of Vortex Beams Using Diffractive Optics In the early 1990s, a couple of groups began to explore the use of diffractive optical components to transform spatially coherent, ﬂat phase beams into beams containing optical vortices. In 1990, Soskin and coworkers recognized that if a diffraction grating was modiﬁed to include an edge dislocation at its center, then the ﬁrst-order diffracted beam contained an optical singularity (Bazenhov, Vasnetsov, and Soskin, 1990). This classic “forked” hologram design is now synonymous with the generation of optical vortices. The forked design can be implemented either as an amplitude or phase grating, both of which can be calculated as the addition modulo 2π of a helical phase exp(iφ) with a diffraction grating (see Figure 9a). In principle, a phase grating can diffract all of the incident energy into the ﬁrst diffraction order. In practise, this is never the case and a fraction of the energy ends up in other orders. For the forked grating, each order is diffracted through a different angle, and consequently it is simple to use a spatial ﬁlter to select the ﬁrst order. Simultaneous with the work by Soskin, Heckenberg et al. (1992) used a variation of the design where the azimuthal phase term was added to a Fresnel lens such that the various diffraction orders were separated radially. (a)

(b)

FIGURE 9 Illustration of a Gaussian beam incident on (a) a hologram with phase modulation mod2π|λθ/2π + αx| where the second term adds a blazed diffraction grating to preferentially diffract light into the positive first order and (b) a spiral phase plate of height φ(n − 1)λ/2π.

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3.4. Generation of Vortex Beams Using Lenses and Spiral Phase Plates The ground-breaking realization of Allen et al. (1992) and Beijersbergen et al. (1993) that beams with optical vortices carried orbital angular momentum was not based on beams produced directly from a laser or transformed by holograms. Instead they developed the use of cylindrical lenses to convert HG into LG modes, extending the earlier work of Tamm and Weiss (1990) and Abramochkin and Volostnikov (1991). Every HG mode with indices m and n is transformed into an LG mode with azimuthal index l = (m − n) and radial index p = min(m, n). The full relationship between HG and LG modes was also clariﬁed and the analogy between them and polarization states established (Beijersbergen et al., 1993). This analogy between transverse modes and polarization states extends to a description of modes and their transformation with an equivalent to the Poincaré sphere (Padgett and Courtial, 1999) and Jones matrices (Allen, Courtial, and Padgett, 1999a). Although cylindrical lenses offer a near lossless transformation between HG and LG modes, they suffer from two drawbacks: ﬁrstly, they require the generation of a high-order HG beam as an input; secondly, any imperfection in the speciﬁcation or alignment of the cylindrical lenses leads to a residual astigmatism in the resulting LG mode. This astigmatism is manifest both as a noncircular beam cross section and in that any higher index vortex splits, upon propagation, into multiple single vortices (Courtial and Padgett, 1999; Dennis, 2006). Woerdman and coworkers (Beijersbergen et al., 1993) pursued a new approach to the generation of vortex-carrying beams. They built optical components termed “spiral phase plates” (see Figure 9b). These are discs of refractive index n with optical thickness t that increases with azimuthal angle, i.e., t = φ(n − 1)λ/2π. Upon transmission, an incident plane wave acquires an exp(iφ) phase term and consequently has an optical vortex along the beam axis. The purpose of this work was to produce a vortex beam, with its associated orbital angular momentum, that was free from astigmatism. However, these spiral phase plates require precise matching of the (n − 1)λ step height at φ = 0, which places extremely high demands on the engineering tolerance—particularly for optical wavelengths. At radio or millimeter wave frequencies, these tolerances are not so extreme (Turnbull et al., 1996). In the original work, the plate was immersed in a temperature-controlled ﬂuid bath, with a slightly different coefﬁcient of refractive index change. Changing the temperature of the bath ingeniously controlled the refractive index mismatch and hence allowed the step height to be tuned to exactly the correct value. Since then, several groups have employed precise micro-machining techniques to manufacture spiral phase plates directly (Tsai, Smith, and Menon, 2007; Watanabe et al., 2004), including a two-photon polymerization technique for application as miniature mode converters within optical tweezers (Knoner et al., 2007).

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311

Great care is usually taken to ensure spiral phase plates have exactly the correct step height. However, they may have interesting properties when the step height has some other value. Spiral phase plates with half-integer step heights have been deliberately fabricated for experiments centered around the quantum entanglement of orbital angular momentum states (Oemrawsingh et al., 2004). Whenever the step height of a spiral phase plate is not an integer, in addition to the on-axis optical vortex, there is a radial phase discontinuity. Closer inspection reveals that this radial line has an intricate vortex structure. For half-integer step heights, there is a chain of vortices with alternating sign on propagation (Berry, 2004a). These vortex points are simply the intersection of vortex lines with the viewing plane that themselves can be viewed by probing successive planes. Doing so reveals that the structure comprises vortex lines which form “hairpins,” with turning points that approach the phase plate (Leach, Yao, and Padgett, 2004b).

3.5. Generation of Vortex Beams Using Spatial Light Modulators Despite the various methods that have been developed for generating vortex-carrying beams, including cylindrical lenses, specially modiﬁed lasers, and spiral phase plates, none matches the ease and ﬂexibility of computer-generated holograms. The popularity of diffractive optical components for generating speciﬁc beams has been massively enhanced by the commercial availability of spatial light modulators (SLMs). These are pixellated devices that can be addressed via a computer with an image that deﬁnes the spatial variation of the phase of the reﬂected light, typically at 1024 × 760 pixels with video rate updates. Such devices are ideal for the implementation of phase holograms that work by reﬂection, and diffraction efﬁciencies typically exceed 50%. SLMs are widely used for the generation of arbitrary beams, including those incorporating optical vortices, for applications ranging from optical tweezers (Curtis, Koss, and Grier, 2002) and atom optics (McGloin et al., 2003) to most recently quantum entanglement (Yao et al., 2006). Until recently, the manufacturing process of the SLM tended to produce some residual astigmatism in the resulting beam; however, it is possible to use a Gerchberg–Saxton algorithm to calculate a correction hologram that can be added to any hologram design (Jesacher et al., 2007). This powerful algorithm was originally developed for crystallography (Gerchberg and Saxton, 1971) but is now used widely in the design of computer-generated holograms. In essence, it relies on the ability to perform high-speed Fourier transforms to calculate the intensity and phase distribution in the far-ﬁeld of any hologram, and inverse transforms to invert the distribution back to the hologram plane. Repeatedly transforming between planes and substituting the intensity corresponding to the illumination and target ﬁelds converges within a few iterations to give the phase of the required hologram, albeit with no control over

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the phase of the target ﬁeld. The algorithm can also be modiﬁed for the generation of LG and other beams (Curtis, Koss, and Grier, 2002). The forked hologram is the diffractive counterpart of a spiral phase plate. The simplest form of each introduces an exp(iφ) phase term to an incident beam while leaving the intensity cross section of the beam unchanged. Consequently, although the transmitted beam has a perfect helical phase structure, the intensity cross section does not match that of a perfect LG mode (Sacks, Rozas, and Swartzlander, 1998). For a Gaussian illumination beam, a single forked hologram ( = ±1) produces a ﬁrstorder diffracted beam that corresponds to a superposition of LG modes, all with = 1 but a range of p. The exact decomposition depends on the waist of the illumination beam for the mode set, but in most cases, the p = 0 (i.e., single ringed) mode is dominant (Figure 10). For most applications, the index is of most importance, because this determines the orbital angular momentum content of the beam. However, for some applications, it is essential that the mode composition be well controlled. The Gouy phase of

1 0.9 Mode coefficient

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

3

8 6, w

w5

50

p5 0 p5 p5 1 p5 2 3 p5 Radial mode p5 4 5 index of decomposition

,5

5,

Hologram ,, and optimum waist size for decomposition

.39 8

0.4 2

0.4 5

w5 ,5

4, ,5

,5

3,

w5

0.5 0

5

9

1 50 .57

,5 2, w

50 .70

1, w

,5

,5

0,

w5

1

0

FIGURE 10 The modal decomposition of the beam produced by a spiral phase plate or forked hologram when illuminated by a fundamental Gaussian beam of beam waist w = 1. The values for w in the decomposition of the final beam have been chosen to maximize the size of the p = 0 component in the superposition.

Creating an Optical Vortex

313

an LG mode depends on its mode order (2p + + 1), and hence interference between any arbitrary superposition means that the resulting beam cross section changes upon propagation—potentially limiting its effectiveness for particle or atom guiding and other applications. A beam type of particular importance, especially within atom optics (Arlt, Hitomi, and Dholakia, 2000), is the so-called “nondiffracting” Bessel beam (Durnin, Miceli, and Eberly, 1987; McGloin and Dholakia, 2005). The terminology ‘nondiffracting’ is unfortunate since diffraction is an inherent property of all physically realizable ﬁelds. In k-space, Bessel beams have a ring spectrum, i.e., every plane wave component has the same radial component, kr . Assuming that the light is monochromatic, it follows that kz = (k02 − kr2 − kφ2 )−1/2 . For a zero-order Bessel beam, each plane wave component (point on the ring in Fourier space) has the same phase, the azimuthal component of the wavevector kφ is zero, and the axial component kz of the wavevector is the same for all the plane wave components; hence their relative phase does not change with propagation. Consequently, the transverse proﬁle of the resulting beam, which results from the interference between the plane wave components, does not change with propagation— the beam is structurally stable. In general, a -order Bessel beam has a strength vortex on the axis. A general Bessel beam is described by

J (kr R) exp(iφ − ikr2 z/2k),

(7)

where kr is the radial wavenumber, k is the overall wavenumber, and J (x) is a Bessel function of the ﬁrst kind (Abramowitz and Stegun, 1965). Such a beam has the appearance of a central maximum or minimum surrounded by an inﬁnite series of bright concentric rings, with each ring containing the same amount of overall energy. Unfortunately, this means that a perfect Bessel beam contains inﬁnite energy, obviously making it unrealizable in a practical setting. In reality, all methods of producing physical Bessel beams result in a ﬁnite aperture containing a ﬁnite number of concentric rings and hence have ﬁnite energy. This corresponds to a slight spread in values of kr and hence a spread in kz and a corresponding change in their proﬁle upon propagation. The distance over which the central maximum of the Bessel beam propagates without signiﬁcant spreading is comparable to the Rayleigh range of a Gaussian beam of the same diameter multiplied by the number of rings within the modiﬁed beam, or Zmax ≈ (kz R0 /kr ), where R0 is the radius of the largest ring. One easy way to produce an approximation to a Bessel beam is to illuminate a lens of conical form (an axicon) with a Gaussian beam from a laser (Herman and Wiggins, 1991) or its holographic equivalent (Paterson and Smith, 1996). Bessel beams of nonzero order = 0 have a uniform, nonzero value of kφ , that is, they are helically phased when there is an azimuthal phase

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Singular Optics: Optical Vortices and Polarization Singularities

dependence of exp(iφ). Such beams carry an associated orbital angular momentum of per photon (see Section 6). One approach to generating such a beam is to illuminate an axicon with a helically phased beam rather than a plane-wave. Atwo-component optical system is therefore required: a forked hologram converts the beam from a fundamental Gaussion to an LG beam, which is then directed through the axicon to give an approximation to a higher order Bessel beam that contains an strength on-axis vortex (Arlt and Dholakia, 2000). Several groups have experimented with both phase and amplitude control of holographically generated beams—while using a phase-only SLM. Once the desired phase function is added to the diffraction grating term to separate the various diffractive orders, then it is important to realize that the exact phase response of the SLM has no bearing on the ﬁdelity of the diffracted beam, only on the relative intensity of the different orders. The phase information of the diffracted beam is effectively encoded in the spatial irregularity of the regular phase pattern of the diffraction grating. Further, the depth of the phase modulation can be used to locally control the intensity of the diffracted light. This was ﬁrst used as a means of reducing the on-axis intensity surrounding an optical vortex so that the near ﬁeld intensity pattern was a closer approximation to an LG mode (Basistiy et al., 2003). The technique was fully exploited in the precise generation of speciﬁc superpositions of LG modes to create vortex loops, links, and knots (Leach et al., 2004a). The ﬁrst step in the algorithm is to calculate the phase cross section of the desired superposition, adding this to a phase grating. If illuminated with a plane wave and with the ﬁrst-order beam reimaged, the resulting beam has the desired phase structure but uniform intensity. In this form, the hologram is a distorted diffraction grating, but with all the lines fully modulated between 0 and 2π. One can then modify this design by using the intensity distribution of the desired superposition to modulate the depth of the phase modulation such that it remains 0 to 2π in the brightest region of the beam and less elsewhere (Leach et al., 2005). This technique allows arbitrary intensity and phase distribution to be created, but note that the hologram is now positioned in the image plane of the desired beam. In another approach, recently it has been demonstrated that a second SLM can be used to correct the phase for a beam produced by a hologram using the Gerchberg–Saxton algorithm (Jesacher et al., 2008).

3.6. Vortex Beams with Low Temporal or Spatial Coherence To date, most works using optical vortex beams have been associated with light that is both temporally and spatially coherent; however this need not necessarily be the case. Naturally occurring optical vortices may arise from the interference between multiple plane waves. Each spectral component creates its own network of phase singularities, and in general, these do

Creating an Optical Vortex

315

not overlap so that the result is a spatially varying spectral distortion rather than isolated points of darkness (Berry, 2002a; Gbur, Visser, and Wolf, 2002). If the vortices corresponding to different spectral components nearly overlap, then there is a characteristic chromatic signature, where various missing wavelengths produce the visual appearance of a complimentary color spectrum (Berry, 2002a). Producing a white-light optical vortex experimentally is not so simple. A standard spiral phase plate only has the correct step height for one spectral component, but proposals have been made to make phase plates in the same fashion as achromatic lenses, namely combining materials with different refractive index dispersion (Swartzlander, 2006). A forked hologram, although correctly introducing a helical phase term to every spectral component, also introduces angular dispersion and if used with white-light illumination, results in a spatial separation between the vortices of each wavelength. By reimaging the plane of the hologram, it is possible to correct for the angular dispersion using either a smallangle prism to introduce the opposite sense of angular dispersion (Leach and Padgett, 2003) or a second SLM (Bezuhanov et al., 2004). Although originally applied to creating a white-light vortex from a spatially ﬁltered discharge bulb, both prism (Leach et al., 2006) and SLMs (Sztul, Kartazayev, and Alfano, 2006) have been used to create broadband optical vortices from super-continuum sources. Most recently, it has been shown that the white-light helically phased beam can transfer its orbital angular momentum to microscopic particles, causing them to rotate around the beam axis (Wright et al., 2008). (The orbital angular momentum of optical beams will be discussed in more detail in Section 6.) A recent development is the study of what happens when a spiral phase plate is illuminated by light which has only partial spatial coherence. Away from the beam axis, the associated energy ﬂow resembles that of a normal, helically phased beam. However, near the beam axis, the spatial incoherence takes over and there is no intensity null nor phase singularity; the result has been termed a Rankine vortex (Swartzlander and HernandezAranda, 2007). This characteristic of a spiral phase plate discriminates a light source of high spatial coherence from an incoherent background (Palacios, Rozas, and Swartzlander, 2002). When the coherent light is an apparent point object, such as a distant star, the spiral phase plate may be used to completely eliminate that object from the image ﬁeld. This latter application uses an optical conﬁguration called an optical vortex coronagraph (Swartzlander et al., 2008), which is being developed to image extrasolar planets without the intense glare of the parent star (Lee et al., 2006). Another application of the hologram technology behind vortex beams is the recent use of helical phase plates and their holographic equivalent in microscopy. Incorporating a forked hologram in the image train of a microscope results in a point spread function corresponding to a

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Singular Optics: Optical Vortices and Polarization Singularities

vortex beam. For an extended object, the resulting interference leads to an edge-enhanced image (Furhapter et al., 2005a), the unique feature being that unlike traditional microscopic techniques, the degree of enhancement does not depend upon the orientation of the edge. Further modiﬁcation of the hologram, removing the helical term from its center, results in a novel form of helical interferometric image in which the fringe pattern associated with a surface feature takes on a spiral form, the sense of which allows bulges in the surface to be distinguished from depressions (Furhapter et al., 2005b). Optical vortices are sure to be a subject of study for many years to come. Although they are a ubiquitous feature of interference, it seems that regarding their deliberate generation within a laboratory, SLMs offer the ﬂexibility and performance that make them the technique of choice for many applications.

4. THE STRUCTURE OF OPTICAL PHASE SINGULARITIES 4.1. The Singular Skeleton of an Optical Field A reason frequently given for studying phase and other singularities is that they organize the global spatial structure of the optical ﬁeld: a “skeleton” on which the phase and intensity structure hangs (Berry and Dennis, 2001b; Nye, 1999; Soskin, Denisenko, and Egorov, 2004). In fact, this aspect of singularities drives their study in other ﬁelds, particularly phase singularities in biology (Winfree, 2001) and engineering (Hesselink, Levy, and Lavin, 1997). Since vortices are zeros, they do not contain this information directly! The information about the rest of the ﬁeld must lie in their spatial conﬁguration, and the local phase and intensity structure near the vortex points and lines. It is this latter aspect that we discuss in this section. We have already seen that phase singularities are vortices of the optical current. Explicitly, they have a vorticity, deﬁned on the vortex lines

≡ 12 ∇ × j = 12 ∇ψ∗ × ∇ψ = ∇ξ × ∇η.

(8)

is directed along the vortex line—it is the direction around which the current circulates in a right-handed sense (Berry, 1998). In 2D ﬁelds, is in the zˆ -direction, = ωzˆ , where

ω = ∂x ξ∂y η − ∂x η∂y ξ.

(9)

sign ω is the sign of the vortex point. If the magnitude of the vortex strength (s in Eq. (2)) is greater than 1, both and ω are zero.

The Structure of Optical Phase Singularities

317

The phase and current landscape have other topological features, such as saddle points in two dimensions, discussed in Section 2.1. More generally, there are critical points where the direction of ∇χ and j is not deﬁned because the gradient is zero, i.e., where ∇χ = j = 0, which are singularities of direction in the vector ﬁelds (Nye, Hajnal, and Hannay, 1988). These singularities in two dimensions may be sources and sinks (corresponding to local maxima and minima of χ), as well as saddles (saddle points of χ) and, of course, circulations (vortices). There is a topological index—the Poincaré index (Firby and Gardiner, 1991)—associated with these vector singularities, namely the number of rotations of ∇χ (or j) in a closed circuit around the singularity: +1 for sources, sinks, and circulations and −1 for saddles. The sign of this index is given by the sign of ∂x jx ∂y jy − ∂x jy ∂y jx (this expression is similar to that of ω, with ξ and η replaced by jx and jy ). Vortices themselves always have Poincaré index +1, regardless of sign or strength. Any 2D complex ﬁeld can be partitioned according to the quadrant of the ψ plane in which each point lies, that is, whether ξ and η are positive or negative. All four quadrants meet at vortices. This leads directly to the so-called sign principle, ﬁrst stated and studied in the context of random ﬁelds by Freund and coworkers (e.g., Freund, 1997; Freund and Shvartsman, 1994, and references therein), and can be stated as follows: vortex points adjacent on a zero contour of ξ and η (or more generally, any contour of constant phase) must have opposite sign. An exception to the sign rule occurs when there is a saddle point on the phase contour line between two vortices: in this case, the vortices have the same strength (Freund, 1995). Optical vortices are the zeros of complex scalar ﬁelds, which must themselves satisfy an optical wave equation. As our emphasis is on monochromatic, stationary ﬁelds, the most important equation is the time-independent wave equation, or the Helmholtz equation

∇ 2 ψ + k 2 ψ = 0,

(10)

satisﬁed by a scalar component of the time-independent free space electric ﬁeld from Maxwell’s equations (Born and Wolf, 1959; Jackson, 1998). However, it can also represent a transverse ﬁeld, where the 3D Laplacian 2 = ∂2 + ∂2 and the wavenumber k is the ∇ 2 is the transverse Laplacian ∇⊥ x y transverse wavenumber kr , in which case Eq. (10) represents an idealized nondiffracting optical ﬁeld (Durnin, Miceli, and Eberly, 1987). In these nondiffracting ﬁelds, the energy never ﬂows from the axial to the transverse direction, meaning that the transverse pattern of nondiffracting ﬁelds has no sources or sinks (Dennis, 2001b; Nye, Hajnal, and Hannay, 1988).

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The other important time-independent wave equation we consider is the paraxial wave equation 2 ∇⊥ ψ + 2ik∂z ψ = 0,

(11)

which is an approximate form of Eq. (10) for waves traveling almost entirely in the z-direction (Lax, Louisell, and McKnight, 1975; Marcuse, 1972). The Laguerre–Gauss and Bessel beams described in Section 3 satisfy Eq. (11), although Bessel beams may also satisfy the Helmholtz equation, if −kr2 z/2k is replaced by kz z in the exponent in Eq. (7) (Berry and Dennis, 2001c). Bessel beams, and LG beams with p = 0, have concentric nodes around the beam axis, in addition to the vortex on the axis. In any transverse plane, these are nodal circles: in three dimensions, they are nodal sheets, topologically cylinders. However, nodal sheets do not occur in general 3D superpositions of these (or other) modes: nodal lines are the most general. The conﬁguration of these nodal lines in space can be very complicated, as we will describe below. However, we will ﬁrst discuss the local structure of vortex points in two dimensions in more detail.

4.2. Local Anisotropy of Vortex Points It is simplest to study the local structure of optical vortices mathematically using a Taylor expansion around the zero, which is assumed to be at the origin:

1 ψ(r) ≈ ∇ψ · r + r · ∇∇ψ · r + · · · 2

(for r near 0),

(12)

where ∇∇ψ is the Hessian matrix of second derivatives of ψ. With this approximation, the current j ≈ × r, that is, the current circulates around the vortex in a perfect circle (Berry and Dennis, 2000). However, both the intensity and the phase have more local structure than this, and this has been described in several different ways (Masajada and Dubik, 2001; Molina-Terriza, Wright, and Torner, 2001b; Schechner and Shamir, 1996). This may be stated quite simply: the intensity contours close to the vortices are elliptical, and around this ellipse, sectors of equal area sweep out equal intervals of phase, in a Keplerian sense (Dennis, 2001c). The ﬁrst part of this statement can be veriﬁed directly (also see below) from Taylor expanding |ψ|2 . The second part follows from manipulations of the above Taylor expansion and the expression for j, and it can be shown that for ρ, χ near the vortex, in cylindrical coordinates R, φ, z,

R2

ρ2 ∂φ = , ∂χ ω(0)

(13)

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The Structure of Optical Phase Singularities

where ω(0) is the vorticity on the vortex. The left-hand side of this equation is therefore constant on the intensity contour, giving an analog of the conservation of angular momentum for a linear central force.5 This predicted phase relationship has been veriﬁed in detail experimentally (Wang et al., 2006). The vortices occurring in the cylindrical modes discussed above are, of course, isotropic. Anisotropic vortices can easily be formed by superposing two vortex beams of strength ±1, such as aψLG+1,0 + bψLG−1,0 . Close to the origin, such a superposition has the form

a(x + iy) + b(x − iy) = (a + b)x + (a − b)iy,

(14)

which is anisotropic if both a, b = 0, and the overall sign depends on which of |a| and |b| is larger. In the sense of the angular momentum conservation rule, this superposition is the generation of elliptical motion by two oppositely signed circular motions. The most general way of describing the local elliptic anisotropy of a vortex follows from the Stokes parameters used to describe elliptic polarization (and described here in Section 7). The elliptical intensity contours are entirely described by the complex gradient vector ∇ψ; as any complex vector (Gibbs, 1928), it has an associated ellipse swept out by Re ∇ψ exp(−iα) as α is varied. With the major and minor axes interchanged, this ellipse centered on the vortex has precisely the same form as the low intensity contours. The ellipse geometry can therefore be completely described by the following parameters, analogous to the Stokes parameters [cf. Born and Wolf (1959) or Eq. (24) later],

T0 = |∇ψ|2 , T2 = 2 Re(∂x ψ∗ ∂y ψ)

T1 = |∂x ψ|2 − |∂y ψ|2 T3 = 2 Im(∂x ψ∗ ∂y ψ) = 2z = 2ω.

(15)

The angle of the major axis of the elliptic intensity contours is therefore π/2 + 12 arg(T1 + iT2 ), and the ellipticity is simply twice the vorticity. Although apparently originally described by Dennis (2001a), this parameterization, and its equivalent Poincaré sphere, was published simultaneously by three different groups considering different problems (Dennis, 2004; Egorov, Fadeyeva, and Volyar, 2004; Roux, 2004). The corresponding Poincaré sphere analog has been used subsequently to identify moving vortices as a way of reconstructing ﬂuid ﬂow (Wang et al., 2007). The parameters T0 –T3 can also be expressed in terms of a and b from Eq. (14) (in a way equivalent to the representation of the Stokes parameters by circular polarization components).

5 Analogous to Kepler’s law for a Hookean harmonic oscillator, not a Newtonian gravitational orbit.

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High-strength vortices, around which the phase changes by 2π times 2 or more, also occur, although they are unstable. The allowed form of such a vortex is determined by the wave equation that the ﬁeld satisﬁes. In particular, any local solution of the Helmholtz equation, or paraxial wave equation in the xy-plane, must also locally be a solution of Laplace’s equation ∇ 2 ψ = 0, since, near the zero, any terms proportional to ψ vanish to a higher order. The most general solution, proportional to ρ|s| , is (with the vortex at the origin, as before)

ψ ≈ a(x + iy)|s| + b(x − iy)|s|

for a strength s vortex,

(16)

where a and b are complex numbers, with |a| > (<)|b| if s > (<)0 (Berry and Dennis, 2001b). This representation therefore generalizes Eq. (14) for strength 1, and anisotropic high-strength vortices may be described by exactly the same parameterization as single-strength ones (Dennis, 2001a). As mentioned previously, ω = 0 for high-strength vortices. Also, to obey Eq. (16) at the vortex, ψ must satisfy |s|(|s| + 1) equations: the ﬁeld and all mixed partial derivatives up to order |s| − 1 vanish, but none of order |s| do so, which requires |s|(|s| + 1) equations to be satisﬁed (Berry and Dennis, 2001a). In terms of the Taylor expansion around the vortex, there can be no terms of order less than |s|, and all the terms of order |s| must appear. Thus high-strength vortices are unstable on perturbation. In a longitudinal plane of a ﬁeld solving the paraxial equation (e.g., the xz-plane), this argument does not hold. In fact, it can be shown (Berry and Dennis, 2001b) that no solution for |s| > 1 is possible: high-order vortices perpendicular to the paraxial propagation direction are forbidden.

4.3. Topological Reactions of Vortex Points: Unfolding, Creation, and Annihilation Most of the previous discussion of this section was focused on singly charged vortices which are structurally stable and generally only change position (and local ellipse structure) on perturbation. A vortex of higher strength s, on the other hand, is unstable, and typically breaks up into |s| vortices of strength sign s. The most common perturbation is just adding a constant, representing a perturbing ﬁeld with no vortices near the perturbed one. When a constant is added to the high-strength vortex given in Eq. (16), it breaks up (unfolds) into |s| vortices, with sign s, regularly spaced on a circle centered on the original vortex. The shape of the anisotropy ellipse of all the perturbed vortices is the same, independent of the strength of the perturbation. There is also an index |s| − 1 saddle point left at the origin (Nye, Hajnal, and Hannay, 1988). This ensures that the topological index of j is conserved (Firby and Gardiner, 1991): the Poincaré index

The Structure of Optical Phase Singularities

321

of the initial vortex is +1, whatever s may be, and the total index of the perturbed vortices is |s|. Other perturbations, such as those involving various elliptic transformations, give rise to different arrangements of vortices on perturbation (Dennis, 2006). Two vortex points of opposite sign can also approach and annihilate, or nucleate as a pair. A simple but general example is

ψ = ax2 − c + iby,

(17)

where a, b, c are real numbers.√When c is positive, this represents a pair of strength ±1 vortices at x = ± c/a, y = 0. The anisotropy of these vortices is proportional to c. As c approaches 0, the vortices approach (becoming more anisotropic), annihilating when c = 0. This point, although a zero of the ﬁeld, has no topological charge, and ω = 0, but is obviously not structurally stable. When c < 0, Eq. (17) has no zeros; so the vortex pair has annihilated. As the two vortices annihilate, two saddles approach, ensuring that the total net Poincaré index of the annihilation is zero before and after. In the special case of a ﬁeld satisfying the 2D Helmholtz equation (10), the saddle points and vortices approach on a small circle (size proportional to c) (Nye, Hajnal, and Hannay, 1988). Other topological possibilities for the annihilation of vortices, such as two vortices (and an attendant saddle) colliding and giving a phase maximum and minimum (plus saddle), have also been investigated (Freund and Kessler, 2001). Fields of the local form of Eq. (17) occur in 3D ﬁelds, where c is proportional to z. In this case, varying z is equivalent to varying the axial position of an imaging plane, and creation/annihilation is equivalent to a vortex line “hairpin” (Berry, 1998).

4.4. Vortex Line Geometry in Three Dimensions In three dimensions, optical vortices are lines, since zero contours of 3D complex scalar ﬁelds are 1D, occurring along the intersections of the zero surface contours of the real and imaginary parts (see Figure 2b). These surfaces can have a very complicated topology, but the vortex line always occurs on the intersection of the two surfaces. The direction of the vortex line, lying in each surface, is normal to both of the surface perpendiculars: this justiﬁes Eq. (8), since the vortex direction = ∇ξ × ∇η (Berry and Dennis, 2000). The vortex conﬁguration therefore is entirely determined by the intersection geometry of the nodal surfaces ξ = 0, η = 0. This intersection geometry and topology can be very complicated, as explained in a series of papers by Winfree and Strogatz (1983a,b,c, 1984), who considered a complicated topological hierarchy of vortex loops, links, knots, etc. (although the physical context was different, the geometry is the same). In the context of the linear waves we are describing here, the

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wave equation can impose restrictions on the nodal surface topology. For instance, in ﬁelds of the form polynomial times plane wave, edge dislocation loops occur in punctured wavefronts rather than extra disks of wavefront (Nye and Berry, 1974; Nye, Hajnal, and Hannay, 1988); this statement is topologically equivalent to the fact that wave vortices do not typically occur as the intersections of a nodal plane and a spheroid. As a parameter is varied in a 3D wave superposition, in general, the vortex lines move (Nye and Berry, 1974) and may topologically interact. Although speciﬁc wave constructions are possible in which vortex lines pass through each other benignly (Bialynicki-Birula, Bialynicka-Birula, and Sliwa, 2000), this is not the generic situation. In general, as one parameter is varied, two topological events may occur, at points where the normals to two nodal surfaces coincide: loop nucleation/annihilation and line reconnection (Berry and Dennis, 2001c; Nye, 2004). These topological events occur when ∇I = 0 on a vortex line and when vortex lines intersect the lines along which = 0 (Berry and Dennis, 2007). The two types of event are shown in Figure 11 as intersections of real and imaginary nodal surfaces. If one nodal surface is (locally) ﬂat, the type of event depends on the sign of Gaussian curvature of the other: nucleation of an elliptical loop if positive (the hill in Figure 11a) and hyperbolic reconnection if negative (the saddle in Figure 11b). In the latter case, the event changes the global topology, as the connectivity of the vortex lines is switched. At the saddle-like point of reconnection, the two straight vortex lines directed inwards are opposite each other, as are the two outwards. At such a point on the axis of a laser beam, where the two inwards-pointing vortices are on-axis, this has the appearance of an inversion of the vortex sign in two dimensions (Molina-Terriza et al., 2001a), although, three-dimensionally, no topological rules are violated. Along a vortex line, the phase surfaces typically are helicoidally twisted (as a screw dislocation). In general, this twist is nonuniform and is related to the rotation of the anisotropy ellipse along the line (Dennis, 2004). If the vortex line is a closed loop, continuity of the ﬁeld requires that there be an (a)

(b)

FIGURE 11 Topological events on 3D vortex lines, intersections of real and imaginary zero surfaces. (a) Sequence showing loop annihilation/nucleation. (b) Sequence showing vortex line reconnection.

The Structure of Optical Phase Singularities

(a)

(b)

323

(c)

FIGURE 12 Vortex line topologies in superpositions of Bessel beams (beam axis is vertical). (a) A straight screw dislocation line threads a single-strength screw dislocation loop. The tubes around the vortices are colored by the local phase. (b) The vortex loops forming the link are threaded by two vortex lines in a superposition of Bessel beams, described by Berry and Dennis (2001b). (c) Vortex braid, described by Dennis (2003a).

integer number of 2π twists of the local phase structure—the screw number of the loop. From the continuity of the complex ﬁeld, this requires that the loop be threaded by vortex lines of exactly the same net strength as the screw number (Berry and Dennis, 2001b; Winfree and Strogatz, 1983b). Thus the local structure of the vortex—its helicoidal phase structure, looped around itself—determines the global topology of the ﬁeld, namely that the loop be threaded. This may be realized optically by a dislocation loop in the waist plane, threaded by an on-axis screw dislocation. This topology is shown in Figure 12a. In general, a vortex loop, threaded by strength m’s worth of other vortices, may itself have strength n > 1; in this case, the phase pattern around the loop rotates m/n times. This is possible in solutions of the Helmholtz equation, and such a conﬁguration was explicitly constructed in superpositions of Bessel beams by Berry and Dennis (2001b). If such a ﬁeld is perturbed by a ﬁeld without vortices, then the high-strength loop unfolds to single-strength strands, but with the same twisted topology, giving knots or links in the torus knot class (Berry and Dennis, 2001b). An example of a vortex link generated in this way is shown in Figure 12b. As mentioned earlier, such a high-strength loop is forbidden mathematically in solutions of the paraxial wave equation. By generalizing this construction, however, it was shown that knots and links can occur in paraxial beam superpositions (Berry and Dennis, 2001c), and the construction was realized experimentally in superpositions of Laguerre–Gauss beams (Leach et al., 2004a, 2005). Other vortex topologies are possible in speciﬁc superpositions of few beams, such as vortex braids, realized theoretically in a superposition of counterpropagating, noncoaxial Bessel beams (Dennis, 2003a).

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There do not appear to be strong restrictions on the allowed topologies of vortex lines in three dimensions—if a certain nodal topological conﬁguration is mathematically possible in a complex scalar ﬁeld, then it is likely that beams can be designed with optical vortices having that conﬁguration. Complicated vortex topologies may not be stable: under large perturbations, knots and links may be dissolved by reconnection (Berry and Dennis, 2001c). Furthermore, just because a certain vortex topology is not forbidden, it is, in general, a hard problem to ﬁnd a wave superposition with correctly weighted modes which realizes it optically. However, as we will see in Section 5, vortex links occur naturally in random wave superpositions.

4.5. Vortices in Simple Superpositions of Plane Waves General statements from the previous discussion can be simply illustrated by considering the vortices which occur in superpositions of a small number of plane waves. As described in Section 2.2, Braunbek (1951) realized that vortices occur generically in superpositions of three plane waves. This was subsequently discussed by Nicholls and Nye (1987) in the context of wave pulses. Masajada and Dubik (2001) considered the distribution of the vortices and their anisotropy as the parameters of the three plane waves were varied. The similar case of vortices in three point-source interference was studied by Ruben and Paganin (2007). The following discussion relies on the use of phasors. Each phasor, ψn , corresponds to a plane wave component of the ﬁeld, ψn = an exp(iθn ), and they are labeled in order of decreasing magnitude a1 ≥ a2 ≥ . . . ≥ aN . Provided that the sum of the two smallest phasors exceeds the amplitude of the largest, i.e., a1 + a2 > a3 , then three-wave interference always results in an array of vortex points in two dimensions or parallel vortex lines in three. Figure 13 shows an example of such a ﬁeld (whose constitutent three plane waves have equal amplitude and are equally spaced in propagation direction). The ﬁgure also shows the phasor addition at two different singularities. It is clear from this diagram that these are the only conﬁgurations possible for destructive interference—any other conﬁguration would lead to the phasor sum failing to close, resulting in a nonzero intensity (whose magnitude is the result of the phasor vector addition). In three dimensions, the phasors cannot rotate with respect to each other— the triangle is rigid and can only rotate as one object around the origin. Thus, in this situation, the phase singularities are parallel straight lines. The direction of the lines is that which results in each of the phasors rotating at the same frequency. This is the direction into which the projection of the wavevectors k1 , k2 , k3 is equal. From Eq. (8), this can easily be shown to be k1 × k2 + k2 × k3 + k3 × k1 (Nicholls and Nye, 1987). In four-wave interference, the phasor addition may only remain closed if the phasors rotate at different and varying frequencies (there

The Structure of Optical Phase Singularities

(a)

(b)

325

(c)

FIGURE 13 Three-wave interference. (a) The only possible phasor arrangements that satisfy (ψ1 + ψ2 + ψ3 ) = 0. Two examples are marked on a phase cross-section (b) of the 3D field and the intensity is shown in (c). Each closed phasor arrangement (corresponding to a vortex with s = ±1) appears in a hexagonal vortex lattice.

is one exception—when all kn have the same kr , i.e., distributed on a ring). The exact path of a vortex line is a result of the particular wavevectors in the superposition. However, due to the geometrical constraints of the quadrilateral conﬁguration of phasors, the topological vortex conﬁguration may be easily obtained for superpositions of four waves. With the magnitudes of the four waves ﬁxed, the relative phases constitute three additional degrees of freedom. All possible phase relationships are explored within the 3D ﬁeld (i.e., all possible shapes of quadrilateral with sides a1 , a2 , a3 , and a4 ). Consequently, a change to the initial phase of any of the superposed waves results only in a spatial translation and does not affect any aspect of the vortex geometry. The result is that vortex topology in such ﬁelds is governed by the real amplitudes of the waves. In fact, two distinct topologies are possible, with a reconnection transition in between (O’Holleran, Padgett, and Dennis, 2006). The topology depends on the following relations:

a1 + a4 < a2 + a3

(lines),

(18)

a1 + a4 = a2 + a3

(reconnections),

(19)

a1 + a4 > a2 + a3

(loops).

(20)

When Eq. (18) is satisﬁed, the phasor addition is similar to the three-wave case but with the addition of a small fourth phasor that can execute full 2π revolutions around the other three. This allows the straight line vortices of the three-wave interference to deviate from their original path to form irregular helices, whose axis is the direction of the straight line conﬁguration if the smallest wave, ψ4 , were not present. However, as the smallest wave amplitude a4 grows in size relative to a1 , Eq. (20) is satisﬁed. Now the quadrilateral formed by the phasors can only

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deform to some maximum angle between phasors. Constant clockwise or counter-clockwise movement between phasors is equivalent to moving in some spatial direction in the ﬁeld. As phasors can only move to a maximum angle, vortex lines can only exist within a ﬁnite space—they are compact, existing within a certain range ( x, y, z). Therefore the vortex lines form loops. The phasor additions are illustrated in Figure 14, and examples of the two possible vortex topologies are shown in Figure 15. It is helpful to imagine that the helices of Eq. (18) become more deformed, with curves becoming less straight as the smallest amplitude, a4 , increases. As the magnitude of this amplitude approaches the others, the curves approach each other, close enough to touch with adjacent lines. As this (a)

(b)

FIGURE 14 Four-wave interference. The amplitudes in (a) satisfy a1 + a4 < a2 + a3 allowing one of the phasors to make full 2π revolutions around the other three, resulting in an array of helical vortex lines. The amplitudes in (b) satisfy a1 + a4 > a2 + a3 restricting the phasors to make maximum angles less than 2π with each other, resulting in an array of vortex loops.

(a)

(b)

z

x

y

z

x

y

FIGURE 15 Possible vortex topologies resulting from four-wave interference. The amplitudes in (a) satisfy a1 + a4 < a2 + a3 resulting in an array of helical vortex lines. The amplitudes in (b) satisfy a1 + a4 > a2 + a3 resulting in an array of vortex loops.

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327

point is passed, each pair of helices dissolves by reconnection into a series of closely spaced, identical loops. The transition between the two topologies is the point at which Eq. (19) is satisﬁed, and it is now possible for the phasor quadrilateral to become ﬂat. The result of this ﬂat phasor conﬁguration is highlighted by looking at four waves of equal amplitude. When the four waves have the same magnitude, the closed loop of phasors forms a rhombus, where (say) ψ1 = −ψ3 and ψ2 = −ψ4 . There are three such pairings of phasors possible. The relative phase of the two paired phasors is locked at π, which, as with destructive two-wave interference, happens along a plane. The three possible pairings result in three planes that intersect along straight lines. There are now three arrays of straight lines intersecting where the rhombus becomes ﬂat. The topology in this scenario is degenerate as the vortex lines cross each other exactly. More generally, when Eq. (19) is satisﬁed, the vortex lines are curved, but still cross at reconnections when the phasor quadrilateral becomes ﬂat. When ﬁve waves interfere, cancellation requires the phasors to lie on a pentagon, with six waves a hexagon, and so on. Each phasor increases the size of the shape space of the closed phasor sum, and each particular superposition only explores a small part of this space (whereas for four waves, there are only three relative phases; enough in three dimensions for all possible conﬁgurations for a given set of amplitudes to be explored). Once the number of plane wave components in a ﬁeld exceeds four, the vortex structure is generally complicated and sensitive to changes in both real amplitude and phase of each component.

5. VORTICES IN RANDOM WAVEFIELDS An extremely important area of singular optics is the study of optical vortices and polarization singularities in random ﬁelds. As described in Section 2.4, singularities were observed in scattered waves as far back as the 1950s, and their observation in the 1970s was a major inﬂuence on the original work by Nye and Berry (1974). In optics, these random waves are most familiar as the mottled speckle patterns, occurring when polarized coherent light—such as that from a laser—is reﬂected or transmitted by a random rough surface. Random ﬁelds are an interesting arena in which to study optical singularities, since by nature, only generic features appear— for instance, singularities of strength higher than one almost never appear in the natural optical ensembles usually considered. A further advantage of the study of fully developed optical speckle patterns is that they are very easy to simulate, as a superposition of many plane waves with independently random phases, amplitudes, and directions (whose precise details ﬁx a particular statistical ensemble). In the limit of inﬁnitely many such plane waves, the random ﬁeld is a Gaussian

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random function, where each point is a normally distributed random variable (Goodman, 1985), and analytic calculations of average quantities are possible. In this way, vortices are studied in random ﬁelds experimentally, numerically, and analytically, with good agreement in general between the three. The interest in zeros of speckle patterns is comparatively recent, although it is now part of mainstream speckle optics (Goodman, 2007). Applications of random vortices in speckle patterns (Wang et al., 2007) have recently been proposed. The random ﬁelds most often studied in the context of optical vortices are the (theoretically) most natural: random ﬁelds are statistically invariant to translation and rotation; so all averages are independent of position and orientation in space. The ﬁrst study of optical vortices in random ﬁelds was made by Berry (1978), who studied the densities of vortex points in two dimensions, for monochromatic and quasimonochromatic light propagating paraxially from a random phase screen. In particular, he found that the density of vortex points in the transverse plane was rather high—several per square coherence area. This theoretical study was pursued further by Baranova et al. (1981), who veriﬁed these theoretical calculations by experiment. Over the subsequent 25 years, the subject has been studied increasingly, particularly in the 1990s in theory and numerics by Freund (1994, 1998, and references therein). The properties of vortices (and other features) in superpositions of random waves agree with experimental speckle patterns and analytic calculations when the superpositions involve a few tens of waves. (The example plotted in Figure 7 involved 729 plane waves.) The properties of these isotropic random ﬁelds are determined completely by their power spectrum, that is, the weighting of the random amplitudes as a function of wavenumber k (since the superpositions are isotropic, all wavevectors with the same wavenumber have equal weighting). The power spectrum determines vortex statistics through its spectral moments kn (Berry and Dennis, 2000), where we assume that the zeroth moment is normalized, k0 = 1. Related to the power spectrum is the two-point correlation function

C(R) = 12 ψ∗ (0)ψ(R) ,

(21)

which is its Fourier transform (Goodman, 1985). Thus spectral moments are related to derivatives of C(R) at R = 0. An example of a physically important spectrum in 2D superpositions is all wavevectors with equal wavenumbers k0 , lying on a ring in Fourier space, for which C(R) = J0 (k0 R), which models a nondiffracting speckle pattern emerging from a random annular aperture of vanishing thickness [and which satisﬁes the 2D Helmholtz equation, Eq. (10)]. A Gaussian spectrum is also important,

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329

for which C(R) = exp(−R2 Kσ2 /2), with Kσ the standard deviation of the Gaussian power spectrum. Superpositions of random waves are mathematically interesting in their own right, as analytic calculations are possible, and theorems can be proved (Adler, 1981). Furthermore, the model is important in other physical systems, including eigenfunctions in random 2D cavities (“quantum billiards”) (Berry, 1977), which may be quantum, electromagnetic, or acoustic in nature. These eigenfunctions are complex when the Hamiltonian of the system is not time-reversal-invariant, such as in the presence of open channels and absorbing walls, and vortices in these situations have been described in detail (e.g., Kuhl, Stöckmann, and Weaver, 2005). In Section 5.1, we will discuss the statistical densities of optical singularities and their properties. We will not discuss the details of the calculations themselves (recently reviewed by Dennis, 2007). We will also describe in some detail the random vortex line topology in 3D paraxial random wave superpositions in Section 5.2. Statistics of polarization singularities in random waves will be discussed later in Section 7.

5.1. Statistical Properties of Vortices in Two- and Three-Dimensional Random Waves The density of vortex points in 2D random ﬁelds, d2 , is the fundamental statistical quantity for vortices in speckle,

d2 =

C (0) k2 =− , 4π 2π

(22)

dependent on the second spectral moment k2 , equivalent to the second derivative C (R) at the origin. It was originally calculated by Berry (1978), and subsequently recalculated and discussed by Baranova et al. (1981), Halperin (1981), and Berry and Dennis (2000). Since each ﬁeld has a statistical weight equal to its conjugate in the ensemble, the densities of s = +1 and −1 are equal. Under any reasonable deﬁnition of coherence length, d2 is of the order of 1. In Section 5.2, we will use the deﬁnition of coherence length = 2π/d2 . The average Poincaré index of any isotropic random ﬁeld must be zero (Longuet-Higgins, 1957). This implies that the density of phase saddles in two dimensions equals the densities of vortices, phase maxima, and minima, and the density depends on k4 and k2 (Dennis, 2001b). The density of maxima, minima, and saddle points has been computed, but is a rather more complicated function of the spectrum (Weinrib and Halperin, 1982). Local properties of vortex points in two dimensions have also been calculated, such as the probability density function of the anisotropy ellipse

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eccentricity ε (Berry and Dennis, 2000; Dennis, 2001a), which has been experimentally veriﬁed (Zhang and Genack, 2007). It is also possible to calculate the vortex two-point density correlation function g(R) and charge correlation function gQ (R). The former is the average of the product of local vortex densities, separated by distance R, and gQ (R) is similar, but with the densities weighted by the vortex sign. gQ (R) was originally calculated by Halperin (1981) in the context of random ﬁelds in condensed matter and cosmology. It has been the object of some subsequent study (e.g., Berry and Dennis, 2000; Dennis, 2003b; Freund and Wilkinson, 1998), and has the remarkably simple form:

1 gQ (R) = 2 ∂R C (0) R

C (R)2 . 1 − C(R)2

(23)

gQ (R) satisﬁes a charge-screening relation—the integral of d2 g(R) over the plane is −1—so the topological charge of each point is statistically “neutralized” by the nearby vortices. This property is similar to charge correlation functions of Coulomb-interacting ionic liquids, although the effect here comes purely from the topology of the random ﬁeld (Foltin, 2003). The number correlation g(R) can be calculated exactly, although it has a rather more complicated form (Berry and Dennis, 2000; Saichev, Berggren, and Sadreev, 2001), which can be written in terms of elliptic integrals (Dennis, 2001c). g(R) and gQ (R) are plotted for two choices of spectrum in Figure 16. Local properties of vortex lines in isotropic 3D random waves may also be calculated. The vortex line density per unit volume is 2d2 , where d2 is the density of the point intersections with any plane (Berry and Dennis, 2000; Dennis, 2007). Distributions of curvature (Berry and Dennis, 2000) and phase twistedness (Dennis, 2004) may also be found. However, the answers to topological questions about the 3D tangle of vortex lines in random spatial ﬁelds, such as the probability of vortex lines being closed (a)

(b)

1

1.25 1

0.5

0.75 0.5

1

0.25 0.5 0.25

2

4

6

8

10

12

2

3

4

5

K␴R

14

k0R

1

FIGURE 16 Vortex correlation functions for two choices of spectrum. Black lines denote g(R), and gray gQ (R). (a) Ring spectrum, C(R) = J0 (k0 R); (b) Gaussian spectrum C(R) = exp(−R2 Kσ2 /2).

Vortices in Random Wavefields

331

loops or inﬁnite lines, whether closed loops can be linked or knotted, etc., cannot be calculated by known techniques. To answer these questions it is necessary to resort to numerical experiments.

5.2. Three-Dimensional Random Wave Simulations: Vortex Fractality and Random Topology We describe here a recent approach (O’Holleran et al., 2008) to the largescale geometry and topology of optical vortex lines in 3D random speckle patterns, by numerically superposing many paraxial plane waves with random complex amplitudes with a Gaussian power spectrum. However, unlike the random waves discussed above where the wavevectors of the superposed waves are random, the k-vectors here lie on a regular square lattice in the transverse k-plane, as shown in Figure 17. The grids chosen are sufﬁciently ﬁne that the density of vortices in the transverse plane agrees with Eq. (22). The random ﬁelds resulting from this superposition are not only transversally periodic but also periodic upon propagation by the Talbot effect (self-imaging effect) (O’Holleran, Padgett, and Dennis, 2006; Talbot, 1836)—3D space is tiled by “Talbot cells.” The vortex line structure in the simulation is therefore ﬁnite and periodic, enabling the total topology to be completely determined: the vortices either form closed loops or extend as inﬁnite, periodic lines. With the data from hundreds of Talbot cells, it may be concluded that the vast majority (> 99%) of vortex lines are closed loops in Cartesian space. However, the remainder are periodic lines that account for 73% of the total vortex line length. This implication for real, nonperiodic but ﬁnite extent ﬁelds is that these lines would traverse the entire bright region of the ﬁeld. On the large scales of interest, it is natural to examine the scaling of the spatial distance R between pairs of points on the same vortex line, with respect to the vortex arc length L between them. Power-law scaling 0 (a)

1

2␲ (b)

␲

21 (c)

1

21 (d)

FIGURE 17 A typical 27 × 27 Gaussian k-space power spectrum on a square grid. (a) Intensity, (b) phase, (c) real part, and (d) imaginary part of the spectrum. A circular aperture has been applied to the Gaussian modulation to remove the corners of the square array.

1

332

Singular Optics: Optical Vortices and Polarization Singularities

(i.e., a straight line in log–log plot) of R against L is the signature of a line’s fractality. The fractal dimension of the line is the reciprocal of the line’s gradient. Figure 18a shows the average straight line distance R against arc length L between points on 100 different vortex lines from 100 different Talbot cells (totaling an entire vortex length of approximately 3 × 105 ). The straight line best ﬁt shown on this plot has a gradient of 0.52 ± 0.01, which is rather close to the gradient 1/2 expected for a Brownian random walk. This suggests that the vortex lines are indeed random walks in the two decades over which the plot is nearly linear. The upper limit to this is a result of the periodicity of the constructed ﬁelds. For large loop sizes, the loop number density decreases with increasing loop size, approximately to the power −2.5, shown in Figure 18. The most common loop size corresponds to slightly less than the Rayleigh resolution limit based on the numerical aperture of the interfering plane waves. The fact that the peak in the size distribution for loops falls below this resolution means that this characteristic length is smaller than the spacing between local maxima in the speckle pattern. Therefore, these small loops occur within the semi-dark regions between bright speckles, and they are abundant in real speckle ﬁelds. In the extensive numerical investigation of O’Holleran et al. (2008), many thousands of vortex line loops were identiﬁed. In addition to their scaling properties, the loops displayed interesting topology. Figure 19 shows some examples of loops threaded by inﬁnite lines and linked loops, although no knots were detected.

6. OPTICAL VORTICES AND ANGULAR MOMENTUM While the modern studies of optical vortices can undoubtedly be considered to have started with the work by Nye and Berry (1974), the more recent rapid growth of the ﬁeld began following the recognition by Allen et al. (1992) that helically phased beams carry orbital angular momentum. This arises from the azimuthal phase structure of the optical beam and hence is totally independent of the spin angular momentum, which is linked to the polarization state. At this point, it is worth clarifying the relationship between optical vortices and orbital angular momentum: two terms that are often, yet incorrectly, used interchangeably. As explained previously in Section 2, an optical vortex is a position in space around which the optical phase advances or retards by a multiple of 2π. At the vortex center, the phase is singular, hence the term phase singularity. However, the intensity at the vortex center is zero, and therefore the vortex itself carries neither linear momentum nor angular momentum. By contrast, it is only in the immediate vicinity of the vortex that the azimuthal phase term means that the phase fronts of the optical ﬁeld are helical (see Figure 20).

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Optical Vortices and Angular Momentum

(a) 2.0 1.5

log(R ) L

1.0 0.5

21

1

P

2

3

P P

4

20.5 21.0

log(L) L

(b) 1

21.0

20.5

0.5

1.0

1.5

2.0

log(N ) L23

21 22

23

24

log(L) L

FIGURE 18 Scaling properties of random vortex lines and loops. (a) Displacement R versus arc length L of vortex lines in Talbot cells (in units of scaled coherence length ). The black line plots the average of 100 different lines, while the colored lines show three individual lines and their displacement versus length. The spots indicate the length of one period for each line. A straight line best fit to the scale log(R)/log(L) from log(L) = 0.5 to log(L) = 2.5 has a gradient of 0.52 ± 0.01 consistent with that of Brownian fractal lines (1/2). (b) Number versus loop length of vortex loops. The vertical dotted lines denote the fitting range, which has gradient −2.46 ± 0.02.

334

Singular Optics: Optical Vortices and Polarization Singularities

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 19 Threaded loops and links found in simulated speckle fields. (a–c) show loops threaded by infinite lines and (d–f) show pairs of linked loops.

FIGURE 20 A diagram showing a helical surface of equal phase with the Poynting vector indicated by a curved line.

The associated Poynting vector (the current j) has an azimuthal component, and hence there is a net ﬂow of both energy and momentum around the vortex line, which in turn gives an angular momentum directed along the line. An optical beam can have an azimuthal phase gradient without a nearby phase singularity, implying that light can still possess an

Optical Vortices and Angular Momentum

335

orbital angular momentum without an optical vortex (Courtial et al., 1997a). Despite this distinction, it is the case that most studies of orbital angular momentum have involved beams with optical vortex lines along the beam axis. For a light beam with an azimuthally symmetric intensity pattern and a helical phase structure described by exp(iφ) (with any integer), the orbital angular momentum is equivalent to per photon. The topic of orbital angular momentum has itself formed the basis of a number of extensive (Allen, Courtial, and Padgett, 1999a) and popular reviews (Padgett, Courtial, and Allen, 2004), and it is not our purpose here to repeat this material. However, given the relevance of optical vortices to this extended body of work, it seems appropriate to cover a number of landmark developments in the ﬁeld. As described in Section 3, the original papers in 1992 and 1993 (Allen et al., 1992; Beijersbergen et al., 1993) not only established a theoretical framework for orbital angular momentum but considered how such beams could be created using cylindrical lenses to transform HG into LG modes (Beijersbergen et al., 1993). It was also recognized within these papers that the generation of optical angular momentum would result in a reaction torque on the optical components, in this case the cylindrical lens modeconverters. Whatever be the details of the laser beam, it is clear that the angular momentum transfer to an optical component can never exceed the linear momentum of the photon multiplied by the radius of the object (Courtial and Padgett, 2000). Additionally, however, the momentum of inertia of a regular solid body scales with the ﬁfth power of radius. These two facts together imply that achieving the optically induced rotation of macroscopic objects is difﬁcult. Indeed, early attempts to observe the rotation of a miniature cylindrical lens mode-converter proved impossible.

6.1. Optical Spanners Despite experimental challenges, it was clear that the need remained for a demonstration that orbital angular momentum was a real angular momentum rather than a mathematical analogy. This ﬁrst demonstration of optical angular momentum transfer was provided by He et al. (1995). They based their experiment around optical tweezers, which normally use a tightly focussed Gaussian mode to generate a gradient force on a dielectric particle, thereby conﬁning it to the focus. However, they took advantage of some of their earlier work with forked diffraction gratings to produce LG beams (Heckenberg et al., 1992), which they could then use as the trapping beam in the optical tweezers. These LG tweezers were used to conﬁne an absorbing particle to the beam axis, where partial absorption of the light and angular momentum led the particle to spin. This spinning was correctly attributed to the transfer of orbital angular momentum from the light to the particle.

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Singular Optics: Optical Vortices and Polarization Singularities

Subsequent to this initial demonstration, the idea was extended to include both orbital and spin angular momentum in the same optical beam. Control of the relative handedness of the two components allowed the spin rate to be increased or decreased (Friese et al., 1996). For a particle conﬁned to the beam axis, the spin and orbital angular momentum were shown to act equivalently. The general technique was christened “optical spanners” to distinguish it from conventional optical tweezers (Simpson et al., 1997). Another signiﬁcant development of optical spanners occurred in 2002 when Padgett and coworkers used a circularly polarized LG beam with a large value of , thereby producing an annular beam of a diameter signiﬁcantly larger than the particle (O’Neil et al., 2002). In such situations, rather than the particle being trapped on the beam axis, the gradient force conﬁnes the particle to the high-intensity annular ring. This conﬁguration of optical spanner reveals a difference in the behavior between orbital and spin angular momentum. While the transfer of spin angular momentum causes the particle to spin around its own axis, the transfer of orbital angular momentum causes the particle to orbit around the beam’s axis (see Figure 21). This orbital motion is a direct consequence of light scattering from the helical phase fronts and the azimuthal component of the Poynting vector (Padgett and Allen, 1995). Optical spanners in various forms have been further developed by numerous groups for various applications ranging from the driving of ﬂuid pumps (Ladvac and Grier, 2004; Leach et al., 2006) and micro-machines (Knoner et al., 2007) to optical probes for measuring viscosity (Bishop et al., 2004). LG beams have also found other uses in optical tweezers simply as annular beams for trapping low-index objects (Gahagan and Swartzlander, 1996) or high-index objects with greater efﬁciency (O’Neil and Padgett, 2001), where their orbital angular momentum is not important.

6.2. Orbital Angular Momentum and Quantum Entanglement Although the transfer of orbital angular momentum to particles held in optical tweezers may have been the largest single sub-ﬁeld, it is not the only impact that orbital angular momentum has had on optical science. In 1996, it was observed that the frequency doubling of an LG beam led to a transformation in its orbital angular momentum state (Dholakia et al., 1996). Frequency doubling is an example of frequency up-conversion where two photons (1,2) of lower frequency are incident on a nonlinear crystal to produce a single photon (3) with frequencies given by ω1 + ω2 = ω3 . If the incoming beams are LG beams described by 1 and 2 , the resulting mode transformation can be understood through a number of complementary arguments ranging from statements of conservation laws or Poynting vector trajectories to the application of phase-matching conditions. The energy ﬂow of an LG mode is represented by the Poynting vector, which

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FIGURE 21 Top: the transfer of orbital angular momentum from an = 3 beam to a particle, causing the particle to rotate about the beam axis. Bottom: the transfer of spin angular momentum, leading to rotation of the particle on its own axis.

also indicates the direction of the momentum ﬂow. For all beams with an exp(iφ) phase term, the skewness of the Poynting vector with respect to the beam axis is /k0 r. If the light is frequency doubled, k0 doubles, meaning that if the Poynting vector is to be continuous then must also double. Since the total number of photons is halved, the doubling of means that the total orbital angular momentum is conserved in the light ﬁelds. More rigorously, at every point in the nonlinear material, the phases of the three ﬁelds are related by φ1 + φ2 − φ3 = constant. This means that for three helically phased beams, 1 + 2 = 3 , equivalent again to the conservation of orbital angular momentum (Courtial et al., 1997b). These early experiments and understanding of the role played by orbital angular momentum in frequency doubling were a precursor to a work of much greater importance within physics, namely the study of orbital angular momentum in parametric down-conversion. Down-conversion is the process by which a single incoming photon (3) incident on a nonlinear

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crystal produces two photons (1,2) of lower energy, ω1 + ω2 = ω3 (Arnaut and Barbosa, 2000; Franke-Arnold et al., 2004). In 2001, Zeilinger and coworkers reported their work on measuring the orbital angular momentum states of the photon pairs produced from parametric down-conversion (Mair et al., 2001). In this work, the measurement of the orbital angular momentum states was accomplished by using a series of interchangeable holograms, each designed as a diffraction grating containing the characteristic -pronged fork on the optical axis (see Section 3). The angular momentum of any incident beam is transformed by the hologram, but only if the result is a fundamental Gaussian mode can it then be efﬁciently coupled into single-mode optical ﬁber. Hence, the orbital angular momentum states of a beam can be deduced by varying the number and sense of the hologram forks, while recording the intensity of the coupled light. This also works for single photons. However, since each photon can only be subjected to a single measurement, the efﬁciency of the process can never exceed the inverse of the number of states to be tested. Despite this limitation in efﬁciency, the Zeilinger study revealed an important shared property of the down-converted photon pairs. When the -fork values of both detection holograms were set independently, the simultaneous detection of both photons is only maximized for 1 + 2 = 3 . When the incident beam has 3 = 0, this gives 1 = −2 . These results were indicative of entanglement of the orbital angular momentum states of the down-converted photon pairs, and could be considered as being the orbital equivalent to the polarization entanglement (i.e., of spin angular momentum) typiﬁed by the work of Aspect, Dalibard, and Roger (1982). Strictly, rather than only measuring the correlation between measurements of orthogonal states, one also needs to measure the reduced correlations between nonorthogonal states. For correlations between polarization states, this simply requires the polarizers to be rotated through various angles, but for orbital angular momentum states, it is slightly more complicated. One option that was explored is to shift the fork position laterally (Vaziri, Weihs, and Zeilinger, 2002) in the hologram so that the measured mode becomes a complex superposition of various modes of lower value. The weighting of the = 0 mode increases as the displacement of the fork is increased. Recently, the use of spiral phase plates in the measurement of the orbital angular momentum state has shown the entanglement of fractional orbital angular momentum states (Oemrawsingh et al., 2005), and that the added dimensionality of the orbital angular momentum Hilbert space increases the dimensionality of the quantum channel (Pors et al., 2008). The recognition that orbital angular momentum is a valid description of single photons and that this quantity can be entangled has led to a signiﬁcant literature on the potential use of orbital angular momentum

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states in both quantum and classical regimes for information processing and data transmission. The full exploitation of the larger dimensionality still awaits a highly efﬁcient scheme for measuring the orbital angular momentum of the photon within a large number of possible states, N, without suffering a 1/N inefﬁciency. Such measurements are possible using interferometric sorting systems based on a phase shift that depends on the rotational symmetry of the orbital angular momentum states (Leach et al., 2002). However, a measurement of N possible states requires N − 1 coupled interferometers, which is certainly possible, but perhaps not highly practical!

6.3. Rotational Frequency Shifts Orbital angular momentum has given fresh insights into the rotational frequency shift of a light beam. In the 1970s, Garetz and Arnold recognized that when a circularly polarized light beam is rotated at an angular frequency around its own beam axis, the corresponding advance or retardation of the rotating electric ﬁeld vector is equivalent to a frequency shift (Garetz and Arnold, 1979); one cycle more or less for each complete revolution of the beam. The sense of circular polarization is described by σ = ±1 meaning that the rotational frequency shift for circularly polarized light is given by ω = σ. Such frequency shifts can also be described in terms of an evolving Berry phase (geometric phase) (Bretenaker and Leﬂoch, 1990). It is not perhaps surprising that a similar frequency shift is obtained from the rotation of helically phased beams. The -fold rotational symmetry of the beam means that one rotation of the beam gives a frequency shift of ω = (Courtial et al., 1998a). What is more surprising is how the spin and the orbital angular momentum can combine. If one plots a cross-section of the vector ﬁelds in a circularly polarized LG beam, one notes that the resulting ﬁeld distribution has a (σ + )-fold rotational symmetry. Consequently, in that case, the result is a single frequency shift, which is proportional to the sum of the spin and orbital angular momentum (Courtial et al., 1998b).

6.4. Fourier and Possible Uncertainty Relationships between Angle and Orbital Angular Momentum Finally, an area where the concept of orbital angular momentum has led to a new understanding and debate is that of angular uncertainty relationships. The Heisenberg uncertainty relationship for linear position and momentum is usually written as x p ≥ /2, where x and p are the standard deviations of the position and linear momentum, respectively. As is well known, the inequality becomes an equality when the probability distributions are Gaussian. Such states are termed intelligent states.

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One obvious question is whether there exists an equivalent expression for angular momentum. The linear uncertainty relationship stems from the Fourier relationship between a wavefunction and the momentum of the particle, as expressed by the de Broglie relationship with respect to the wavelength. Obviously, there exists a similar relationship between an angular position distribution and its Fourier cylindrical harmonics. For light beams, these harmonics are exactly the orbital angular momentum states of the light. Consequently, if an angular aperture is illuminated by an optical beam of a given value of , then the transmitted light will be described by a distribution of orbital angular momentum states, described by the Fourier series of the 2π periodic function describing the azimuthal aperture (Yao et al., 2006). This Fourier relationship between the azimuthal aperture function and orbital angular momentum states is not disputed. More subtle is whether this Fourier relationship between a periodic angle and discrete angular momentum can be re-expressed as an angular uncertainty relationship. A point of concern is that the periodic nature of the angular variable means ˇ that its standard deviation is ill-deﬁned (Reháˇ cek et al., 2008). One solution to this problem is to evaluate the angular standard deviation only over the −π to π range with the origin of the angular variable positioned at the center of the aperture. Deﬁning the angular standard deviation in this way means that for narrow azimuthal apertures, one can write θ L ≥ /2, where L is the angular momentum, i.e., L = . As the aperture widens, the formula no longer holds. In the extreme where the beam is unrestricted while θ is still ﬁnite, the corresponding value of L is zero, meaning the angular inequality is apparently violated. The resolution of this is the inclusion of an additional term on the right-hand side θ L ≥ (1 − 2πP(−π))/2, where P(−π) is the probability of the aperture function at the edge of the integration limit (Franke-Arnold et al., 2004). This modiﬁed form of an angular uncertainty relation is in close agreement with observed experimental results. However, the restriction on the deﬁnition of angle has led to other approaches being proposed (Hradil et al., 2006). Despite these alternatives to the expression of an uncertainty relationship, the underlying analog of the Fourier relationship between angle and orbital angular momentum is beyond doubt and has most recently also been shown to be valid for pairs of entangled photons (Jha et al., 2008).

7. POLARIZATION SINGULARITIES 7.1. Introduction: Polarization Singularities in Daylight Had our species evolved to see the polarization of light as well as its intensity and color, polarization singularities would be extremely familiar to

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us, as they occur naturally in the blue daylight sky. They also occur in many other optical situations where the state of polarization varies with position. The polarization pattern in the sky results from Rayleigh scattering of sunlight in the atmosphere. Multiple scattering effects mean that the degree of polarization, and the direction in which the polarization is maximum, varies with position: the polarization degree is maximum on the arc at 90◦ to the sun’s direction, and zero at the so-called neutral points. Such a neutral point occurs approximately 15◦ above the sun, discovered in 1840 by Babinet; one was discovered earlier by Arago, a few degrees above anti-solar point (the point in the sky diametrically opposite the sun), yet another a few degrees below the sun by Brewster, diametrically opposite Arago’s point (Brewster, 1847, 1863a). Since in usual conditions, only a hemisphere of the sky is visible, only the Babinet point and one of the Brewster or Arago points is visible. All three are visible simultaneously in observations from a higher elevation, as well as a fourth neutral point opposite the Babinet point, below the Arago point (Horvath et al., 2002). The observed polarization, together with a theoretical ﬁt (Berry, Dennis, and Lee, 2004b), is shown in Figure 22. The skylight polarization pattern with its neutral points has been the subject of several observational studies in various atmospheric circumstances (Coulson, 1988; Gal et al., 2001; Horvath et al., 1998; Lee, 1998), including its possible role in Viking navigation (Hegedüs et al., 2007). The phenomenon was explained quantitatively with radiative transfer theory by Chandrasekhar and Elbert (1951, 1954). A simple mathematical ansatz of the skylight polarization pattern was proposed by Berry, Dennis, and Lee (2004b), inspired by the connection with crystal optics, as anticipated by Brewster (1863b), to which we will return in Section 7.3. The cosmic sky also has a polarization pattern from the cosmic microwave background radiation, and the associated polarization singularities have been investigated (Dolgov et al., 1999; Huterer and Vachaspati, 2005). Although historically, the interest of neutral points was as nodes of degree of polarization, they are also singularities of the direction of polarization azimuth θ, which is deﬁned as the direction in which the electric vibration is maximum. This direction θ, in addition to being undeﬁned in unpolarized light, is also undeﬁned for circular polarization. θ is not the direction of a vector, rather a director: it has a 180◦ rotational symmetry (and is represented by an undirected line rather than an arrow). As with the phase of optical vortices, all values of this polarization azimuth direction (between 0◦ and 180◦ ) occur in the neighborhood of a neutral point, leading to a half-integer quantized topological index for polarization direction. Topological singularities and defects with half-integer index occur in other ﬁelds described by a position-dependent director (Mermin, 1979; Nelson, 2002), notably disclinations in liquid crystals (Frank, 1958),

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FIGURE 22 The polarization pattern in the sky. The disk is the full visible sky hemisphere, in fisheye projection, with the sun occulted to prevent overexposure. Superimposed are the measured directions of (partial) polarization direction (short straight lines), and the thick curves are the mathematical ansatz for these directions of Berry, Dennis, and Lee (2004b). The opacity of the curves is proportional to the degree of polarization, which goes to zero at the neutral points, where the polarization direction is undefined. The Babinet point occurs toward the center of the figure, “above” the sun, and the Arago point is just above the horizon, toward the bottom of the figure. The photograph was taken by R. L. Lee Jr, and the observational and theoretical data plots are based on Figures 5b, and 7b of Berry, Dennis, and Lee (2004b).

ﬁngerprints (Kucken and Newell, 2004; Penrose, 1979), and umbilic points in the curvature of surfaces (Berry and Hannay, 1977; Porteous, 2001). This last case is particularly important as the umbilic point morphologies (which are connected to the umbilic catastrophes discussed earlier) have very similar morphological forms to polarization singularities. Integer index polarization singularities have also received attention recently, particularly in radially or azimuthally polarized laser beams, which may be thought of as analogs of = 1 LG modes [cf. Eq. (6)], where the constituent HG (TEM) modes have the same phase but orthogonal linear polarizations (rather than the same polarization and a quarter-cycle phase difference). These beams are of interest as their focus is tighter than a conventional fundamental Gaussian beam (Dorn, Quabis, and Leuchs, 2003) and they have possible applications such as imaging (Sheppard and Choudhury, 2004) and trapping (Zhan, 2004). Numerous methods

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(ii)

(i) (i)

(ii)

(iii)

(iii)

FIGURE 23 A typical field of position-varying elliptical polarization. Regions of opposite handedness are shaded differently and are separated by L lines. C points are the circles, whose shading denote morphology: star (dark gray), monstar (light grey), lemon (white), elliptic (black outline), and hyperbolic (gray outline). Insets show (i) a lemon, (ii) a monstar, and (iii) a star.

for their generation have been suggested (e.g., Davidson and Bokor, 2004; Shoham, Vander, and Lipson, 2006), although they are not always structurally stable on propagation (Niv et al., 2006). Here, we will describe the polarization singularities that naturally appear in completely polarized ﬁelds whose state of elliptic polarization varies with position. Such ﬁelds, in which the polarization ellipses vibrate in the transverse plane, are shown in Figure 23. The polarization is circular at points (where the azimuth θ is not deﬁned) called C points (Nye, 1983b) and linear along L lines (originally called S lines), where the polarization handedness is not deﬁned (Nye, 1983a). These singularities are structurally stable upon perturbation and propagation, and in this sense are the natural counterpart to vortices in scalar optics. In three dimensions, circular polarization occurs along C lines, and linear along L surfaces (provided the polarization ellipses are transverse to the propagation direction). In the next subsection, we will discuss the various morphologies of polarization singularities in transverse ellipse patterns, and in Section 7.3, how polarization singularities occur in crystal optics. We then move to more general polarization ﬁelds: in Section 7.4, ﬁelds where the ellipses are not transverse to the propagation direction, and ﬁnally, in Section 7.5, to singularities in relativistic electromagnetic ﬁelds.

7.2. Parametrizing Polarization Singularities Amonochromatic time-varying electric vector ﬁeld at a point is speciﬁed by a complex vector (Jones vector) E = p + iq = (Ex , Ey ), where the real, timevarying ﬁeld is E(re) (t) = Re{E exp(−iωt)}. The locus traced out by E(re) (t) is,

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of course, an ellipse, determined completely by E [a general mathematical fact appreciated by Gibbs, in his original work on vectors (Gibbs, 1928)]. E can also be speciﬁed by its circular components ER ≡ √1 (Ex − iEy ), EL ≡ 2

√1 (Ex 2

+ iEy ). The ellipse geometry can easily be understood in terms of the four Stokes parameters (Born and Wolf, 1959; Brosseau, 1998):

S0 = I = |Ex |2 + |Ey |2 = |ER |2 + |EL|2 , S1 = I0◦ − I90◦ = |Ex |2 − |Ey |2 = 2 Re(E∗R EL), S2 = I45◦ − I135◦ = 2 Re(E∗x Ey ) = 2 Im(E∗R EL),

(24)

S3 = IR − IL = 2 Im(E∗x Ey ) = |ER |2 − |EL|2 , where I denotes the intensity (with subscripts denoting the appropriate cartesian or circular components). There is a natural complex scalar ﬁeld whose phase gives the polarization azimuth:

σ ≡ S1 + iS2 = E∗R EL = |σ| exp(iθ/2).

(25)

σ is a complex scalar function whose nodes are the C points; the half-integer index of these singularities comes from the fact that arg σ = θ/2. Of course, right-handed and left-handed C points and lines occur, and from Eq. (25), it is clear that the C points are simply the vortices in the left- and right-circular components. σ is very useful in the study of C points and lines in transverse complex vector ﬁelds (Dennis, 2002; Freund, Soskin, and Mokhun, 2002b; Konukhov and Melnikov, 2001); sometimes it is convenient to work with σ divided by S0 + S3 , which gives EL/ER , i.e., stereographic projection of the Poincaré sphere to the complex plane (Azzam and Bashara, 1977; Berry and Dennis, 2003). Since circular polarization occurs when both S1 and S2 are zero, the alternate name “Stokes vortex” has been proposed (Freund, 2001; Freund et al., 2002a). By analogy with Eq. (9) for vortices, the C point index is given by the sign of

DI = S1,x S2,y − S1,y S2,x ,

(26)

where x and y subscripts denote spatial derivatives. Many of the aspects of vortex morphology described previously, such as the sign rule, the relationship with saddle points, creation, and annihilation, are the same as for vortices (Mokhun, Soskin, and Freund, 2002), although σ does not satisfy a wave equation. The richer geometry of ellipse ﬁelds means that C points have additional classiﬁcations (Nye, 1983b, 1999), based on morphologies previously studied for ellipse ﬁelds around umbilic points (Berry and Hannay, 1977).

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The ﬁrst, associated with θ, can be understood by considering the “polarization lines,” the curves whose tangents give the polarization azimuth. On a C point, different numbers of polarization lines may terminate: one only on a “lemon,” three on a “star,” or inﬁnitely many, with three straight, on a “monstar.” Stars have index −1/2, whereas lemons and monstars have index +1/2 (Nye, 1983b). The polarization lines around the three C points of Figure 23 are shown in Figure 24. The monstar has a transitional nature: it has the same index as a lemon and the same number of terminating lines as a star, hence its name (le)monstar (Berry and Hannay, 1977). The line classiﬁcation (Darbouxian, or L classiﬁcation), distinguishing three terminating lines (star, monstar) from one (lemon), is determined by the sign of (Dennis, 2002).

DL = (2S1y + S2x )2 − 3S2y (2S1x − S2y ) (2S1x − S2y )2 + 3S2x (2S1y + S2x ) − (2S1x S1y + S1x S2x − S1y S2y + 4S2x S2y )2 .

(27)

This complicated expression can be simpliﬁed somewhat in terms of natural geometric parameters of the singularities (Dennis, 2008). A creation/annihilation event of a C point pair occurs between a star and a monstar (Nye, 1983b).

(a)

(b)

(c)

(d)

(e)

(f)

FIGURE 24 Morphologies of the three inset C points of Figure 23: (a) and (d) from (i); (b) and (e) from (ii); (c) and (f) from (iii). Polarization lines are shown around (a) lemon, (b) monstar, and (c) star with one or three straight terminating lines on the singularity. Double cones of ellipse axis lengths are shown in (d) (elliptic), (e) and (f) (hyperbolic).

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The axes of the polarization ellipse are equal on C points but not elsewhere; in general, the pair of ellipse axis lengths is a double-valued function of position, intersecting as a double cone at C points (as shown in Figure 24d–f). The nature of this intersection determines the C classiﬁcation (catastrophe or contour classiﬁcation), determined by

DC = D2I − (S1,x S0,y − S1,y S0,x )2 − (S0,x S2,y − S0,y S2,x )2 ,

(28)

with the singularity called elliptic (hyperbolic) when DC > 0 (< 0). At hyperbolic C points, the cones are tipped below/above the height of their intersection; this is not so for elliptic points. Each of stars, monstars, and lemons may be of elliptic or hyperbolic type (Dennis, 2002), and all six morphologies have been seen experimentally in random optical ﬁelds (Denisenko, Egorov, and Soskin, 2004; Egorov, Soskin, and Freund, 2006; Flossmann et al., 2008), with probabilities agreeing with theoretical predictions (Dennis, 2002). σ is not the only complex scalar quantity whose nodes give C singularities: they are also zeros of

ϕ ≡ E · E = p2 − q2 + 2ip · q = ER EL

(29)

(Berry and Dennis, 2001a). arg ϕ, as the sum of the phases of the circular components, is independent of arg σ, the difference of them (Hajnal, 1987a). The L lines occur when S3 = 0, that is, the ellipse handedness—given by the discrete sign S3 —is undeﬁned, and these lines separate regions of predominantly right-handed polarization from those of left-handed (Hajnal, 1987a; Nye, 1983a). Along L lines, the well-deﬁned azimuth θ may change and is precisely 0◦ , 90◦ , 45◦ , and 135◦ to the horizontal at zeros of the other complex Stokes ﬁelds S1 + iS3 , S2 + iS3 , with the Stokes vortex sign related to the sign of the polarization azimuth change through these points (Angelsky et al., 2002). A feature of the polarization ﬁelds described in this scheme is that there are no zeros of the complex ﬁeld E itself (this occurs at points in four dimensions generically). However, there are time-dependent zeros of codimension two in the instantaneous ﬁeld E(re) (t), called disclinations (Nye, 1983a). These disclination points and lines move along the L singularities as time varies, possibly annihilating. As zeros of a vector ﬁeld, they have Poincaré index ±1 and are source, sink, spiral, or saddle type; on a closed L line loop in two dimensions, the total index of disclination points equals that of the C points enclosed by the loop (Hajnal, 1987a). C points and L lines have been studied in a variety of optical and electromagnetic experiments, including those in random ﬁelds (Flossmann et al., 2008; Soskin, Denisenko, and Egorov, 2004; Vasil’ev and Soskin, 2008), those from a microchip laser (Chen, Lu, and Huang, 2006), and in polarized microwave ﬁelds (Hajnal, 1987b; Zhang et al., 2007). However,

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it is the particular case of polarization singularities in crystal optics that we now consider.

7.3. Polarization Singularities in Anisotropic Media, and Skylight Revisited We previously described uniformly polarized scalar ﬁelds as being structurally stable on propagation—of course, this is only true in isotropic media (which includes free space). In general, the polarization of an initially polarized light beam changes on propagation through a birefringent (anisotropic) medium. Each ray (plane wave) is resolved into two components, with different orthogonal polarizations, different directions, and propagating with different speeds. A vortex in a scalar beam, on propagating through a crystal, unfolds into a complicated network of polarization singularities with a 3D structure (transverse x and y, with the third dimension the phase shift between the two components) (Flossmann et al., 2005; Volyar and Fadeeva, 2003). More signiﬁcantly, the main body of classical crystal optics—that is, the propagation of plane waves in homogeneous, birefringent media—may be formulated in terms of singular polarization optics (Berry and Dennis, 2003). For a plane wave propagating with direction s in an anisotropic medium, it is natural to work with the displacement vector D, as divergencelessness implies D · s = 0. The electric vector is related to D by E(s) = η · D(s), where η is the reciprocal dielectric tensor describing the material. Elimination of E and B from Maxwell’s equations gives the following equation, satisﬁed by the D eigenpolarizations:

P⊥ (s) · η · P⊥ (s) · D(s) =

1 n2 (s)

D(s),

(30)

where P⊥ (s) is the matrix projecting into the plane perpendicular to ray direction s, and n(s) denotes the direction-dependent refractive index. The optic axes of the crystal are those directions s for which the non-null part of P⊥ (s) · η · P⊥ (s) is degenerate, and the eigenvector direction is undeﬁned. This description holds in the simple case of a nonabsorbing, nonchiral medium where η is real and symmetric. It can be extended to absorbing and chiral materials, for which η is nonsymmetric and nonhermitian (Berry and Dennis, 2003), which display more complicated polarization singularity phenomena. The physical effect of the polarization singularity on diffraction in the crystal is the celebrated “conical refraction” of Hamilton and its generalizations (Berry and Jeffrey, 2007). Berry, Dennis, and Lee (2004b) suggested that the directional dependence of Eq. (30) (in a slightly different, complex variable form), with η a real symmetric matrix, was a good ﬁt to the skylight polarization pattern. This ansatz was justiﬁed by Hannay, who realized that the relevant matrix

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was the 3D polarization matrix, for which the form of Eq. (30) is exact for a “canopy” model of the atmosphere (Hannay, 2004), and a good approximation for the radiative transfer equations in the limit of a thin atmosphere (Hannay, 2007). Complicated patterns of polarization and polarization singularities have also been studied in light propagating through liquid crystal cells (Kiselev, 2007), and when the anisotropy is weak, they can be analyzed with semianalytic methods (Bliokh et al., 2008).

7.4. Polarization Singularities in Nontransverse Fields The most general free space monochromatic electric ﬁelds do not have an overall propagation direction z. This means that ∇ · E = 0 does not imply E · zˆ = 0. In this case, the E ﬁeld has three components, and the usual Stokes parameters cannot be deﬁned. Nevertheless, E is still represented by a complex vector and associated ellipse. The normal to the ellipse plane is

N ≡ Im E∗ × E = 2p × q,

(31)

where E = p + iq. In transverse ﬁelds, |N| = |S3 | with the direction of N parallel or antiparallel to the propagation direction, according to the sign of the ellipse handedness. In this kind of ﬁeld, the ellipses are circular on lines in three dimensions, on C lines (or CT lines) as before, but the polarization is now linear on lines also, on L lines (or LT lines) (Nye and Hajnal, 1987). (The superscript T is used to denote “true,” i.e., circular or linear in all three vector components.) Although Stokes parameters and hence σ [as deﬁned in Eq. (25)] cannot be deﬁned in this case, ϕ of Eq. (29) can, and true C lines are phase singularities of the complex ﬁeld ϕ (Berry and Dennis, 2001a). The vanishing of the real and imaginary parts of ϕ as circular polarization can be interpreted geometrically in terms of the vectors p and q on the ellipse: they must have equal length (p2 = q2 ) and be orthogonal (p · q = 0) (of course, this also applies transversely). The direction of a C line is given by Im ∇ϕ∗ × ∇ϕ, just as with vortices in three dimensions [cf. Eq. (8)]. The topological index of the C point is sign(Im ∇ϕ∗ × ∇ϕ) · N (Berry, 2004d; Berry and Dennis, 2001a; Nye and Hajnal, 1987). Numerous other singularities, dependent on line of sight, can also be considered (Freund, 2004). Linear polarization occurs when the ellipse normal N = 0, i.e., when p and q are (anti-)parallel. The direction of an L line in three dimensions was calculated by Berry and Dennis (2001a) and Berry (2004d),

DL =

1 ˆ ∇a × ∇b (N a × N b · p), 2

(32)

where a and b are labels denoting where the ∇ operators act. Local to each point on the L line, the nearby N vectors are all coplanar, in the plane

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normal to the direction of p and q. It is therefore a singularity in a (locally) 2D vector ﬁeld and can be a source, sink, circulation, spiral, or saddle. The index switches sign when DL · eP = 0 (Nye and Hajnal, 1987); there are numerous other ways of writing this expression in terms of the ﬁeld derivatives near the L singularity (Berry, 2004d). In reality, of course, all electric ﬁelds have a z-component, and so the “true” polarization singularities described here are the most general, albeit hard to measure experimentally. Careful experimentation has revealed the true C and L lines near their transverse counterparts (Hajnal, 1990), which has been analyzed theoretically (Berry, 2004b). These studies also reveal an interesting feature of the electromagnetic ﬁeld: C and L singularities coincide in transverse waves but do not in general. The electric and magnetic C lines and L lines in nonparaxial electromagnetic ﬁelds are quite independent (Nye and Hajnal, 1987), and a local direction of propagation was proposed for such ellipse ﬁelds by Nye (1991).

7.5. Relativistic Electromagnetic Singularities: Riemann-Silberstein Vortices and Poynting Stagnation Lines The C and L lines deﬁned in the previous section are purely geometric features of 3D complex vector ﬁelds (not even necessarily optical). This geometry has been used to deﬁne line singularity structures in the relativistic (time-dependent) electromagnetic ﬁeld, which can be described by the so-called Riemann–Silberstein vector (Bialynicki-Birula and BialynickaBirula, 2003)

V ≡ E(re) (t) + iH (re) (t),

(33)

which can be constructed in a mathematically natural way from the electromagnetic Faraday tensor Fμ,ν (Penrose and Rindler, 1986). (We set physical constants from here to be 1.) The time-dependent vector V combines the electric and magnetic ﬁelds in a natural, gauge-independent way. The real and imaginary parts of V · V , namely |E(re) |2 − |H (re) |2 and 2E(re) · H (re) , are the relativistic invariants of the instantaneous electromagnetic ﬁeld: they are the same for all relativistic observers. In particular, all observers agree when they are both zero, on so-called Riemann–Silberstein vortices (Bialynicki-Birula and Bialynicka-Birula, 2003): these vortices are therefore relativistic invariants of a general electromagnetic ﬁeld, with several interesting properties. The analog of the vector N of Eq. (31) is proportional to the instantaneous electromagnetic Poynting vector,

S = E(re) × H (re) =

1 Im V ∗ × V . 2

(34)

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Thus, in any frame of reference, the analogs of the L lines are instantaneous lines where the electromagnetic energy ﬂow, as deﬁned by S, is zero: Poynting stagnation lines (Dennis, 2001a). Morphologically, these are identical to L lines, and so in the neighborhood of such a line, the Poynting vector can be a vortex, or a source, sink, or saddle. However, S is not relativistically invariant, and its value at a given spacetime point changes under Lorentz transformation. In fact, provided the two relativistic invariants do not vanish, a Lorentz transformation exists, which moves a Poynting stagnation line to that point, i.e., for points on a Riemann–Silberstein vortex, the current never vanishes (Bialynicki-Birula, 2004; Bialynicki-Birula and Bialynicka-Birula, 2003; Dennis, 2001a). If an electromagnetic ﬁeld is a superposition of helicity states (i.e., a superposition of circularly polarized plane waves all of the same handedness), then the Riemann–Silberstein vortices, the electric C lines, and magnetic C lines coincide (Kaiser, 2004). For paraxial electromagnetic ﬁelds, the Riemann–Silberstein vortices have a simple form which is time-dependent (Berry, 2004c). It does not seem, however, that Riemann– Silberstein vortices are particularly useful in optics, mainly due to their time dependence (since, under Lorentz transformations, all sources acquire a Doppler shift). The time-averaged Poynting vector (Born and Wolf, 1959) E∗ × H (where E and H are now complex) does not have any natural vortex structures, although it has other interesting geometry related to the ﬁeld (Berry, 2009). Furthermore, since most optical phenomena take place in a natural frame of reference—the lab frame—the advantage of relativistic invariance is not strongly manifest. It is interesting to observe that the two relativistic electromagnetic invariants are zero exactly for a plane wave (whatever its polarization). Every point in a plane waves satisﬁes the Riemann–Silberstein vortex conditions. Thus, for relativistic electromagnetic singular optics, the singularity is to be like a plane wave—quite a contrast from optical vortices!

8. CONCLUSIONS Since the time of Newton, optical singularities have, on a number of occasions, come close to the mainstream of optical physics. However, it was not until the work of Nye and Berry in the 1970s that vortices became a key area of modern optical research. The widespread availability of spatially and temporally coherent laser sources makes the production of optical vortices inevitable in any experiment involving scattered laser light. The realization of quantized vortices is, however, not speciﬁc to optics: these objects occur in all spatial scalar ﬁelds. Although optical vortices are often referred to as points of phase singularity within a cross section of the ﬁeld, physical optical ﬁelds extend over three dimensions, and the phase singularities are

Conclusions

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actually lines of perfect destructive interference that are embedded in the volume ﬁlled by the light. These lines are continuous, either forming closed loops or inﬁnite, unbounded lines. Although most readily observed in randomly scattered monochromatic light (e.g., laser speckle), singularities are present within every spectral component, and, unless all the singularities coincide, tend to go unnoticed by the casual observer. The azimuthal phase gradient of the light around the singularity results in an azimuthal component to the Poynting vector. This can lead to an angular momentum additional to the spin angular momentum associated with circularly polarized light. This orbital angular momentum is, however, not a property of the optical vortex itself, but rather of the light that surrounds it. The recognition by Allen and coworkers in 1992 that helically phased light carried this angular momentum generated a huge interest both in itself and more widely in optical vortices. However, it is important to realize that it is not a new property of light. Everything it predicts could, albeit with signiﬁcant additional complexity, be predicted from coherent superpositions of Hermite-Gaussian modes. Orbital angular momentum and helical beams provide a basis through which a physical phenomenon can be understood better or indeed realized for the ﬁrst time. For example, the possibility of rotating particles using optical beams (optical spanners), with azimuthal phase gradient, had apparently never been proposed prior to the quest to observe orbital angular momentum. Similarly, the entanglement of orbital angular momentum states shed new light on the entanglement of spatial modes and ghost-imaging. The convenience of using the helically phased basis sets leads to fresh insights into rotational phase and frequency shifts as well as angular uncertainty relationships. Optical vortices are examples of the singularity lines within all complicated scalar ﬁelds. By comparison, electromagnetic vector ﬁelds do not generally have nodes in all components simultaneously. However, vector ﬁelds possess singularities associated with the parameterization of elliptical and partial polarization rather than phase. Polarization singularities are present in many situations, ranging from sunlight to the light transmitted by birefringent materials. Their descriptors are more complicated than their scalar counterpart in that they have both handedness and additional categorization. The study of optical vortices and orbital angular momentum has led to a recognition that the energy ﬂow, characterized by the Poynting vector, has features not immediately apparent from the intensity alone, nor from global properties of a beam. The exploitation of this ﬁne structure of optical energy ﬂow is still in its infancy. The last 30 years have seen an explosion of interest in optical singularities of all types: maybe the next 30 years will see their application to areas as diverse as planet spotting, novel forms of

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microscopy, and micro-manipulation, and to new approaches in quantum information processing.

ACKNOWLEDGMENTS We are indebted to Michael Berry, John Nye, Grover Swartzlander, and members of the Glasgow Optics Group for their careful reading and comments. We take responsibility for any remaining imperfections. We are very grateful to Caroline Turner and Ulrich Schwarz for providing German translations. This work was supported by the Leverhulme Trust and the Royal Society of London.

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CHAPTER

6 Quantum Feed-Forward Control of Light Ulrik L. Andersen* and Radim Filip†

Contents

1 Introduction 2 Theoretical Feed-Forward Tools 2.1 Quantum Representations 2.2 Basic Gaussian Operations 2.3 Quantum States 2.4 Homodyne Detection and Feed-Forward 2.5 Illustrative Example 3 Experimental Feed-Forward Tools 3.1 Defining the Quantum State 3.2 The Interaction Hamiltonian 3.3 Electro-Optic Feed-Forward 4 Applications of Feed-Forward 4.1 Homodyne Detection and Displacement 4.2 Homodyning and Heralding 4.3 Heterodyning and Displacement 4.4 More Complex Applications of Feed-Forward 5 Conclusions Acknowledgments References

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1. INTRODUCTION Feed-forward or feedback control is the process of monitoring a physical system and subsequently use the attained information to change the system so as to control it toward a certain target state. Such a control system is an indispensable component in nearly all modern technologies, e.g.,

* Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark † Department of Optics, Palacký University, 77200 Olomouc, Czech Republic Progress in Optics, Volume 53, ISSN: 0079-6638, DOI: 10.1016/S0079-6638(08)00206-0. Copyright © 2009 Elsevier B.V. All rights reserved.

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the circuitry in modern electronic devices are often loaded with classical feedback systems, and in the modern optical laboratory, classical feedback systems are, for example, used to stabilize the lengths of interferometers and temperatures of various components. Over the last 15 years, there has been a growing interest in extending the ideas of classical control theory to a regime that is only describable by quantum mechanics (Wiseman, 1994; Wiseman and Milburn, 1993). In this regime, the goal is to optimally monitor quantum observables with extreme sensitive detectors and thereby control quantum system with ultra-high precision in real-time. Such quantum-enhanced control has been used to steer the dynamics of various different quantum systems, ranging from atomic systems for the construction of a quantum memory (Sherson et al., 2006) to the cooling of single trapped ions (Bushev et al., 2006) and resonators (Kleckner and Bouwmeester, 2006; LaHaye et al., 2004). In this review, however, we focus only on a single but very powerful quantum system, namely the light ﬁeld. The application of quantum control of light can basically be categorized into three groups: the control system can be used for the preparation of a particular quantum state; it can be used to enable a speciﬁc quantum operation (that is, a quantum informational protocol); it can be used to optimize the measurement of a certain observable. These applications on the control of light are based on either a feedback system or a feed-forward system. To understand the difference between feedback and feed-forward, we consider two light sequences, E1 and E2 , where E2 is ahead of E1 in time. In feedback systems, the ﬁeld E2 is measured and the information is fed back onto the consecutive light sequence, namely E1 . On the contrary, in feed-forward systems, when E2 is measured, the information is fed forward onto the same light sequence, that is E2 . The main technical difference between feedforward and feedback is that the latter is restricted in bandwidth due to the ﬁnite delay time involved, whereas the former is in principle bandwidth unlimited. Quantum feedback on light has mainly been used to implement measurements of certain quantum observables at the ultimate quantum limit. For example, feedback has been used to physically implement the phase projector, that is, to construct a physical apparatus capable of accessing the phase information of a light beam nearly at the quantum limit (Armen et al., 2002). Furthermore, feedback has been used to enable the optimal discrimination strategy between two coherent states of light (Cook, Martin, and Geremia, 2007). However, in this review, we focus on quantum feed-forward systems. The ﬁrst account on quantum feed-forward appeared in a study by Björk and Yamamoto (1988). In this study, they introduced the idea of using a linear quantum feed-forward scheme for the preparation of different

Introduction

367

non-classical light beams from a pair of entangled light beams. They envisioned the production of entangled beams—called the signal and idler beams—from a non-degenerate optical parametric ampliﬁer, followed by a phase quadrature measurement of the idler beam and a conditional, but linear, phase quadrature displacement of the signal beam by means of feedforward control. This ultimately forces the signal state into a quadrature eigenstate (inﬁnitely squeezed state) if the initial state is maximally entangled and the detectors are perfect. They also showed that by changing the measurement device and the feed-forward strategy, various different quantum states of light can be produced. The idea of preparing a squeezed state through measurement of an entangled state was implemented by Mertz et al. (1990). This experiment constituted the ﬁrst demonstration of a quantum feed-forward system, and it belongs to the preparation category as deﬁned above. In this review, we mainly address the quantum feed-forward systems that are used to implement a continuous variable (CV) quantum operation (or quantum informational protocol) rather than a quantum preparation. To the best of our knowledge, the ﬁrst study addressing the issue of using quantum feed-forward for the implementation of a CV quantum operation was done by Vaidman (1994). He introduced the idea of sending a CV quantum state through classical channels by means of CV entanglement and linear feed-forward, which is a strategy known as CV teleportation. This scheme was further elaborated by Braunstein and Kimble (1998) and by Ralph and Lam (1998), and experimentally implemented for the ﬁrst time by Furusawa et al. (1998). However, the ﬁrst experimental realization of a CV quantum operation based on a feed-forward system appeared in 1997 (Lam et al., 1997). In this work, it was shown that a simple experimental setup consisting of a single beam splitter and a linear feed-forward loop (composed of a beam splitter, an amplitude quadrature detector, feed-forward electronics, and an amplitude modulator) enables noise-less ampliﬁcation of the amplitude quadrature of a light beam. This extraordinary simple set-up has proven to be very powerful; by changing the input auxiliary states, the detector, the feed-forward gain, and the displacement operation, a large range of different quantum informational protocols can be accomplished at the ultimate quantum limit e.g., it has recently been shown that by introducing some modiﬁcations to this feed-forward system, a universal squeezing operation as well as a quantum-limited phaseinsensitive ampliﬁer can be realized; see Filip, Marek, and Andersen (2005). These basic quantum protocols have recently been experimentally demonstrated [see Yoshikawa et al. (2007) and Josse et al. (2006)], and they have been shown to be the main building blocks in various protocols ranging from quantum cloning (Andersen, Josse, and Leuchs, 2005) to quantum

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error correction coding (Braunstein, 1998a) and cluster state computation (Menicucci et al., 2006). Due to the importance of the simple feed-forward system, its description will constitute the core of this review. In Section 2, we introduce theoretical tools and concepts that exhaustively describe the function of the feedforward loop on a theoretical level using various approaches. In Section 3, we describe the simple feed-forward loop from an experimental point of view by addressing the function of the various important components. To get a better understanding of the function of the components, we carefully address the deﬁnition of a quantum state from an experimental viewpoint. In Section 4, we discuss the main applications of the simple feed-forward system by reviewing a number of theoretical proposals and experimental implementations.

2. THEORETICAL FEED-FORWARD TOOLS A generic feed-forward transformation is shown in Figure 1: two input states are injected into an operating box in which an arbitrary transformation between the input states takes place. This results in two outputs, one of which is measured destructively with a detector. Based on the measurement outcome, the remaining part of the state is controlled through a continuous or discrete displacement operation. Such a quantum operation can be described theoretically using many different strategies, the preferred one being dependent on different factors, such as the type of input states, the type of measurement, and the type of quantum operations that are to be considered. One must therefore decide whether the function of the system is most easily described by determining the dynamics of the quadrature eigenstates, the Fock states or the Wigner functions (which corresponds to the Schrödinger picture), or by determining the dynamics of the observables (corresponding to the Heisenberg picture). In the following, we describe brieﬂy the different pictures and base states and use these theoretical concepts to describe the function of the various components in the generic feed-forward system in Figure 1.

Input 1 Input 2

Processing

Interaction

FIGURE 1 Schematic of a generic feed-forward setup. Two modes interact in an arbitrary Hamiltonian that results in two output modes, one of which is measured. The bipartite state is accordingly reduced to a single system.

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2.1. Quantum Representations In quantum optics, an arbitrary quantum state is usually expanded either in the discrete Fock-state basis |n, where n refers to the photon number, or in the continuous quadrature (or coordinate) basis |x, where x is the quadrature. The Fock-state basis {|n} consists of states with no quantum noise in the energy, i.e., they are eigenstates of the photon number operator, n, which can be expressed as a function of the annihilation and creation ﬁeld operators, a and a† : n = a† a. The Fock states are orthogonal m|n = δnm , and expand the complete Hilbert space: I = n |nn|; see Louisell (1973) for more details. The quadrature basis {|x} is a continuous set of inﬁnitely squeezed states (each of which is possessing inﬁnite energy). They are eigenstates of the amplitude quadrature operator X; thus they exhibit no quantum noise in this variable. The dimensionless amplitude quadrature can be written as X = a + a† and the conjugate quadrature, the phase, can likewise be written as P = −i(a − a† ). Since X and P are canonically conjugate quadratures, they obey the commutation relation [X, P] = 2i and the corresponding Heisenberg uncertainty relation, 2 X2 P ≥ 1, where unity represents a normalized unit of vacuum. Both the amplitude and phase quadrature orthogonal [x|x = δ(x − x ) and p|p = δ(p − p )] and complete basesare ∞ ∞ I = −∞ |xx|dx = −∞ |pp|dp . The two quadrature bases are mutually ∞ coupled through Fourier transformations |p = √1 −∞ eipx/2 |xdx and 4π √ ∞ |x = √1 −∞ e−ipx/2 |pdp and their overlap is x|p = exp(ipx/2)/( 4π). 4π It is often useful to simultaneously visualize the system behavior in both complementary quadrature representations. For this purpose, the Wigner function representation of the density matrix is the most convenient; see Leonhardt (1997). The Wigner function of a state with density matrix ρ is deﬁned as

1 W(x, p) = 2π

∞

x − x |ρ|x + x exp(ix p)dx ,

(1)

−∞

and it is a real function and normalized to unity. The Wigner function can attain negative values for some highly non-classical states (e.g. Fock states); thus it is not a regular probability distribution. A special class of states is the so-called Gaussian states deﬁned by having a Gaussian Wigner function. For the general case of a state consisting of N modes, where each mode is described by a pair of quadratures, x and p, the Wigner function reads

W(q) =

1 1 exp − (q − d)T V −1 (q − d) , √ 2 2π DetV

(2)

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Quantum Feed-Forward Control of Light

where q = (x1 , p1 , . . . , xN , pN ) is a vector of quadrature arguments. The ﬁrst moments are represented be the vector d = (x1 , p1 , . . . , xN , pN ), whereas all the symmetrized second-moments are given by the covariance matrix Vij = 12 qi qj + qj qi − qi qj (Simon, Mukunda, and Dutta, 1994; Simon, Sudarshan, and Mukunda, 1987). The Heisenberg uncertainty principle gives fundamental constraints on any physical covariance matrix; V + i ≥ 0, where = ω⊕N and ω = ((0, 1), (−1, 0)).

2.2. Basic Gaussian Operations Quantum evolution or, equivalently, unitary operations are described either in the Schrödinger or in the Heisenberg picture. In the former picture, the state of the system undergoes time evolution and the operators associated with the system are time-independent, whereas in the latter picture it is the opposite; see Louisell (1973). The evolution is governed by a Hamiltonian, which can be decomposed into two parts, H = H0 + HI , where H0 is the free evolution Hamiltonian and HI is the interaction Hamiltonian. If the unitary quantum dynamics is governed by a Hamiltonian, which is quadratic in the operators X and P, it is by far the easiest to carry out the calculations in the Heisenberg picture and subsequently apply the solutions for the quadrature to the arguments of the Wigner function. A general solution for the evolution of X and P in the Heisenberg picture is given by the matrix Q(t) = T(t)Q(0), where Q(t) = (X1 (t), P1 (t), . . . , XN (t), PN (t)) is a vector of the quadratures at time t and T(t) is an unitary transformation matrix. The Wigner function of the state after evolution is easily obtained by substituting the quadrature arguments of the initial Wigner function W0 (q) with T −1 q:

W0 (q) → Wt (q) = W0 (T −1 q).

(3)

Due to the assumed linearity of the quadrature transformations, the operation preserves the Gaussian statistics of an input state. Since such transformations are mapping Gaussian states onto other Gaussian states, they are known as Gaussian operations. The elementary, unitary Gaussian operations in quantum optics are the single-mode phase shift, the single-mode displacement, the squeezing, and the two-mode beam splitting operation. These basic Gaussian operations sufﬁce to perform an arbitrary Gaussian transformation (Bartlett et al., 2002; Braunstein and Loock, 2005). In the following we describe these components one by one. The single-mode phase shift operation is simply introducing a phase shift to the light beam using the unitary operator, UPS = exp(iφa† a), where φ

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is the rotation angle. In the Heisenberg picture, the phase shift operation can be represented by the phase space rotation, X = X cos φ + P sin φ and P = P cos φ − X sin φ, and in the Schrödinger picture the Wigner function is transformed as W(x, p) → W(x cos φ − p sin φ, p cos φ + x sin φ). The coherent displacement operation is shifting the quadratures. In the ¯ and P = Heisenberg picture, it makes the transformation X = X + X ¯ where (X, ¯ P) ¯ ∈ R are the quadrature displacements. To incorpoP + P, rate such a transformation, the quantum ﬁeld has to interact with a “classical” driving ﬁeld α during time interval (0, t) and the interaction I = i(αa† − α∗ a) = /2(δxP − δpX), is described by the Hamiltonian HD where α = (δx + iδp)/2. In the coordinate representation, the displacement of X is written as |x → |x + x¯ , whereas in the Wigner-function formalism, the transformation reads W(x, p) → W(x − x¯ , p − p¯ ), where x¯ = δxt and p¯ = δpt. The single-mode squeezer makes a scaling transformation of the quadrature; for example, it can transform the X quadrature as X → sX, where s is a real scaling factor, and simultaneously transform the P quadrature as P → P/s. For s < 1, the X quadrature is squeezed (de-ampliﬁed), whereas the P quadrature is ampliﬁed by the inverse factor. To achieve such a transformation, the interaction Hamiltonian has to take the form, HSI = iκs (a2 − a†2 )/2 = −κs (XP + PX)/4, where κs is the effective nonlinear interaction constant. This Hamiltonian yields the above transformations in X and P for s = exp(−κs t), where t is the effective interaction time. In the coordinate representation, the squeezing transformation is simply given by |x → |sx and the corresponding Wigner function changes as W(x, p) → W(x/s, sp), because of the inverse transformation rule. The beam splitter interaction passively mixes together two different modes. In such a beam splitter, the quadratures transform as X1 = √ √ √ √ √ √ TX1 + 1 − TX2 , P1 = TP1 + 1 − TP2 and X2 = TX2 − 1 − TX1 , √ √ P2 = TP2 − 1 − TP1 , where the beam splitter transmission, T, is in the interval 0 ≤ T ≤ 1, and the indices 1 and 2 refer to the two modes. To achieve this transformation, the interaction Hamiltonian has to be I = iκ (a† a − a† a )/2 = κ (X P − P X )/2, where κ is the linear couHBS 2 1 2 1 l 2 1 l l 1 2 pling constant. The relation between T and κl is T = cos2 (κl t), where t is the effective interaction time. In the coordinate representation, the coupling is √ √ √ described by the transformation |x |x → | Tx + 1 − Tx | Tx − 1 1 2 2 1 2 1 2 √ 1 − Tx1 2 , and in the formalism of Wigner functions, the transformation rule is

√ √ √ Tx1 − 1 − Tx2 , Tp1 − 1 − Tp2 √

√ √ √ × W2 Tx2 + 1 − Tx1 , Tp2 + 1 − Tp1 .

W1 (x1 , p1 )W2 (x2 , p2 ) →W1

√

(4)

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Quantum Feed-Forward Control of Light

A more general linear coupling can be build from a sequence of beam splitters; see Reck et al. (1994). Another important two-mode non-linear operation is the so-called quantum non-demolition interaction. It couples together the two input modes in an asymmetric manner, for example, as X1 = X1 , P1 = P1 − GP2 and X2 = X2 + GX1 , P2 = P2 , where G is a non-linear interaction gain. The I = κq X1 P2 , where Hamiltonian for such active transformation is HQND G = κq t and t is the effective interaction time. In the coordinate representation, the coupling is described by the transformation |x1 1 |x2 2 → |x1 1 |x2 + Gx1 2 andthe Wigner function evolves as W1 (x1 , p1 )W2 (x2 , p2 ) → W1 x1 , p1 + Gp2 W2 x2 − Gx1 , p2 .

2.3. Quantum States Having introduced the basic Gaussian operations, in this section we proceed with a very short description of the most important optical quantum states. The minimum uncertainty single-mode state, which saturates the Heisenberg uncertainty principle, is a pure Gaussian state. It can be simply written in the coordinate representation

|x0 , p0 , VX =

ix p ∞ exp − 04 0 1

(2πVX ) 4

−∞

(x − x0 )2 p0 x |xdx, exp − +i 4VX 2

(5)

fulﬁlling VX VP = 1, independent of the reference frame. For a coherent state, the variances of all quadratures, in particular X and P, are identical and normalized to 1, and in the expression (5) x0 and p0 are substituted with the mean values X and P, respectively. For VX < 1 the variance of quantum noise in the X quadrature is reduced below the variance of the vacuum noise and the minimum uncertainty state corresponds to a squeezed state. For the minimum uncertainty state, the Wigner function has the form

1 (x − x0 )2 (p − p0 )2 W(x, p) = exp − − , √ 2VX 2VP 2π VX VP

(6)

with the constraint VX VP = 1. The Wigner function of such a state is always positive. In many experiments, however, the states under interrogation are often mixed; thus VX VP > 1. The form of the Wigner function (6) still remains valid but now with the variances decomposed into a term that represents the variance of the quantum noise and one that represents the

Theoretical Feed-Forward Tools

quant

373

quant

excess noise: VX = VX + VXclas and VP = VP + VPclas . If, for example, VX < 1, the state is a mixed squeezed state with an excess noise variance of VPclas = VP − 1/VX . Another key resource for some continuous variable feed-forward operations is the highly non-Gaussian Fock state, |n. It can be written in the coordinate representation in the following form

x2 x|n = √ exp − 4 2πn!2n 1

Hn

x √ , 2

(7)

where Hn (z) is Hermite polynomial. The corresponding Wigner function is

1 x 2 + p2 n 2 2 Wn = (−1) Ln (x + p ) exp − , 2π 2

(8)

where Ln (z) is a Laguerre polynomial. While the quadrature-squeezed state exhibits quantum noise suppression in a quadrature variable, the ideal Fock state is inﬁnitely squeezed in the number of photons.

2.4. Homodyne Detection and Feed-Forward In almost all the continuous variables, feed-forward schemes, which are considered in this review, are based on homodyne detection. Such a detector is measuring a quadrature of the light ﬁeld, and in the ideal case of unit quantum efﬁciency and no electronic noise, a homodyne measurement is described by the projector x = |xx| or p = |pp| depending on which quadrature is being probed. These projectors ideally reduce the measured quantum state to an inﬁnitely squeezed state. For an amplitude quadrature measurement, the post-measurement state is given by ρ → (x ρx )/p(x), where p(x) = Tr(x ρx ) is the probability distribution for x. First we describe the homodyne measurement and feed-forward in the quadrature eigen-basis of the input state. For example, consider the homodyne ∞ measurement of an arbitrary pure state, −∞ G(x1 , x2 )|x1 , x2 dx1 dx2 . If the coordinate quadrature X2 of mode 2 is measured and the measurement outcome is x¯ 2 , the transformation is as follows:

∞

∞ G(x1 , x¯ 2 )|x1 dx1 .

G(x1 , x2 )|x1 , x2 dx1 dx2 → −∞

−∞

(9)

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Quantum Feed-Forward Control of Light

The measured value x¯ 2 could then be used to displace the amplitude quadrature of the post-measurement state of mode 1. Such transformation yields the state

∞

∞ G(x1 , x¯ 2 )|x1 + gx¯ 2 dx1 =

−∞

G(x1 − gx¯ 2 , x¯ 2 )|x1 dx1 ≡ | x¯ 2 (x1 ).

−∞

(10)

∞ and ﬁnally one must trace over x¯ 2 if it is still present, −∞ | x¯ 2 (x1 ) x¯ 2 (x1 )|d¯x2 , to obtain the ﬁnal state. In the Wigner function formalism, a homodyne measurement can be simulated by replacing the measured quadrature with the measurement outcome and can be integrated over the conjugate variable for which no information is obtained. For example, consider the homodyne measurement of a two-mode Wigner function. If the amplitude quadrature X2 of mode 2 is measured and the measurement outcome is x¯ 2 , the argument of the Wigner function is changed as x2 → x¯ 2 and the function is integrated over p2

∞ W(x1 , p1 , x¯ 2 , p2 )dp2 ≡ Wx¯ 2 (x1 , p1 ),

W(x1 , p1 , x2 , p2 ) →

(11)

−∞

where Wx¯ 2 (x1 , p1 ) is the reduced Wigner function of the remaining system conditioned by the measurement result x¯ 2 . The probability distribution p(¯x2 ) of the measured results can be obtained after integrating the remaining variables x1 , p1 . The displacement operation in X1 is simulated as

Wx¯ 2 (x1 , p1 ) → Wx¯ 2 (x1 − gx¯ 2 , p1 ).

(12)

To get the ﬁnal state, one needs to integrate over the possible outcomes x¯ 2 . If all outcomes are used for displacement, the integral runs from −∞ to ∞. In some cases, however, the post-measurement state is heralded based on a post-selection of the measurement outcomes. Such a selection process is conveniently described by an appropriate integration of the Wigner function. As an example, we consider the selection of the post-measurement state based on outcomes that fall within certain intervals deﬁned by the set B = (I1 , . . . , IN ), where Ii = [ai , bi ]. The post-measurement state is then found by an integration of the measurement outcomes over B:

Wx¯ 2 (x1 , p1 ) →

Wx¯ 2 (x1 , p1 )d¯x2 ≡ Wx¯ 2 ∈B (x1 , p1 ). B

(13)

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The probability of success is given by

∞ PB =

Wx¯ 2 ∈B (x1 , p1 )dx1 dp1 .

(14)

−∞

If the states of the system before and after measurement are Gaussian, the detection process can be more simply described with the covariance matrix formalism. Consider a two-mode Gaussian state represented by the vectors of mean values, D1 = (X1 , P1 ), D2 = (X2 , P2 ), and the 4 × 4 covariance matrix, V12 = (A, C), (CT , B) , where A, B, and C are 2 × 2 matrices. The projection is described by a 2 × 2 matrix (see below), and as a result of the measurement, the mean value vector and the covariance matrix are changed as

D1 → D1 + C

1 1 (DM − D2 ), V12 → B − C CT , A+M A+M (15)

where DM is a vector of measured values. For homodyne detection of X2 , MX = limv→0 ((v, 0), (1/v, 0)) and DM = (¯x, 0), whereas the measurement of P2 is represented by MP = limv→0 ((1/v, 0), (v, 0)) and DM = (0, p¯ ). Note that the covariance matrix transformation does not depend on the measurement value DM . This approach can be easily generalized to the multi-mode case. Finally, in many cases, the homodyne measurement followed by conditional displacement is described by input–output relations for the quadratures. Presuming that the output quadratures are functions ( fX1 (X1 , X2 ), fP1 (P1 , P2 ), fX2 (X1 , X2 ), fP2 (P1 , P2 )) of the input quadratures, the homodyne measurement of the amplitude quadrature of mode 2 yields the number fX¯ 2 (X1 , X2 ) and erases the information in fP2 (P1 , P2 ). A conditional linear amplitude displacement of mode 1 then gives

fX1 (X1 , X2 ) → fX1 (X1 + gfX¯ 2 (X1 , X2 ), X2 ) fP1 (P1 , P2 ) → fP1 (P1 , P2 ). If the functions are linear in X and P, the states have Gaussian statistics and can be fully characterized by their ﬁrst and second moments. These input– output relations are widely used to describe the operation of a feed-forward system with only Gaussian states.

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2.5. Illustrative Example We have now introduced various theoretical tools that are useful for describing the function of a quantum feed-forward system. To get some practice in using the tools, we apply them to a speciﬁc feed-forward scheme. We consider the simple set-up consisting of a single beam splitter and an electro-optic feed-forward loop based on a homodyne detector (see Figure 2). Suppose there are two input modes; a signal mode, denoted by index 1, and an ancillary mode, denoted by the index 2. The two modes are mixed at the beam splitter with transmittivity T; one of the outputs is measured by the detector and the outcome is used either to deterministically displace a quadrature (similar to the measured quadrature) of the postmeasurement state or to probabilistically herald the state. This setup will now be described theoretically using the different methods that have been discussed in the previous sections. In particular, the evolution will be investigated by using the Schrödinger picture (in the quadrature eigenbasis), the Heisenberg picture, and the Wigner function formalism. There are basically three different stages in the set-up; beam splitter interaction (BS), homodyne detection (HD), and displacement (D) or heralding (H). All these transformations of the set-up will be indicated by an arrow with the particular action denoted atop the arrow. First we consider the evolution of a quadrature eigenstate |x through the system assuming that the auxiliary state takes the form |p = 0 ∝ |zdz:

∞ |x1

∞ √ √ √ √ |z2 dz → | Tx + 1 − Tz1 | Tz − 1 − Tx2 dz BS

−∞

−∞

∞ √ √ √ √ → | Tx + 1 − Tz1 δ(¯x − Tz + 1 − Tx)dz M

−∞

Processing

Input 1

Input 2

FIGURE 2 Schematic of the simple feed-forward setup. Two input modes interfere on beam splitter and one output is measured. The resulting photo-currents are processed and subsequently used to transform the other output state.

Theoretical Feed-Forward Tools

1 = |√ x + T 1 → |√ x + T D

377

1−T x¯ 1 T

(16)

1−T x¯ + gx¯ 1 . T

(17)

Depending on the particular task to be implemented, the gain g is accordingly optimized. Knowing how the system will interact with every quadrature eigenstate, it is possible to predict the evolution of any state since the quadrature eigenbasis is complete and thus any state can be expanded on that basis. This approach is convenient to illustrate the qualitative performance of the feed-forward system if some of the involved states are non-Gaussian. If the involved states are known to be Gaussian, the simplest way to describe the system is to follow the evolution of conjugate quadrature observables in the Heisenberg picture:

√ √ √ √ TX1 + 1 − TX2 X1 TX1 + 1 − TX2 √ √ √ P1 BS TP1 + 1 − TP2 HD √ → √ → TP1 + 1 − TP2 √ √ √ X2 TX2 − 1 − TX1 TX2 − 1 − TX1 √ √ P2 TP2 − 1 − TP1 √ √ √ √ D ( T − g 1 − T)X1 + ( 1 − T + g T)X2 √ → √ TP1 + 1 − TP2 . Full information about the output states can be easily found by evaluating the ﬁrst- and second-order moments of the quadratures. This approach is widely used to evaluate the expected quantitative performance of a CV feed-forward loop and will be extensively used in Section 4. For non-Gaussian states, the approaches in the Heisenberg picture are not convenient. For such states, it is better to investigate the evolution of the Wigner function directly. Moreover, the heralding procedure is easily described in the Wigner function formalism. If the two input states are given by the Wigner functions, W(x1 , p1 ) and W(x2 , p2 ), the evolution through the system is as follows: BS

W(x1 , p1 )W(x2 , p2 ) → W1

√

× W2

Tx1 −

√

√

Tx2 +

1 − Tx2 , √

√ √ Tp1 − 1 − Tp2

1 − Tx1 ,

√ √ Tp2 + 1 − Tp1

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Quantum Feed-Forward Control of Light

= W (x1 , p1 , x2 , p2 ) HD

∞

→

dp2 Wx¯ (x1 , p1 , p2 ). 2

(18)

−∞

Now if the measurement outcome is used to displace the remaining state, it transforms as

∞

dp2

−∞

∞

d¯x2 Wx¯ (x1 + gx¯ 2 , p1 , p2 ), 2

(19)

−∞

and if the outcomes are selected in an interval to herald the state, one gets

∞ −∞

dp2

d¯x2 Wx¯ (x1 , p1 , p2 ). 2

(20)

B

For some cases, the evolved Wigner function can be derived analytically, but in many cases one must resort to numerical methods. The ﬁndings of this section will be used in Section 4 to explain the function of the various feed-forward protocols in quantum optics and quantum information.

3. EXPERIMENTAL FEED-FORWARD TOOLS In this section we describe some of the basic physical components that are used to implement the electro-optical feed-forward loop. We put emphasis on the technical details that are of importance when designing the loop. We also slightly extend the single-mode description in the previous section to a more realistic multi-mode description. Most experiments on quantum optical feed-forward with continuous variables have been carried out on pairs of sidebands relative to the optical carrier. Therefore, only a multimode treatment will adequately describe the function of the system.

3.1. Defining the Quantum State The way a state is deﬁned in the laboratory depends mainly on the light sources being used as well as on the characteristics of the measurement devices. The sources are normally operated in either the pulsed regime or the continuous wave (CW) regime. Usually when pulsed light is generated, the quantum state of the system is associated with a single pulse. Such a quantum state can have different shapes depending on the laser source;

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it can be a secant-hyperbolic-shaped femto second or picosecond pulse directly emitted from a mode-locked laser or it can be a nearly squareshaped pulse produced by chopping a CW laser beam using for example a fast acousto-optic modulator. For CW light beams, the deﬁnition of the quantum state is usually given by the detection bandwidth, thus assuming that a quantum state is received every inverse bandwidth time. That is, if the bandwidth of the detector is 1 MHz, a new quantum state arrives every 1 μs and the associated averaged measurement outcome in 1 μs corresponds to the eigenvalue of the observable being measured. The CW signal is therefore turned into separate quantum states by slicing the signal into various time slots. It is however also possible to extend the time slots with respect to the detector response so that a number of measurement outcomes will be acquired within a single state. The measured eigenvalue associated with that state is then found by averaging the outcomes obtained within the time slot. For some experiments on quantum feedback, where the aim is to quickly act on the state being measured, the pulse should be chosen much longer than the response time of the detector. The length of the time slot is, however, limited by the coherence time of the laser. In cases where the states are deﬁned as pulses from a laser, one must assure that the bandwidth of the detector is larger than the repetition rate of the laser to be able to resolve the individual pulses thus obtaining at least one measurement outcome per pulse arrival. If many outcomes are obtained within a single pulse (which happens when the detector bandwidth is much larger than the repetition rate), the eigenvalue is found by averaging the outcomes. On the other hand, if the detector is very slow with respect to the laser repetition rate, the pulses are completely smeared out in the detection process and the observed quantum state (and therefore the deﬁnition of the quantum state) is effectively identical to the deﬁnition of the quantum state of a CW system. Generally, the output electric currents from the detector reﬂecting the quantum state is a convolution between the optical state and the response function, (τ), of the detector

∞ i(t) ∝

dτI(t − τ) (τ),

(21)

0

where I(t − τ) is the intensity of the state at time t − τ. An inﬁnitely fast detector will have a δ response, whereas a realistic detector has a non-delta response function with a width determined by the inverse bandwidth of the detector and the associated electronics. In quantum mechanics, the state being measured is a quantum state and the intensity is an operator, which is a function of the creation and annihilation operators. The detection process can be elucidated by the photoelectric effect. It means that the

380

Quantum Feed-Forward Control of Light

randomness of the photoelectric events is reﬂected in the ﬂuctuations of the generated photocurrents. The quantum statistics of the intensity operator is therefore mapped onto the photocurrents in the detector (Ou and Kimble, 1995). In many experiments, it is common to resolve the evaluated quantities spectrally. The spectrally measured photocurrent is given by

i() ∝ I() (),

(22)

where (ω) is the frequency response of the detector. Many of the feedforward experiments presented in this review have involved the measurement of the mean and variances in frequency domain. The spectral density is given by

∞ S(ω) ∝

i(ω)i(−ω )dω

(23)

−∞

and is easily measured with an electronic spectrum analyzer. We also note in passing that the state (or mode) of the optical ﬁeld is not only determined by its temporal (or frequency) dependence but also by its spatial and polarization degrees of freedom. The complete deﬁnition of the state therefore includes, in addition to the temporal dependence, also a description of its polarization mode and its spatial structure.

3.1.1. Quantum States as Sideband Modes There is yet another way of deﬁning the quantum state of a light beam, namely in terms of sideband modes. This deﬁnition is related to the above deﬁnition of a state for CW beams in the sense that the state is deﬁned by the measurement scheme. In the sideband picture, however, further ﬁltration of the state is made, which transforms the system into a composite system consisting of two localized frequency modes (Bachor and Ralph, 2004). This will be detailed in this section. The mode of a laser beam is normally quite noisy at low frequencies due to mechanical instabilities and other detrimental effects that occur in real systems and that cannot be easily overcome, e.g., the state of a commonly stabilized Ti:Sapphire laser possesses excess noise up to some kilohertz due to mechanical vibrations, whereas the laser beam generated by a monolithic Nd:YAG is infected by excess noise up to around 10 MHz due to the relaxation oscillation of the laser. However, beyond these frequencies, the laser modes are normally accommodated with vacuum modes. Therefore, by considering only these modes, the state of the system is very quiet

Experimental Feed-Forward Tools

Pure frequency modes

Excess noise

381

Pure frequency modes

center Composite quantum system

FIGURE 3 Sideband representation illustrated in frequency space. The quantum state is defined as a pair of a collection of frequency modes shown by the shaded boxes. Excess noise evades the system around the central mode, but above a certain frequency (illustrated by vertical dashed lines) the frequency modes are pure.

and pure, and since most quantum information protocols and other quantum mechanical experiment require that the involved states are strictly pure, it has been common to ﬁlter away the low frequency, thus noisy, part of the spectrum. More speciﬁcally, the quantum state of the system is often deﬁned as being consisting of a pair of adjacent sideband frequency modes relative to the optical carrier as illustrated in Figure 3. If the modes are residing at frequencies above the frequencies of the excess noise, the deﬁned state is in a pure quantum state although the laser beam itself is noisy (and bright). Usually, the sidebands are at radio frequencies relative to the frequency of the carrier and their bandwidths often range from a few kilohertz to some megahertz. The ﬁeld operator for the pair of sideband modes can be described by the relation

a(t) =

a(−ω)e

−iωt

a(ω)eiωt dω,

dω +

B

(24)

B

where the ﬁrst and second terms are associated with the upper and lower sideband modes, respectively. For example, a(ω) (a(−ω)) is the annihilation (creation) operator corresponding to the mode at frequency ω (−ω) relative to the carrier frequency, which is set to zero. The integration runs over the bandwidth, B, of the deﬁned sideband modes. In the dirac notation, the vacuum state of the system can be written as

|0 =

g(ω)|0ω dω ⊗

B

g(ω)|0−ω dω,

(25)

B

where g(ω) is the amplitude function of the state, and for a single pair of frequencies, the state is simply |0 = |0ω ⊗ |0−ω . We have now seen how a true vacuum state can be deﬁned as sideband modes. Another state of high importance that can be deﬁned as sideband

382

Quantum Feed-Forward Control of Light

modes is the coherent state. This state is deﬁned by the composite system of two coherent state excitations residing at adjacent frequency modes, |α = |αω ⊗ |α−ω . A pure coherent state composed of two sidebands can be easily prepared by modulating the laser beam with an electro-optical modulator (EOM). The modulator transfers photons from the carrier mode to the two sideband modes, thereby creating pure coherent excitations at these modes. The phase and amplitude of the composite coherent state are controlled by independent phase and amplitude modulators. The vacuum and coherent state sidebands are examples of uncorrelated sideband modes. Quantum-correlated sideband modes can however occur by using non-linear optical processes. For example, the squeezed state of light is a manifestation of quantum-correlated or entangled sideband modes. An interesting question that we would like to address at the end of this section is how far and in what sense the sideband picture departs from the more standard deﬁnitions of a quantum state. Various deﬁnitions of a quantum state was discussed in the time domain in Section 3.1. In the frequency domain, these states can be deﬁned as a collection of frequency modes around the central frequency given by the Fourier transform. In this case, as opposed to the sideband deﬁnition, the frequency interval is not broken into two separate spectra; all frequency modes around the center are considered as being a part of the quantum state. However, one can still consider the frequency spectrum as being made up of two sideband modes that have merged at the central frequency. Indeed, the vacuum state and the squeezed state are deﬁned similarly in the two presentations. The deﬁnition of a coherent state, however, deviates in the two pictures. In the sideband picture, it is a composite system of two coherent excitations, whereas in the other picture it is associated with a single excitation at the central frequency. Another advantage of deﬁning the state as sidebands is that the carrier mode can be very bright and thus can be used as an auxiliary mode either for measuring the sidebands or for stabilizing interferometers in the optical set-up.

3.2. The Interaction Hamiltonian After having deﬁned the notion of a quantum state, we now proceed with a short description of the interaction Hamiltonian in Figure 1. Although the interaction Hamiltonian could in principle be arbitrarily complex, in the following we restrict our discussion to Gaussian transformations. As mentioned in Section 2, Bloch-Messiah’s reduction theorem tells us that an arbitrary multi-mode Gaussian transformation can be implemented with a linear multi-port interferometer followed by single-mode squeezers and another multi-port interferometer. In addition, it has been shown that the squeezing transformations can be accomplished by using

Experimental Feed-Forward Tools

383

/2

PBS

FIGURE 4

PBS

An arbitrary beam splitter transformation.

off-line squeezed vacuum states, which are fed into a linear network (consisting solely of beam splitters) and electro-optic feed-forward. It means that the beam splitter transformation constitutes only interaction Hamiltonian for accomplishing an arbitrary complex Gaussian transformation, assuming one has a source of auxiliary squeezed vacua. Let us therefore describe the function of a beam splitter. The experimental realization of a beam splitter with tunable beam splitting ratio is depicted in Figure 4. It consists of a single half-wave plate, which is sandwiched between two polarizing beam splitters (PBSs). The relative phase of the two input modes can be controlled by impinging one of the modes on a piezo-controlled mirror. To concentrate the two modes into a single spatial mode but orthogonal polarization modes, they merge on a PBS with orthogonal linear polarizations. The two modes are subsequently mixed on the second PBS with a mixing degree controlled by the half-wave plate, placed in between the two PBSs. The reﬂection coefﬁcient is then given by R = sin2 2θ, where θ is the angle of the half-wave plates’ optical axis with respect to the axis of the PBS. To ensure a nice spatiotemporal overlap, the two input modes must have the same frequency spectrum and the same transverse mode. The relative phase between the two input modes is stabilized through a feedback loop where the piezomirror is actively controlled based on a measurement of the output modes (or a part of the output modes).

3.3. Electro-Optic Feed-Forward In this section, we will consider the experimental details of the electro-optic feed-forward loop. The ﬁrst important component in the arrangement is the detector. At this stage of the set-up, the quantum state is reduced into some classical numbers, the values of which are normally random due to the inherent quantum ﬂuctuations of the measured quantum state. However, the outcomes can be biased by the type of measurement employed; e.g., in the extreme case where the state is an eigenstate of the detector, the outcomes will be predictable. In this section, we ﬁrst describe some measurement devices used to probe the quadratures of the afore-mentioned sideband modes, and second, we address the physics of the feed-forward mechanism.

384

Quantum Feed-Forward Control of Light

3.3.1. Quadrature Detectors The signal produced by an ideal photo-detector is given by the intensity operator:

I(t) ∝ a† (t)a(t),

(26)

where prefactors have been ignored. We assume that a(t) has the mean value α = a(t), thus writing (24) as the superposition

a(t) = α +

δa(−ω)e−iωt dω +

B

δa(ω)eiωt dω.

(27)

B

By inserting this expression into (26), it is permissable to linearize if the power of the mean value is much larger than the noise at the considered sidebands. The amount of noise at the sidebands depends partly on the excess noise above the shot noise level per unit bandwidth and partly on the chosen bandwidth. Therefore, the sideband noise should be small and slow. In that case the intensity operator reads

2

I(t) ∝ α +

α(δXu (ω) + δXl (ω)) cos(ωt)dω +

B

α(δPu (ω) B

− δPl (ω)) sin(ωt)dω,

(28)

where δXu (ω)(δPu (ω)) and δXl (ω)(δPl (ω)) are the amplitude (phase) quadratures at the upper and lower sideband relative to the carrier. From this expression, it is clear that direct amplitude detection is measuring a combination of the quadratures of the sideband modes: the sum of the amplitude quadratures for the upper and lower sideband modes as well as the difference of the phase quadratures of the adjacent sideband modes. The carrier mode therefore probes simultaneously the quadratures of the upper and lower sideband modes with a bandwidth B. It is also clear from this expression that intensity-squeezing is a result of anti-correlation and correlation between the amplitude and phase quadratures of adjacent sidebands, respectively. The amplitude quadrature for the state is usually deﬁned as

δX(t) = B

(δXu (ω) + δXl (ω)) cos(ωt)dω + (δPu (ω) − δPl (ω)) sin(ωt)dω B

(29)

Experimental Feed-Forward Tools

385

so that the intensity operator can be succinctly written as I(t) = α2 + αX(t); thus, the time-varying part of the intensity reﬂects the amplitude quadrature of the measured quantum state. At this point, we note that many of the experiments conducted on quantum feed-forward (and those reviewed in the next section) can be described in a single-mode picture as done in Section 2. The approximation is valid as long as the action on the different frequency modes (either within a single collection of sideband or adjacent sidebands) is the same. This is the case if the considered bandwidth of the quantum state is so small that the transformations are invariant within this bandwidth, and if, in addition, the action is invariant with respect to symmetrically placed sidebands around the carrier. A more ﬂexible quadrature detector is the homodyne detector, which is capable of measuring an arbitrary quadrature: the state under interrogation is mixed on a symmetric beam splitter with a strong local oscillator, the two outputs are detected using the afore-described detectors and the difference of the electronic outputs is found. By using the linearization approach, the output is then I− (t) ∝ αδXθ (t) = δX(t) cos θ + δP(t) sin θ, where θ is the phase of the local oscillator and δP(t) is the phase quadrature. Thus by controlling the phase of the local oscillator any quadrature can be measured, which is seen to be a combination of quadratures of the upper and lower sideband modes. It should be noted, however, that only the part of the signal overlapping with the spatio-temporal mode of the local oscillator will be probed. It means that the local oscillator must be carefully aligned to obtain a large overlap with the signal mode. Moreover, since the power of the local oscillator effectively acts as a noise-less ampliﬁer for the quantum ﬂuctuations of the signal beam, the optical ﬂuctuations seen by the detector can be made much larger than the detrimental electronic ﬂuctuations of the detector. The electronic noise can be included by adding a linear stochastic variable, which is independent of the amplitude of the local oscillator. Another detector imperfection of importance is the non-ideal photon–electron conversion performance of real diodes. This non-unit quantum efﬁciency of the detectors leads to the intrusion of vacuum noise in the measurement, thus washing away the quantumness of the state. This imperfection can however almost be completely overcome by using special fabricated diodes, and quantum efﬁciencies of more than 99% have been reported. Another important measurement device is the heterodyne detector, which accesses conjugate variables simultaneously, e.g., it may execute measurements of the amplitude and phase quadrature simultaneously. Such a measurement can be carried out by splitting the signal on a symmetric beam splitter and placing a homodyne detector at each of the two outputs of the beam splitter. The two homodyne detectors are set to measure conjugate variables, e.g., by choosing local oscillator phases of

386

Quantum Feed-Forward Control of Light

0 and π/2 corresponding to measurements of the amplitude and phase quadratures, respectively. Since one cannot perform simultaneously the sharp measurements of conjugate quadratures according to the laws of quantum mechanics, the accuracy with which the quadratures are determined is intrinsically limited. Vacuum noise is introduced in such a measurement and it can be traced back to the quantum noise entering the empty port of the beam splitter directing the signal to the two homodyne detectors. If the signal being measured is relatively bright (that is, has a strong carrier component), there exists a simpler scheme for heterodyning the state at the sidebands. The bright signal is mixed on a symmetric beam splitter with an auxiliary beam with a carrier power matching the power of the signal beam but with vacuum at the sidebands. By measuring the two resulting output beams with intensity detectors, the sum of the photocurrents is proportional to the amplitude quadrature of the signal, whereas the difference is proportional to the phase quadrature. Such a scheme is simpler to implement, since it requires only a single beam splitter and two detectors in contrast to the traditional scheme with three beam splitters and four detectors. In some cases the signal beam is too bright to be used in a homodyne detector and in some cases it is not easily matched to a local oscillator. An alternative way of accessing quadrature information of bright beams without the use of a local oscillator is to use the dispersive features of an empty cavity (Galatola et al., 1991): by reﬂecting the beam off a high ﬁnesse cavity, the carrier is phase-rotated while the sideband modes are unaffected. In direct detection of the reﬂected beam, different quadratures are then probed with the speciﬁc quadrature being determined by cavity detuning. Yet another approach to access information of conjugate quadratures at a certain sideband frequency is to use an asymmetric Mach–Zehnder interferometer. In this approach, the dispersion appearing in free propagation is used to rotate the carrier with respect to sidebands; see Glöckl et al. (2004). There exists of course a wealth of other types of detectors such as the binary detectors which can discriminate between vacuum and some photons as well as photon-counting detectors that can resolve the number of photons in a state. Moreover, it is possible to combine the detectors with some unitary transformation or some feedback mechanics to measure a certain property of the light ﬁeld. This is a very interesting topic but its description is outside the scope of this review.

3.3.2. Feed-Forward Once the measurement is performed, the acquired classical data undergo classical data processing in the feed-forward electronics and the outputs are subsequently used to control an operation of the post-measurement

Experimental Feed-Forward Tools

387

state. The design of this feed-forward algorithm naturally depends on the task to be executed but it also depends on the measurement process. In general, the measurement together with the feed-forward algorithm must be optimized to successfully execute a speciﬁc task. In what follows we discuss a few simple but important feed-forward algorithms. To accomplish a high-speed complex operation in the feed-forward loop, one may use a programmable logic device. Such devices are capable of making a complex non-linear transformation with a large bandwidth, and they have been widely used in fast feedback loops. In many applications, including those discussed in this review, the programmable logic device is unnecessarily complex, and standard analog circuits sufﬁce to accomplish the feed-forward operation. This is, for example, the case when the feedforward control is a linear function of the measurement outcomes. Such simple feed-forward transformations have been standard in most quantum information processing experiments. The effect of the electronics in the feed-forward loop is a convolution of the time-dependent part of the measurement outcome, δX(t), and the impulse response of the electronics, h(τ). It can be written as

t δXout =

h(τ)δXin (t − τ)dτ,

(30)

o

and in the frequency domain it reads

δXout (ω) = h(ω)δXin (ω),

(31)

which is obtained by a Fourier transform. In most experiments on continuous variables quantum feed-forward, the frequency response function, h(ω), has been either a constant or a discrete function. If the function is constant, every single acquired data in the measurement process undergoes identical ampliﬁcation and ﬁltering steps, and the output of the processor is a linear function of the input. The resulting outcome of the feed-forward electronics is normally used to linearly displace the remaining optical beam. Although this operation is easily described by a simple displacement operator, it can be physically implemented using at least two different strategies. One of them is to directly phase- and amplitude-modulate the remaining beam by driving two independent EOMs with the outcome of the electronics. However, since the amplitude modulator is rather lossy, it is common to use a different strategy (see Figure 5): an auxiliary mode is amplitude- and phase-modulated based on the electronic output signal and subsequently combined with the quantum state in a largely asymmetric beam splitter,

388

Quantum Feed-Forward Control of Light

Displacement

operation

PM

Input

AM /4

EOM

PBS EOM EOM

Output

FIGURE 5 Displacement operation. The displacement is imprinted onto the signal via the modulation of an auxiliary beam. This beam traverses several electro-optic modulators (EOMs) to ensure stable and clean amplitude and phase modulation.

say 99:1; 99% of the fragile quantum state is transmitted through the beam splitter and coupled with 1% of the signal carried by the auxiliary beam. In that way the information of the feed-forward loop is carried over to the quantum state via an auxiliary state in a lossless manner. To avoid cross coupling of conjugate quadratures, it is of paramount importance that the modulators are producing pure amplitude and pure phase modulations. The modulations are usually performed by using the arrangement shown in Figure 5. The ﬁrst EOM modulates the phase of the beam and is oriented with its optical axis in the polarization direction of the linearly polarized input beam. The second modulator induces a stable and pure amplitude modulation and it consists of two EOMs sandwiched between a quarter wave plate, and a PBS. The amplitude modulation is produced through interference of orthogonal polarization modes on the PBS. Temperature drifts change the refractive index of the crystals and thus give rise to low-frequency ﬂuctuations in the amplitude modulation. To circumvent such effect, a second identical crystal rotated 90 degree with respect to the ﬁrst EOM is inserted. Since this crystal will have the same temperature ﬂuctuations, the phase noise of the orthogonal polarization modes will be classically correlated such that the amplitude will be stable upon interference. In the linear feed-forward process just described, all quantum states are kept, thus rendering the overall protocol deterministic. On the contrary, some feed-forward experiments are probabilistic in the sense that the measurement data are used to determine whether the remaining state should be kept or discarded. Therefore, for such protocols the outcome of the feed-forward electronics is binary, the value of which is determined by the measurement outcomes. As an example, the response function might be set to h1 (ω) if the measurement outcomes are within the intervals I, whereas it is set to h2 (ω) if the measurement outcomes are outside these intervals:

δXout (ω) =

h1 (ω) if δXin (ω) ∈ I . /I h2 (ω) if δXin (ω) ∈

(32)

Applications of Feed-Forward

389

The output of the electronics is then used to control the transmittivity of a beam splitter through which the remaining beams are traversing. If the electronic output is h1 (ω), the beam splitter is, say, fully reﬂective (thus discarding the state), and if the output is h2 (ω), the beam splitter transmission is set to be fully transmittive (thus keeping the state). Such a discrete electro-optic feed-forward loop is technically demanding to implement, and in previous implementations of probabilistic continuous variable protocols, the selection of states have been carried out purely electronically. Instead of heralding the real optical quantum states with a fast beam splitter, the “state” is heralded electronically after it has been measured. It basically means that both beams are detected and one of the resulting data sets is heralded based on post-selection of the other data set. For ideal operation of the feed-forward loop, the detector and associated feed-forward electronics must have an inﬁnite bandwidth. In other words, the delay time associated with the electronic feed-forward loop with respect to the non-measured beam should be zero for optimal performance for all frequency components of the mode. However, in practice, the feed-forward loop will be non-ideal and thus will have a ﬁnite response bandwidth. This basically means that the feed-forward manipulations will not work on all frequency components. In time domain, this can be interpreted as the quantum state (or pulse of light) being smeared out. If, however, the duration of the quantum state (being it a light pulse or a speciﬁc time slot in a CW beam) is much larger than the delay of the feedforward electronics, or equivalently, if the bandwidth of the state is much smaller than that of the electronics, the feed-forward correction will be successful.

4. APPLICATIONS OF FEED-FORWARD We have now described in detail the theoretical concepts and experimental techniques that are useful for understanding the basics of the simple feed-forward loop. Therefore, we are now ready to discuss the wealth of applications in which the simple feed-forward loop is the engine. The list of applications is very long; thus we cannot describe them all here. Instead we focus on some few applications that we believe are good representatives for many other applications, which are only brieﬂy mentioned. The ﬁrst three sections will be devoted to the applications of the most simple feed-forward loop based on two inputs, a beam splitter, a detector, and some feed-forward corrections as shown schematically in Figure 6. In the three subsections, we consider applications relying on homodyne measurements and linear displacements (Section 4.1), homodyne measurements and selective heralding (Section 4.2), and ﬁnally, heterodyne measurements

390

Quantum Feed-Forward Control of Light

Processing

Detector - Homodyne - Heterodyne - Photon counting

- Linear gain - Post-selection

Input 1 - Signal state - Linear displacement - Heralding

Input 2 - Vac. state - Sqz. state

FIGURE 6

Schematic of the simple feed-forward set-up.

and linear displacements (Section 4.3). The experimentally implemented feed-forward protocols relying on the simple feed-forward scheme in Figure 6 are summarized in Table 1. In Section 4.4, we address more complex feed-forward set-ups that ﬁnd use in CV quantum information processing.

4.1. Homodyne Detection and Displacement In this section, we will address applications that rely on the homodyne detection of a partially reﬂected quantum state and the subsequent deterministic and linear displacement of the remaining ﬁeld.

4.1.1. Noise-Less Amplification The feed-forward scheme, illustrated in Figure 6, was used for the ﬁrst time in 1997 (to the best of our knowledge) by Lam et al. (1997) [see also Lam et al. (1998)] to implement a noise-less ampliﬁer. Such an ampliﬁer is phase-sensitive and is capable of amplifying a single quadrature of the light ﬁeld such that the signal-to-noise ratio is conserved. The scheme was surprisingly simple, relying only on a single beam splitter, an amplitude quadrature detector, some linear feed-forward electronics, and an amplitude modulator. The evolution of a Gaussian state through such a system is easily determined by Eq. (5) derived in Section 2.5, and we see that if the feed-forward gain is chosen as g = − (1 − T)/T, the transformation of the amplitude quadrature is

X1 →

1 X1 . T

(33)

It clearly demonstrates noise-less ampliﬁcation with an optical gain factor of G = 1/T. The system performance in this case is quantiﬁed in

Coherent Coherent Bright beam Coherent Vacuum X-sqz Coherent Two-photon Mixed-sqz Mixed-sqz Mixed Coherent Coherent

Noise-less ampliﬁcation

Universal sqz

Sqz transfer

Non-unitary QND

Quantum erasing

Sqz puriﬁcation

Probabilistic sqz

Cat state generation Sqz distillation I Sqz distillation II Filtration

Phase-insensitive amp

MDM

Vacuum

Vacuum

Vacuum Vacuum Mixed-sqz Vacuum

P-sqz

Vacuum

P-sqz

P-sqz

X-sqz

P-sqz

Vacuum

Input 2

|αα|

1 2

1−T T

2(1−T) T

Post-selection Post-selection Post-selection Post-selection

|xx| |pp| |xx| |xx| |αα|

Post-selection

X and P displacement

X and P displacement

Heralding Heralding Heralding Heralding

Heralding

P displacement

X displacement

1−T T

T(1−T)(1−VX2 ) 1−T(1−VX2 )

P displacement

1−T T

|xx|

|pp|

−

|xx|

−

X displacement

1−T T

X displacement

X displacement

Post-measurement operation

1−T T

T(1−T)(1−VX2 ) 1−T(1−VX2 )

|pp|

|xx|

−

|xx|

−

Feed-forward gain

|xx|

Measurement

The observables mentioned in the table refer to the description in the text and not necessarily to the actually executed experiment.

Input 1

Summary of feed-forward experiments based on the simple feed-forward system illustrated in Figure 6.

Function

TABLE I

Applications of Feed-Forward

391

392

Quantum Feed-Forward Control of Light

terms of the transfer coefﬁcient, which is deﬁned as the ratio between the signal-to-noise ratios of the input and the output states, TX = X SNRX out /SNRin . Taking into account the detection efﬁciency of η, the transfer coefﬁcient is given by TX = η. In the experiment, the input state was deﬁned as sideband modes at 20 MHz relative to the carrier (as discussed in Section 3.1.1). The tapped-off signal was measured with an amplitude quadrature detector and fed forward onto an amplitude modulator, through which the ﬁnal state was prepared. For veriﬁcation, the signal-to-noise ratio of the amplitude quadrature was measured. Two different beam-splitting ratios were tested resulting in optical gains at the sidebands of G = 3.4 ± 0.06 dB and G = 13.4 ± 0.5 dB. The measured transfer coefﬁcients were TX = 0.86 ± 0.02 and TX = 0.88 ± 0.02, respectively. Noise-less ampliﬁcation of a squeezed 10 MHz sideband was also carried out, and a transfer coefﬁcient of gain of TX = 0.75 ± 0.02 for a gain of G = 9.3 ± 0.2 dB was measured. The system was extended to include power ampliﬁcation using an injection-locked laser (Huntington et al., 1998). Such combination of power ampliﬁcation and noise-less signal ampliﬁcation allows for the robust transfer of signals in lossy channels (Ralph, 1997). Using a similar set-up with a phase quadrature detector and a phase modulator in replacement of the amplitude detector and the amplitude modulator, Buchler, Hungtington, and Ralph (1999) experimentally demonstrated noise-less ampliﬁcation of the phase quadrature. For the phase quadrature, the measured transfer coefﬁcient was TP = 0.9 ± 0.14 for a gain of 10 dB.

4.1.2. Squeezing Operation The afore-mentioned experiments on noise-less ampliﬁcation are not unitary; the quadratures, which are conjugate to the ampliﬁed ones, are not transformed correctly so as to ensure a unitarity. It is however possible to use another feed-forward approach to implement a unitary and noiseless phase-sensitive ampliﬁer as suggested by Filip, Marek, and Andersen (2005). In fact, such an amplifying operation is a squeezing operation, which in principle can be accomplished by using optical parametric processes such as the Kerr effect in a ﬁber or the second-order nonlinearity in parametric ampliﬁers. However, in practice, it can be difﬁcult to obtain a clean experimental execution of such an operation, since the injection of a signal into the squeezer (e.g., a ﬁber or a cavity with a nonlinear crystal) can be quite lossy, thus severely degrading the efﬁciency of the operation. A much cleaner squeezing operation can be accomplished by using a feed-forward-based scheme. The core of the squeezing experiment is again the simple feed-forward system illustrated in Figure 6. The ancillary mode 2 is occupied by a

Applications of Feed-Forward

393

squeezed vacuum state, which is generated off-line (that is, the signal need not be injected into the squeezer), and the feed-forward detector measures the most noisy quadrature corresponding to the anti-squeezed quadrature. The dynamics is easily addressed by evolving a quadrature eigen-state in the Schrödinger picture, as done in the transformation (16). It is easily

seen that by displacing the quadrature variable by the amount gx¯ = − (1 − T)/T x¯ , the result is an √ ideally squeezed state where every eigenstate |x is transformed as |x/ T. In the Heisenberg picture, assuming that the input ancillary mode is squeezed in P2 , the evolved quadratures transform as

X1 → P1 →

1 X1 , T

√ √ TP1 + 1 − TP2 ,

(34)

which approaches the ideal squeezing operation as the variance of P2 decreases. Note that the squeezing operation is deterministic and universal in the sense that it works every time and it works equally well on any unknown input state. The deterministic, universal squeezing operation was experimentally veriﬁed for coherent state inputs by Yoshikawa et al. (2007). In this experiment, the coherent states were deﬁned as sideband modes at 1 MHz (and 30 kHz bandwidth) and the squeezed state was produced in an optical parametric oscillator based on a periodically poled KTiOPO4 (KTP) crystal. The two states merged on a beam splitter with a variable beam-splitting ratio (see Section 3.2) and one output was measured with a high-efﬁciency homodyne detector set to measure the anti-squeezed quadrature. The displacement was imposed by modulating the phase of an auxiliary beam that was coupled to the signal beam via a highly asymmetric beam splitter (99:1) as discussed in Section 3.3.2. The process ﬁdelity, which determines the phase space overlap between the ideally squeezed state and the actual squeezed state, was used to quantify the performance of the system. They measured ﬁdelities of 78 ± 2%, 89 ± 1%, and 94 ± 1% for output squeezing degrees of −2.5 dB (with T = 0.25), −1.6 dB (with T = 0.5), and −0.7 dB (with T = 0.75), and anti-squeezing degrees of 5.8, 3.0, and 1.3 dB, respectively. In the experiment, the mean values√of the coherent state was trans√ formed ideally (that is, X1 → X1 / T and P1 → TP1 ), which ensure universal operation. If, however, one is willing to relax the universality criterion, the electronic feed-forward gain can be optimized to minimize the squeezing variance for a given transmission of the beam splitter. To enable such an optimization strategy, the squeezed quadrature is measured and the variance of the input state must be

394

Quantum Feed-Forward Control of Light

apriori tailor the electronic feed-forward gain; known to appropriately g = T (1 − T) VX1 − VX2 / (1 − T)VX1 + TVX2 . Optimization yields an output squeezing variance of

V X1 →

VX1 VX2 , (1 − T)VX1 + TVX2

(35)

which is always smaller than the variance obtained for the universal squeezer. However, the expectation value of the amplitude quadrature √ is transformed to X → TV X 1 X2 1 /(TVX2 + (1 − T)VX1 ) and P1 → √ √ TP1 + 1 − TP2 , which differs from the ideal (text book) squeezing operation. The advantage of the present transformation, however, is that it ensures perfect purity of the squeezed output state if the input states are pure. This is not the case for the universal squeezer unless the squeezed vacuum is a unphysical quadrature eigenstate. This technique was experimentally used by Lam et al. (1999) to transfer squeezing from a vacuum-squeezed state to a bright laser beam.

4.1.3. Quantum Non-Demolition Coupling By using Bloch–Messiah’s decomposition theorem, Braunstein (2005) showed that any multi-mode Gaussian operation can be implemented by placing a squeezing operation in each of the paths of a multi-mode interferometer. In other words, the feed-forward-based universal squeezing operations as introduced in the previous section together with linear optics sufﬁce to enable any Gaussian operation. As an example of a simple two-mode Gaussian operation we consider in this section a quantum non-demolition (QND) coupling. QND measurements were initially proposed to allow for increased accuracy in the detection of gravitational waves (Braginsky and Voronstsov, 1974; Caves et al., 1980). The idea of a QND measurement is to measure nondestructively a QND observable, for example, a certain quadrature of a light ﬁeld, say X1 , which commutes with the systems’ Hamiltonian and thus is a constant of motion. The measurement conserves the measured observable, X1 → X1 , and places the involved back action noise onto a conjugate observable, P1 → P1 − GP2 , where G is the QND gain and P2 is the phase quadrature of the ancillary state (also known as the probe state). In this coupling process, the signal information is transferred to the probe beam, which undergoes the transformation X2 → X2 + GX1 and P2 → P2 . Thus, information about X1 is now contained in the observable X2 . Note also that P2 is a QND variable in this transformation. The QND coupling based on the universal squeezer was proposed by Filip, Marek, and Andersen (2005) [see also Wilde et al. (2007)], and a

Applications of Feed-Forward

395

Processing

Input 2

Output 2 Output 1

Input 1

Processing

FIGURE 7 Schematic of the quantum non-demolition interaction. The scheme is based on two feed-forward loops placed inside a two-mode interferometer (Mach–Zehnder interferometer).

schematic of the set-up is shown in Figure 7. The transformation of the two pairs of quadratures are

1−R Xsqz1 , 1+R 1−R 1−R Xsqz1 , X2 → X2 + √ X1 + R 1+R R 1−R 1−R Psqz2 , P1 → P1 − √ P2 + R 1+R R 1−R Psqz2 , P2 → P2 + 1+R

X1 → X1 −

(36) (37) (38) (39)

where R is the reﬂectivity of the beam splitters in the universal squeezers (see Figure 7), and Xsqz and Psqz represent the quadratures of the squeezed vacua. For inﬁnite squeezing, the last terms can be neglected, which √ renders the QND transformation ideal with a gain of G = (1 − R)/ R. QND operation based on this scheme has been recently implemented by Yoshikawa et al. (2008). In this experiment, two feed-forward-based universal squeezers were built into a Mach–Zehnder interferometer. The squeezed vacua in the universal squeezing operations were prepared by two sub-threshold optical parametric oscillators based on periodically poled KTP crystals, and each device produced about 5 dB of squeezing. Quantum information was encoded onto sidebands at 1.25 MHz on a bright laser beam with a wavelength of 860 nm. All the beam splitters making

396

Quantum Feed-Forward Control of Light

up the Mach–Zehnder interferometer as well as the beam splitters in the universal squeezers were made tunable using the experimental procedure discussed in Section 3.2. The usual criteria for verifying a QND coupling are 1 < T1 + T2 ≤ 2 and V1|2 < 1, where T1 and T2 are the transfer coefﬁcients from signal input to signal output and from signal input to probe output, respectively, and V1|2 is the variance of the signal output conditioned on a measurement of the probe. In the experiment by Yoshikawa et al. (2008), they measured T1 + T2 = 1.42 ± 0.06 and V1|2 = 0.61 ± 0.01 for the amplitude quadrature, and T1 + T2 = 1.27 ± 0.05 and V1|2 = 0.61 ± 0.01 for the phase quadrature. The QND gain for these measurements was G = 1.5. These results demonstrated that a true QND coupling was obtained for conjugate quadratures using simple feed-forward systems and vacuum-squeezed states. Such a QND operation is the core of the cluster state computation scheme, which will be addressed in Section 4.4.3. Optimization of the transfer coefﬁcients and the conditional variance for just a single QND quadrature can be achieved by using a simpler feedforward system as proposed almost 10 years ago by Buchler, Lam, and Ralph (1999) and demonstrated by Buchler et al. (2001). In this experiment, the QND coupling was enabled by a squeezed light beam splitter (Bruckmeier et al., 1997), and further enhanced by using the electrooptic feed-forward system. They measured a total transfer coefﬁcient of T1 + T2 = 1.62 ± 0.02 and a conditional variance of V1|2 = 0.54 ± 0.03. These values are deep inside the QND domain, thus clearly demonstrating QND coupling for one QND observable. By quantifying the QND measurements in terms of the transfer coefﬁcient and the conditional variance, it is actually possible to enable a “QND” measurement by full destructive detection of the state followed by recreation, that is, by using the simplest possible feed-forward scheme. This was ﬁrst demonstrated in the experiments by Goobar, Karlson, and Björk (1993) and Roch, Poizat, and Grangier (1993), where an optical quantum state was completely measured and the resulting photocurrent was partly used to estimate the measured parameter and partly used to drive a light-emitting diode. A similar scheme but based on the feed-forward onto a squeezed light beam has been also implemented and demonstrated to fulﬁll the QND criteria for a single quadrature (Schneider et al., 2005).

4.1.4. Quantum Erasing Quantum feed-forward can be also used to implement a protocol known as quantum erasing, which is an idea that initially was put forward by Scully and Drühl (1982): It is a well-known fact that in a double-slit experiment the determination of which-way information and the observation of interference fringes are mutually exclusive as a result of Bohr’s complementarity

Applications of Feed-Forward

397

principle. If one attempts to gain which-way information, the interference fringes are lost. However, as suggested by Scully and Drühl, under certain circumstances it is possible to erase the gained which-way information through a unitary operation and thereby recover the lost interference. This idea was extended to continuous variables by considering the amplitude and phase quadratures of light as being the complementary variables (Filip, 2002). Instead of preparing a state with two possible outcomes (one of two different paths), in the continuous variable version a vacuum mode consisting of a continuum of quadrature eigenstates is prepared. Filip (2003) suggested to measure the continuous which-state information in the P1 variable (see Section 2.5) through a simple QND coupling by mixing the prepared state with a state squeezed in the P2 variable on a beam splitter (Bruckmeier et al., 1997). For an inﬁnitely squeezed state, the obtained which-state information (P2 ) is perfect, whereas the complementary information (interference), X1 , is very noisy. The idea of quantum erasing is now used to erase completely the which-state information by performing a simple unitary π/2 phase rotation of the quantum state before it is measured. Due to the phase rotation, the whichstate detector measures the complementary information, the outcomes of which is fed forward (after an appropriate scaling of g = − (1 − T)/T) to the remaining quantum state to correct the interference information. After erasing the which-state information, the interference information can in principle be perfectly recovered independent of the degree of squeezing. Experimental demonstration of the feed-forward-based quantumerasing protocol was carried out by Andersen et al. (2004). In this experiment, the QND coupling was enabled by a squeezed light beam splitter as discussed earlier. The squeezed light was produced by exploiting the Kerr effect in an optical ﬁber. Femto-second light pulses at 1530 nm were launched into ﬁber, which is placed inside an asymmetric Sagnac interferometer, to yield amplitude squeezed light pulses (Schmitt et al., 1998). The considered state was a vacuum state located at a sideband with frequency 20.5 MHz relative to the optical carriers. Due to the brightness of the pulses, phase information was not easily accessed by the standard method of homodyne detection. A new scheme based on an asymmetric Mach–Zehnder interferometer was therefore employed to access the phase information (Glöckl et al., 2004). The interferometer was capable of rotating the carriers of the light pulse with respect to the sidebands such that information of the phase quadrature could be accessed. The protocol was quantiﬁed in terms of added noise to the input vacuum state. If the amplitude information was measured, the added noise in the which-state variable was V = 0.54 ± 0.02, whereas the added noise in the complementary quadrature was V = 455 ± 7. Hence, as a result of the which-state measurement, the interference information was

398

Quantum Feed-Forward Control of Light

completely demolished. If, however, the which-state information is erased by measuring the complementary information and subsequently feeding forward the result to the remaining state through linear displacement, the added noise was reduced to V = 0.14 ± 0.02. Thus, by exploiting the simple feed-forward scheme, the idea of quantum erasing for continuous variables was experimentally realized. It has been shown theoretically that quantum erasing of continuous variables can be used to protect an unknown quantum state inside a cavity (Marek and Filip, 2004) and to decouple modes in a quantum communication scheme (Filip, Mišta, and Marek, 2005).

4.1.5. Purification of a Squeezed State Presently available squeezed states are often highly mixed due to decoherence and dissipation in the preparation sources. Due to the requirement of having pure squeezed states for the implementation of various quantum information protocols, it is important to develop effective puriﬁcation techniques for these states. Also here the simple feed-forward system can be used. It was shown by Glöckl et al. (2006) that by tapping off a small part of a squeezed state, measuring the noisy quadrature, and feed-forwarding the result onto the remaining part of the squeezed state, it is possible to optimally purify a mixed Gaussian-squeezed state. √ Consider a Gaussian-squeezed state with the purity given by 1/ VX VP , where VX is the variance of the squeezed quadrature and VP is the variance of the anti-squeezed√quadrature. Suppose VP >> 1 so that the state is highly mixed; thus 1/ VX VP << 1. This is for example the case for squeezed states generated in optical ﬁbers where the scattering effect such as guided acoustic wave Brillouin scattering introduces excess noise in the phase quadrature (Shelby et al., 1986). The optimal puriﬁcation technique was deﬁned as the one that minimizes VP while minimally disturbing the squeezing, VX . They found that the optimal transformation yields a minimal variance of

min VP = 1

VP2 (1 − VX2 ) , (1 − VX ) + VP2 (VX − VX2 ) 1

(40)

1

where primes denote the transformed variances. It turned out that the optimal puriﬁcation scheme could be implemented by the simple homodynebased feed-forward scheme at input port 2 and by setting the with vacuum feed-forward gain to g = T(1 − T) VX1 − VX2 / (1 − T)VX1 + TVX2 . The puriﬁcation scheme was demonstrated for a 20.5 MHz squeezed sideband by using an experimental set-up similar to the quantumerasing set-up discussed in the previous section. For 90% beam splitter

Applications of Feed-Forward

399

transmission, they measure a tenfold reduction of the thermal noise while the squeezing was only degraded by 11%. Furthermore, they discussed the potentials of using the purifying feed-forward scheme to improve the performance of quantum informational protocols such as dense coding and entanglement generation.

4.2. Homodyning and Heralding Now we consider feed-forward protocols that rely on post-selection of the measurement outcomes and heralding of the post-measurement state. We discuss a probabilistic squeezing operation, a squeezed state distillation operation, and a ﬁltration operation.

4.2.1. Probabilistic Squeezing Operation The deterministic squeezing operation discussed in Section 4.1.2 can also be implemented probabilistically as proposed by Lance et al. (2006) [see also Jeong et al. (2006)]. They suggested to use the simple feedforward scheme with a squeezed ancillary state (as for the deterministic squeezing operation), but in replacement of the deterministic and measurement-induced displacement operation, they proposed to perform a measurement-induced heralding transformation. Despite the probabilistic nature of this squeezing operation, it has the advantage of enabling a pure squeezing transformation that achieves the result in (35) without having knowledge about the input variances VX1 and VX2 . Therefore, by using the probabilistic approach, the universality and determinism are traded for a pure squeezing transformation of an unknown input state. As an example, we consider the probabilistic squeezing of a single photon state, |1. By using the theory of Section 2, we get (up to normalization)

∞ 2 ∞ x1 x22 1 x1 |x1 dx1 |1|sqz = exp − exp − |x2 dx2 1 1 4 4VA (2π) 4−∞ (2πVA) 4−∞ 1

∞ → −∞

∞ ∝ −∞

x22 x12 exp − − x1 δ(y1 )|y2 dx1 dx2 4 4VA

x12 x12 x1 x1 | √ dx1 , exp − − 4 4TVA/(1 − T) T

where√VA is the anti-squeezing variance of the input ancillary state and √ √ √ y1 = Tx2 − 1 − Tx1 and y2 = Tx1 + 1 − Tx2 . After the substitution

400

Quantum Feed-Forward Control of Light

√ x1 / T = x, we reach the ﬁnal state (after normalization)

|φ =

∞

1 (2πVA)

1 4

−∞

x2 exp − 4VA

1 1−T x . |xdx, where = T + VA VA VA (41)

This is clearly a pure single photon state squeezed in the P quadrature by an amount VS = T + (1 − T)VS , where VS is the squeezed variance of the ancillary state. This state was selected only if the measurement outcome was X = 0, which yields a very low success rate. It is therefore common to select states for which the measurement outcomes fall within a nonvanishing interval to increase the success rate. To deduce the expected output state, it is for such cases more appropriate to use the Wigner function representation rather than the quadrature representation as described in Section 2. The main motivation for squeezing the single photon state is that the transformed state is similar to a superposition of small coherent states (sometimes known as a kitten state), which in turn might be useful in quantum information processing (Ralph et al., 2003). As a proof of principle, Lance et al. (2006) have demonstrated the probabilistic squeezing operation on a coherent state. The transformation corresponds to the modiﬁed squeezing operation discussed at the end of Section 4.1.2 (except that the latter transformation is deterministic). An advantage of the former transformation, however, is that no apriori knowledge about the variance of the input state is needed, as was the case for the deterministic transformation. In the experiment, they used an ancillary squeezed state with 4.50 dB squeezing and 8.50 dB anti-squeezing produced by an sub-threshold optical parametric oscillator at 1064 nm. The coherent state inputs were deﬁned as displaced sideband modes at 6.8 MHz of the laser ﬁeld. They achieved the best performance by setting the beam splitter transmission to 25% for which they measured optical gains of 0.50 and 0.71 and variances of −3 and 6.7 dB for the amplitude and phase quadratures, respectively. Note that for the universal squeezer, the gains for such a transformation would have been 1/2 and 2, thus rendering the probabilistic squeezer non-universal. Nevertheless, a high ﬁdelity of 90% between the actually squeezed state and the expected ideally transformed state was observed for this particular input state. Moreover, the output state was measured to have a purity of 0.81, which is higher than the purity of the squeezed ancillary state. As mentioned earlier, by sending a single photon through the feedforward circuit, the resulting output state mimics an optical cat state with small excitations. It was however realized by Ourjoumtsev et al. (2007) that a similar task can be accomplished without the use of squeezed vacuum;

Applications of Feed-Forward

401

it can in fact be replaced by vacuum. Furthermore, they showed that the larger the input Fock state is, the larger will be the output cat state. If the input ancillary state is a two-photon Fock state, the transformation in Fock state representation can be written as

√ √ √ † √ † 2 1/ 2 a†2 2 |01 |02 = 1/ 2( Ta2 + 1 − Ta1 ) |01 |02

√ = 1/ 2(Ta2†2 + 2 T(1 − T)a2† a1† + (1 − T)a1†2 )|01 |02 √ = T|01 |22 + 2 T(1 − T)|11 |12 + (1 − T)|21 |02 , √ using the inverse transformation for the beam splitter a1 = Ta1 − √ √ √ 1 − Ta2 and a2 = Ta2 + 1 − Ta1 . The feed-forward system now selects the states for which the quadrature measurement outcome was x = 0, which means a projection onto the inﬁnitely squeezed state |x = 0 ∝ n |2n. Using the measurement operator ∝ n,n |2n2n |, the resulting state is proportional to a superposition between the vacuum state and the two-photon state: T|01 + (1 − T)|21 . The amplitude ratio between the two states can be easily tuned by varying the beam splitter transmittivity T, and thus they can be tailored to approach with a very√high ﬁdelity the coherent state √ superposition state: | + = (|α + | − α)/ N ≈ 1/(Coshα)(α|0 + α2 / 2|2 for small α, where N = 2(1 + exp(−2α2 )). This feed-forward procedure has been experimentally realized by Ourjoumtsev et al. (2007). The implementation used ultra-short light pulses prepared as two-photon Fock states: a highly pure two-mode squeezed state containing a correlated number of photons was produced in a spatially nondegenerate optical parametric ampliﬁer by single-pass down-conversion in a very thin non-linear crystal. One beam was split between two avalanche photodiodes (APDs) and coincidence of counts projects the other beam into a two-photon Fock state as the parametric gain was small. The prepared Fock state was then split on a 50/50 beam splitter, the reﬂected mode was measured by a homodyne detector, and the remaining state was kept only if measurement outcome was |p| < 0.1. The Wigner function of the heralded state was reconstructed and interference in phase space was observed. It resembled a coherent state superposition state with an amplitude α ≈ 1.61.

4.2.2. Distillation of Squeezing The deterministic puriﬁcation transformation discussed in Section 4.1.5 was not capable of increasing the amount of squeezing. In fact, it has been proven that it is impossible to increase the degree of squeezing of a Gaussian state (that is, to perform squeezing distillation) using only linear optics and feed-forward (Kraus et al., 2003). Moreover, there is a No-Go theorem saying that distillation of entanglement of two-mode Gaussian states is

402

Quantum Feed-Forward Control of Light

impossible using only Gaussian local operations and feed-forward. On the other hand, if the state under interrogation is non-Gaussian, it has been shown by Heersink et al. (2006) and by Franzen et al. (2006) that squeezing distillation is indeed possible by linear optics and feed-forward, and very recently it has been shown to be the case also for entanglement (Dong et al., 2008; Hage et al., 2008). Moreover, it has been shown theoretically that a similar protocol can be used to purify a coherent state superposition state which has undergone Gaussian attenuation in a lossy channel (Suzuki et al., 2006). In the squeezed state distillation experiment of Heersink et al. (2006), the pre-distilled state was a non-Gaussian mixture of two Gaussian-squeezed states with the Wigner function

x12 pa p2 W(x1 , p1 ) = exp − − √ 2VX1a 2VP1a 2π VX1a VP1a pb (x − x0 )2 (p − p0 )2 + √ exp − − . (42) 2VX1b 2VP1b 2π VX2a VP2a The distillation scheme employs the simple feed-forward set-up, where a small part of the non-Gaussian state is tapped off using an asymmetric beam splitter, measured using a homodyne detector, and subsequently post-selected to transform the state into a Gaussian-squeezed state. The function of the homodyne detector was to discriminate between the two constituents of the mixture, and based on the conclusion, either keep or discard the post-measurement state. If one of the two states is perfectly discriminated, the state is perfectly Gaussiﬁed and the maximally obtainable squeezing for this distillation process is recovered. In the proof-of-principle experiment performed by Heersink et al. (2006), a simple balanced mixture of two displaced squeezed states was generated. Without any distillation processing, the smallest quadrature variance was Vsq = 1.4 ± 0.3 dB. After tapping off 7% of the beam and performing homodyne detection and post-selection, the state was distilled and the = −2.6 ± 0.3 dB. The success probability smallest variance was then Vsq −2 was approximately 10 . A similar protocol was implemented by Franzen et al. (2006) (see also Fiurášek et al., 2007; Hage et al., 2007; Marek et al., 2007) to distill a squeezed state from a phase-randomized mixture of squeezed states

1 W(x1 , p1 ) = √ 2π VX VP

∞

P(φ) exp −

−∞

xφ2 2VX

−

p2φ 2VP

dφ,

(43)

Applications of Feed-Forward

403

where xφ = x cos φ + p sin φ, pφ = p cos φ − x sin φ, and P(φ) denotes the probability distribution of the random phase ﬂuctuations. However, as opposed to the single copy distillation protocol, this protocol relies on the interference of at least two copies of a mixed state. The idea is to interfere two copies of the phase-randomized squeezed state and subsequently project the state onto the inﬁnitely squeezed vacuum state through a quadrature measurement and post-selection on null detection. As above, to obtain an appreciable success rate, the state is transformed based on postselection of the measurement outcomes lying in a non-vanishing interval around zero. Note that this protocol is similar to the Gaussiﬁcation protocol proposed by Browne et al. (2003). The efﬁciency of the protocol depends on the number of copies of the squeezed state and it has been shown that the protocol approaches ideal distillation for an inﬁnite number of copies.

4.2.3. Filtration of Quantum Information We saw in the preceding section that a probabilistic feed-forward scheme can be used to distill squeezed and entangled states, which in turn can be used to transport quantum information via teleportation. The feed-forward-based distillation protocol can also be used directly for the quantum states that carry information. This was shown experimentally by Wittmann et al. (2008) and the protocol was named quantum ﬁltration. In the experiment, coherent states were sent through an erasure channel in which the state was either fully transmitted or completely lost, thus resulting in the mixed state, (1 − p)|αα| + p|00|, where p is the erasure probability. Using the distillation approach outlined earlier, it was shown that the vacuum could be ﬁltered out from the mixture. It was furthermore shown that perfect ﬁltration can be obtained by using an APD in replacement of the homodyne detector in the feed-forward system. Feed-forward can also be used to probabilistically purify Gaussian mixed states with an unknown degree of mixedness (Marek and Filip, 2007). In this scheme, two copies of the mixed state interfere on a symmetric beam splitter; one output is measured by heterodyne detection (or by APD), and the state is probabilistically transformed to yield a single state of higher purity than the original states.

4.3. Heterodyning and Displacement In this section, we consider some protocols that also rely on the feedforward set-up shown in Figure 6 but with a heterodyne detector in replacement of the homodyne detector. We discuss the implementation of a phase-insensitive ampliﬁer, a cloning protocol, and a minimum disturbance measurement protocol.

404

Quantum Feed-Forward Control of Light

4.3.1. Phase-Insensitive Amplifier As mentioned in Section 4.1.3, a scheme for phase-insensitive ampliﬁcation can be accomplished by using a Mach–Zehnder interferometer, two off-lineprepared squeezed vacua, and two feed-forward schemes (Filip, Marek, and Andersen, 2005). This scheme was used to enable QND operation. If however the goal of the ampliﬁer is to phase-insensitively amplify an input state at the quantum limit, then a much simpler feed-forward set-up, which is not relying on any non-linearities, can be used. Such a non-linear-free phase-insensitive ampliﬁer was proposed and experimentally accessed by Josse et al. (2006). It was based on the simple feed-forward approach in Figure 6 where the detector was simultaneously measuring conjugate quadratures and the resulting two photocurrents were used to displace the corresponding two quadratures using two modulators. The transformation of the quadratures for this amplifying operation are given by

√ √ GX1 + G − 1X2 √ √ P1 → GP1 − G − 1P2 (44)

provided that the electronic gain is set to g = 2(1 − T)/T. Vacuum modes responsible for the inevitable noise in the ampliﬁcation is represented by the modes X2 and P2 , and the optical gain, G, of the ampliﬁer is a simple function of the transmittivity of the beam splitter, T; G = 1/T. The transformations in (44) perfectly match those of a quantum-noise-limited ampliﬁer. The feed-forward ampliﬁer was demonstrated by characterizing the ampliﬁcation of a 14.3 MHz sideband of a bright laser beam. Phaseinsensitive ampliﬁcation is typically quantiﬁed by the noise ﬁgure, which is deﬁned as NF = SNRout /SNRin , where SNRin(out) is the signal-to-noise ratio of the input (output) ﬁeld. For ideal phase-insensitive ampliﬁcation, the noise ﬁgure is NF = G/(2G − 1). In the experiment, the noise ﬁgure was found for different gains, for example, a gain of G = 1.5 exhibiting a noise ﬁgure of 0.7 was demonstrated. This is very close to the theoretically ideal value, which is 0.75, and thus the ampliﬁer operates very close to the optimal ampliﬁer. X1 →

4.3.2. Quantum Cloning Phase-insensitive ampliﬁcation close to the ultimate quantum limit is known to be the enabling technology for quantum cloning. Actually, perfect cloning of an unknown quantum state is not possible as is formulated in the no-cloning theorem (Wootters and Zurek, 1982). However, by the use of a near perfect phase-insensitive ampliﬁer, approximative clones with high ﬁdelity to the original quantum state can be produced. The ideal ﬁdelity of cloning an unknown coherent state is 2/3 and by using the

Applications of Feed-Forward

405

feed-forward ampliﬁer discussed earlier, ﬁdelities as high as 64 and 65% have been experimentally accessed (Andersen, Josse, and Leuchs, 2005). This feed-forward-based cloning scheme has been further extended to enable the cloning of a pair of phase conjugate coherent states. Such a scheme was experimentally realized by Sabuncu, Andersen, and Leuchs (2007), and ﬁdelities of three clones were measured to be as high as 0.89, which was very close to the theoretical value of 0.9. There have been various further extensions to the cloning scheme based on feed-forward. An asymmetric multi-copy coherent state cloner has been devised (Fiurášek and Cerf, 2007), the cloning of entanglement has been discussed (Weedbrook et al., 2008), the cloning of Gaussian state with prior information about the second moments has been analyzed (Olivares, Paris, and Andersen, 2006), and the reversibility of cloning was addressed (Filip, Fiurášek, and Marek, 2004). Moreover, telecloning of coherent states have been addressed theoretically (van Loock and Braunstein, 2001; Zhang, Xie, and Peng, 2007; Zhang et al., 2008) and experimentally (Koike et al., 2006). Interestingly, the feed-forward-based ampliﬁer has also been discussed in terms of cloning of polarization qubits (Hofmann and Ide, 2006; Ide and Hofmann, 2007).

4.3.3. Minimum Disturbance Measurement The optical feed-forward circuit used for phase-insensitive ampliﬁcation and cloning has also been the engine for the implementation of a minimumdisturbance measurement for coherent states as shown by Andersen et al. (2006). A minimum-disturbance measurement is the task of making a guess about an unknown coherent state and at the same time altering the state as little as possible. Obviously, there is a trade-off stating the balance between information gain and state disturbance, which are quantiﬁed by the estimation ﬁdelity, H, and the transfer ﬁdelity, F, respectively. It was found that this trade-off can be saturated for coherent states by using the simple feed-forward scheme in which the information acquired in the heterodyne detector is used to estimate the state, and the remaining state after displacement serves as the post-measurement quantum state. The optimal trade-off between estimation and disturbance is given by the relation

F≤

H , √ 2 1 − H − (1 − H)(1 − 2H)

(45)

and different trade-offs can be easily accessed by changing the transmittivity of the beam splitter (and the electronic feed-forward gain correspondingly). The scheme was experimentally tested using the same set-up as discussed in Section 4.3.1. A whole range of different trade-offs were accomplished, and the experimental ﬁndings were close to the optimal

406

Quantum Feed-Forward Control of Light

trade-offs only limited by the non-unity quantum efﬁciency of the heterodyne detector. Furthermore, the minimum-disturbance measurement in terms of minimizing the added noises was addressed theoretically and experimentally by Sabuncu et al. (2007). It was realized that such a minimum-disturbance measurement provides the upper bound on the eavesdropper information in a quantum key distribution system based on coherent states and heterodyne detection; see Lodewyck and Grangier (2007).

4.4. More Complex Applications of Feed-Forward In the previous section, we introduced a number of quantum circuits that rely on the simple feed-forward approach. Some of these protocols, in particular the feed-forward squeezer and the phase-insensitive ampliﬁer, have solidiﬁed the foundation for more complex continuous variable quantum protocols such as quantum error correction coding, quantum secret sharing, and cluster state computation. However, before addressing these protocols, we discuss one of the most important feed-forward-based protocols, namely the quantum teleporter.

4.4.1. Quantum Teleportation The idea of quantum teleportation was ﬁrst put forward in the context of spin particles and later it was extended to also include the continuous degrees of freedom, ﬁrst by Vaidman (1994) and further elaborated upon by Braunstein and Kimble (1998), Hofmann et al. (2000) and Ralph and Lam (1998) (and many others). Teleportation is associated with the task of sending quantum information in classical channels (a feed-forward channel) by means of shared entanglement between the sender and the receiver. Teleportation goes as follows (see Figure 8). A two-mode entangled state is prepared and one half is sent to a person A and the other half is sent to another person B. The quantum state to be teleported from A to B is then measured jointly with the part of the entangled state, which is at A. This is done by interfering them at a symmetric beam splitter and measuring the amplitude quadrature of one output and the phase quadrature of the other output. Such a joint measurement is known as a continuous variable Bell state measurement, since it ideally projects the state onto one of the inﬁnite numbers of Bell states (maximally entangled states). The resulting outcomes are then fed forward onto B who uses this information to linearly displace the other half of the entangled state and thereby ﬁnalizes the protocol. Under the assumption that the initially distributed quantum state was maximally entangled, the manipulated entangled state will be turned into an exact copy of the input state. In practice, however, the degree of entanglement is always ﬁnite and the protocol has to be quantiﬁed. A standardly

Applications of Feed-Forward

407

Bell measurement

A Input

Processing

Output B

EPR Gate

FIGURE 8 Schematic of the teleportation scheme. Einstein–Podolsky–Rosen (EPR) is the bipartite entangled state, which is distributed among A and B. At A a Bell measurement is performed and the mode at B is conditionally displaced. A gate operation can be teleported onto the input state by placing the gate in the entangled state at B.

used parameter to quantify the performance of the teleporter is the ﬁdelity, F, which states the similarity between the input and the output state. To demonstrate the potential success of teleportation, one must assure that the ﬁdelity is larger than the ﬁdelity that is obtainable by a classical strategy. This classical approach is associated with the optimal measurement followed by an optimal recreation method, which yields a ﬁdelity of 50% for unknown coherent states at the input. In addition to using the ﬁdelity as a measure, it has been suggested by Ralph and Lam (1998) to use transfer coefﬁcients and conditional variances as measures when coherent states are teleported. Continuous variable quantum teleportation was demonstrated for the ﬁrst time by Furusawa et al. (1998). In this experiment, the entangled state was generated by interfering on a beam splitter two squeezed vacua produced in a sub-threshold optical parametric oscillator in counter propagating directions. The quantum information was encoded as coherent state sidebands, and the resulting ﬁdelity between the input sideband and the output sideband was measured to be F = 0.58 ± 0.02. Being well above 50%, the ﬁdelity shows that quantum teleportation was carried out. Reﬁned experimental methods have later enabled the demonstration of teleportation of coherent state with higher ﬁdelities [F = 0.61 by Zhang et al. (2003); F = 0.64 by Bowen et al. (2003); F = 0.7 by Takei et al. (2005); F = 0.83 by Yukawa, Benichi, and Furusawa (2008)], teleportation of single and two-mode squeezed states by Yonezawa, Braunstein, and Furusawa (2007) and Takei et al. (2005), and teleportation in a network by Yonezawa, Aoki, and Furusawa (2004) and in a linear sequence by Yonezawa, Furusawa, and van Loock (2007).

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The standard teleportation scheme can be envisaged as an operation that implements the identity gate, since what comes in also goes out. It is however possible to modify the entangled state to enforce a desired transformation of the input state (Bartlett and Munro, 2003). The advantage of such a teleportation-based operation is that the difﬁcult transformation is applied off-line to the entangled resources and therefore can be done probabilistically, whereas the transformation of the unknown quantum state is done through deterministic teleportation. The teleportation-based quantum operation is shown in Figure 8.

4.4.2. Quantum Error Correction and Secret Sharing One of the most important quantum protocols is quantum error correction (QEC) coding, since this is probably the technology that eventually will enable fault tolerant quantum computing and communication in the same way as classical error correction coding does it in classical systems. The main idea of QEC is to encode the quantum information into a multimode entangled state, which protects the information from speciﬁc noise sources. After processing of the special encoded quantum information, the information is decoded using feed-forward: the error that might have occurred is diagnosed through a so-called syndrome measurement and the outcome is used to make a corrective transformation. The measurement that retrieves information about the error is not disrupting the quantum information due to a special multi-mode encoding. A circuit for implementing a continuous variable QEC code was ﬁrst suggested by Braunstein (1998a,b). He presented a scheme based solely on linear optical components, feed-forward, and squeezed vacuum states. More speciﬁcally, the proposed scheme relies on the interference of eight highly squeezed states and the signal in an array of beam splitters that produce an output state with the quantum information encoded in an eight-partite entangled state. After transmission in eight channels, the information is decoded through interference in another array of beam splitters followed by homodyne measurements and feed-forward correction. By following such an encoding and decoding strategy, any error on a single mode can in principle be perfectly corrected for inﬁnite squeezing in the resources. The homodyne-measurement-induced corrective transformation can also be used to correct displacement errors of continuous variable representation of qubits (Glancy and Knill, 2006; Gottesman, Kitaev, and Preskil, 2001). A code for protecting coherent states from complete erasure noise was suggested by Niset, Cerf, and Andersen (2008). The quantum error erasure protocol thus allows for the faithful transmission of coherent states through channels, which either erase the information or perfectly transmit it. (Note that this channel has been discussed already in Section 4.2.3.)

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409

The scheme encodes two coherent states into an bi-partite entangled state through linear interference at beam splitters. The transmitted state is corrected by reversing the beam splitter transformations, performing two homodyne measurements of conjugate quadratures, and ﬁnally executing a feed-forward corrective transformation. This transformation, however, depends on the location of the error. If, instead, the deterministic feedforward correction is substituted with a probabilistic heralding correction, it becomes independent of the error location. A similar idea exploiting feed-forward correction has also been introduced in the context of continuous variables quantum secret sharing (Lance et al., 2003). The idea of this protocol was to share a coherent state secret among three parties in a network with one of the members being untrustworthy. By neglecting the information from the unreliable member, the remaining two members were (in principle) still able to retrieve the full information that was distributed among all of them. The information retrieval was carried out by homodyne measurements and feed-forward. This protocol has been implemented experimentally (Lance et al., 2004, 2005).

4.4.3. Cluster-State Operations In a recent work by Menicucci et al. (2006), it was found that universal quantum computation can be implemented simply by using an off-lineprepared entangled Gaussian state (known as a cluster state) together with measurements and feed-forward. In particular, it was realized that once the Gaussian cluster state has been created, homodyne detection and linear feed-forward sufﬁce to implement any multi-mode Gaussian operation. Moreover, they showed that by introducing one single-mode non-Gaussian measurement (such as photon counting), universal computation can be realized. Likewise, it was shown by Filip, Marek, and Andersen (2005) (and discussed in previous sections) that a universal Gaussian operation can be implemented with off-line-prepared squeezed vacuum states, feedforwards, and linear displacements. The main difference of the two approaches appears in the function by which the coupling strengths of the Gaussian operation are controlled. In the approach by Filip et al., the coupling strength of the operation is controlled by the linear optical interactions (beam splitter ratios) as well as in the homodyne detectors and feed-forward gains. This is in contrast to the cluster state approach, where the coupling strength is solely controlled by the homodyne detector and feed-forward gain. As an example of a cluster state operation, van Loock (2006) considered the implementation of a squeezing operation based on a ﬁve-mode cluster state constructed from four elementary gate operations as shown

410

Quantum Feed-Forward Control of Light

Processing

Input

QND

QND

QND

QND

QND

Output

Infinitely squeezed ancillas

FIGURE 9 Schematic of a five-mode cluster state transformation. A cluster state is formed by the interaction of the input state with five infinitely squeezed states in five fixed QND gates. The transformation is subsequently entirely controlled in the measurement process.

in Figure 9. Each of these elementary operations correspond to the unitary QND operation introduced in Section 4.1.3, and by placing four of them in series, a ﬁve-mode linear cluster state is formed. By performing homodyne detection of four of the ﬁve cluster modes and conditionally displacing the last mode, a squeezing gate can be approached.

5. CONCLUSIONS Quantum electro-optics feed-forward have made a major impact on the ﬁeld of quantum information processing. Feed-forward has enabled the implementation of quantum teleportation, a universal squeezing operation, and a computational SUM gate, and it is foreseen that it will enable the development of QEC-coding protocols and one-way quantum computation. Needless to say, feed-forward control will be a very important part of future quantum informational systems. Many of these complex computational protocols rely on a very simple feed-forward arrangement based on a beam splitter, quadrature detectors, and displacements. Therefore, this system has been investigated in great detail in this review, both from a theoretical and an experimental perspective. With these theoretical and experimental tools at our disposal, we have been in a good position to review a number of important proposal and implementations that make use of the simple feed-forward scheme. In particular, we described the function of a feed-forward-based noise-less ampliﬁer, which constitutes the core of all Gaussian quantum information operations: any Gaussian operation can be constructed from a set-up composed of off-line-prepared squeezed vacuum states, a multiport interferometer (solely composed of beam splitters), homodyne detectors, and feed-forward displacements. The crucial point to note is that the fragile input quantum states only propagate through noise-less and easy beam splitter operations.

References

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We have been mainly addressing the use of feed-forward in quantum information operations, since this has been the main aim of this review. However, we have also addressed a few quantum state preparation protocols relying on quantum feed-forward whenever they appropriately have been describing a particular interesting property of the feed-forward system. For example, the idea of heralding based on post-selected measurement outcomes has been widely used for the preparation of quantum states, such as the preparation of highly squeezed states from an ensemble of less squeezed states as well as the preparation of highly non-Gaussian states. In recent years, there has been a substantial and growing interest in the investigation of quantum feedback and feed-forward system on both the experimental and the theoretical sides. The ﬁeld of quantum feedback and feed-forward has now turned into its own research discipline, and new methods and applications are continuously being developed and explored. In particular, the importance of quantum feedback/forward in quantum information science cannot be overestimated: futuristic quantum information processor will inevitably be based on a large number of quantum control systems. We therefore believe that further development of quantum feedback/forward is essential to facilitate the construction of future quantum technologies.

ACKNOWLEDGMENTS We acknowledge the support from the EU under project no. 212008 (COMPAS) and from the Danish Agency for Science, Technology and Innovation (no. 274-07-0509). ULA also acknowledges the Lundbeck Foundation and RF acknowledges Czech Ministry of Education (for grants MSM 6198959213 and LC 06007), GACR (202/08/0224), and the Alexander von Humbold Foundation.

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AUTHOR INDEX FOR VOLUME 53

A Abad, H., 212 Abbe, E., 1 Abdolvand, A., 212 Abedin, K.S., 236, 240 Ablowitz, M.J., 78, 79, 140 Abraham, M., 73 Abramochkin, E., 310 Abramowitz, M., 313 Adler, R., 240 Adler, R.J., 329 Agard, D.A., 2, 51, 54, 55 Agarwal, G.S., 113n Aggarwal, R.L., 228 Agranat, A.J., 156, 159, 170, 191, 193 Agrawal, G., 141 Aharonov, Y., 133, 135, 136 Akhmanov, S.A., 216 Akiyama, Y., 224 Alcock, A.J., 235, 239, 243 Alfano, R.R., 216, 315 Allen, L., 296, 310, 332, 335, 336 Allwood, R.L., 227 Alpiner, B., 240 Alu, A., 73, 143 Amimoto, S.T., 240 Ammann, E.O., 211 Amzallag, E., 211 Andersen, U.L., 367, 392, 394, 397, 404, 405, 408, 409 Anderson, D.Z., 159, 189, 190 Ando, H., 24 Andrés, P., 24 Andryunas, K., 212 Angelsky, O., 346 Aoki, T., 407 Appold, G., 227 Arizonis, P.V., 271 Arlt, J., 313, 314 Armandill, E., 235 Armen, M.A., 366 Arnaut, H.H., 338 Arnold, S., 339

Arsenault, H., 28n, 30 Aspect, A., 338 Audibert, M.M., 210 Averbakh, V.S., 222 Azzam, R.M.A., 344

B Babiker, M., 296 Bachor, H.A., 380 Bailey, B., 53 Baldeck, P.L., 216 Ball, P., 129 Balmain, K.G., 70 Barak, S., 226 Barakat, R., 24, 28 Baranova, N.B., 328, 329 Barbosa, G.A., 338 Barends, P., 2 Barker, D.L., 210 Barnes, W.L., 73 Barnett, S.M., 304 Bartlett, S.D., 370, 408 Bashara, N.M., 344 Basiev, T.T., 211 Basistiy, I.V., 314 Basov, N.G., 222 Battjes, J.A., 136 Bazenhov, V.Y., 309 Beijersbergen, M.W., 310, 335 Bekenstein, J.D., 141 Belic, M.R., 187 Benabid, F., 213, 223 Benichi, H., 407 Berberich, P., 240 Berggren, K.-F., 330 Bernage, P., 224 Berriel-Valdos, L.R., 28 Berry, A.J., 213 Berry, M.V., 70, 134, 296–298, 301–305, 307, 311, 315, 316, 318, 320–324, 327–330, 332, 341, 342, 344, 345, 347–350 Bespalov, V.I., 248

415

416

Author Index for Volume 53

Bewersdorf, J., 62 Bezuhanov, K., 315 Bialynicka-Birula, S., 322, 349, 350 Bialynicki-Birula, I., 322, 349, 350 Birrell, N.D., 141 Bishop, A.I., 336 Björk, G., 366, 396 Bjorklund, G.C., 271 Blaikie, R.J., 125, 128 Blanca, C.M., 2, 28, 62 Bliokh, K.Y., 348 Block, S.M., 73 Bloembergen, B., 216 Bloembergen, N., 243 Blom, P., 2 Bloom, D.M., 271 Blum, F.A., 227 Bocquet, H., 224 Bohm, D., 133, 135, 136 Boissel, P., 263 Boivin, A., 28n, 30, 300 Bokor, N., 343 Bol´shov, M.A., 244 Booth, M.J., 33 Borghi, R., 303 Born, M., 4, 7, 20, 70, 79, 113, 124–126, 128, 142, 301, 317, 319, 344, 350 Bortfeld, D.P., 209–211 Bortolotti, E., 71, 73 Bourne, O.L., 235, 243 Bouwmeester, D., 366 Bowen, W.P., 407 Boyd, R.W., 123, 203, 240, 246, 274, 276 Boyer, G., 43 Boyer, K., 235, 243 Boyer, T.H., 128, 129 Brabec, T., 141 Brakenhoff, G.J., 2 Brasseur, J.K., 210 Braunbek, W., 300, 301, 324 Braunstein, S.L., 367, 370, 394, 405–408 Brekhovskikh, G.L., 272, 274, 275 Bret, G., 244 Bretenaker, F., 339 Breuninger, T., 52 Brewer, R.G., 235, 240 Brewster, D., 341 Briane, M., 143 Bridges, T.J., 209, 213 Brillouin, L., 228 Bronk, H., 235 Brosseau, C., 344 Brout, R., 136, 141 Browne, D.E., 403 Bruch, L.W., 298

Bruckmeier, R., 396, 397 Brueck, S.R.J., 209, 227, 228 Bruesselbach, H., 208, 275 Bryant, C.H., 215 Buchler, B.C., 392, 396 Buckland, J.R., 305 Burleﬁnger, E., 235, 239 Burnham, R., 224 Burzynski, R., 203, 252, 253, 260 Bushev, P., 366 Byer, R.L., 222, 223

C Caballero, M.T., 62 Cai, W.S., 119 Caloz, C., 70 Calvo, G.F., 191 Campillo, A.J., 213 Canalias, C., 211 Capasso, F., 129 Carlsten, J.L., 210, 223, 224 Carman, R.L., Jr., 215 Carr, I.D., 276 Carraz, O., 213 Cartwright, D.E., 298, 298n Carvalho, M.I., 155, 160, 179, 180 Casimir, H.B.G., 128 Castillo, M.D.I., 155 Cerf, N.J., 405, 408 Cerny, P., 211 Chabay, I., 213 Chan, C.T., 143 Chan, H.B., 129 Chandrasekhar, S., 341 Chang, R.K., 211, 213 Chauvet, M., 191 Chen, F.Z., 226 Chen, H.Y., 143 Chen, Y.F., 211, 346 Chen, Z.G., 191 Chiao, R.Y., 202, 216, 217, 228, 236 Chilukuri, S., 224 Cho, A., 137 Cho, C.W., 248 Choudhury, A., 2, 342 Chraplyvy, A.R., 213 Christodoulides, D.N., 155, 160, 179, 180 Ciattoni, A., 158–160, 182–187, 191, 193 Clark, G.L., 210, 222 Clayton, C.M., 275 Cogswell, C.J., 31 Colles, M.J., 209 Cook, G., 238 Cook, R.J., 117, 133

Author Index for Volume 53

Cook, R.L., 366 Corle, T.R., 34 Coskun, T.H., 179 Cotter, D., 224, 226, 236 Coullet, P., 308 Coulson, K.L., 341 Couny, F., 213 Courtial, J., 310, 335, 337, 339 Cragg, G.E., 48 Cremer, C., 48, 51 Crosignani, B., 155, 156, 160, 161, 188, 193 Cui, Y., 261 Cummer, S.A., 143 Curtis, J.E., 311, 312 Czarnetzki, U., 226

D Dainese, P., 236 Dalibard, J., 338 Damen, T.C., 228 Damour, T., 140 Damzen, M.J., 235, 239, 240, 243 Dane, C.B., 238 Darée, K., 248 Dari-Salisburgo, C., 171 Davidson, N., 343 Davies, P.C.W., 141 Debye, P., 20 Decker, C.D., 209, 211 de Fermat, P., 70, 74 Degasperis, A., 154 de Jong, M.P.C., 136 de Juana, D.M., 24 DelRe, E., 156, 158–160, 169–171, 174, 175, 179, 181–184, 191–194 DeMartini, F., 210 De Martino, A., 209, 210 DeMichelis, C., 239 Demidovich, A.A., 212 Denariez, M., 244 Denisenko, V.G., 316, 346 Denk, W., 3, 62 Dennis, M.R., 297, 304, 307, 316–325, 328–331, 341, 342, 344–348, 350 Dennis, R.B., 227 De Nunzio, G., 222 Denz, C., 156, 159 de Oliveira, C.A.S., 236 D’Ercole, A., 158, 174, 179, 181, 190 Dereux, A., 73 Dernis, R.B., 227 DeSilets, C.S., 228 Desyatnikov, A.S., 297 Deykoon, A.M., 303

Dheer, M.K., 209 Dholakia, K., 73, 313, 314, 336 Diaspro, A., 34 Díaz, A., 24 Dietz, D.R., 236 Ding, S., 211 Di Porto, P., 156 Dirac, P.A.M., 298 Dittrich, P.S., 62 Djeu, N., 224 Doebele, H.F., 226 Dolgolenko, D.A., 263 Dolgov, A.D., 341 Dong, R., 402 Dorkenoo, K.D., 246 Dorn, R., 24, 342 Dow, J., 300 Dransfeld, K., 240 Drühl, K., 396, 397 Duardo, J.A., 210, 222 Dubik, B., 318, 324 Dubinskii, M.A., 236 Ducuing, J., 210 Dudovich, N., 64 Duignan, M.T., 235, 275 Dunn, P.C., 224 Duree, G.C., 155, 156, 159, 160 Durnin, J., 313, 317 Dutta, B., 370 Dutton, Z., 298 Dzyaloshinskii, I.E., 129

E Ebbesen, T.W., 73 Eberly, J.H., 313, 317 Eckhardt, G., 202, 209–211 Eco, U., 298 Egorov, R.I., 316, 346 Egorov, Y.A., 313 Eichler, H.J., 211, 212, 236, 237 Elbert, D., 341 Eleftheriades, G.V., 70, 125, 128 Emmett, J.L., 235, 238 Eng, R.S., 228 Engelhardt, J., 60 Engheta, N., 70, 73, 143 Eremenko, A.S., 210 Escobar, I., 33 Eversole, J.D., 213

F Fabelinskii, I.L., 249 Fadeeva, T.A., 347

417

418

Author Index for Volume 53

Fadeyeva, T.A., 319 Fang, N., 125, 128, 143 Faris, G.W., 236, 240 Fearn, H., 117, 133 Fedosejevs, R., 235, 243 Feldman, B.J., 235, 275 Ferrier, J.L., 274 Fetterman, H.R., 228 Feynman, R.P., 75 Ficz, G., 3 Filip, R., 367, 392, 394, 397, 398, 403–405, 409 Filippo, A.A., 238 Findeisen, J., 212 Findlay, J.W., 298, 305 Firby, P.A., 317, 320 Fischer, I., 210 Fiurášek, J., 402, 405 Fizeau, H., 117, 133 Flossmann, F., 346, 347 Fokas, A.S., 78, 79, 140 Foltin, G., 330 Frank, F.C., 341 Franke-Arnold, S., 338, 340 Franzen, A., 402 French, W.G., 212 Fresnel, A.J., 117, 133 Fressengeas, N., 171, 174 Freund, I., 317, 321, 328, 330, 344, 346, 348 Frey, R., 209, 210 Friberg, A.T., 58 Frieden, B.R., 16 Friese, M.E.J., 336 Froehly, C., 245 Frohn, J.T., 48, 51 Furhapter, S., 316 Furusawa, A., 367, 407

G Gad, G.M.A., 211 Gahagan, K.T., 336 Gal, J., 341 Galatola, P., 386 Garcia, N., 125 Gardiner, C.F., 317, 320 Garetz, B.A., 339 Garmire, E., 216, 217, 235, 240 Garrett, W.R., 226 Gatz, S., 160, 190, 193 Gauthier, D.J., 123, 240 Gazengel, J., 215, 216 Gbur, G., 73, 305, 315 Geller, M., 209–211

Genack, A.Z., 330 Gerchberg, R.W., 311 Geremia, J.M., 366 Ghaziaskar, H.S., 209 Ghiglia, D.C., 305 Gibbs, J.W., 319, 344 Gil, I., 308 Giordmaine, J.A., 211, 216, 221 Glancy, S., 408 Glöckl, O., 386, 397, 398 Godfried, H.P., 235 Goebel, C.J., 298 Goldblatt, N.R., 240 Goldstein, H., 89 Golombok, M., 215 Gómez-Sarabia, C.M., 5 Goobar, E., 396 Goodman, J.W., 5, 328 Goos, F., 299 Göppert-Mayer, M., 3, 62 Gordon, J.P., 154 Gordon, W., 73, 116, 131 Gottesman, D., 408 Grangier, P., 396 Grangier, Ph., 406 Grassl, H.P., 211 Gray, M.A., 202, 246 Grbic, A., 125, 128 Green, M.B., 141 Greenleaf, A., 73, 117, 122, 143 Greger, K., 52 Greiner-Mothes, M.A., 242 Grier, D.G., 311, 312, 336 Grimm, M.A., 45n Grunnet-Jepsen, A., 155, 161 Gu, M., 17, 22, 33, 73 Günter, P., 155 Guo, A., 191 Guo, H., 224 Gupta, S.D., 113n Gustafsson, M.G.L., 2, 48, 51, 54, 55

H Haeberlé, O., 33 Häfele, H.G., 227 Hage, B., 402 Hagenlocker, E.E., 208, 222, 235 Hajnal, J.V., 297, 298, 317, 320–322, 346–349 Hakuta, K., 213 Halperin, B.I., 329, 330 Halpern, P., 70 Han, X.F., 226 Hänchen, H., 299

Author Index for Volume 53

Hanna, D.C., 213, 224, 276 Hannay, J.H., 117, 133, 298, 317, 320–322, 342, 344, 345, 348 Hanson, F., 222 Happer, W., 224 Harris, A.L., 224 Harrison, R.G., 227, 236 Hart, J.B., 94, 104 Hasegawa, A., 154 Hasman, E., 297 Hawking, S.W., 140, 141 He, G.S., 203, 206, 216, 217, 249–253, 258, 260, 261, 263, 268, 271–274, 276 He, H., 239 Heckenberg, N.R., 309, 335 Heersink, J., 402 Hegedüs, R., 341 Hegedus, Z.S., 26 Heiman, D., 228 Heinicke, W., 235, 240 Heintzmann, R., 3, 48, 51 Heitler, W., 204 Hell, S.W., 2, 3, 28, 34, 57, 60, 62, 64 Hellwarth, R.W., 206, 271 Henesian, M.A., 222 Herman, R.M., 202, 246, 313 Hernandez-Aranda, R., 315 Herrmann, J., 160, 190, 193 Hesselink, L., 316 Heuer, A., 236 Heupel, T., 239 Hickman, A.P., 236, 240 Hikspoors, H.M.J., 226 Hirschfelder, J.O., 298 Hitomi, T., 313 Ho, P.P., 216 Hofmann, H.F., 405, 406 Hogervorst, W., 243 Holz, D.T., 211 Hon, D.T., 243 Horvath, G., 341 Houdart, R., 224 Hovis, F.E., 235 Howls, C.J., 302, 303 Hradil, Z., 400 Hsu, H., 240 Hu, D.-W., 201 Huang, K.F., 346 Huignard, J.P., 155 Hulliger, J., 211 Hunklinger, S., 240 Huntington, E.H., 392 Huntley, J.M., 305 Huo, Y., 224 Hutchinson, M.H.R., 235, 240, 243

419

Huterer, D., 341 Huysken, J., 34

I Ibáñez-López, C., 43 Ide, T., 405 Ikeda, M., 236 Ippen, E.P., 212, 236 Iqbal, Z., 224 Irrera, F., 212 Irslinger, C., 227 Ishaaya, A.A., 309 Itoh, T., 70 Itzkan, I., 224 Ivlena, L., 212

J Jackson, J.D., 99, 107, 108, 113, 113n, 117, 119, 126, 296, 317 Jacobson, T., 136 Jacquinot, P., 28 Jaywant, S.M., 224 Jeffrey, M.R., 291, 347 Jeong, H., 399 Jesacher, A., 311, 314 Jha, A.K., 340 Joffrin, C., 210 Johnson, A.M., 215 Johnson, F.M., 210, 222 Johnson, R., 276 Jordan, C., 221 Josse, V., 367, 404, 405 Jovin, T.M., 51 Jusinski, L.E., 236, 240 Juskaitis, R., 4, 52

K Kaiser, G., 350 Kaiser, W., 203, 215, 216, 221, 235, 236, 248 Kalogerakis, K.S., 240 Kaminskii, A.A., 211, 212 Karlson, A., 396 Karman, G.P., 300 Karpov, V.B., 263 Karpukhin, S.N., 210 Kartazayev, V., 315 Kärtner, F.X., 141 Kaser, W., 209 Kavage, W., 240 Kelley, J.D., 235

420

Author Index for Volume 53

Kerl, R.J., 227 Kessler, D., 321 Khokhlov, R.V., 216 Kildal, H., 209 Kildishev, A.V., 143 Kim, M.K., 52 Kimble, H.J., 367, 380, 406 Kino, G.S., 34 Kiselev, A.D., 348 Kiss, T., 304 Kitaev, A., 408 Kivshar, Y.S., 297 Kleckner, D., 366 Kmetik, V., 238, 241 Knapp, H.F., 48, 51 Knill, E., 408 Knoner, G., 310, 336 Koike, S., 405 König, F., 141 Konov, V.I., 211 Konukhov, A.I., 344 Koprinkov, I.G., 210 Korn, G., 210 Korobkin, V.V., 263 Korpel, A., 240 Kos, K., 157, 169 Koss, B.A., 311, 312 Kovacs, M.A., 210 Kovalev, V.I., 235 Kralikova, B., 275 Kraus, B., 401 Krausz, F., 141 Krishnamurti, V., 53 Krishnan, K.S., 204 Kroekel, D., 226 Krolikowski, W., 156, 160, 191 Krowne, C.M., 70 Kruse, P.W., 228 Krylov, V., 210 Krzewina, L.G., 52 Kubarev, A.M., 248 Kucken, M., 342 Kudryavtseva, A.D., 272, 274, 275 Kugel, G., 171 Kugler, N., 276 Kuhl, U., 329 Kuleshov, N.V., 212 Kung, R.T.V., 224 Kurbasov, S.V., 222 Kurtz, S.K., 211 Kushawaha, V., 210 Kuwahara, K., 243 Kyte, R.G., 305 Kyzylasov, Yu.P., 249

L Ladvac, K., 336 Lagatsky, A.A., 212 LaHaye, M.D., 366 Lai, E.P.C., 209 Lallemand, P., 243 Lam, P.K., 367, 390, 394, 396, 406, 407 Lamoreaux, S.K., 128, 129 Lan, S., 174, 175 Lance, A.M., 399, 400, 409 Landau, L.D., 71, 101, 113, 113n, 114, 117, 128, 130, 138 Landgrave, J.E.A., 28 Lang, M.C., 60 Lang, P.T., 210 Lankard, J.R., 224 Lanni, F., 53 Lassas, M., 73, 117 Laukien, G., 301 Laurell, F., 211 Lavin, Y., 316 Lax, M., 318 Leach, J., 311, 314, 315, 323, 336, 339 Lee, J., 315 Lee, R.L., Jr., 341, 342, 347 Lefebvre, M., 276 LeFebvre, M.J., 276 Leﬂoch, A., 339 Leiderer, P., 240 Leighton, R.B., 75 Leonhardt, U., 70, 73, 76, 78, 79, 112, 114, 116–118, 121n, 123, 124, 126, 128–131, 133, 135, 140–143, 304, 369 Leuchs, G., 342, 367, 405 Levy, Y., 316 Li, Y., 28 Lifshitz, E.M., 71, 101, 113, 113n, 114, 117, 128–130, 138 Lin, C., 212 Lin, H.-B., 213 Lin, T.-C., 203, 263 Linden, S., 71 Lipson, S.G., 343 Litovitz, T.A., 240 Liu, D., 217, 272, 273, 276 Liu, J., 212 Liu, S.-H., 217, 272 Lodewyck, J., 406 Lohmann, A.W., 45n, 48, 53 Longuet-Higgins, M.S., 329 Loree, T.R., 210 Losev, L.L., 222 Lou, Q., 210, 224

Author Index for Volume 53

Louisell, W.H., 318, 369, 370 Lu, T.H., 346 Lu, W., 263 Lu, X., 263 Lu, Y.Q., 224 Ludewigt, K., 226 Luk, T.S., 226 Lukosz, W., 5, 44 Luneburg, R.K., 126 Luther-Davies, B., 156 Lysiak, R.J., 274

M Mack, M.E., 210, 222 Madhavan, D., 209 Madigosky, W.M., 240 Mahajan, V.N., 32 Maier, M., 203, 209, 211, 215, 216, 221, 235 Maillotte, H., 245 Mair, A., 338 Makarov, A., 222 Maker, P.D., 216, 217 Malcuit, M.S., 274, 276 Mandel, L., 128, 141 Manners, J., 224 Marcus, G., 243 Marcuse, D., 318 Marechand, M., 44 Marek, P., 367, 392, 394, 398, 402–404, 409 Mariyenko, I.G., 307 Marqués, R., 70, 71, 125, 143 Marshall, L.R., 224 Martin, F., 70, 71, 125, 143 Martin, P.J., 366 Martínez-Corral, M., 27, 28, 62 Masajada, J., 318, 324 Mash, D.I., 202, 236, 243 Masters, B.R., 34 Mattiuzzo, L., 212 Maufoy, J., 171 Maxwell, J.C., 126 May, P.G., 213 Mays, R., Jr., 274 McCutchen, C.W., 2, 19, 23, 25n, 26 McElhenny, J.E., 240 McGloin, D., 311, 313 McIntyre, I.A., 235, 243 McKen, D.C.D., 235, 243 McKnight, W.B., 318 Meixner, H., 240 Mellish, R.G., 227 Melnikov, L.A., 344 Melville, D.O.S., 125, 128

421

Mendlovic, D., 48 Menicucci, N.C., 368, 409 Menon, R., 310 Menzel, R., 236, 239 Merkle, L.D., 236 Mermin, N.D., 341 Mertz, J., 367 Miao, J., 2 Miceli, J.J., 313, 317 Michael, A., 210 Midorikawa, K., 223 Milburn, G.J., 366 Miller, D.A.B., 143 Miller, E.J., 246, 274, 276 Miller, R.E., 94, 104 Mills, J.P., 24 Mills, R.L., 94, 104 Milonni, P.W., 117, 123, 125, 128, 133 Milton, G.W., 70, 71, 73, 143 Minck, R.W., 210, 222, 235 Minkel, J.R., 125 Minkowski, H., 73 Minsky, M., 2 Misner, C.W., 70, 71, 106, 112, 113, 145, 147 Mišta, L., 398 Mitchell, M., 160, 173 Moh, K.J., 307 Mokhun, A.I., 344 Molina-Terriza, G., 318, 322 Mollenauer, L.F., 154 Moncrief, V., 136 Monkewicz, A.A., 240 Monneret, J., 245 Mooradian, A., 227, 228 Moore, M.A., 226 Mori, K., 226 Morin, M., 160 Moriwaki, H., 221 Mukunda, N., 370 Mullen, R.A., 242 Mullett, W.M., 209 Munch, J., 276 Munro, W.J., 408 Murray, J.R., 222 Murray, J.T., 211

N Nacher, J.C., 143 Nagorni, M., 60 Nassisi, V., 222 Nazarkin, A., 210 Needham, T., 78, 79 Nehari, Z., 77, 79, 127n, 128

422

Author Index for Volume 53

Neil, M.A.A., 43, 52 Nelson, D.R., 341 Nemarich, J., 228 Neshev, D., 243 Neuman, K.C., 73 Newell, A.C., 342 Ng, W.K., 202 Niay, P., 224 Nicholls, K.W., 307, 324 Nicorovici, N.-A.P., 73, 143 Nieto-Vesperinas, M., 125 Niset, J., 408 Niv, A., 343 Nizienko, Yu., 242 Norton, P., 228 Nosach, O.Yu., 275 Novello, M., 141 Nye, J.F., 136, 296–298, 300, 302–305, 307, 316, 317, 320–322, 324, 327, 332, 343–346, 348, 349

O Ochiai, T., 143 Oemrawsingh, S.S.R., 311, 338 Offerhaus, H.L., 235 Ohde, H., 224 O’Holleran, K., 325, 331, 332 Ojeda-Castañeda, J., 5, 24 O’Key, M.A., 238, 242 Okida, M., 309 Olivares, S., 405 Olver, F.W.J., 296 Omatsu, T., 309 O’Neil, A.T., 336 Oron, D., 64 Osborne, M.R., 238, 242 Ostermeyer, M., 239 Ou, Z.Y., 380 Ourjoumtsev, A., 400, 401

P Padgett, M.J., 296, 310, 311, 315, 325, 331, 335, 336 Paganin, D.M., 324 Palacios, D.M., 305, 315 Palange, E., 158, 160, 171, 174, 175, 179, 181–184 Panzera, C., 222 Paris, M.G.A., 405 Park, H., 235 Pascher, H., 227 Pasiskevicius, V., 211

Pasmanik, G., 243 Pasmanik, G.A., 248 Patel, C.K.N., 226–228 Paterson, C., 313 Pawley, J.B., 2 Payne, M.G., 226 Peacock, J.A., 70 Pearcey, T., 292, 296 Pearl, S., 243 Pecoraro, A., 222 Peierls, R., 73 Pendry, J.B., 70, 73, 117, 121–126, 128, 142, 143 Peng, K., 405 Penrose, R., 342, 349 Perrone, M.R., 222, 238 Peshkin, M., 133 Peuser, P., 212 Pfeifer, S., 276 Philbin, T.G., 70, 112, 114, 117, 118, 121n, 123, 124, 128–130, 137, 141 Phu Xuan, N., 215 Pierangelo, A., 195, 196 Piestun, R., 28 Pine, A.S., 235, 238 Pini, R., 212 Pinnick, R.G., 214 Piper, J.A., 224 Pismennaya, K., 160 Pitaevskii, L.P., 129 Piwnicki, P., 113, 117, 130, 131, 133, 135 Plebanski, J., 73, 116, 148 Pohl, D., 236 Pointer, D.J., 213 Poirier, P., 222 Poizat, J.P., 396 Pons, A., 62 Pors, J.B., 338 Porteous, I.R., 342 Poston, T., 302, 304 Pozza, D., 212 Pradere, F., 209, 210 Prasad, P.N., 203, 250–253, 258, 260, 261, 263, 268, 274 Pratt, D.J., 213 Preskill, J., 408 Pritt, M.D., 299 Proch, D., 235 Prokhorov, A.M., 211 Puell, H., 235

Q Qian, S.-X., 213, 214 Qiu, M., 263

Author Index for Volume 53

Quabis, S., 342 Quan, P.M., 131

R Rado, W.G., 208, 210, 222, 235 Ralchenko, V.G., 211 Ralph, T.C., 367, 380, 392, 396, 400, 406, 407 Ralston, J.M., 211 Raman, C.V., 204 Rank, D.H., 202, 235, 243, 247 Rao, D.R., 209 Rao, D.V.G.L.N., 235 Read, W.T., 306 Reck, M., 372 Reece, P., 73 ˇ Reháˇ cek, J., 340 Reif, J., 226 Reilly, J.P., 210 Repasky, K.S., 210 Rhodes, C.K., 226, 235, 243 Ridley, K.D., 238 Rieckhoff, K.E., 235 Riess, J., 298 Rindler, W., 349 Rivoire, G., 215, 246 Robinowitz, P., 223 Rocca, F., 308 Roch, J.F., 396 Roger, G., 338 Roizen-Dossier, B., 28 Rokni, M., 224, 226 Rong, H., 211 Röntgen, W.C., 117 Rosu, H.C., 300 Rother, W., 235 Rothschild, M., 212 Rousseaux, G., 140 Roux, F.S., 319 Roux, P., 134 Rovelli, C., 141 Roy, D.N.G., 235 Rozas, D., 312, 315 Ruan, Z., 122 Ruben, G., 324 Rufﬁni, R., 140 Ruostekoski, J., 298 Russell, P., 72, 141 Rytov, S.M., 71, 73

S Sabuncu, M., 405, 406 Sacchi, C.A., 245 Sacks, Z.S., 312

423

Sadreev, A.F., 330 Saichev, A.I., 330 Saito, T.T., 235 Salamo, G., 191 Sales, T.R.M., 2 Sali, E., 213 Samokhina, M.A., 213 Sands, M., 75 Sapondzhyan, S.O., 224 Sarkisyan, D.G., 224 Sarychev, A.K., 70, 71, 125, 143 Sasaki, Y., 212 Sattler, J.P., 228 Saxton, W.O., 311 Schaefer, J.C., 213 Schawlow, A.L., 235, 238 Schechner, Y.Y., 318 Schelonka, L.P., 275, 276 Schiemann, S., 243 Schleich, W., 71 Schmit, J., 305 Schmitt, S., 397 Schneider, J., 396 Schouten, H.F., 305 Schrader, M., 60 Schroeder, W.A., 235, 240 Schultz, S., 71, 125 Schultz, T., 210 Schurig, D., 70, 71, 73, 117, 119, 121–123, 142, 143 Schutz, B.F., 80n, 88 Schützhold, R., 141 Schwarz, J.H., 141 Schwertner, M., 33 Schwille, P., 62 Scott, J.F., 228 Scully, M.O., 71, 396, 397 Sedat, J.W., 2, 51, 54, 55 Segev, M., 155, 156, 159, 160, 167, 169, 170, 173, 174, 176 Seidel, S., 276 Sekreta, E., 210 Sentrayan, K., 210 Serdyukov, A., 117 Shaham, Y.J., 215 Shahidi, M., 226 Shalaev, V.M., 70, 71, 125, 143 Shamir, J., 28, 318 Shapiro, S.L., 216 Sharma, A., 224 Shaw, E.D., 226–228 Shelby, R.A., 71, 125 Shelby, R.M., 398 Shen, Y.R., 203, 215, 216

424

Author Index for Volume 53

Sheppard, C.J.R., 2, 13, 17, 22, 23, 26, 31, 33, 43, 342 Sherif, S.S., 24 Sherson, J.F., 366 Sheu, F.W., 191 Shih, M.F., 156, 159, 160, 167, 169, 174, 176, 190, 191 Shilov, A.A., 235, 243 Shimazu, M., 224 Shirafuji, J., 228 Shoham, A., 343 Shvartsman, N., 317 Sibbett, W., 213 Sihombing, R.S.D., 213 Sihvola, A.H., 117 Silberberg, Y., 64 Simon, R., 370 Simpson, N.B., 336 Singh, S.R., 155, 179, 180 Slatkine, M., 235, 275 Sliwa, C., 322 Smith, D.R., 70, 71, 73, 117, 121–123, 125, 142, 143 Smith, H., 310 Smith, R., 313 Smith, S.D., 227 Snow, J.B., 213 So, P.T.C., 48 Sobel’man, I.I., 249 Sokolovskaya, A.I., 272, 274, 275 Solymar, L., 155, 161 Sommerfeld, A.J.W., 298, 301, 301n Sonehara, T., 236 Sooy, W.R., 209 Sorokin, P.P., 224 Sorolla, M., 70, 71, 125, 143 Soskin, M.S., 297, 299, 303, 307, 309, 316, 344, 346 Soukoulis, C.M., 71 Spektor, B., 28 Stachel, J., 134 Staliunas, K., 297 Starunov, V.S., 243, 249 Stegeman, G.I., 155 Stegun, I.A., 313 Stelzer, E.H.K., 3, 34, 52, 57 Stemmer, A., 48, 51 Stepanov, A.I., 210 Stewart, I., 302, 304 Stöckmann, H.-J., 329 Stoicheff, B.P., 202, 217, 228, 236 Stolen, R.H., 154, 212, 215, 236 Stoll, R.R., 89, 98 Stratton, J.A., 299 Streibl, N., 18

Strickler, J.H., 3, 62 Strogatz, S.H., 321, 323 Strohaber, J., 307 Su, H., 239 Suastika, I.K., 136 Suda, A., 223 Sudarshan, E.C.G., 370 Sukhorukov, A.P., 216 Sulc, J., 212 Sulem, C., 154 Sulem, P., 154 Suzuki, S., 402 Svelto, O., 245 Swartzlander, G.A., 303, 305, 312, 315, 336 Swift, C.D., 222 Sze, R.C., 210 Sztul, H., 315

T Takei, N., 407 Takubo, Y., 224 Talanov, V.I., 222 Talbot, H.F., 331 Tamburrini, M., 156, 191 Tamm, C., 308, 310 Tamm, I.Y., 73 Tang, C.L., 232, 235 Tannenwald, P.E., 211 Tappert, F., 154 Tashiro, H., 223 Taylor, D.L.A., 53 Tcherniega, N., 209 Terhune, R.W., 210, 216, 217 Thompson, B.J., 24 Thorne, K.S., 70, 71, 106, 112, 113, 145, 147 Threllfall, D.C., 305 Tilley, D.R., 298 Tilley, J., 298 Tomov, I.V., 235, 243, 274 Tonomura, A., 133 Toraldo di Francia, G., 24 Torner, L., 297, 318 Török, P., 24, 31, 33 Torruellas, W., 155 Tosi-Belefﬁ, G.M., 171 Townes, C.H., 202, 216, 228, 235, 236, 240 Trillo, S., 155 Tromberg, B.J., 64 Trutna, W.R., 223 Tsai, H., 310 Tsuchiya, M., 224 Turnbull, G.A., 310 Turner, S.R.E., 305 Tynes, A.R., 212

Author Index for Volume 53

U Ubachs, W., 243 Uesugi, N., 236 Uetake, S., 213, 214 Uhlmann, G., 73, 117 Uiterwaal, G.C.G.J., 307 Unruh, W.G., 136, 141 Upstill, C., 297, 302

V Vachaspati, T., 341 Vaidman, L., 367, 406 Valanju, A.P., 125 Valanju, P.M., 125 Valley, G.C., 156, 159, 167, 169, 176 Van Bladel, J., 70 Vander, R., 343 van Loock, P., 370, 405, 407, 409 Varga, P., 33 Vasil’ev, V., 346 Vasnetsov, M., 297 Vasnetsov, M.V., 297, 299, 307, 309 Vaughan, J.M., 308 Vaziri, A., 338 Verkhovskii, V.S., 224 Verma, R.D., 224 Verne, J., 298 Veselago, V.G., 70, 124–126 Visser, M., 136, 141 Visser, T.D., 305, 315 Vivanco, F., 134 Vodchits, A.I., 210 Volostnikov, V., 310 Volovik, G.E., 141 Volyar, A.V., 319, 347 Volynkin, V.M., 241 von der Linde, D., 209 von Wonderen, A.J., 246 Vrehen, Q.H.F., 226

W Waggoner, A.S., 53 Wagner, F.E., 72 Wald, R.M., 116 Walda, G., 235 Walder, J., 235, 240 Walford, M.E.R., 305 Walser, R.M., 125 Walther, H., 226 Wang, C.C., 215 Wang, D., 246

425

Wang, H.Z., 263 Wang, Q.Z., 226 Wang, W., 58, 305, 307, 319, 328 Watanabe, T., 310 Weaver, R., 329 Webb, D.J., 155, 161 Webb, W.W., 3, 62 Weber, B.A., 228 Wecht, K.W., 216 Weedbrook, Ch., 405 Wegener, M., 71 Weihs, G., 338 Weinberg, D.L., 211 Weingarten, R.A., 224 Weinrib, A., 329 Weiss, C.O., 308, 310 Welling, H., 226 Wendl, G., 216 Wenzel, R.G., 223 Wheeler, J.A., 70, 71, 106, 112, 113, 145, 147 Whewell, W., 298 Whitney, W.T., 235, 275 Wichmann, J., 64 Wick, R.V., 243 Wiggins, T.A., 236, 239, 243, 248, 313 Wilkerson, C.W., Jr., 210 Wilkinson, M., 330 Willetts, D.V., 308 Willis, J.R., 143 Wilson, T., 10, 13, 33, 52 Winfree, A.T., 296, 298, 316, 321, 323 Winterling, G., 235, 240 Wirth, F.H., 242 Wiseman, H.M., 366 Witte, K.J., 242 Witteman, W.J., 235 Witten, E., 141 Wittmann, C., 403 Wittmann, M., 210 Wojak, U., 226 Wolf, E., 4, 7, 20, 28, 58, 70, 79, 113, 124–126, 128, 141, 142, 300, 301, 305, 315, 317, 319, 344, 350 Wolfersberger, D., 171 Wolter, H., 298, 299, 300n Wong, G.K.N., 239 Woodbury, E.J., 202 Wootters, W.K., 404 Wright, A., 405 Wright, E.M., 318 Wright, F.J., 303, 307 Wu, C.Y.R., 226 Wuerker, R., 276 Wyatt, R., 224 Wynne, J.J., 224

426

Author Index for Volume 53

X

Z

Xia, X.W., 224 Xie, Ch., 405 Xu, G.C., 263

Zaitsev, G.I., 243, 248 Zalevsky, Z., 48 Zapka, W., 224 Zaraga, F., 245 Zeiger, H.J., 217 Zeilinger, A., 338 Zel’dovich, B.I., 249 Zel’dovich, B.Ya., 271, 274 Zhan, Q.W., 342 Zhang, J., 405 Zhang, S., 330, 347 Zhang, T.C., 407 Zhang, Y., 70 Zheng, Q., 268, 274 Zhou, J.Y., 263 Zhu, Z., 240 Ziolkowski, R.W., 70 Zolin, V.F., 213 Zoumi, A., 64 Zozulya, A.A., 159, 189–191 Zurek, W.H., 404 Zverev, P.G., 211

Y Yamamoto, Y., 366 Yao, E., 311, 340 Yao, Y.B., 238 Yariv, A., 157, 169, 189, 271 Yasuda, K., 228 Yatsiv, S., 224, 226 Ye, C., 235 Yeh, A., 64 Yeh, P., 155, 169 Yonezawa, H., 407 Yoshida, H., 235, 242 Yoshikawa, J., 367, 393, 395, 396 Yoshimura, M., 236 Young, T., 294 Yu, Z.X., 263 Yui, H., 202 Yukawa, M., 407

SUBJECT INDEX FOR VOLUME 53

A Abraham-Minkowski controversy, 73 Aharonov-Bohm effect, 74, 133, 136 –– – –, gravitational, 134 –– – –, optical, 113, 117, 132, 135 Airy-disc proﬁle, 12 anti-Stokes Raman scattering, 205, 230 atom optics, 311

B beam propagation, nonlinear, 165 – splitter interaction, 371 Bekenstein-Hawking entropy, 141 benzene, 209, 260, 263 Berry phase, 339 Bessel beam, 313, 318, 323 black hole, 141 Bloch-Messiah’s decomposition theorem, 394 Born approximation, 7 Bose-Einstein condensate, 298 Bragg grating, 264 – phase grating, 266

C calcite, 210 Casimir force, 128, 129 Cauchy-Riemann differential equations, 79 caustic, 302 Christoffel symbol, 92–99, 102, 105 cloaking device, 122, 123 – –, electromagnetic, 70 cluster state, 409 coherent anti-Stokes Raman spectroscopy, 204 – transfer function, 15 – state, 393, 400, 404, 408 constitutive equations, 117 contraction, 86 control theory, 366 covariant derivative, 90, 93, 95, 104, 112

– vector, 86 Cramer’s rule, 98

D Debye scalar integral representation, 20 diffraction, 301 – catastrophe, 302, 303 Doppler effect, 265 double-slit experiment, 396

E Euclidean metric, 83 Euler-Lagrange equations, 101, 102, 115 Einstein range convention, 81 electro-optic effect, 169 entangled beam, 367 – state, 367, 406 ethanol, 213 evanescent wave, 125 event horizon, 135

F Fabry-Perot interferometer, 239 feedback control, 365 feed-forward control, 365 –– –, elctro-optic, 383 Fermat’s principle, 70, 71, 74–77, 79, 112–115, 130 Fock state, 368, 369, 373, 401 Foucault’s pendulum, 104 four-wave frequency mixing, 204, 217 –– – – –, Raman-enhanced, 218 Fraunhofer pattern, 303 Fresnel diffraction formula, 45 – dragging coefﬁcient, 133 – lens, 309

G Gaussian beam, 342 general relativity, 80

427

428

Subject Index for Volume 53

geodesic, 100–103, 106 – equation, 102–104, 114 geometrical optics, 113, 115 Gerchberg-Saxton algorithm, 314 Goos-Hanchen shift, 299 Gouy phase, 312 gyromagnetic ratio, 227

Landau levels, 227 Laplace’s equation, 320 Laplacian, 99, 100 Leibniz rule, 91, 95 Levi-Civita tensor, 88–90, 98, 107, 11, 112, 115, 147 Lorentz transformation, 350

H

M

Hamilton-Jacobi equation, 114 Hawking radiation, 141 – temperature, 141 Helmholtz equation, 77–79, 166, 317, 318, 320 Hermite-Gaussian mode, 308, 351 – polynomial, 373 Hessian matrix, 318 hologram, 272, 315 –, computer-generated, 311 homodyne detection, 373, 385, 390, 402 – measurement, 374 Huygenian source, 23

Mach-Zehnder interferometer, 386, 395, 397, 404 Magnetic monopole, 298 Maxwell’s equations, 79, 106–108, 110, 111, 116, 126, 145, 146, 231, 317, 347 metamaterial, 70, 71, 73, 117, 129 methane, 210 metric tensor, 83, 86, 87, 97 microscope, confocal scanning, 2 –, optical, 1, 24, 28 microscopic imaging, 1 – –, basic theory of, 3–18 microscopy, axially oriented, 53 –, confocal scanning, 34–63 –, electron, 2 –, excitation ﬂuorescence, 3 –, image interference, 53, 54 –, optical, 2 –, – sectioning, 33, 51 –, saturated-patterned excitation, 50 –, selective-plane illumination, 34 –, standing-wave ﬂuorescence, 53 –, structured scanning, 43 –, two-photon excitation scanning, 62 –, X-ray, 2 Minkowski metric, 115

I imaging, three-dimensional, 8 invisibility device, 121 ionosphere, 305

J Jones matrix, 310

K Kerr effect, 155, 168, 209, 392 – –, optical, 251, 252 – -like nonlinearity, 170, 178 – liquid-core ﬁber system, 249 – medium, 260 – scattering, cross-section for, 253 – self-focusing effect, 170 kitten state, 400 Kukhtarev model, 155

N negative-index material, 125 no-cloning theorem, 404 noise-less ampliﬁcation, 390, 392 non-classical state, 369

O L Lagrangian, 74, 101 Laguerre-Gaussian beam, 307, 318, 323 –– – mode, 295, 308 – polynomial, 373

optical confocal mapping, 78, 79 – parametric oscillator, 395 – solid-core ﬁber, 212 – spanner, 335 – transfer function, 14 – tweezer, 310, 311, 335

Subject Index for Volume 53

P parabolic equation, 166 parallel transport, 93, 94, 102 parametric down-conversion, 337 paraxial approximation, 5, 24, 28 – beam, 185 – propagation, 166 – wave equation, 318, 320 Pearcey function, 302, 303 Peltier junction, 158 Penrose diagram, 116 perfect lense, 124 phase-conjugate wave, 203, 269, 273 – -insensitive ampliﬁer, 404 – -sensitive ampliﬁer, 392 – singularity, 305, 306, 316 photoelectric effect, 379 photonic crystal, 72 – – ﬁber, 223 photorefractive crystal, 157 Poincare sphere, 310, 319 point-spread function, 3 polarization singularity, 340, 343, 347, 348 Poynting vector, 296, 334, 336, 337, 349

Q quadrature detector, 384 quantum cloning, 367, 404 – entanglement, 311, 336 – erasing, 396, 397 – – of continuous variables, 398 – error correction, 368, 408 – feed-forward system, 366, 367, 378, 396 – information, 366, 367, 390, 395, 400, 403 – memory, 366 – non-demolition interaction, 372, 394

R refraction, negative, 124, 128 Raman-active materials, 209 – scattering cross section, 206, 208 – –, spontaneous, 206, 208 – shift, 211, 222 Rayleigh scattering, backward, 266 – –, spontaneous, 248, 264 ray optics, 302 Ricci tensor, 113 Riemann tensor, 106, 112, 113

S sampling theorem, axial, 30 Schrodinger equation, nonlinear, 154, 177

429

secret sharing, 408 self-focusing, 171, 215, 216 –– –, Raman enhanced, 216 Snell’s law, 31 soliton, 1D, 177 –, 2D, 187, 193 –, Kerr-like, 167 –, optical, 154, 155 –, photorefractive, 155, 156, 159–161, 181, 189, 194 – -soliton scattering, 191 –, spatial, 155 –, temporal, 154 spatial light modulator, 311 squeezed state, 367, 373, 382, 393, 396, 400, 402 – –, puriﬁcation of, 398 – vacuum, 383, 400 stimulated Brillouin scattering, 202, 228 – electronic Raman scattering, 223 – hyper-Raman scattering, 225 – Kerr scattering, 249–263 – Raman scattering, 202, 204–228 – – -Kerr scattering, 203, 251–254, 256 – Rayleigh-Bragg scattering, 203, 263–269 – – -Kerr scattering, 203, 251–253, 255, 257 – – -wing scattering, 202, 243 – spin-ﬂip Raman scattering, 226 – rotational Raman scattering, 222 – thermal Rayleigh scattering, 202, 243, 246 Stokes parameter, 344, 348 – scattering, 229 – shift, 36 sub-resolution imaging, 125

T teleportation, continuous-wave, 367, 407 –, quantum, 406 three-wave mixing, 271 transformation media, 109, 117–142 – –, spatial, 118 two-photon absorption, 263 –– – experiment, 294

V vortex beam, 308–311, 314 – in diffraction, 300 – – random optical ﬁeld, 304, 327, 329 – – superposition, 300 –, optical, 307, 308, 332 –, Rankine, 315

430

Subject Index for Volume 53

–, Riemann-Silberstein, 349 –, Stokes, 344

– –, two-mode, 374 Wolter’s vortex, 299

W

Z

which-way information, 397 Wigner function, 368–372, 376–378, 402

Zeeman splitting, 226 zero-point energy, 128

CUMUL ATIVE INDEX – VOLUMES 1–53

Abdullaev, F. and J. Garnier: Optical solitons in random media Abdullaev, F.Kh., S.A. Darmanyan and J. Garnier: Modulational instability of electromagnetic waves in inhomogeneous and in discrete media Abelès, F.: Methods for determining optical parameters of thin ﬁlms Abella, I.D.: Echoes at optical frequencies Abitbol, C.I., see Clair, J.J. Abraham, N.B., P. Mandel and L.M. Narducci: Dynamical instabilities and pulsations in lasers Aegerter, C.M. and G. Maret: Coherent backscattering and Anderson localization of light Agarwal, G.S.: Master equation methods in quantum optics Agranovich, V.M. and V.L. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.P., see Essiambre, R.-J. Allen, L. and D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett and M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefringent networks Andersen, U.L. and R. Filip: Quantum feed-forward control of light Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A. and A.W. Smith: Experimental studies of intensity ﬂuctuations in lasers Arnaud, 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., see Glesk, I. Baltes, H.P.: On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, V.D. Ozrin and A.I. Saichev: Enhanced backscattering in optics Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

48, 35 44, 303 2, 249 7, 139 16, 71 25,

1

52, 1 11, 1 9, 235 26, 163 37, 185 9, 179 39, 291 9, 123 53, 365 41, 97 36, 179 35, 257 6, 211 11, 247 34, 183 37, 57 39, 1 39, 291 45, 53 13,

1

29, 65

443

444

Cumulative Index – Volumes 1–53

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 and R. Winston: Nonimaging optics for ﬂux concentration Beckmann, P.: Scattering of light by rough surfaces Benisty, H. and C. Weisbuch: Photonic crystals Beran, M.J. and J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M. Berry, M.V. and C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M. and C. De Mol: Super-resolution by data inversion Bertolotti, M., see Chumash, V. Bertolotti, M., see Mihalache, D. Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Biener, G., see Hasman, E. Björk, G., A.B. Klimov and L.L. Sánchez-Soto: The discrete Wigner function Bloembergen, N.: From millisecond to attosecond laser pulses Bloom, A.L.: Gas lasers and their application to precise length measurements Bokor, N. and N. Davidson: Curved diffractive optical elements: Design and applications Bokor, N., see Davidson, N. Bouman, M.A., W.A. Van De Grind and P. Zuidema: Quantum ﬂuctuations in vision Bousquet, P., see Rouard, P. Boyd, R.W. and D.J. Gauthier: “Slow” and “fast” light Braat, J.J.M., S. van Haver, A.J.E.M. Janssen and P. Dirksen: Assessment of optical systems by means of point-spread functions Brambilla, E., see Gatti, A. Brosseau, C. and A. Dogariu: Symmetry properties and polarization descriptors for an arbitrary electromagnetic waveﬁeld Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W. and H. Paul: Theory of optical parametric ampliﬁcation and oscillation Bryngdahl, O.: Applications of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., T. Scheermesser and F. Wyrowski: Digital halftoning: synthesis of binary images Bryngdahl, O. and F. Wyrowski: Digital holography – computer-generated holograms Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing

1, 67 21, 217 12, 287 27, 161 6, 53 49, 177 33, 319 35, 61 18, 257 36, 129 36, 1 27, 227 16, 357 36, 245 47, 215 51, 469 50, 1 9, 1 48, 107 45, 1 22, 77 4, 145 43, 497 51, 349 51, 251 49, 315 23, 1 35, 61 15, 1 4, 37 11, 167 33, 389 28, 1 2, 73 19, 211

445

Cumulative Index – Volumes 1–53

Bužek, V. and P.L. Knight: Quantum interference, superposition states of light, and nonclassical effects

Cagnac, B., see Giacobino, E. Cao, H.: Lasing in disordered media Carmichael, H.J., G.T. Foster, L.A. Orozco, J.E. Reiner and P.R. Rice: Intensity-ﬁeld correlations of non-classical light Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi and M. Mansuripur: Principles of optical disk data storage Casasent, D. and D. Psaltis: Deformation invariant, space-variant optical pattern recognition Cattaneo, S., see Kauranen, M. Ceglio, N.M. and D.W. Sweeney: Zone plate coded imaging: theory and applications Cerf, N.J. and J. Fiurášek: Optical quantum cloning Chang, R.K., see Fields, M.H. Charnotskii, M.I., J. Gozani, V.I. Tatarskii and V.U. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T. and Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y. and A.M. Steinberg: Tunneling times and superluminality Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, W.M. Christov, I.P.: Generation and propagation of ultrashort optical pulses Chumash, V., I. Cojocaru, E. Fazio, F. Michelotti and M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass ﬁlms Clair, J.J. and C.I. Abitbol: Recent advances in phase proﬁles generation Clarricoats, P.J.B.: Optical ﬁbre waveguides – a review Cohen-Tannoudji, C. and A. Kastler: Optical pumping Cojocaru, I., see Chumash, V. Cole, T.W.: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.J.: Quantum jumps Courtès, G., P. Cruvellier and M. Detaille: 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 Crosignani, B., see DelRe, E. Cruvellier, P., see Courtès, G. Cummins, H.Z. and H.L. Swinney: Light beating spectroscopy

Dainty, J.C.: The statistics of speckle patterns Dändliker, R.: Heterodyne holographic interferometry Darmanyan, S.A., see Abdullaev, F.Kh.

34,

1

17, 85 45, 317 46, 355 41, 97 16, 289 51, 69 21, 287 49, 455 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 20, 63 28, 361 20, 1 26, 349 11, 223 53, 153 20, 1 8, 133

14, 1 17, 1 44, 303

446

Cumulative Index – Volumes 1–53

Dattoli, G., L. Giannessi, A. Renieri and A. Torre: Theory of Compton free electron lasers Davidson, N. and N. Bokor: Anamorphic beam shaping for laser and diffuse light Davidson, N., see Bokor, N. Davidson, N., see Oron, R. De Mol, C., see Bertero, M. De Sterke, C.M. and J.E. Sipe: Gap solitons Decker Jr, J.A., see Harwit, M. Delano, E. and R.J. Pegis: Methods of synthesis for dielectric multilayer ﬁlters DelRe, E., B. Crosignani and P. Di Porto: Photorefractive solitons and their underlying nonlocal physics Demaria, A.J.: Picosecond laser pulses Dennis, M.R., K. O’Holleran and M.J. Padgett: Singular optics: optical vortices and polarization singularities DeSanto, J.A. and G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Desyatnikov, A.S., Y.S. Kivshar and L.L. Torner: Optical vortices and vortex solitons Detaille, M., see Courtès, G. Dexter, D.L., see Smith, D.Y. Di Porto, P., see DelRe, E. Dirksen, P., see Braat, J.J.M. Dogariu, A., see Brosseau, C. Domachuk, P., see Eggleton, B.J. 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 Dušek, M., N. Lütkenhaus and M. Hendrych: Quantum cryptography Dutta, N.K. and J.R. Simpson: Optical ampliﬁers Dutta Gupta, S.: Nonlinear optics of stratiﬁed media

Eberly, J.H.: Interaction of very intense light with free electrons Eggleton, B.J., P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel and M.J. Steel: Laboratory post-engineering of microstructured optical ﬁbers Englund, J.C., R.R. Snapp and W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity Ennos, A.E.: Speckle interferometry Erez, N., see Greenberger, D.M. Essiambre, R.-J. and G.P. Agrawal: Soliton communication systems Etrich, C., F. Lederer, B.A. Malomed, T. Peschel and U. Peschel: Optical solitons in media with a quadratic nonlinearity

31, 321 45, 1 48, 107 42, 325 36, 129 33, 203 12, 101 7, 67 53, 153 9, 31 53, 293 23,

1

47, 291 20, 1 10, 165 53, 153 51, 349 49, 315 48, 1 37, 1 43, 433 12, 163 14, 161 49, 381 31, 189 38, 1

7, 359

48,

1

21, 355 16, 233 50, 275 37, 185 41, 483

Cumulative Index – Volumes 1–53

Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C., see Reynaud, S. Facchi, P. and 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. and C.K. Hitzenberger: Optical coherence tomography Ficek, Z. and H.S. Freedhoff: Spectroscopy in polychromatic ﬁelds Fields, M.H., J. Popp and R.K. Chang: Nonlinear optics in microspheres Filip, R., see Andersen, U.L. Fiorentini, A.: Dynamic characteristics of visual processes Fiurášek, J., see Cerf, N.J. Flytzanis, C., F. Hache, M.C. Klein, D. Ricard and Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics Focke, J.: Higher order aberration theory Forbes, G.W., see Kravtsov, Yu.A. Foster, G.T., see Carmichael, H.J. Françon, M. and S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlídal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, V.D. and 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 and M. Vampouille: Shaping and analysis of picosecond light pulses Fry, G.A.: The optical performance of the human eye Fu, Z., see Chen, R.T.

447

37, 95 30, 1 42, 147 22, 341 36, 1 44, 215 40, 389 41, 1 53, 365 1, 253 49, 455

29, 321 4, 1 39, 1 46, 355 6, 71 41, 181 40, 389 30, 137 9, 311 42, 325 20, 63 8, 51 41, 283

Gabitov, I.R., see Litchinitser, N.M. Gabor, D.: Light and information Gallion, P., F. Mendieta and S. Jiang: Signal and quantum noise in optical communications and cryptography Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H. and G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media

51, 1 1, 109

34, 333

Gantsog, Ts., see Tana´s, R. Gao, W., see Yin, J. García-Ojalvo, J., see Uchida, A. Garnier, J., see Abdullaev, F. Garnier, J., see Abdullaev, F.Kh. Gatti, A., E. Brambilla and L. Lugiato: Quantum imaging Gauthier, D.J.: Two-photon lasers

35, 355 45, 119 48, 203 48, 35 44, 303 51, 251 45, 205

52, 149 3, 187

448

Cumulative Index – Volumes 1–53

Gauthier, D.J., see Boyd, R.W. Gbur, G.: Nonradiating sources and other “invisible” objects Gea-Banacloche, J.: Optical realizations of quantum teleportation Ghatak, A. and K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E. and B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., see Reynaud, S. Giannessi, L., see Dattoli, G. Ginzburg, V.L.: Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Ginzburg, V.L., see Agranovich, V.M. Giovanelli, R.G.: Diffusion through non-uniform media Glaser, I.: Information processing with spatially incoherent light Glesk, I., B.C. Wang, L. Xu, V. Baby and P.R. Prucnal: Ultra-fast all-optical switching in optical networks Gniadek, K. and J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, J., see Charnotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, V.D. Greenberger, D.M., N. Erez, M.O. Scully, A.A. Svidzinsky and M.S. Zubairy: Planck, photon statistics, and Bose–Einstein condensation Grillet, C., see Eggleton, B.J. 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.: The geometric phase Hariharan, P. and B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M. and J.A. Decker Jr: Modulation techniques in spectrometry Hasegawa, A., see Kodama, Y. Hasman, E., G. Biener, A. Niv and V. Kleiner: Space-variant polarization manipulation Hasman, E., see Oron, R. He, G.S.: Stimulated scattering effects of intense coherent light Heidmann, A., see Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Hendrych, M., see Dušek, M. Herriot, D.R.: Some applications of lasers to interferometry

43, 497 45, 273 46, 311 18, 1 13, 169 17, 85 30, 1 31, 321

32, 267 9, 235 2, 109 24, 389 45, 53 9, 281 8, 1 32, 203 12, 233 30, 137 50, 275 48, 1 29, 321 29, 1 20, 263 24, 103 48, 149 36, 49 12, 101 30, 205 47, 215 42, 325 53, 201 30, 1 38, 85 10, 289 49, 381 6, 171

Cumulative Index – Volumes 1–53

449

Hitzenberger, C.K., see Fercher, A.F. Horner, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images

44, 215 38, 343 10, 1

Ichioka, Y., see Tanida, J. Imoto, N., see Yamamoto, Y. Ishii, Y.: Laser-diode interferometry Itoh, K.: Interferometric multispectral imaging Iwata, K.: Phase imaging and refractive index tomography for X-rays and visible rays

40, 77 28, 87 46, 243 35, 145

Jacobsson, R.: Light reﬂection from ﬁlms of continuously varying refractive index Jacquinot, P. and B. Roizen-Dossier: Apodisation Jacquod, Ph., see Türeci, H.E. Jaeger, G. and A.V. Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection Jamroz, W. and B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation Janssen, A.J.E.M., see Braat, J.J.M. Javidi, B. and J.L. Horner: Pattern recognition with nonlinear techniques in the Fourier domain Jiang, S., see Gallion, P. Jones, D.G.C., see Allen, L. Joshi, A. and M. Xiao: Controlling nonlinear optical processes in multi-level atomic systems Kartashov, Y.V., V.A. Vysloukh and L. Torner: Soliton shape and mobility control in optical lattices Kastler, A., see Cohen-Tannoudji, C. Kauranen, M. and S. Cattaneo: Polarization techniques for surface nonlinear optics Keller, O.: Local ﬁelds in linear and nonlinear optics of mesoscopic systems Keller, O.: Optical works of L.V. Lorenz Keller, O.: Historical papers on the particle concept of light Keller, U.: Ultrafast solid-state lasers Khoo, I.C.: 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. Kivshar, Y.S., see Desyatnikov, A.S. Kivshar, Y.S., see Saltiel, S.M.

47, 393

5, 247 3, 29 47, 75 42, 277 38, 419 20, 325 51, 349 38, 343 52, 149 9, 179 49, 97

52, 63 5, 1 51, 69 37, 257 43, 195 50, 51 46, 1 26, 105 41, 97 20, 155 42, 1 4, 85 28, 87 47, 291 47, 1

450

Cumulative Index – Volumes 1–53

Klein, M.C., see Flytzanis, C. Kleiner, V., see Hasman, E. Klimov, A.B., see Björk, G. Klyatskin, V.I.: The imbedding method in statistical boundary-value wave problems Knight, P.L., see Bužek, V. Kodama, Y. and 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. and L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Forbes and A.A. Asatryan: Theory and applications of complex rays Kravtsov, Yu.A., see Barabanenkov, Yu.N. Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrný and B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities Labeyrie, A.: High-resolution techniques in optical astronomy Lakhtakia, A., see Mackay, T.G. Lean, E.G.: Interaction of light and acoustic surface waves Lederer, F., see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N. and J. Upatnieks: Recent advances in holography Leonhardt, U. and T.G. Philbin: Transformation optics and the geometry of light Letokhov, V.S.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H. and C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Litchinitser, N.M., I.R. Gabitov, A.I. Maimistov and V.M. Shalaev: Negative refractive index metamaterials in optics Lohmann, A.W., D. Mendlovic and Z. Zalevsky: Fractional transformations in optics Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L., see Gatti, A.

29, 321 47, 215 51, 469 33, 34,

1 1

30, 205 7, 1 3, 1 4, 281 6, 331 42, 93 26, 227 36, 179 39, 1 29, 65 1, 211 40, 343 42, 93 14, 47 51, 121 11, 123 41, 483 16, 119 6, 1 53, 69 16, 1 39, 373 8, 343 41, 97 5, 287 51,

1

38, 263 40, 271 35, 61 51, 251

Cumulative Index – Volumes 1–53

451

Lugiato, L.A.: Theory of optical bistability Luis, A. and L.L. Sánchez-Soto: Quantum phase difference, phase measurements and Stokes operators Lukš, A. and V. Peˇrinová: Canonical quantum description of light propagation in dielectric media Lukš, A., see Peˇrinová, V. Lukš, A., see Peˇrinová, V. Lütkenhaus, N., see Dušek, M.

21, 69

Machida, S., see Yamamoto, Y. Mackay, T.G. and A. Lakhtakia: Electromagnetic ﬁelds in linear bianisotropic mediums Mägi, E.C., see Eggleton, B.J. Mahajan, V.N.: Gaussian apodization and beam propagation Maimistov, A.I., see Litchinitser, N.M. Mainfray, G. and C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas Malacara, D.: Optical and electronic processing of medical images Malacara, D., see Vlad, V.I. Mallick, S., see Françon, M. Malomed, B.A.: Variational methods in nonlinear ﬁber optics and related ﬁelds Malomed, B.A., see Etrich, C. Malomed, B.A., see Kurizki, G. Mandel, L.: Fluctuations of light beams Mandel, L.: The case for and against semiclassical radiation theory Mandel, P., see Abraham, N.B. Mansuripur, M., see Carriere, J. Manus, C., see Mainfray, G. Maradudin, A.A., see Shchegrov, A.V. Marchand, E.W.: Gradient index lenses Maret, G., see Aegerter, C.M. Martin, P.J. and R.P. Netterﬁeld: Optical ﬁlms produced by ion-based techniques Martínez-Corral, M. and G. Saavedra: The resolution challenge in 3D optical microscopy 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 Méndez, E.R., see Shchegrov, A.V. Mendieta, F., see Gallion, P. Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process

41, 419 43, 295 33, 129 40, 115 49, 381 28, 87 51, 121 48, 1 49, 1 51, 1 32, 313 22, 1 33, 261 6, 71 43, 71 41, 483 42, 93 2, 181 13, 27 25, 1 41, 97 32, 313 46, 117 11, 305 52, 1 23, 113 53, 1 22, 145 21, 1 15, 77 8, 373 46, 117 52, 149 38, 263 40, 271 30, 261

452

Cumulative Index – Volumes 1–53

Meystre, P., see Search, C.P. Michelotti, F., see Chumash, V. Mihalache, D., M. Bertolotti and C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L.: Self-focusing media with variable index of refraction Mikaelian, A.L. and M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mills, D.L. and K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, P.W.: Field quantization in optics Milonni, P.W. and B. Sundaram: Atoms in strong ﬁelds: photoionization and chaos Miranowicz, A., see Tana´s, R. Miyamoto, K.: Wave optics and geometrical optics in optical design Mollow, B.R.: Theory of intensity dependent resonance light scattering and resonance ﬂuorescence Murata, K.: Instruments for the measuring of optical transfer functions Musset, A. and A. Thelen: Multilayer antireﬂection coatings Nakwaski, W. and M. Osinski: ´ Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., see Carriere, J. Narducci, L.M., see Abraham, N.B. Navrátil, K., see Ohlídal, I. Netterﬁeld, R.P., see Martin, P.J. Nguyen, H.C., see Eggleton, B.J. Nishihara, H. and T. Suhara: Micro Fresnel lenses Niv, A., see Hasman, E. Noethe, L.: Active optics in modern large optical telescopes Novotny, L.: The history of near-ﬁeld optics Nussenzveig, H.M.: Light tunneling Ohlídal, I. and D. Franta: Ellipsometry of thin ﬁlm systems Ohlídal, I., K. Navrátil and M. Ohlídal: Scattering of light from multilayer systems with rough boundaries Ohlídal, M., see Ohlídal, I. O’Holleran, K., see Dennis, M.R. Ohtsu, M. and T. Tako: Coherence in semiconductor lasers Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback Okamoto, T. and T. Asakura: The statistics of dynamic speckles Okoshi, T.: Projection-type holography Omenetto, F.G.: Femtosecond pulses in optical ﬁbers Ooue, S.: The photographic image Opatrný, T., see Kurizki, G.

47, 139 36, 1 27, 227 17, 279 7, 231 19, 45 50, 97 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201

38, 165 41, 97 25, 1 34, 249 23, 113 48, 1 24, 1 47, 215 43, 1 50, 137 50, 185 41, 181 34, 249 34, 249 53, 293 25, 191 44, 1 34, 183 15, 139 44, 85 7, 299 42, 93

Cumulative Index – Volumes 1–53

Opatrný, T., see Welsch, D.-G. Oron, R., N. Davidson, A.A. Friesem and E. Hasman: Transverse mode shaping and selection in laser resonators Orozco, L.A., see Carmichael, H.J. Orrit, M., J. Bernard, R. Brown and B. Lounis: Optical spectroscopy of single molecules in solids Osinski, ´ M., see Nakwaski, W. Ostrovskaya, G.V. and Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.I. and V.P. Shchepinov: Correlation holographic and speckle interferometry Ostrovsky, Yu.I., see Ostrovskaya, G.V. Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, V.D., see Barabanenkov, Yu.N. Padgett, M.J., see Allen, L. Padgett, M.J., see Dennis, M.R. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D. and G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., see Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., see Brunner, W. Pegis, R.J.: The modern development of Hamiltonian optics Pegis, R.J., see Delano, E. Peiponen, K.-E., E.M. Vartiainen and T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C., see Carriere, J. Peˇrina Jr, J. and J. Peˇrina: Quantum statistics of nonlinear optical couplers Peˇrina, J.: Photocount statistics of radiation propagating through random and nonlinear media Peˇrina, J., see Peˇrina Jr, J. Peˇrinová, V. and A. Lukš: Quantum statistics of dissipative nonlinear oscillators Peˇrinová, V. and A. Lukš: Continuous measurements in quantum optics Peˇrinová, V., see Lukš, A. Pershan, P.S.: Non-linear optics Peschel, T., see Etrich, C. Peschel, U., see Etrich, C. Petite, G., see Shvartsburg, A.B. Petykiewicz, J., see Gniadek, K. Philbin, T.G., see Leonhardt, U. Picht, J.: The wave of a moving classical electron Pollock, C.R.: Ultrafast optical pulses

453

39, 63 42, 325 46, 355 35, 61 38, 165 22, 197 30, 87 22, 197 24, 165 33, 319 29, 65 39, 291 53, 293 32, 1 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67 37, 57 41, 97 41, 359 18, 127 41, 359 33, 129 40, 115 43, 295 5, 83 41, 483 41, 483 44, 143 9, 281 53, 69 5, 351 51, 211

454

Cumulative Index – Volumes 1–53

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, P.R., see Glesk, I. Psaltis, D. and Y. Qiao: Adaptive multilayer optical networks Psaltis, D., see Casasent, D.

34, 159 45, 53 31, 227 16, 289

Qiao, Y., see Psaltis, D. Qiu, M., see Yan, M.

31, 227 52, 261

Raymer, M.G. and I.A. Walmsley: The quantum coherence properties of stimulated Raman scattering Reiner, J.E., see Carmichael, H.J. Renieri, A., see Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino and C. Fabre: Quantum ﬂuctuations in optical systems Ricard, D., see Flytzanis, C. Rice, P.R., see Carmichael, H.J. Riseberg, L.A. and M.J. Weber: Relaxation phenomena in rare-earth luminescence Risken, H.: Statistical properties of laser light Roddier, F.: The effects of atmospheric turbulence in optical astronomy Rogister, F., see Uchida, A. 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. and J.L. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena

31, 139 41, 1 27, 315

28, 181 46, 355 31, 321 30, 1 29, 321 46, 355 14, 89 8, 239 19, 281 48, 203 3, 29 25, 279 35, 1 13, 69

Rouard, P. and P. Bousquet: Optical constants of thin ﬁlms Rouard, P. and A. Meessen: Optical properties of thin metal ﬁlms Roussignol, Ph., see Flytzanis, C. Roy, R., see Uchida, A. Rubinowicz, A.: The Miyamoto–Wolf diffraction wave

24, 39 4, 145 15, 77 29, 321 48, 203 4, 199

Rudolph, D., see Schmahl, G.

14, 195

Saavedra, G., see Martínez-Corral, M. Saichev, A.I., see Barabanenkov, Yu.N. Saito, S., see Yamamoto, Y. Sakai, H., see Vanasse, G.A.

53, 1 29, 65 28, 87 6, 259

455

Cumulative Index – Volumes 1–53

Saleh, B.E.A., see Teich, M.C. Saltiel, S.M., A.A. Sukhorukov and Y.S. Kivshar: Multistep parametric processes in nonlinear optics Sánchez-Soto, L.L., see Björk, G. Sánchez-Soto, L.L., see Luis, A. Sanders, B.C., see Hariharan, P. Scheermesser, T., see Bryngdahl, O. Schieve, W.C., see Englund, J.C. Schirripa Spagnolo, G., see Paoletti, D. Schmahl, G. and D. Rudolph: Holographic diffraction gratings Schubert, M. and B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G.: Aspheric surfaces Schulz, G. and J. Schwider: Interferometric testing of smooth surfaces Schwefel, H.G.L., see Türeci, H.E. Schwider, J.: Advanced evaluation techniques in interferometry Schwider, J., see Schulz, G. Scully, M.O. and K.G. Whitney: Tools of theoretical quantum optics Scully, M.O., see Greenberger, D.M. Search, C.P. and P. Meystre: Nonlinear and quantum optics of atomic and molecular ﬁelds Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework Sergienko, A.V., see Jaeger, G. Shalaev, V.M., see Litchinitser, N.M. Sharma, S.K. and D.J. Somerford: Scattering of light in the eikonal approximation Shchegrov, A.V., A.A. Maradudin and E.R. Méndez: Multiple scattering of light from randomly rough surfaces Shchepinov, V.P., see Ostrovsky, Yu.I. Shvartsburg, A.B. and 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 De Sterke, C.M. Sipe, J.E., see Van Kranendonk, J. Sittig, E.K.: Elastooptic light modulation and deﬂection Sizmann, A. and 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. and D.L. Dexter: Optical absorption strength of defects in insulators Smith, R.W.: The use of image tubes as shutters Snapp, R.R., see Englund, J.C. Sodha, M.S., A.K. Ghatak and V.K. Tripathi: Self-focusing of laser beams in plasmas and semiconductors

26,

1

47, 1 51, 469 41, 419 36, 49 33, 389 21, 355 35, 197 14, 195 17, 163 25, 349 13, 93 47, 75 28, 271 13, 93 10, 89 50, 275 47, 139 16, 413 42, 277 51, 1 39, 213 46, 117 30, 87 44, 143 27, 227 31, 189 33, 203 15, 245 10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355 13, 169

456

Cumulative Index – Volumes 1–53

Somerford, D.J., see Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices Soskin, M.S. and M.V. Vasnetsov: Singular optics Spreeuw, R.J.C. and J.P. Woerdman: Optical atoms Steel, M.J., see Eggleton, B.J. Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Steinvurzel, P., see Eggleton, B.J. Stoicheff, B.P., see Jamroz, W. Stone, A.D., see Türeci, H.E. Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy Subbaswamy, K.R., see Mills, D.L. Suhara, T., see Nishihara, H. Sukhorukov, A.A., see Saltiel, S.M. Sundaram, B., see Milonni, P.W. Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams Svidzinsky, A.A., see Greenberger, D.M. Sweeney, D.W., see Ceglio, N.M. Swinney, H.L., see Cummins, H.Z. Tako, T., see Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets Tana´s, R., A. Miranowicz and Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J. and R.Q. Twiss: Michelson stellar interferometry Tanida, J. and Y. Ichioka: Digital optical computing Tatarskii, V.I. and V.U. Zavorotnyi: Strong ﬂuctuations in light propagation in a randomly inhomogeneous medium Tatarskii, V.I., see Charnotskii, M.I. Taylor, C.A., see Lipson, H. Teich, M.C. and 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 Torner, L., see Kartashov, Y.V. Torner, L.L., see Desyatnikov, A.S. Torre, A.: The fractional Fourier transform and some of its applications to optics Torre, A., see Dattoli, G. Tripathi, V.K., see Sodha, M.S.

39, 213 27, 109 42, 219 31, 263 48, 1 5, 145 37, 345 48, 1 20, 325 47, 75 9, 73 2, 1 19, 45 24, 1 47, 1 31, 1 12, 1 50, 275 21, 287 8, 133 25, 191 23, 63 35, 355 17, 239 40, 77 18, 204 32, 203 5, 287 26, 1 7, 231 8, 201 7, 169 18, 1 23, 183 52, 63 47, 291 43, 531 31, 321 13, 169

Cumulative Index – Volumes 1–53

Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency ﬁltering Türeci, H.E., H.G.L. Schwefel, Ph. Jacquod and A.D. Stone: Modes of wave-chaotic dielectric resonators Turunen, J., M. Kuittinen and F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J. Uchida, A., F. Rogister, J. García-Ojalvo and R. Roy: Synchronization and communication with chaotic laser systems Upatnieks, J., see Leith, E.N. Upstill, C., see Berry, M.V. Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids

457

2, 131 47, 75 40, 343 17, 239

48, 203 6, 1 18, 257 19, 139

Vampouille, M., see Froehly, C. Van De Grind, W.A., see Bouman, M.A. van Haver, S., see Braat, J.J.M. Van Heel, A.C.S.: Modern alignment devices Van Kranendonk, J. and J.E. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vanasse, G.A. and H. Sakai: Fourier spectroscopy Vartiainen, E.M., see Peiponen, K.-E. Vasnetsov, M.V., see Soskin, M.S. Vernier, P.J.: Photoemission Vlad, V.I. and D. Malacara: Direct spatial reconstruction of optical phase from phase-modulated images Vogel, W., see Welsch, D.-G. Vysloukh, V.A., see Kartashov, Y.V.

20, 63 22, 77 51, 349 1, 289

33, 261 39, 63 52, 63

Walmsley, I.A., see Raymer, M.G. Wang Shaomin, and L. Ronchi: Principles and design of optical arrays Wang, B.C., see Glesk, I. Weber, M.J., see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weisbuch, C., see Benisty, H. Weiss, G.H., see Gandjbakhche, A.H. Welford, W.T.: Aberration theory of gratings and grating mountings

28, 181 25, 279 45, 53 14, 89 29, 293 49, 177 34, 333 4, 241

Welford, W.T.: Aplanatism and isoplanatism Welford, W.T., see Bassett, I.M. Welsch, D.-G., W. Vogel and T. Opatrný: Homodyne detection and quantum-state reconstruction Whitney, K.G., see Scully, M.O. Wilhelmi, B., see Schubert, M. Winston, R., see Bassett, I.M.

13, 267 27, 161

15, 245 6, 259 37, 57 42, 219 14, 245

39, 63 10, 89 17, 163 27, 161

458

Cumulative Index – Volumes 1–53

Woerdman, J.P., see Spreeuw, R.J.C. Wolf, E.: The inﬂuence of Young’s interference experiment on the development of statistical optics Wolinski, ´ T.R.: Polarimetric optical ﬁbers and sensors Wolter, H.: On basic analogies and principal differences between optical and electronic information Wynne, C.G.: Field correctors for astronomical telescopes Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Turunen, J.

31, 263

1, 155 10, 137 28, 1 33, 389 40, 343

Xiao, M., see Joshi, A. Xu, L., see Glesk, I.

49, 97 45, 53

Yan, M., W. Yan and M. Qiu: Invisibility cloaking by coordinate transformation Yan, W., see Yan, M. Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y., S. Machida, S. Saito, N. Imoto, T. Yanagawa and M. Kitagawa: 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., see Carriere, J. Yin, J., W. Gao and Y. Zhu: Generation of dark hollow beams and their applications Yoshinaga, H.: Recent developments in far infrared spectroscopic techniques Yu, F.T.S.: Principles of optical processing with partially coherent light Yu, F.T.S.: Optical neural networks: architecture, design and models

52, 261 52, 261

50, 251 40, 1

22, 271 6, 105 8, 295

28, 87 28, 87 32, 145 41, 97 45, 119 11, 77 23, 221 32, 61

Zalevsky, Z., D. Mendlovic and A.W. Lohmann: Optical systems with improved resolving power Zalevsky, Z., see Lohmann, A.W. Zavorotny, V.U., see Charnotskii, M.I. Zavorotnyi, V.U., see Tatarskii, V.I. Zhu, Y., see Yin, J. Zubairy, M.S., see Greenberger, D.M.

40, 271 38, 263 32, 203 18, 204 45, 119 50, 275

Zuidema, P., see Bouman, M.A.

22, 77