Progress in Optics, Volume 47 (Progress in Optics)

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PROGRESS IN OPTICS VOLUME 47

EDITORIAL ADVISORY BOARD

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H. Walther

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PROGRESS IN OPTICS VOLUME 47

EDITED BY

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

Contributors G. Biener, A.S. Desyatnikov, E. Hasman, K. Iwata, Ph. Jacquod, Y.S. Kivshar, V. Kleiner, P. Meystre, A. Niv, S.M. Saltiel, H.G.L. Schwefel, Ch.P. Search, A.D. Stone, A.A. Sukhorukov, L. Torner, H.E. Türeci

2005

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Preface We are pleased to present, in this volume, six review articles which cover a broad range of topics of current interest in modern optics. The first article by S.M. Saltiel, A.A. Sukhorukov and Y.S. Kivshar presents an overview of various types of parametric interactions in nonlinear optics which are associated with simultaneous phase-matching of several optical processes in quadratic nonlinear media, the so-called multi-step parametric interactions. A number of possibilities of double and multiple phase-matching in engineering structures with the sign-varying second-order nonlinear susceptibilities are discussed. The most important experimental results on multi-step parametric processes are summarized. Multicolor optical parametric solitons generated by the parametric interactions are also discussed. The second article by H.E. Tureci, H.G.L. Schwefel, Ph. Jacquod and A.D. Stone reviews the progress that has been made in recent years in the understanding of modes in wave-chaotic systems. Dielectric optical micro-resonators and microlasers are realizations of such systems, whose modes exhibit rich spatial structure. An efficient numerical method for calculating quasi-bound modes of dielectric resonators is introduced. The relationship between classical phase-space studies and modes is indicated via the Husimi projection technique. The next article by C.P. Search and P. Meystre reviews some important recent developments in nonlinear optics and in quantum optics. After an elementary review of the formalism of second quantization it is shown that attractive two-body interactions are the analogue of the de Broglie waves of a self-focusing medium in optics, while repulsive interactions correspond to defocussing. Review of research on focusing and defocussing of coherent atomic matter waves and the generation of dark and bright solitons is then discussed. Further, four-wave mixing and phase conjugation of Fermions matter waves is considered as is the molecular analog of the cavity QED micromaser. The fourth article by E. Hasman, G. Biener, A. Niv and V. Kleiner discusses space-variant polarization manipulation. This technique, which is being exploited in a variety of applications, includes polarization encoding of data, neural networks, optical computing, optical encryption, tight focusing, imaging polarimev

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try, material processing, atom trapping and optical tweezers. The article reviews both theoretical analysis and experimental techniques. The article which follows, by A.S. Desyatnikov, L. Torner and Y.S. Kivshar presents an overview of recent researches on optical vortices and phase singularities of electromagnetic waves in different types of nonlinear media, with emphasis on the properties of vortex solitons. A self-focusing nonlinearity leads, in general, to the azimuthal instability of a vortex-carrying beam, but it can also support novel types of stable or meta-stable self-trapped beams carrying nonzero angular momentum, such as ring-like solitons, necklace beams, and soliton clusters. Vortex solitons created by multi-component beams, by parametrically coupled beams in quadratic nonlinear media, and in partially coherent light, as well as discrete vortex solitons in periodic photonic lattices are also discussed. The concluding article by K. Iwata presents a review of imaging techniques with X-rays and visible light in which phase of the radiation that penetrates through a transparent object plays an important part. Using the phase information one can tomographically reconstruct the refractive index distribution within the object. This technique produces images with higher contrast than does conventional tomography for weakly absorbing objects. Emil Wolf Department of Physics and Astronomy and the Institute of Optics University of Rochester Rochester, NY 14627, USA April 2005

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1. Multistep parametric processes in nonlinear optics, Solomon M. Saltiel (Canberra, Australia/Sofia, Bulgaria), Andrey A. Sukhorukov (Canberra, Australia) and Yuri S. Kivshar (Canberra, Australia) . .

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§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . § 2. Single-phase-matched processes . . . . . . . . . . . § 3. Multistep phase-matched interactions . . . . . . . . 3.1. Third-harmonic multistep processes . . . . . . 3.2. Wavelength conversion . . . . . . . . . . . . . 3.3. Two-color multistep cascading . . . . . . . . . 3.4. Fourth-harmonic multistep cascading . . . . . 3.5. OPO and OPA multistep parametric processes 3.6. Other types of multistep interactions . . . . . . 3.7. Measurement of the χ (3) -tensor components . § 4. Phase matching for multistep cascading . . . . . . . 4.1. Uniform QPM structures . . . . . . . . . . . . 4.2. Nonuniform QPM structures . . . . . . . . . . 4.3. Quadratic 2D nonlinear photonic crystals . . . § 5. Multi-color parametric solitons . . . . . . . . . . . . 5.1. Third-harmonic parametric solitons . . . . . . 5.2. Two-color parametric solitons . . . . . . . . . 5.3. Solitons due to wavelength conversion . . . . 5.4. Other types of multi-color parametric solitons § 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Modes of wave-chaotic dielectric resonators, H.E. Türeci (New Haven, CT, USA), H.G.L. Schwefel (New Haven, CT, USA), Ph. Jacquod (Genève, Switzerland) and A. Douglas Stone (New Haven, CT, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. § 2. § 3. § 4. § 5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Failure of eikonal methods for generic dielectric resonators Ray dynamics for generic dielectric resonators . . . . . . . Formulation of the resonance problem . . . . . . . . . . . . Reduction of the Maxwell equations . . . . . . . . . . . . . vii

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§ 6. Scattering quantization – philosophy and methodology . . . . . . . . . . . § 7. Root-search strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Zero deformation – case of the rotationally symmetric dielectric . . . 7.2. Deformed dielectric resonators . . . . . . . . . . . . . . . . . . . . . . § 8. The Husimi–Poincaré projection technique for optical dielectric resonators § 9. Far-field distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 10. Mode classification: theory and experiment . . . . . . . . . . . . . . . . . . § 11. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Numerical implementation issues . . . . . . . . . . . . . . . . . . . Appendix B: Lens transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3. Nonlinear and quantum optics of atomic and molecular fields, Chris P. Search (Hoboken, NJ, USA) and Pierre Meystre (Tucson, AZ, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . § 2. Field quantization . . . . . . . . . . . . . . . . . . § 3. Quantum-degenerate atomic systems . . . . . . . 3.1. Bose–Einstein condensates . . . . . . . . . . 3.2. Quantum-degenerate Fermi systems . . . . . § 4. Collisions . . . . . . . . . . . . . . . . . . . . . . 4.1. s-wave scattering . . . . . . . . . . . . . . . 4.2. Feshbach resonances . . . . . . . . . . . . . 4.3. Photoassociation . . . . . . . . . . . . . . . § 5. Mean-field theory of Bose–Einstein condensation 5.1. The Gross–Pitaevskii equation . . . . . . . . 5.2. Quasiparticles . . . . . . . . . . . . . . . . . § 6. Degenerate Fermi gases . . . . . . . . . . . . . . . 6.1. Normal Fermi systems . . . . . . . . . . . . 6.2. Superfluid Fermi systems . . . . . . . . . . . § 7. Atomic solitons . . . . . . . . . . . . . . . . . . . § 8. Four-wave mixing . . . . . . . . . . . . . . . . . . 8.1. Bosonic four-wave mixing . . . . . . . . . . 8.2. Fermionic four-wave mixing . . . . . . . . . 8.3. Fermionic phase conjugation . . . . . . . . . § 9. Three-wave mixing . . . . . . . . . . . . . . . . . 9.1. Nonlinear mixing of quasi-particles . . . . . 9.2. Coherent molecule formation . . . . . . . . 9.3. The molecular ‘micromaser’ . . . . . . . . . § 10. Outlook . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . Appendix A: Feshbach resonances . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. Space-variant polarization manipulation, Erez Hasman (Haifa, Israel), Gabriel Biener (Haifa, Israel), Avi Niv (Haifa, Israel) and Vladimir Kleiner (Haifa, Israel) . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Formation of space-variant polarization-state manipulations . . . . . . . . . . . . . . . 2.1. Space-variant polarization-state manipulation by use of sub-wavelength gratings 2.2. Space-variant vectorial fields obtained by using interference methods . . . . . . 2.3. Space-variant vectorial fields obtained by using liquid-crystal devices . . . . . . 2.4. Alternative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 3. Geometrical phase in space-variant polarization-state manipulation . . . . . . . . . . . 3.1. Continuous Pancharatnam–Berry-phase optical elements . . . . . . . . . . . . . 3.2. Quantized Pancharatnam–Berry-phase diffractive optics . . . . . . . . . . . . . . 3.3. Polarization Talbot self-imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . § 4. Applications of space-variant polarization manipulation . . . . . . . . . . . . . . . . . 4.1. Polarization measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Spatial polarization scrambling . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Polarization encryption and polarization encoding . . . . . . . . . . . . . . . . . 4.4. Space-variant polarization-dependent emissivity . . . . . . . . . . . . . . . . . . § 5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5. Optical vortices and vortex solitons, Anton S. Desyatnikov (Canberra, Australia), Yuri S. Kivshar (Canberra, Australia) and Lluis Torner (Barcelona, Spain) . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . § 2. Self-trapped vortices in Kerr-type media . . . . . 2.1. Vortices in defocusing nonlinear media . . . 2.2. Ring-like beams in focusing nonlinear media 2.3. Azimuthal modulational instability . . . . . § 3. Composite spatial solitons with phase dislocations 3.1. Soliton-induced waveguides . . . . . . . . . 3.2. Higher-order vector solitons . . . . . . . . . 3.3. Multi-component vortex solitons . . . . . . . 3.4. Partially coherent vortices . . . . . . . . . . § 4. Multi-color vortex solitons . . . . . . . . . . . . . 4.1. Model . . . . . . . . . . . . . . . . . . . . . 4.2. Frequency doubling with vortex beams . . . 4.3. Families of the vortex solitons . . . . . . . . 4.4. Spontaneous break-up: azimuthal instability 4.5. Induced break-up: soliton algebra . . . . . . 4.6. Dark multi-color vortex solitons . . . . . . . § 5. Stabilization of vortex solitons . . . . . . . . . . . 5.1. Cubic-quintic nonlinearity . . . . . . . . . . 5.2. Quadratic-cubic nonlinearity . . . . . . . . . 5.3. Spatiotemporal spinning solitons . . . . . . . § 6. Other optical beams carrying angular momentum . 6.1. Soliton spiraling . . . . . . . . . . . . . . . . 6.2. Optical necklace beams . . . . . . . . . . . . 6.3. Soliton clusters . . . . . . . . . . . . . . . . § 7. Discrete vortices in two-dimensional lattices . . . § 8. Links to vortices in other fields . . . . . . . . . . .

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8.1. Vortices in dissipative optical systems . . . 8.2. Vortices in matter waves . . . . . . . . . . 8.3. Optical vortices and quantum information . § 9. Concluding remarks . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6. Phase imaging and refractive index tomography for X-rays and visible rays, Koichi Iwata (Osaka, Japan) . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Properties of X-ray and visible ray . . . . . . . . . . . . . . . . . . . . . . 2.1. Absorption coefficient and refractive index . . . . . . . . . . . . . . 2.2. Deviation of ray direction . . . . . . . . . . . . . . . . . . . . . . . § 3. Formation of intensity and phase images . . . . . . . . . . . . . . . . . . 3.1. Intensity image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Phase image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 4. Phase imaging methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Shadowgraphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Schlieren method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Other phase imaging methods . . . . . . . . . . . . . . . . . . . . . § 5. Reference type interferometers . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Visible ray region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. X-ray region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 6. Shearing type interferometers . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Visible ray region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. X-ray region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 7. Refractive index tomography . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Reference type interferometers . . . . . . . . . . . . . . . . . . . . . 7.2. Shearing type interferometers . . . . . . . . . . . . . . . . . . . . . § 8. Discussion on interferometers and refractive index tomography . . . . . . 8.1. Comparison between the interferometers . . . . . . . . . . . . . . . 8.2. Consideration for interferometers with a conventional X-ray source 8.3. Diffraction tomography . . . . . . . . . . . . . . . . . . . . . . . . . § 9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Author index for Volume 47 . . . . Subject index for Volume 47 . . . . Contents of previous volumes . . . Cumulative index – Volumes 1–47

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E. Wolf, Progress in Optics 47 © 2005 Elsevier B.V. All rights reserved

Chapter 1

Multistep parametric processes in nonlinear optics by

Solomon M. Saltiel Nonlinear Physics Group and Center for Ultra-high bandwidth Devices for Optical Systems (CUDOS), Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia Faculty of Physics, University of Sofia, 5 J. Bourchier Bld, Sofia BG-1164, Bulgaria E-mail address: [email protected] (S. Saltiel)

and

Andrey A. Sukhorukov, Yuri S. Kivshar Nonlinear Physics Group and Center for Ultra-high bandwidth Devices for Optical Systems (CUDOS), Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia E-mail addresses: [email protected] (A.A. Sukhorukov), [email protected] (Y.S. Kivshar)

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)47001-8 1

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

§ 2. Single-phase-matched processes . . . . . . . . . . . . . . . . . . . . .

5

§ 3. Multistep phase-matched interactions . . . . . . . . . . . . . . . . . .

8

§ 4. Phase matching for multistep cascading . . . . . . . . . . . . . . . . .

39

§ 5. Multi-color parametric solitons . . . . . . . . . . . . . . . . . . . . .

53

§ 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

2

§ 1. Introduction Energy transfer between different modes and phase-matching relations are fundamental concepts in nonlinear optics. Unlike nonparametric nonlinear processes such as self-action and self-focusing of light in a nonlinear Kerr-like medium, parametric processes involve several waves at different frequencies and they require special relations between the wave numbers and wave group velocities to be satisfied, the so-called phase-matching conditions. Parametric coupling between waves occurs naturally in nonlinear materials without inversion symmetry, when the lowest-order nonlinear effects are presented by quadratic nonlinearities, often called χ (2) nonlinearities because they are associated with the second-order contribution (∼χ (2) E 2 ) to the nonlinear polarization of a medium. Conventionally, the phase-matching conditions for most parametric processes in optics are implemented either by using anisotropic crystals (the so-called perfect phase matching), or in fabricated structures with a periodically reversed sign of the quadratic susceptibility (the so-called quasiphase matching or QPM). The QPM technique is one of the leading technologies today, and it employs spatial scales (∼1–30 µm) which are compatible with the operational wavelengths of optical communication systems. Nonlinear effects produced by the quadratic intensity-dependent response of a transparent dielectric medium are usually associated with parametric frequency conversion, such as second harmonic generation (SHG). The SHG process is among the most intensively studied parametric interactions which may occur in a quadratic nonlinear medium. Moreover, recent theoretical and experimental results demonstrate that quadratic nonlinearities can also produce many of the effects attributed to nonresonant Kerr nonlinearities via cascading of several second-order parametric processes. Such second-order cascading effects can simulate third-order processes, in particular those associated with the intensitydependent change of the medium refractive index (Stegeman, Hagan and Torner [1996]). Importantly, the effective (or induced) cubic nonlinearity resulting from a cascaded SHG process in a quadratic medium can be several orders of magnitude higher than that usually measured in centrosymmetric Kerr-like nonlinear media, and it is practically instantaneous. 3

4

Multistep parametric processes in nonlinear optics

[1, § 1

The simplest type of phase-matched parametric interaction is based on the simultaneous action of two second-order parametric sub-processes that belong to a single second-order interaction. For example, the so-called two-step cascading associated with type I SHG includes the generation of the second harmonic (SH), ω + ω = 2ω, followed by the reconstruction of the fundamental wave through the down-conversion frequency-mixing process, 2ω − ω = ω. These two subprocesses depend on only a single phase-matching parameter k. In particular, for nonlinear χ (2) media with a periodic modulation of the quadratic nonlinearity, for QPM periodic structures, we have k = k2 − 2k1 + Gm , where k1 = k(ω), k2 = k(2ω) and Gm is the reciprocal vector of the periodic structure, Gm = 2πm/Λ, where Λ is the lattice spacing and m is an integer. For a homogeneous bulk χ (2) medium, we have Gm = 0. Multistep parametric interactions and multistep cascading represent a special type of second-order parametric processes that involve several different secondorder nonlinear interactions; they are characterized by at least two different phasematching parameters. For example, two parent processes of the so-called thirdharmonic cascading are: (i) second-harmonic generation, ω + ω = 2ω, and (ii) sum-frequency mixing, ω + 2ω = 3ω. Here, we may distinguish five harmonic sub-processes, and the multistep interaction results in their simultaneous action. Different types of multistep parametric processes include third-harmonic cascaded generation, two-color parametric interaction, fourth-harmonic cascading, difference-frequency generation, etc. Various applications of multistep parametric processes have been mentioned in the literature. In particular, multistep parametric interaction can support multi-color solitary waves, it usually leads to larger accumulated nonlinear phase shifts in comparison with simple cascading, it can be employed effectively for the simultaneous generation of higher-order harmonics in a single quadratic crystal, and it can be employed for the generation of a cross-polarized wave and frequency shifting in fiber-optics gratings. In general, simultaneous phase matching of several parametric processes cannot be achieved by traditional methods such as those based on the optical birefringence effect. However, the situation becomes different for media with a periodic sign change of the quadratic nonlinearity, as occurs in QPM structures or two-dimensional nonlinear photonic crystals. In this review, we describe the basic principles of simultaneous phase matching of two (or more) parametric processes in different types of one- and twodimensional nonlinear quadratic optical lattices. We divide the different types of phase-matched parametric processes studied in nonlinear optics into two major classes, as shown in fig. 1, and discuss different types of parametric interac-

1, § 2]

Single-phase-matched processes

5

Fig. 1. Different types of parametric processes in nonlinear optics, and the specific topics covered by this review. SHG: second-harmonic generation; SFG and DFG: sum- and difference-frequency generation, THG: third-harmonic generation; FWM: four-wave mixing; DFWM: degenerate FWM; OPO: optical parametric oscillator.

tions associated with simultaneous phase matching of several optical processes in quadratic (or χ (2) ) nonlinear media, the so-called multistep parametric interactions. In particular, we provide an overview of the basic principles of double and multiple phase matching in engineered structures with sign-varying secondorder nonlinear susceptibility, including different types of QPM optical superlattices, noncollinear geometry, and two-dimensional nonlinear quadratic photonic crystals (which can be considered two-dimensional QPM lattices). We also summarize the most important experimental results on the multi-frequency generation due to multistep parametric processes, and survey the physics and basic properties of multi-color optical solitons generated by these parametric interactions.

§ 2. Single-phase-matched processes One of the simplest and first-studied parametric processes in nonlinear optics is second-harmonic generation (SHG). The SHG process is a special case of a more general three-wave mixing process which occurs in a dielectric medium with a quadratic intensity-dependent response. The three-wave-mixing and SHG processes require only one phase-matching condition to be satisfied and, therefore, can both be classified as single-phase-matched processes.

6

Multistep parametric processes in nonlinear optics

[1, § 2

In this section, we briefly discuss these single-phase-matched processes, and consider parametric interaction between three continuous-wave (CW) waves with electric fields Ej = 12 [Aj exp(−ikj · r + iωj t) + c.c.], where j = 1, 2, 3, with the three frequencies satisfying the energy-conservation condition, ω1 + ω2 = ω3 . We assume that the phase-matching condition is nearly satisfied, with a small mismatch k between the three wave vectors; i.e., k = k3 (ω3 ) − k1 (ω1 ) − k2 (ω2 ). In general, the three waves do not propagate in the same direction, and the beams may walk off from each other as they propagate inside the crystal. If all three wave vectors point in the same direction (e.g., in the case of QPM materials), the waves have the same phase velocity and exhibit no walk-off. The theory of χ (2) -mediated three-wave mixing is available in several books devoted to nonlinear optics (Shen [1984], Butcher and Cotter [1992], Boyd [1992]). The starting point is the Maxwell wave equation written as ∇ ×∇ ×E+

1 ∂ 2P 1 ∂ 2E = − , c2 ∂t 2 ε0 c2 ∂t 2

(2.1)

where ε0 is the vacuum permittivity and c is the speed of light in vacuum. The induced polarization is written in the frequency domain as ˜ + ε0 χ (2) E˜ E ˜ + ··· , ˜ ω) = ε0 χ (1) E P(r,

(2.2)

where the tildes denote the Fourier transform. Using the slowly-varying-envelope approximation and neglecting the walk-off, one can derive the following set of three coupled equations describing the parametric interaction of three waves under type II phase matching: dA1 − ω12 Γ A3 A∗2 e−ikz = 0, dz dA2 − ω22 Γ A3 A∗1 e−ikz = 0, ik2 dz dA3 − ω32 Γ A1 A2 eikz = 0, ik3 dz ik1

(2)

(2)

(2.3) (2.4) (2.5)

where Γ = deff /c2 , and deff is a convolution of the second-order suscepti(2) bility tensor χˆ (2) and the polarization unit vectors of the three fields, deff = 1 (2) 2 e3 χˆ e1 e2 . In the case of type I SHG, only a single beam at the pump frequency ω1 is incident on the nonlinear crystal, and a new optical field at the frequency 2ω1 is generated during the SHG process. We can adapt eqs. (2.3)–(2.5) to this case with minor modifications. More specifically, we set ω3 = 2ω1 and A1 = A2 . The first two equations then become identical, and one of them can be dropped. The type I

1, § 2]

Single-phase-matched processes

7

SHG process is thus governed by the following set of two coupled equations: dA1 − k1 σ A3 A∗1 e−ikz = 0, dz dA3 ik3 − 2k1 σ A21 eikz = 0, dz

ik1

(2)

(2.6) (2.7)

where σ = (ω1 /n1 c)deff is the nonlinear coupling parameter and k = k3 − 2k1 is the phase-mismatch parameter. Both three-wave mixing and SHG are examples of a single-phase-matched parametric process, because they are controlled by a single phase-matching parameter k. This kind of parametric process can be described as a two-step cascading interaction which includes: (i) the generation of the second-harmonic (SH) wave, ω + ω = 2ω, followed by (ii) the reconstruction of the fundamental wave through the down-conversion frequency mixing process, i.e. 2ω − ω = ω. Respectively, the first sub-process is responsible for the generation of the SH field, with the most efficient conversion observed at k = 0, while the second subprocess, also called cascading, can be associated with an effective intensitydependent change of the phase of the fundamental harmonic (∼d 2 /k), which is similar to that of the cubic nonlinearity (DeSalvo, Hagan, Sheik-Bahae, Stegeman, Vanstryland and Vanherzeele [1992], Stegeman, Hagan and Torner [1996], Assanto, Stegeman, Sheik-Bahae and Vanstryland [1995]). This latter effect is responsible for the generation of the so-called quadratic solitons: two-wave parametric solitons composed of mutually coupled fundamental and second-harmonic components (Sukhorukov [1988], Torner [1998], Kivshar [1997], Etrich, Lederer, Malomed, Peschel and Peschel [2000], Torruellas, Kivshar and Stegeman [2001], Boardman and Sukhorukov [2001], Buryak, Di Trapani, Skryabin and Trillo [2002], and references therein). The multistep parametric interactions and multistep cascading effects discussed in this review are represented by different types of phase-matched parametric interactions in quadratic (or χ (2) ) nonlinear media which involve several different parametrically interacting waves, as in the case of frequency mixing and sum-frequency generation. However, all such interactions can also be associated with the two major physical mechanisms of the wave interaction discussed for the SH process above: (i) parametric energy transfer between waves determined by the phase mismatch between the wave vectors of the interacting waves, and (ii) phase changes due to this parametric interaction. Below, we discuss these interactions for a number of physically important examples.

8

Multistep parametric processes in nonlinear optics

[1, § 3

§ 3. Multistep phase-matched interactions In this section we consider nonlinear parametric interactions that involve several processes, with each of the processes described by an independent phasematching parameter. In the early days of nonlinear optics, the motivation to study this kind of parametric interactions was to explore various possibilities for the simultaneous generation of several harmonics in a single nonlinear crystal (see, e.g., Akhmanov and Khokhlov [1964, 1972]) as well as to use the cascading of several parametric processes for measuring higher-order susceptibilities in nonlinear optical crystals (see, e.g., Yablonovitch, Flytzanis and Bloembergen [1972], Akhmanov, Dubovik, Saltiel, Tomov and Tunkin [1974], Akhmanov [1977], Kildal and Iseler [1979], Bloembergen [1982]). More recently, these processes were proved to be efficient for higher-order harmonic generation, for building reliable standards for third-order nonlinear susceptibility measurements (see, e.g., Bosshard, Gubler, Kaatz, Mazerant and Meier [2000] and references therein), and also for generating multi-color optical solitons. Additionally, it is expected that multistep parametric processes and multistep cascading will find application in optical communication devices, for wavelength shifting and all-optical switching (to be discussed below). Another class of applications of multistep phasematched parametric processes is the construction of optical parametric oscillators (OPOs) and optical parametric amplifiers (OPAs) with complementary phasematched processes in the same nonlinear crystal where the main phase-matched parametric process occurs. In this way, OPOs and OPAs may possess additional coherent tunable outputs (see Section 3.5). On the basis of the third-harmonic multistep cascading process {ω + ω = 2ω; ω + 2ω = 3ω}, real advances have been made in the development of nonlinear optical systems for division by three. Multistep parametric interactions governed by several phase-mismatch parameters can also occur in centrosymmetric nonlinear media (see, e.g., Akhmanov, Martynov, Saltiel and Tunkin [1975], Reintjes [1984], Astinov, Kubarych, Milne and Miller [2000], Crespo, Mendonca and Dos Santos [2000], Misoguti, Backus, Durfee, Bartels, Murnane and Kapteyn [2001]). The first proposal for simultaneous phase matching of two parametric processes can be found in the pioneering book by Akhmanov and Khokhlov (Akhmanov and Khokhlov [1964, 1972]) who derived and investigated the condition for the THG process to occur in a single crystal with a quadratic nonlinearity through the combined action of the SHG and SFG parametric processes. For efficient frequency conversion, both parametric processes, i.e. SHG and SFG, should be phase matched simultaneously. Some earlier experimental attempts to simultaneously achieve the two phase-matching conditions for the SHG and SFG processes

1, § 3]

Multistep phase-matched interactions

9

Table 1 Examples of multistep parametric interactions involving SHG No. 1 2 3 4 5

Multistep parametric process

Cascading χ (2) steps

Equivalent high-order parametric process

Type I third-harmonic multistep process Type II third-harmonic multistep process 3:1 frequency conversion and division Fourth-harmonic multistep process

ω + ω = 2ω ω + 2ω = 3ω ω + ω = 2ω ω⊥ + 2ω = 3ω 3ω → 2ω + ω ω = 2ω − ω ω + ω = 2ω 2ω + 2ω = 4ω ω + ω = 2ω 2ω − ω = ω⊥ ω + ω = 2ω ω⊥ + ω⊥ = 2ω ω + ω = 2ω 2ω − ωa = ωb ωp → ωi + ωs ωs + ωs = ωs,SH ωp → ωi + ωs ωs + ωp = ωSFG ωp/2 + ωp/2 = ωp ωp → ωi + ωs

ω + ω + ω = 3ω

7

Type I & type II two-color multistep process Type I & type I two-color multistep process wavelength conversion

8

Self-doubling OPO

9

Self-sum-frequency generation OPO Internally pumped OPO

6

10

ω + ω + ω⊥ = 3ω

ω + ω + ω + ω = 4ω ω + ω − ω = ω⊥

ω + ω − ωa = ωb

In the table, the symbol ω⊥ stands for a wave polarized in the plane perpendicular to that of the wave with the main carrier frequency ω.

were not very successful (Sukhorukov and Tomov [1970], Orlov, Sukhorukov and Tomov [1972]). The recent development of novel techniques for efficient phase matching, including the QPM technique, make many of such multistep processes readily possible. The multistep parametric processes investigated so far can be divided into several groups, as shown in fig. 1 and Table 1. These processes include: thirdharmonic multistep cascading; fourth-harmonic multistep cascading; two-color multistep cascading; FWM and DFWM in χ (2) media, OPO and OPA multistep cascading. We will review these groups separately. Table 1 does not provide a complete list of all possible types of multistep parametric processes. Rather, we present only several examples for each type of multistep process. As can be seen in Table 1, some parametric processes simulate some known higher-order processes but occur through several steps. However, other multistep parametric processes have no such simpler analogues. In this review we discuss some of these processes in more detail. For example, THG multistep cascading is considered in Section 3.1, while two-color multistep cascading is

10

Multistep parametric processes in nonlinear optics

[1, § 3

discussed in Section 3.3. The last three rows of Table 1 present examples of multistep cascading processes in optical parametric oscillators and amplifiers, and these will be discussed in Section 3.5. To illustrate the physics that induced the use of the terminology “multistep interaction” or “multistep cascading”, we just point out the simultaneous action of SHG and SFG (row 1 of Table 1); this three-wave interaction involves five simpler parametric sub-processes in a quadratic medium. In order to describe, for example, the nonlinear phase shift of the fundamental wave accumulated in this interaction, we should consider the following chain of parametric interactions: SHG (ω + ω = 2ω), SFG (ω + 2ω = 3ω), DFM (3ω − ω = 2ω), and, finally, another DFM (2ω − ω = ω).

3.1. Third-harmonic multistep processes The multistep parametric interaction involving the THG process is one of the most extensively studied multistep cascading schemes (see fig. 2). The two simpler parametric interactions are the SHG process, ω + ω = 2ω, and the SFG process, ω + 2ω = 3ω. Each of these sub-processes is characterized by an independent phase-matching parameter, namely kSHG = k2ω − 2k1ω and kSFG = k3ω − k2ω − kω . This kind of multistep cascading appears in many schemes of parametric interactions in nonlinear optics, including: (i) efficient generation of the third harmonic in a single quadratic crystal; (ii) measurement of unknown χ (3) tensor components using known χ (2) components of the crystal as a reference; (iii) accumulation of a large nonlinear phase shift by the fundamental wave; (iv) propagation

Fig. 2. Schematic of the THG multistep cascading.

1, § 3]

Multistep phase-matched interactions

11

of multi-color solitons; (v) frequency division; (vi) generation of entangled and squeezed photon states, etc. 3.1.1. Efficient generation of a third-harmonic wave First we consider the process of efficient generation of a third-harmonic wave. In the plane-wave approximation, this process is described by the following system of coupled equations for the slowly varying amplitudes A1 , A2 , and A3 of the fundamental, second-, and third-harmonic waves, respectively: dA1 = −iσ1 A2 A∗1 e−ikSHG z − iσ3 A3 A∗2 e−ikSFG z , dz dA2 = −iσ2 A21 eikSHG z − iσ4 A3 A∗1 e−ikSFG z , (3.1) dz dA3 = −iσ5 A2 A1 eikSFG z − iγ A31 eikTHG z , dz where kSHG = k2 − 2k1 + Gp and kSFG = k3 − k2 − k1 + Gq , σ1,2 = (2π/λ1 n1,2 )deff,I and σj = (ωj −2 /ω1 )(2π/λ1 nj −2 )deff,II (where j = 3, 4, 5). Here, the parameters deff,I and deff,II are the effective quadratic nonlinearities corresponding to the two steps of the multistep parametric process; their values depend on the crystal orientation (see, e.g., Dmitriev, Gurzadyan and Nikogosyan [1999]) and the method of phase matching (PM). The parameter γ is found as (3) (3) γ = (3π/4λ1 n3 )χeff (for calculation of χeff in crystals see Yang and Xie [1995]). The complementary wave vectors Gp and Gq are two vectors of the QPM structure that can be used for achieving double phase matching. A solution of this system (see, e.g., Kim and Yoon [2002], Qin, Zhu, Zhang and Ming [2003]) gives maxima for THG in several different situations, when (i) kSHG → 0; (ii) kSFG → 0; (iii) kTHG = kSHG + kSFG = k3 − 3k1 → 0; and (iv) simultaneously kSHG → 0 and kSFG → 0. The latter condition, for which the SHG and SFG steps should be phase matched simultaneously, corresponds to the highest efficiency for THG. The intensity of the third-harmonic (TH) wave is proportional to the fourth power of the crystal length,


A3 (L)
2 = 1 σ 2 σ 2 |A1 |6 L4 . (3.2) 4 2 5 This expression should be compared with the analogous result for centrosymmetric media:


A3 (L)
2 = γ 2 |A1 |6 L2 . (3.3) The advantage of the single-crystal phase-matched cascaded THG becomes clear if we note that, on average, (σ2 σ5 L)2 is 104 –106 times larger than γ 2 , even for a sample length as small as 1 mm.

12

Multistep parametric processes in nonlinear optics

[1, § 3

Introducing normalized efficiency (measured in units of W−1 cm−2 ) for the first and second steps in separate crystals as η0,1 and η0,2 , respectively, we obtain 1 η0,1 η0,2 P12 L4 . (3.4) 4 The results for THG in a single quadratic crystal under the condition of double phase matching were reported for the efficiency exceeding 20%, as shown in Table 2. Figure 3 shows the phase-matching curves in the experiment (Zhang, Wei, Zhu, Wang, Zhu and Ming [2001]) where 27% THG efficiency has been achieved. The two PM curves are not perfectly overlapping. We may expect that with an improvement of the superlattice structure the achieved efficiencies will be higher. Numerical solution of the system (3.1) shows that, in general, the third-harmonic output has an oscillating behavior as a function of the length or input power, and generally it does not reach 100% efficiency. However, as shown by (Egorov and Sukhorukov [1998], Chirkin, Volkov, Laptev and Morozov [2000], and Zhang, Zhu, Yang, Qin, Zhu, Chen, Liu and Ming [2000]), the total conversion of the fundamental wave into the third-harmonic wave is possible if the ratio σ2 /σ5 is optimized. In other cases, where only one phase-matched condition is satisfied, the THG conversion efficiency is not so large, and it should be compared to that of the direct THG process in a cubic medium. The efficiency of a cascaded THG process is inversely proportional to the square of the wave-vector mismatch of the unmatched process. It is important to note that in such cases, even when only one of the phase-matched parameters kSHG or kSFG is close to zero (as can easily be realized in birefringent phase matching in bulk quadratic media or in a uniform QPM structure) the THG process behaves similarly to the phase-matched process with the characteristic dependence [sin(x)/x]2 for the TH intensity vs. tuning and a cubic dependence on the input intensity. The dependence on the sample thickness is quadratic but not a periodic function as for the totally non-phase-matched processes. Situation (iii), that is, tuning where kSHG + kSFG = 0, corresponds to the condition for direct THG where the fundamental wave is converted directly into a third-harmonic wave. In such a case, both the cascade process and the direct process k3 = 3k1 contribute to the third-harmonic wave. However, the relative contributions of the two processes can be different. In some cases the direct THG process is stronger (see, e.g., Feve, Boulanger and Guillien [2000]), in other cases the cascading THG process is dominant (see, e.g., Banks, Feit and Perry [2002], Bosshard, Gubler, Kaatz, Mazerant and Meier [2000]). One may wonder why the whole process is phase matched when both steps are mismatched. The reason is that situation (iii) is similar to that of the quasiη3ω =

1, § 3]

Multistep phase-matched interactions

13

Table 2 Experimental results on cascaded THG processes Nonlinear crystal

λω [µm]

Phasematched steps

Phasematching method

L [cm]

Regime

η [%]

Refs.

LiTaO3

1.44

QPOS

1.5

[a]

1.570

QPOS

0.8

23%

[b, c]

LiTaO3

1.342

1.8

19.3%

[d]

LiTaO3

1.342

kSHG ≈ 0; kSFG ≈ 0

1.2

Pulsed (90 ns)

10.2%

[e]

β-BBO

1.055

0.3

Pulsed (350 fs)

6%

[f, g]

LiTaO3

1.442

QPOS

0.6

[h]

1.8

5%

[i]

LiTaO3

1.064

2.8%

[j]

KTP

1.618

Pulsed (10 ns) Pulsed (35 ps) QuasiCW (150 ns) Pulsed (22 ps)

5.8%

KTP

kTHG ≈ 0; kSFG = −kSHG = 0 kSHG ≈ 0; kSFG ≈ 0 kSHG = 0; kSFG ≈ 0 kSHG ≈ 0; kSFG ≈ 0

SHG-QPM 1st-ord. SFG-QPM 3rd-ord. SHG-QPM 1st-ord. SFG-QPM 3rd-ord. BPM

Pulsed (8 ns) Pulsed (8 ns) QuasiCW (30 ns)

27%

LiTaO3

kSHG ≈ 0; kSFG ≈ 0 kSHG ≈ 0; kSFG ≈ 0 kSHG ≈ 0; kSFG ≈ 0

2.4%

[k, l]

SBN

1.728

1.6%

[m]

d-LAP

1.055

1.2%

[f, g]

β-BBO

1.05

KTP waveguide KTP

1.234

LiNbO3 waveguide

1.619

LiNbO3

1.534

1.32

kTHG ≈ 0; kSFG = −kSHG = 0 kSHG ≈ 0; kSFG ≈ 0 kTHG ≈ 0; kSFG = −kSHG = 0 kTHG ≈ 0; kSFG = −kSHG = 0 kSHG ≈ 0; kSFG ≈ 0 kSHG = 0; kSFG ≈ 0 kSHG ≈ 0; kSFG ≈ 0 kSHG = 0; kSFG ≈ 0

BPM PR-QPM

1.2

BPM

0.11

QPOS

0.75

BPM

0.1

BPM

0.72

Pulsed (5 ps)

0.8%

[n]

QPM

0.26

0.4%a

[o]

BPM

0.47

Pulsed (9 ns) Pulsed (200 fs) Pulsed (7 ps)

0.17%

[p, q]

0.055%

[r]

Pulsed (9 ns)

0.016%a

[s]

SHG-QPM 1st-ord. SFG-QPM 3rd-ord. QPM

0.6

Pulsed (15 ps) Pulsed (350 fs)

(continued on next page)

14

Multistep parametric processes in nonlinear optics

[1, § 3

Table 2 (Continued) Nonlinear crystal

λω [µm]

Phasematched steps

Phasematching method

L [cm]

Regime

η [%]

Refs.

KTP waveguide

1.65

kSHG ≈ 0; kSFG ≈ 0

0.35

Pulsed (6 ps)

0.011%

[t]

β-BBO

1.053

0.7

Pulsed (45 ps)

0.007%

[u]

LiNbO3 2D-NPC LiNbO3

1.536

2D-QPM

1

[v]

QPM

2

Pulsed (5 ns) CW

0.01%b

3.561

KTP

1.55

QPOS

1

CW

Y:LiNbO3

1.064

kTHG ≈ 0; kSFG = −kSHG = 0 kSHG ≈ 0; kSFG ≈ 0 kSHG ≈ 0; kSFG ≈ 0 kSHG ≈ 0; kSFG ≈ 0 kSHG ≈ 0; kSFG ≈ 0

SHG-QPM 1st-ord. SFG-QPM 3rd-ord. BPM

SHG-QPM 9th-ord. SFG-QPM 33rd-ord.

0.5

Pulsed (100 ns)

10−4 [w] [% W−2 ] 3 × 10−5 [x] [% W−2 cm−2 ] [y] 10−5 %

References: [a] Zhang, Wei, Zhu, Wang, Zhu and Ming [2001]. [b] Qin, Zhu, Zhu and Ming [1998]. [c] Zhu, Zhu and Ming [1997]. [d] He, Liu, Luo, Jia, Du, Guo and Zhu [2002]. [e] Luo, Zhu, He, Zhu, Wang, Liu, Zhang and Ming [2001]. [f] Banks, Feit and Perry [1999]. [g] Banks, Feit and Perry [2002]. [h] Chen, Zhang, Zhu, Zhu, Wang and Ming [2001]. [i] Takagi and Muraki [2000]. [j] Liu, Du, Liao, Zhu, Zhu, Qin, Wang, He, Zhang and Ming [2002]. [k] Feve, Boulanger and Guillien [2000]. [l] Boulanger, Feve, Delarue, Rousseau and Marnier [1999]. [m] Zhu, Xiao, Fu, Wong and Ming [1998]. [n] Qiu and Penzkofer [1988]. [o] Gu, Makarov, Ding, Khurgin and Risk [1999]. [p] Mu, Gu, Makarov, Ding, Wang, Wei and Liu [2000]. [q] Ding, Mu and Gu [2000]. [r] Baldi, Trevino-Palacios, Stegeman, Demicheli, Ostrowsky, Delacourt and Papuchon [1995]. [s] Gu, Korotkov, Ding, Kang and Khurgin [1998]. [t] Sundheimer, Villeneuve, Stegeman and Bierlein [1994b]. [u] Tomov, Van Wonterghem and Rentzepis [1992]. [v] Broderick, Bratfalean, Monro, Richardson and de Sterke [2002]. [w] Pfister, Wells, Hollberg, Zink, Van Baak, Levenson and Bosenberg [1997]. [x] Fradkin-Kashi, Arie, Urenski and Rosenman [2002]. [y] Volkov, Laptev, Morozov, Naumova and Chirkin [1998]. (continued on next page)

1, § 3]

Multistep phase-matched interactions

15

Table 2 (Continued) Abbreviations: QPOS – quasi-periodical optical superlattices. BPM – birefringence phase matching. PR-QPM – phase reversed QPM structure (see Chou, Parameswaran, Fejer and Brener [1999]). 2D-NPC – two-dimensional photonic crystals. 2D-QPM – two-dimensional QPM structure. a Backward THG. b Type II cascaded THG.

Fig. 3. (Top) Schematic of the experimental setup. (Bottom) Average powers of the second harmonic (right-hand scale) and third harmonic (left-hand scale) vs. temperature. Average power of the fundamental wave is 4.8 mW (Zhang, Wei, Zhu, Wang, Zhu and Ming [2001]).

phase-matching effect (see, e.g., Reintjes [1984], Banks, Feit and Perry [1999], Durfee, Misoguti, Backus, Kapteyn and Murnane [2002]). Indeed, if the first step is mismatched, inside a nonlinear medium it generates a periodically modulated polarization at frequency 2ω (P (2ω)) and the modulation period is exactly the

16

Multistep parametric processes in nonlinear optics

[1, § 3

mismatch of the second step. Thus, in the regions where the phase of the polarization P (2ω) is reversed, we have the minimum generation of second- and third-harmonic waves, and the generated TH components interfere constructively with the propagating third-harmonic wave along the length of the crystal. This leads to a quadratic dependence of the THG efficiency on the crystal length. For the cascaded THG processes, optimal focusing is an important issue when the goal is maximum conversion efficiency. If only one of the steps is phase matched, the optimal focusing is at the input face when SFG is phase matched, but at the output face when SHG is phase matched (Rostovtseva, Sukhorukov, Tunkin and Saltiel [1977], Rostovtseva, Saltiel, Sukhorukov and Tunkin [1980]). If both steps are phase matched, the optimum focusing position is at the center of the nonlinear medium (Ivanov, Koynov and Saltiel [2002]). In Table 2 we summarize the experimental results obtained for efficient singlecrystal THG processes. Conditions at which the third-harmonic wave included in double-phase-matched interaction can be transformed with 100% efficiency into the 2ω wave or ω wave were found by Komissarova and Sukhorukov [1993], Komissarova, Sukhorukov and Tereshkov [1997], and Egorov and Sukhorukov [1998]. Volkov and Chirkin [1998] have shown that the same parametric interaction can be used for 100% conversion of a 2ω wave into a 3ω wave. Considering situations with nonzero SH and TH boundary conditions it has to be taken into account that, as shown by Alekseev and Ponomarev [2002], the spatial evolution of three light waves participating simultaneously in SHG and SFM under the conditions of QPM double phase matching becomes chaotic at large propagation distances for many values of the complex input wave amplitudes. Thus, the possibility of transition to chaos exists if an additional pump at frequency 2ω is applied in order to increase the efficiency of THG (Egorov and Sukhorukov [1998]). As shown by Longhi [2001b] for the multistep cascading process {ω + ω = 2ω; ω + 2ω = 3ω} in a cavity, the formation of spatial patterns is possible. Another application of multistep cascading interaction and, in particular, THG multistep cascading is the generation of entangled and squeezed quantum states (see Section 3.5). The conditions for the simultaneous phase matching of both SHG and SFG processes have been considered by Pfister, Wells, Hollberg, Zink, Van Baak, Levenson and Bosenberg [1997] and Grechin and Dmitriev [2001a] for uniform QPM structures; by Zhu and Ming [1999] for quasi-periodic optical superlattices; by Fradkin-Kashi and Arie [1999] and Fradkin-Kashi, Arie, Urenski and Rosenman [2002] for generalized Fibonacci structures; and by Saltiel and Kivshar [2000a] for two-dimensional nonlinear photonic crystals. This topic will be discussed in Section 4.

1, § 3]

Multistep phase-matched interactions

17

3.1.2. Nonlinear phase shift in multistep cascading The multistep parametric process, that combines two phase-matched interaction SHG (ω + ω = 2ω) and SFG (2ω + ω = 3ω), is described by the system of eqs. (3.1), and it can be used for all-optical processing and the formation of parametric solitons. Indeed, the nonlinear phase shift (NPS) accumulated by the fundamental wave in the multistep cascading process is several times larger than that in the standard cascading interaction that involves only one phase-matched process (Koynov and Saltiel [1998]). To illustrate this result, we consider a fundamental wave with frequency ω entering a second-order nonlinear media under appropriate phase-matching conditions. In the first step, a wave with frequency 2ω is generated via the type I SHG process, and then, in the second step, the third-harmonic wave is generated via the SFG process (2ω + ω = 3ω). Both the SHG and the SFG process are assumed to be nearly phase matched. The generated second- and third-harmonic waves are down-converted to the fundamental wave ω via the processes (2ω−ω), (3ω−2ω), and (3ω − ω, 2ω − ω), all contributing to the nonlinear phase shift of the fundamental wave. As follows from numerical calculations, the total NPS is a result of the simultaneous action of two-, three-, and four-step χ (2) cascading, and it can exceed the value of π for relatively low input intensities. The possible channels for the phase modulation and NPS of the fundamental wave are shown in fig. 4. The interpretation of the analytical result obtained in the fixed-intensity approxi-

Fig. 4. Schematic of the possible channels for the phase modulation of the fundamental wave (Koynov and Saltiel [1998]).

18

Multistep parametric processes in nonlinear optics

[1, § 3

mation (Koynov and Saltiel [1998]) shows that the effective cascaded fifth-order and higher-order nonlinearities are involved in the accumulation of the total NPS, and the signs of the contributions from different processes can be controlled by a small change of the phase-matching conditions. Schematically, the role of the multistep cascading with three and four steps can be interpreted as equivalent to a contribution from the higher-order nonlinear corrections n4 and n6 to the refractive index, and they can be linked to the cascaded χ (5) and χ (7) processes, respectively. Indeed, for a relatively low intensity of the fundamental wave, the refractive index can be written as the expansion n(E) = n0 + n2 E 2 + n4 E 4 + n6 E 6 + · · · .

(3.5)

The advantage of multistep cascading over conventional two-step cascading for accumulating a large nonlinear phase shift is illustrated in fig. 5, obtained by numerical integration of the system (3.1). This result is also useful for studying the multi-color solitons supported by this type of multistep parametric interaction (Kivshar, Alexander and Saltiel [1999], Huang [2001]). We note again that THG multistep cascading can be efficient only if the two processes can be phasematched simultaneously. Due to dispersion, it is generally impossible to achieve

Fig. 5. Nonlinear phase shift and depletion of the fundamental wave as a function of its normalized input amplitude: solid line, multistep cascading; dashed line, type I SHG case (Koynov and Saltiel [1998]).

1, § 3]

Multistep phase-matched interactions

19

double phase matching in a bulk material. However, owing to the recent progress in the design of QPM structures, double phase matching can be achieved in nonlinear photonic materials (Zhang, Wei, Zhu, Wang, Zhu and Ming [2001], Luo, Zhu, He, Zhu, Wang, Liu, Zhang and Ming [2001], Fradkin-Kashi, Arie, Urenski and Rosenman [2002]), although a number of technical problems remain to be solved.

3.2. Wavelength conversion So far we have discussed only the multistep cascading process for generating the third harmonic by combining the SHG and SFG phase-matched parametric interactions in a single crystal. Let us now consider another single-crystal multistep process that combines SHG and DFM and mimics in this way the four-wave mixing (FWM) process with two input waves ωp and ωsig , resulting in the generation of a signal at ωout = 2ωp − ωsig . The principle of this type of multistep cascading is illustrated in fig. 6. Usually, the difference λsig − λp is less than 50 nm (with λp ∼ 1550 nm), leading to the result that when the SHG process is phase matched the DFM process is very close to exact phase matching. Here we have the situation of double-phase-matched multistep cascading. This allows a very efficient conversion from λsig to 2λp − λsig which is 104 –105 times larger than what can be obtained with the direct FWM process. More specifically, let us consider the parametric interaction shown in fig. 6 which is described by the following system

Fig. 6. Wavelength conversion of optical communication channels in a periodically poled nonlinear crystal using SHG/DFG multistep cascading.

20

Multistep parametric processes in nonlinear optics

[1, § 3

of parametrically coupled equations: dAp = −iσ1 A2 A∗p e−ikSHG z , dz dA2 = −iσ2 A2p eikSHG z − iσ3 Asig Aout eikDFG z , dz

(3.6) dAout ∗ −ikDFG z = −iσ4 A2 Asig e , dz dAsig = −iσ5 A2 A∗out e−ikDFG z , dz where kSHG = k2 − 2k1 + Gm and kDFG = k2 − ksig − kout + Gm , and σ1 through σ5 are coupling coefficients proportional to the second-order nonlinearity parameter deff . The vector Gm is one of the QPM vectors used for achieving phase matching (Fejer, Magel, Jundt and Byer [1992]) in the QPM structure. If birefringence phase matching is used, then Gm = 0. Two phase-matching parameters, i.e. kSHG and kDFG , are involved in this parametric cascaded interaction. However, if the signal wavelength is sufficiently close to that of the pump, i.e. |λsig − λp | λp , the tuning curves for the two processes are practically coincident, and in this situation, if one of the parametric processes is phase matched, the other one is also phase matched. In other words, in this case the signal wavelength is sufficiently close to that of the pump, and we work under the conditions of double phase matching. If we neglect depletion, the amplitude of the phasematched SH wave can be found as A2 (z) = −iσ2 A2p z. Then, the output signal can be written in the form 4




Aout (L)
2 4σ 2 σ 2
A2
2 |Asig |2 sin (kDFG L/2) . p 2 4 (kDFG )4

(3.7)

In the limit kDFG → 0, the efficiency becomes




Aout (L)
2 = 1 σ 2 σ 2
A2
2 |Asig |2 L4 , p 2 4 4

(3.8)

or ηout =

1 2 2 4 η P L , 4 0 p

(3.9)

where η0 is the normalized efficiency measured in W−1 cm−2 ; it depends on the overlap integral between the interacting modes and the effective nonlinearity deff , and it is the same normalized value that describes the efficiency of the first step, the SHG process (Chou, Brener, Fejer, Chaban and Christman [1999]). From eq. (3.7) it follows that the output signal is a linear function of the input signal – an important property for communications. Also, the efficiency is proportional

1, § 3]

Multistep phase-matched interactions

21

to the fourth power of the crystal length L. Detailed theoretical descriptions of this cascading process can be found in articles by Gallo, Assanto and Stegeman [1997], Gallo and Assanto [1999], and Chen, Xu, Zhou and Tang [2002]. For the DFG process, the width of the phase-matching curve depends on the type of crystal, its length, and the phase-matching method. Several proposals have been made for increasing the width of the phase-matching region, including the use of phase-reversed QPM structures (Chou, Parameswaran, Fejer and Brener [1999]); pump deviation from exact phase matching (Chou, Brener, Parameswaran and Fejer [1999]); periodically chirped (phase-modulated) QPM structures (Asobe, Tadanaga, Miyazawa, Nishida and Suzuki [2003]); and the phaseshifting domain (Liu, Sun and Kurz [2003]). In particular, Gao, Yang and Jin [2004] report that the use of sinusoidally chirped QPM superlattices provides broader bandwidth and a flatter response compared to homogeneous and segmented QPM structures. Novel cascaded χ (2) wavelength-conversion schemes are based on the SFG and DFG processes and the use of two pump beams, as proposed and demonstrated by Xu and Chen [2004], Chen and Xu [2004]. The conversion efficiency is enhanced by 6 dB, as compared with the conventional cascaded SHG + DFG wavelength conversion configuration. The cascading steps in this SFG + DFG wavelengthconversion method are ωSF = ωp1 + ωp2 and ωout = ωSF − ωsig , so that the effective third-order interaction is the totally nondegenerate FWM process: ωout = ωp1 + ωp2 − ωsig . Second-order cascading wavelength conversion is one of the best examples of the richness of the multistep cascading phenomena. In the initial stage of development, this concept was experimentally demonstrated in media employing two types of phase-matching techniques, i.e. birefringence phase matching in a bulk crystal (e.g., Tan, Banfi and Tomaselli [1993], Banfi, Datta, Degiorgio, Donelli, Fortusini and Sherwood [1998]) and periodically poled nonlinear LiNbO3 crystal (Banfi, Datta, Degiorgio, Donelli, Fortusini and Sherwood [1998]). Since then, this parametric process has been extensively investigated not only as an interesting multistep parametric effect that may occur in different nonlinear media (see Table 3), but also as a tool for realizing all-optical communication devices (Chou, Brener, Fejer, Chaban and Christman [1999], Chou, Brener, Lenz, Scotti, Chaban, Shmulovich, Philen, Kosinski, Parameswaran and Fejer [2000], Kunimatsu, Xu, Pelusi, Wang, Kikuchi, Ito and Suzuki [2000]). Today, there is real progress in suggesting this device for use in the optical communication industry (Cardakli, Gurkan, Havstad, Willner, Parameswaran, Fejer and Brener [2002], Cardakli, Sahin, Adamczyk, Willner, Parameswaran and Fejer [2002]), because of its clear advantages over all other devices used for wavelength shifting. Indeed, one such

22

Multistep parametric processes in nonlinear optics

[1, § 3

Table 3 Experimental works on the multistep SHG and DFM cascading Nonlinear crystal

λp (λsmax ) [µm]

L [cm]

Regime

Phasematching method

Refs.

BBO

1.064 (1.090) 1.064 (1.090) 1.533 (1.535) 1.8 (1.863)

1

Pulsed (30 ps) Pulsed (30 ps) CW & pulsed (7 ps) Pulsed (20 ps) Pulsed (20 ps) CW

BPM

[a]

BPM

[b]

QPM

[c]

QPM

[d]

NCPM

[e]

QPM

[f]

Pulsed (20 ps) CW

NCPM

[g, h]

QPM

[i]

CW & pulsed (6 ps) CW

QPM

[j]

QPM

[k]

CW & pulsed CW

QPM

[l]

QPM

[m]

6

CW

QPM

[n]

6

CW

QPM

[o]

2

CW

QPM

[p]

4.5

CW

QPM

[q]

3.4

CW

[r]

5

CW

PM QPMa QPM

3

CW

QPMb

[t]

5

CW

QPM

[u]

MBA-NP LiNbO3 waveguide LiNbO3 NPP LiNbO3 waveguide Ti:LiNbO3 waveguide LiNbO3 waveguide Ti:LiNbO3 waveguide LiNbO3 waveguide LiNbO3 waveguide LiNbO3 waveguide LiNbO3 waveguide Ti:LiNbO3 waveguide LiNbO3 waveguide LiNbO3 waveguide LiNbO3 waveguide MgO:LiNbO3 waveguide Ti:LiNbO3 waveguide LiNbO3 waveguide

1.148 (1.158) 1.562 (1.600) 1.103 (1.107) 1.545 (1.580) 1.556 (1.565) 1.565 (1.585) 1.542 (1.562) 1.550 (1.560) 1.553 (1.565) 1.557 (1.553) 1.532 (1.565) 1.537 (1570) 1.558..1.568 (1.600) 1.543 (1.573) 1.55 (1.62) 1.545 (1.580)

0.32 1 1.9 0.28 4 5.8 5 7.8 (8.6) 2 1

[s]

(continued on next page)

1, § 3]

Multistep phase-matched interactions

23

Table 3 (Continued) References: [a] Tan, Banfi and Tomaselli [1993]. [b] Nitti, Tan, Banfi and Degiorgio [1994]. [c] Trevino-Palacios, Stegeman, Baldi and De Micheli [1998]. [d] Banfi, Datta, Degiorgio and Fortusini [1998]. [e] Banfi, Datta, Degiorgio, Donelli, Fortusini and Sherwood [1998]. [f] Chou, Brener, Fejer, Chaban and Christman [1999]. [g] Cristiani, Banfi, Degiorgio and Tartara [1999]. [h] Banfi, Christiani and Degiorgio [2000]. [i] Chou, Brener, Lenz, Scotti, Chaban, Shmulovich, Philen, Kosinski, Parameswaran and Fejer [2000]. [j] Schreiber, Suche, Lee, Grundkotter, Quiring, Ricken and Sohler [2001]. [k] Cristiani, Liberale, Degiorgio, Tartarini and Bassi [2001]. [l] Ishizuki, Suhara, Fujimura and Nishihara [2001]. [m] Cardakli, Sahin, Adamczyk, Willner, Parameswaran and Fejer [2002]. [n] Harel, Burkett, Lenz, Chaban, Parameswaran, Fejer and Brener [2002]. [o] Cristiani, Degiorgio, Socci, Carbone and Romagnoli [2002]. [p] Zeng, Chen, Chen, Xia and Chen [2003]. [q] Zhou, Xu and Chen [2003]. [r] Asobe, Tadanaga, Miyazawa, Nishida and Suzuki [2003]. [s] Bracken and Xu [2003]. [t] Gao, Yang and Jin [2004]. [u] Sun and Liu [2003]. a Phase modulated QPM. b Three types of QPM grating are compared: homogeneous, segmented, and sinusoidally chirped.

device can simultaneously shift several channels. As shown in fig. 6, the spectrum of the shifted signal is a mirror image of the original. This feature can be used to invert the signal chirp for dispersion management in transmission systems. A successful experimental demonstration of this property has been reported by Kunimatsu, Xu, Pelusi, Wang, Kikuchi, Ito and Suzuki [2000]: a 600-fs pulse transmission over 144 km using midway frequency inversion with this type of second-order cascaded wavelength conversion resulted in a negligible pulse distortion. Waveguides made in LiNbO3 crystals (see fig. 7) are at present the most suitable nonlinear structures for the cascaded simulation of FWM wavelength shifting. The important features of this device are an almost perfect linear dependence between the input and output signals for more than 30 dB of the dynamic range, instantaneous memoryless transparent wavelength shifting that can be used at rates of several teraHertz, and transparent crosstalk-free operation (Cardakli, Sahin, Adamczyk, Willner, Parameswaran and Fejer [2002]). Additionally, Couderc, Lago, Barthelemy, De Angelis and Gringoli [2002] demonstrated that the wavelength-conversion multistep cascading system can support

24

Multistep parametric processes in nonlinear optics

[1, § 3

Fig. 7. (Right) Periodically poled LiNbO3 crystal for the cascaded wavelength conversion. (Left) Wavelength conversion with 1545 nm pump and four inputs in the range 1555–1560 nm (Chou, Brener, Lenz, Scotti, Chaban, Shmulovich, Philen, Kosinski, Parameswaran and Fejer [2000]).

parametric solitons in the waveguiding regime; we discuss these results in more detail in Section 5.3. Experimental works reporting second-order cascaded wavelength conversion are summarized in Table 3. 3.3. Two-color multistep cascading By two-color multistep cascading in a quadratic medium we mean multi-phasematched parametric interaction between several waves of only two frequencies (or wavelengths). One way to introduce a parametric process involving more than one phase-matched interaction with two wavelengths is to consider vectorial interaction between waves with different polarizations, or degenerate interaction between allowed modes in a waveguide. We denote two waves at the fundamental frequency (FF) (at λ = λfund ) by A and B, and the two waves of the secondharmonic (SH) field (at λsh = λfund /2), by S and T. Each pair of eigenmodes [(A, B) and (S, T)] can be, for example, two orthogonal polarization states or two different waveguide modes at the fundamental and second-harmonic wavelengths, respectively. There exists a finite number of possible multistep parametric interactions that can couple these waves. For example, if we consider the AA–S & AB–S cascading, then the multistep cascading is composed of the following subprocesses. First, the fundamental wave A generates the SH wave S via the type I SHG process. Then, by the down-conversion process SA–B, the other fundamental eigenmode B is generated. Finally, the initial FF wave A is reconstructed by the processes SB–A or AB–S, SA–A. When we deal with two orthogonal polarizations, the two principal second-order processes AA–S and AB–S are governed by two different components (or two different combinations of components) of the χ (2) susceptibility tensor, thus introducing additional degrees of freedom into the parametric interaction. The classification of different types of multistep paramet-

1, § 3]

Multistep phase-matched interactions

25

Table 4 Two-color multistep cascading processes Multistepcascading schemes

No of waves

SHG processes

Equivalent cascading schemes

WC/SC

Refs.

AA–S:AB–S

3

Type I & type II

SC

[a, b, c]

AA–S:AB–T

4

Type I & type II

WC

[d, e]

AA–S:BB–S AA–S:AA–T AB–S:AB–T AA–S:BB–S:AB–S

3 3 4 3

Type I & type I Type I & type I Type II & type II Type I & type II

BB–S:AB–S; AA–T:AB–T; BB–T:AB–T BB–S:AB–T; AA–T:AB–S; BB–T:AB–S AA–T:BB–T BB–S:BB–T

WC SC SC

[f, g, h, i] [h, j]

AA–T:BB–T:AB–T

[k, l, m]

References: [a] Saltiel and Deyanova [1999]. [b] Saltiel, Koynov, Deyanova and Kivshar [2000]. [c] Petrov, Albert, Etchepare and Saltiel [2001], Petrov, Albert, Minkovski, Etchepare and Saltiel [2002]. [d] DeRossi, Conti and Assanto [1997]. [e] Pasiskevicius, Holmgren, Wang and Laurell [2002]. [f] Assanto, Torelli and Trillo [1994]. [g] Kivshar, Sukhorukov and Saltiel [1999]. [h] Grechin, Dmitriev and Yur’ev [1999], Grechin and Dmitriev [2001b]. [i] Grechin and Dmitriev [2001b]. [j] Trevino-Palacios, Stegeman, Demicheli, Baldi, Nouh, Ostrowsky, Delacourt and Papuchon [1995]. [k] Trillo and Assanto [1994]. [l] Towers, Sammut, Buryak and Malomed [1999], Towers, Buryak, Sammut and Malomed [2000]. [m] Boardman and Xie [1997], Boardman, Bontemps and Xie [1998].

ric interactions has been introduced by Kivshar, Sukhorukov and Saltiel [1999] and Saltiel, Koynov, Deyanova and Kivshar [2000]. The different types of multistep parametric interactions can be divided into two major groups. The first group consists of parametric interactions with two common waves in both cascading processes, and the two processes are strongly coupled (SC). For the other group, the two parametric processes have one common wave, and these processes are weakly coupled (WC). The same classification can be applied to other types of multistep parametric interactions. In Table 4 we present an updated classification of two-color multistep parametric processes and the publications in which they are analyzed. The first analysis of multistep parametric interactions of this type has been carried out by Assanto, Torelli and Trillo [1994] and Trillo and Assanto [1994]. These authors studied the simultaneous phase matching of two SHG processes of

26

Multistep parametric processes in nonlinear optics

[1, § 3

the type AA–S and BB–S, where S denotes the SH wave whereas A and B stand for FF waves polarized in perpendicular planes. The two orthogonal FF fields interact through the generated SH wave. All-optical operations and polarization switching can be performed on the base of this scheme. Kivshar, Sukhorukov and Saltiel [1999] demonstrated that this two-color parametric interaction can support two-color spatial solitary waves. Trevino-Palacios, Stegeman, Demicheli, Baldi, Nouh, Ostrowsky, Delacourt and Papuchon [1995] considered the interference between the parametric processes AA–S and AA–T, where S and T are two different modes of the waveguide at frequency 2ω. In both cases mentioned above, two different parametric processes share one and the same fundamental wave. However, it is possible that the multistep cascading interaction involves the SHG process of the type AB–S. Such a four-wave multistep cascading process was considered by DeRossi, Conti and Assanto [1997]. The two SHG processes were AB–S and AA–T, where (A, B) and (S, T) are pairs of modes at frequencies ω and 2ω, respectively. DeRossi, Conti and Assanto [1997] concluded that a device based on this type of multistep cascading can operate as an all-optical modulator or as an all-optical switch, with good switching contrast at 1.55 µm. Another interesting multistep interaction process was considered by Boardman and Xie [1997] and Boardman, Bontemps and Xie [1998] who studied parametric mode coupling in a nonlinear waveguide placed in a magnetic field. In this case, the simultaneous coexistence of six SHG processes is possible, namely, ooo, ooe, oee, eee, eeo, and eoo. The exact number of allowed parametric interactions depends on the symmetry point group of the material. It was shown that, by controlling the ratio of the input fundamental components, one of the SH components can be controlled and switched off. Importantly, repulsive and collapsing regimes for the interacting parametric solitons can be produced by switching the direction of the magnetic field. Saltiel and Deyanova [1999] considered the possibility of efficient polarization switching in a quadratic crystal that supports simultaneous phase matching for both type I and type II SHG processes (e.g., ooo and oeo). Under certain conditions, the fundamental beam involved in such a process can accumulate a large nonlinear phase shift at relatively low input power (Saltiel, Koynov, Deyanova and Kivshar [2000]). A SHG process that can be realized by two possible pairs of simultaneously phase-matched processes (ooo, ooe) and (ooe, eee) was studied theoretically by Grechin, Dmitriev and Yur’ev [1999]. SHG by simultaneous phase matching of three parametric processes (ooe, eee, and oee) in a crystal of LiNbO3 was considered theoretically by Grechin and Dmitriev [2001b]. It was shown that, under certain conditions, one can obtain polarization-insensitive SHG.

1, § 3]

Multistep phase-matched interactions

27

Also, we would like to mention several experimental studies of two-color multistep cascading interactions. Trevino-Palacios, Stegeman, Demicheli, Baldi, Nouh, Ostrowsky, Delacourt and Papuchon [1995] observed an interplay of two type I SHG processes with a common fundamental wave. Petrov, Albert, Etchepare and Saltiel [2001] and Petrov, Albert, Minkovski, Etchepare and Saltiel [2002] performed an experiment with a BBO crystal in which, as a result of the simultaneous action of type I SHG and type II SHG interactions, the generation of a wave orthogonally polarized to the input fundamental wave was observed. Couderc, Lago, Barthelemy, De Angelis and Gringoli [2002] demonstrated that multistep cascading interaction can support parametric solitons in the waveguiding regime. In this latter case, the multistep interaction simulates an effective nearly degenerate FWM process and, in this sense, it is almost “two-color”. Pasiskevicius, Holmgren, Wang and Laurell [2002] realized experimentally a simultaneous SHG process that gives two SH waves with orthogonal polarizations in the blue spectral region by use of type II and type I QPM phase matching in a periodically poled KTP crystal. As a simple example of two-color multistep parametric interaction, we consider the cascading of type I and type II SHG processes according to the scheme BB–S and AB–S. Here, the SH wave is generated by two interactions. Thus, we can expect an increase of the SHG efficiency when both A and B waves (i.e. the ordinary and extraordinary waves) are involved. However, if only one of the waves, say A, is launched at the input, this type of double-phase-matched interaction will lead to the generation of a wave perpendicular to the input wave, through the parametric process SA → B. Additionally, the fundamental wave A accumulates a strong nonlinear phase shift. This parametric interaction is described by the following equations for plane waves: dA = −iσ1 SA∗ e−ikSHG z − iσ3 SB ∗ e−ikDFG z , dz dS = −iσ2 A2 eikSHG z − iσ4 ABeikDFG z , dz dB = −iσ5 SA∗ e−ikDFG z , dz

(3.10)

where A, S, and B are the complex amplitudes of the input fundamental wave, the second-harmonic wave and the orthogonally polarized wave at the fundamental frequency, σ1 and σ2 have been defined above, and   2πdeff,II ω1,2,1 . σ3,4,5 = λ1 nA,2,B ω1

28

Multistep parametric processes in nonlinear optics

[1, § 3

If we neglect dispersion of the index of refraction, i.e. nA nB n2 , then we can assume that σ1 σ2 and σ3 σ5 σ4 /2. The phase-mismatch parameters are defined as kSHG = k2 − 2kA + Gp and kDFG = k2 − kA − kB + Gq , where Gp and Gq are two QPM vectors used for the phase matching. A solution of this system, with respect to the amplitude of wave B, has been obtained by Saltiel and Deyanova [1999] in the approximation of a nondepleted pump, with the following result: B(L) =

iσ1 σ3 |A|2 A sin(D− L)eiD+ L , D− kSHG

(3.11)

where D± =

σ12 1 (kSHG − kDFG ) ± |A|2 . 2 kSHG

Equation (3.11) shows that the generation of component B by this multistep interaction mimics a FWM process of the type AAA* –B, governed by the cascaded cubic nonlinearity. This effective cubic nonlinearity leads to the accumulation of a nonlinear phase shift by the wave A that, similar as for the cascaded THG process, includes a contribution of high-order (> 3) nonlinearities (Saltiel, Koynov, Deyanova and Kivshar [2000]). The existence of two-color multistep parametric solitons and waveguiding effects is the result of the nonlinear phase shift collected by the interacting waves (Kivshar, Sukhorukov and Saltiel [1999]).

3.4. Fourth-harmonic multistep cascading Another fascinating example of multistep cascading interaction is fourth-harmonic generation (FHG) in a single crystal with a second-order nonlinearity. Two possible second-order parametric processes lead to the generation of a fourth harmonic. In both cases the fourth-harmonic (FH) amplitude is proportional to the factor (χ (2) )3 : 1. SHG + SHG: ω + ω = 2ω; 2ω + 2ω = 4ω; 2. SHG + SFG + SFG: ω + ω = 2ω; 2ω + ω = 3ω; 3ω + ω = 4ω. The phase-matching conditions define which of these processes will be more effective. Obviously, the former process is easier to realize technically because it requires only two phase-matching conditions to be fulfilled simultaneously, and therefore many papers deal with this case. The first experiment on the generation of a fourth-harmonic wave by cascading was reported by Akhmanov, Dubovik, Saltiel, Tomov and Tunkin [1974], where

1, § 3]

Multistep phase-matched interactions

29

the FHG cascaded process in a lithium formate crystal was studied. As pointed out by the authors, the generation is due to the simultaneous action of two and three processes with the involvement of the quadratic and cubic nonlinearities. The FHG process in CdGeAs2 was observed by Kildal and Iseler [1979]. Both these pioneer papers were motivated by the idea to estimate the magnitude of the direct fourth-order nonlinearity in terms of (χ (2) )3 . One of the first studies that reported the efficiency of the cascaded FHG process was by Hooper, Gauthier and Madey [1994], where an efficiency of 3.3 × 10−4 % was obtained in a single LiNbO3 crystal. Another paper, by Sundheimer, Villeneuve, Stegeman and Bierlein [1994b], reported an efficiency of 0.012%. However, in those experiments only the first step ω + ω = 2ω was phase-matched while the other step was not matched, leading to the low overall conversion efficiency. A somewhat larger efficiency of 0.066% was reported by Baldi, Trevino-Palacios, Stegeman, Demicheli, Ostrowsky, Delacourt and Papuchon [1995], who used a periodically poled LiNbO3 waveguide. Both steps of the multistep parametric interaction, namely ω + ω = 2ω and 2ω + 2ω = 4ω, were phase matched: the first step was realized through the first-order QPM process, while the second step was realized through the 7th-order QPM process. As was shown by Norton and de Sterke [2003b] and Sukhorukov, Alexander, Kivshar and Saltiel [2001], if both steps are phase matched, the resulting efficiency should be close to 100%. In the latter paper, the existence and stability of the normal modes for such a multistep cascading system were studied. Some possibilities for the double-phase-matched FHG process for certain input wavelengths in single crystals of LiNbO3 , LiTaO3 , KTP and GaAs have been shown by Pfister, Wells, Hollberg, Zink, Van Baak, Levenson and Bosenberg [1997] and Grechin and Dmitriev [2001a]. The possibility of FHG by double phase matching in a broader spectral region by use of a phase-reversed QPM structure was discussed by Sukhorukov, Alexander, Kivshar and Saltiel [2001]. In a very interesting experiment, Broderick, Bratfalean, Monro, Richardson and de Sterke [2002] demonstrated cascaded FHG in a 2D nonlinear photonic crystal, with efficiency 0.01% in a (not optimized) 2D planar QPM structure. Several useful efficient schemes for FHG in 2D nonlinear photonic crystals have been proposed by Saltiel and Kivshar [2000a] and de Sterke, Saltiel and Kivshar [2001]. The optimal design of 2D nonlinear photonic crystals for achieving the maximum efficiency of FHG has been discussed by Norton and de Sterke [2003a, 2003b]. The use of FHG multistep interaction for the frequency division schemes (4:1) and (4:2) has been discussed by Dmitriev and Grechin [1998] and Sukhorukov, Alexander, Kivshar and Saltiel [2001].

30

Multistep parametric processes in nonlinear optics

[1, § 3

The basic equations describing the FHG parametric process in a double-phasematched QPM structure for the interaction of plane waves can be written in the form (see, e.g., Hooper, Gauthier and Madey [1994]) dA1 = −iσ1 A2 A∗1 e−ikSHG z , dz dA2 = −iσ2 A21 eikSHG z − iσ6 A4 A∗2 e−ik4 z , dz dA4 = −iσ7 A22 eik4 z , dz

(3.12)

where A1 , A2 , and A4 are the complex amplitudes of the fundamental, secondharmonic, and fourth-harmonic waves, respectively. The parameters σ1 and σ2 have been defined above, and σ6,7 =

4π deff,II . λ1 n2,4

As before, if we neglect the index of refraction dispersion (n1 n2 n4 ) then we can accept that σ1 σ2 and σ6 σ7 . The phase-mismatch parameters are kSHG = k2 − 2k1 + Gp and k4 = k4 − 2k2 + Gq , where Gp and Gq are two QPM reciprocal vectors. The contributions of high-order nonlinearities are not included in (3.12) since they are rather small when one works in conditions close to double or triple phase matching. A solution of the system (3.12) can be found neglecting the depletion effects. It reveals that a phase-matched FHG wave is generated when either one of the following phase-matching conditions is satisfied: 1. kSHG → 0; 2. k4 → 0; 3. kSHG + k4 = k4 − k2 − 2k1 + Gp + Gq → 0; or 4. 2kSHG + k4 = k4 − 4k1 + 2Gp + Gq → 0. Of these, the case kSHG → 0 gives the strongest phase-matching process, and the generated FH wave exceeds by several orders of magnitude the FH generated by other schemes. For the double-phase-matching process (when kSHG → 0 and k4 → 0) the squared amplitude of the fourth harmonic is given by the expression (de Sterke, Saltiel and Kivshar [2001])


A4 (L)
2 = 1 σ 4 σ 2 |A1 |8 L6 . 9 2 6

(3.13)

Again introducing the normalized efficiency (measured in W−1 cm−2 ) for the first and second steps as η0,1 and η0,2 , respectively, we can obtain the efficiency

1, § 3]

Multistep phase-matched interactions

31

of the cascaded FHG process: η4ω =

1 2 η η0,2 P13 L6 . 9 0,1

(3.14)

Thus, the efficiency of the cascaded FHG process in a single χ (2) is proportional to the 6th power of the length and the 3rd power of the pump, and it can be estimated with the known efficiencies for the separated steps. For pump intensities at which the depletion effect for the fundamental and second-harmonic waves can not be neglected, the system (3.12) should be solved numerically. As shown by Zhang, Zhu, Zhu and Ming [2001], a 100% conversion of the fundamental wave into the fourth-harmonic wave is possible independently of the ratio of the nonlinear coupling coefficients σ2 /σ6 . This behavior is in contrast with the χ (2) -based cascaded THG process where 100% conversion is possible only for a specific ratio of the nonlinear coupling coefficients (see Section 3.1).

3.5. OPO and OPA multistep parametric processes Many authors dealing with optical parametric oscillators (OPOs) and optical parametric amplifiers (OPAs) have observed, in addition to the expected signal and idler waves, other waves at different wavelengths that are coherent and are generated with good efficiency. These waves were identified as being the result of additional phase-matched (or nearly phase-matched) second-order parametric processes in OPOs and OPAs. In some cases, up to five additional waves emerging from the nonlinear crystal were detected and measured. By tuning the wavelength of the signal and idler waves simultaneously, the additional outputs from OPO can be frequency-tuned as well. An example is shown in fig. 8, adapted from Zhang, Hebling, Kuhl, Ruhle, Palfalvi and Giessen [2002]. The study of simultaneous phase matching in OPOs was started more than 30 years ago. As shown by Ammann, Yarborough and Falk [1971], in many nonlinear crystals the phasematching tuning curves for OPO and SHG of the signal and the idler cross, and experimental observation of the corresponding double-phase-matching parametric processes is possible. A plane-wave single-pass theory of the new frequency generation with OPO with the simultaneous frequency doubling of the signal wave in the OPO crystal was presented by Aytur and Dikmelik [1998]. The case of OPO with simultaneous frequency mixing between the pump and signal waves was dealt with by Dikmelik, Akgun and Aytur [1999] and Morozov and Chirkin [2003]. Moore, Koch, Dearborn and Vaidyanathan [1998] considered theoretically a simultaneous phase-matched tandem of OPOs in a single nonlinear crystal. The signal wave of

32

Multistep parametric processes in nonlinear optics

[1, § 3

Fig. 8. The tuning characteristics of OPO with the simultaneously phase-matched SHG and SFG processes. (a) SFG between the signal and pump, and (b) SHG of the signal. (Inset) Output power vs. wavelength (Zhang, Hebling, Kuhl, Ruhle, Palfalvi and Giessen [2002]).

the first OPO process becomes the pump wave of the second process. The simultaneous action of the parametric generation, ωp → ωi +ωs , and SFG between the pump and signal wave, ωSF = ωp + ωs , was also analyzed by Huang, Zhu, Zhu and Ming [2002], whose main goal was to show that this type of multistep cascading can be used for the simultaneous generation of three fundamental colors. In Table 5 we have summarized different experimental works on multistep cascading processes observed in OPOs and OPAs. For each of these works, we also mention the additional second-order parametric processes.

1, § 3]

Multistep phase-matched interactions

33

Table 5 Experimental works on the OPO and OPA multistep cascading Nonlinear crystal

λpump [µm]

Phase-matching processes

Phasematching method

L [cm]

Regime

Refs.

ADP

1.06

ω1 = ωp + ωs

BMP

5

[a]

LiNbO3

1.06

BMP

0.38

LiNbO3

1.064

BPM

3 (5)

KTP

0.79

BPM

0.115

LiNbO3 BBO

0.532 0.74–0.89

ω1 = ωs + ωs or ω2 = ωi + ωi ω1 = ωs + ωs ω2 = ω1 − ωi ω1 = ωs + ωs ω2 = ωp + ωs ω3 = ωp + ωi ωp = 12 ωp + 12 ωp ω1 = ωp + ωi

Pulsed (Q pulse) Pulsed (Q pulse) Pulsed (35 ps) Pulsed (115 fs)

NCPM BPM

3.2 0.4

[e] [f]

KTP

0.82–0.92

NCPM

0.2

LiNbO3

1.064

QPM

1.5

Pulsed (7 ns)

[h]

LiNbO3 LiNbO3

0.532 0.78–0.80

NCPM QPM

0.75 0.6

CW Pulsed (2 ps)

[i] [j]

LiNbO3

0.793

ω1 = ωs + ωs ω2 = ωp + ωs ω3 = ωp + ωi ω1 = ωp + ωp ω2 = ωp + ωs ω3 = ωp + 2ωs ωp = 12 ωp + 12 ωp ω1 = ωp + ωp ω2 = ωs + ωs ω3 = ωp + ωs ω4 = ωp + ωi ω5 = ωs + ωs + ωs ω1 = ωp + ωs

CW Pulsed (150 fs) Pulsed (120 fs)

QPM

0.08

[k]

β-BBO

0.53

BPM

0.8

KTP

0.74–0.76

ω1 = ωs + ωs ωs1 = ω1 − ωi ωi1 = ωp − ωs1 ω2 = ωs1 + ωs1 ωs2 = ω2 − ωi ωi2 = ωp − ωs2 . . . ω1 = ωs + ωs

Pulsed (85 fs) Pulsed (1 ps)

BPM

0.5

LiNbO3

1.064

ωs → ωs2 + ωi2

QPM

2.5

LiNbO3

0.79–0.81

ω1 = ωs + ωs

QPM

0.1

KTP

0.827

ω1 = ωp + ωs

BPM

0.5

KTP

0.76–0.84

0.05

0.532

= ωs + ωs = ωp + ωs = ω1 + ωi = ωp + ωs

QPM

LiTaO3

ω1 ω2 ω3 ω1

QPOS

2

[b] [c] [d]

[g]

[l]

Pulsed (150 fs) Pulsed (43 ns) Pulsed (100 fs) Pulsed (170 fs) Pulsed (30 fs)

[m]

Pulsed (40 ps)

[s]

[n] [o] [p] [q, r]

(continued on next page)

34

Multistep parametric processes in nonlinear optics

[1, § 3

Table 5 (Continued) Nonlinear crystal

λpump [µm]

Phase-matching processes

LiNbO3

0.8

LiNbO3

1.064

KTA

0.796

ω1 ω2 ω1 ω2 ω3 ω4 ω5 ω1

LiNbO3

0.79

β-BBO

0.405 (0.81)

Phasematching method

L [cm]

Regime

Refs.

QPM

0.05

[t]

QPM

2

Pulsed (40 fs) Pulsed (17.5 ns)

BPM

2

[v]

ω1 = ωs + ωs

APQPM

1.8

ω1 = 12 ωp + ωs ω2 = ωi + ωi

BPM

0.2

Pulsed (140 fs) Pulsed (5 ns) Pulsed (90 fs)

= ωp + ωs = ωs + ωs = ωs + ωs = ωs + ωp = ωs + ωs + ωs = ωp + ωp = ωp + ω1 = ωs + ωs

References: [a] Andrews, Rabin and Tang [1970]. [b] Ammann, Yarborough and Falk [1971]. [c] Bakker, Planken, Kuipers and Lagendijk [1989]. [d] Powers, Ellingson, Pelouch and Tang [1993]. [e] Schiller and Byer [1993]. [f] Petrov and Noack [1995]. [g] Hebling, Mayer, Kuhl and Szipocs [1995]. [h] Myers, Eckardt, Fejer, Byer, Bosenberg and Pierce [1995]. [i] Schiller, Breitenbach, Paschotta and Mlynek [1996]. [j] Butterworth, Smith and Hanna [1997]. [k] Burr, Tang, Arbore and Fejer [1997]. [l] Varanavicius, Dubietis, Berzanskis, Danielius and Piskarskas [1997]. [m] Kartaloglu, Koprulu and Aytur [1997]. [n] Vaidyanathan, Eckardt, Dominic, Myers and Grayson [1997]. [o] McGowan, Reid, Penman, Ebrahimzadeh, Sibbett and Jundt [1998]. [p] Koprulu, Kartaloglu, Dikmelik and Aytur [1999]. [q] Zhang, Hebling, Kuhl, Ruhle and Giessen [2001]. [r] Zhang, Hebling, Kuhl, Ruhle, Palfalvi and Giessen [2002]. [s] Du, Zhu, Zhu, Xu, Zhang, Chen, Liu, Ming, Zhang, Zhang and Zhang [2002]. [t] Zhang, Hebling, Bartels, Nau, Kuhl, Ruhle and Giessen [2002]. [u] Xu, Liang, Li, Yao, Lin, Cui and Wu [2002]. [v] Kartaloglu and Aytur [2003]. [w] Kartaloglu, Figen and Aytur [2003]. [x] Lee, Zhang, Huang and Pan [2003]. Abbreviations: BPM – birefringence phase matching. NCPM – noncritical phase matched. QPM – uniformly poled quasi-phase-matched structure. QPOS – quasi-periodical optical superlattices. APQPM – aperiodically poled QPM structure.

[u]

[w] [x]

1, § 3]

Multistep phase-matched interactions

35

In the special case when all frequencies become phase-related, i.e. ωp : ωs : ωi = 3:2:1, OPOs display unique properties (Kobayashi and Torizuka [2000], Longhi [2001b]). As a matter of fact, this case corresponds to third-harmonic multistep cascading (see Section 3.1) but realized in a cavity. The second harmonic of the idler simultaneously generated in the OPO crystal (or by external frequency doubling), ω2i = ωi + ωi , will interfere with the signal wave of frequency ωi . The resulting beat signal can be used for locking the OPO at this particular tuning point and for realizing the frequency divisions (3:2) and (3:1), or vice versa. More details about frequency division with these types of OPO can be found in the papers by Lee, Klein, Meyn, Wallenstein, Gross and Boller [2003], Douillet, Zondy, Santarelli, Makdissi and Clairon [2001], Slyusarev, Ikegami and Ohshima [1999], Zondy, Douillet, Tallet, Ressayre and Le Berre [2001], Zondy [2003]. For double-phase-matched parametric interactions with frequencies 3ω; 2ω; ω in the OPA scheme, total conversion from a 3ω wave to a 2ω wave was predicted for the pump being the 3ω wave (Komissarova and Sukhorukov [1993]), and from a 2ω wave to a 3ω wave for the pump being the 2ω wave (Volkov and Chirkin [1998]). Additionally, the theoretical studies by Longhi [2001b, 2001a, 2001c] show that OPOs, when the signal and idler wave have frequencies ω and 2ω, respectively, can produce different types of interesting patterns, including spiral and hexagonal patterns. The internally pumped OPOs, due to the simultaneous action of the SHG process and parametric down-conversion, also show the formation of spatial patterns and parametric instability dynamics (Lodahl and Saffman [1999], Lodahl, Bache and Saffman [2000, 2001]). Additionally, multistep cascaded OPOs and OPAs can be an efficient tool for generating entangled and squeezed photon states. In particular, Smithers and Lu [1974] considered the quantum properties of light generated during the simultaneous action of the parametric processes ωp → ωi + ωs and ω1 = ωi + ωp . The theoretical predictions by Marte [1995b, 1995a] and Eschmann and Marte [1997] suggest that, due to the multistep cascading in an internally pumped OPO, this type of OPO can be an excellent source for generating twin photons with sub-Poissonian statistics, and the generated SH wave exhibits a perfect noise reduction. Several other phase-matched schemes for generating entangled and squeezed photon states have been proposed by Chirkin [2002], Nikandrov and Chirkin [2002b], and Chirkin and Nikandrov [2003] who utilized OPA with two simultaneously phase-matched processes, 2ω = ω + ω and 3ω = 2ω + ω. In the theoretical study by Nikandrov and Chirkin [2002a], the possibility of generating squeezed light using a single quadratic crystal was compared with that using two different crystals for the steps. The conclusion is that the single-crystal cascaded method is more efficient. New possibilities for the generation of squeezed

36

Multistep parametric processes in nonlinear optics

[1, § 3

polarized light have been discussed by Dmitriev and Singh [2003], who considered five-wave generation via four simultaneously phase-matched parametric processes.

3.6. Other types of multistep interactions Multistep parametric interactions permit the construction of compact frequency converters with several visible beams as the output. He, Liao, Liu, Du, Xu, Wang, Zhu, Zhu and Ming [2003] reports on the simultaneous generation of all three “traffic signal lights”. The simultaneous generation of a pair of blue and green waves has been achieved by Capmany, Bermudez, Callejo, Sole and Dieguez [2000] in exploring self-doubling and self-frequency mixing active media. Recently, several studies have presented successful attempts to obtain the simultaneous generation of red, green and blue radiation (the so-called RGB radiation) from a single nonlinear quadratic crystal. This is an important target for building compact laser-based projection displays. Theoretically, the parametric process for achieving the generation of three primary colors as OPO outputs was considered by Huang, Zhu, Zhu and Ming [2002]. Liao, He, Liu, Wang, Zhu, Zhu and Ming [2003] used a single aperiodically poled LiTaO3 crystal for generating 671, 532 and 447 nm (see fig. 9) with three simultaneously phase-matched processes: SHG of 1342 and 1064 nm (output of a dual-output Nd:YVO4 laser) and SFG of 671 and 1342 nm. Jaque, Capmany and Garcia Sole [1999] realized a different method for achieving the three-wavelength output. In their experiments, the nonlinear medium was a Nd:YAl3 (BO3 )4 crystal pumped by a Ti:sapphire laser. The red signal at 669 nm was obtained by self-frequency doubling of the

Fig. 9. (Left) The phase-matching curves for three simultaneously phase-matched processes in a single aperiodically poled LiTaO3 crystal. (Right) Visible red, green, and blue outputs from a nonlinear crystal diffracted by a prism (Liao, He, Liu, Wang, Zhu, Zhu and Ming [2003]).

1, § 3]

Multistep phase-matched interactions

37

fundamental laser line. The green signal at 505 nm and a blue signal at 481 nm were obtained by self-SFG of the fundamental laser radiation at 1338 nm and the pump radiation (807 nm, for green, and 755 nm, for blue). All three processes were simultaneously phase matched by birefringence phase matching owing to the exceptional situation that the three phase matchings appear extremely close to each other and their tuning curves overlap. The generation of red, green and blue signals by triple phase matching in LiNbO3 and KTP periodically poled waveguides was reported also by Sundheimer, Villeneuve, Stegeman and Bierlein [1994a] and Baldi, Trevino-Palacios, Stegeman, Demicheli, Ostrowsky, Delacourt and Papuchon [1995]. The experimental efforts to build optical devices with the simultaneous generation of several visible harmonics are summarized in Table 6. Table 6 Experiments on the simultaneous generation of several visible beams Nonlinear crystal

λp [µm]

Colors

PM method

L [cm]

Regime

Refs.

KTP waveguide KTP waveguide LiNbO3 waveguide NYAB

1.023 0.716 1.650

Red, green, blue

BPM/QPM

0.45

CW

[a]

Red, green, blue

QPM

0.35

[b]

1.620

Red, green, blue

QPM

1.338 0.807 0.755 1.084 0.744 1.372 1.084 0.744 1.34 0.88 1.342 1.064 1.342 1.064

Red, green, blue

BPM

0.5

Pulsed (6 ps) Pulsed (7 ps) CW

[d]

Green, blue

APQPM

0.095

CW

[e]

Red, orange, green, blue (2)

APQPM

0.3

CW

[f]

Red, green, blue

APQPM

0.7

CW

[g]

Red, green, blue

APQPM

1

CW

[h]

Red, yellow, green

APQPM

1

CW

[i]

Nd:LiNbO3 Nd:LiNbO3

SBN LiTaO3 LiTaO3

References: [a] Laurell, Brown and Bierlein [1993]. [b] Sundheimer, Villeneuve, Stegeman and Bierlein [1994b]. [c] Baldi, Trevino-Palacios, Stegeman, Demicheli, Ostrowsky, Delacourt and Papuchon [1995]. [d] Jaque, Capmany and Garcia Sole [1999]. [e] Capmany, Bermudez, Callejo, Sole and Dieguez [2000]. [f] Capmany [2001]. [g] Romero, Jaque, Sole and Kaminskii [2002]. [h] Liao, He, Liu, Wang, Zhu, Zhu and Ming [2003]. [i] He, Liao, Liu, Du, Xu, Wang, Zhu, Zhu and Ming [2003].

[c]

38

Multistep parametric processes in nonlinear optics

[1, § 3

3.7. Measurement of the χ (3) -tensor components An important possible application of multistep parametric processes in nonlinear optics is making a calibration link between the second- and third-order nonlinearities in a nonlinear medium. Both THG and FWM processes in noncentrosymmetric nonlinear media can be used for this purpose. Measurements can be done in the phase-matched or non-phase-matched regimes. The non-phasematched regime allows higher accuracy, however, it is more complicated since the signal is weak and additional care should to be taken to avoid the influence of the respective third-order effect in air. The basic idea of these types of measurements is to compare the THG or FWM signal in several configurations that include a proper choice of the direction and polarization of the input and output waves. In some of the configurations the output signal is generated as a result of the direct contribution of the inherent cubic nonlinearity of the sample; in other cases, the signal is generated due to the cascade contribution, while in the third group, the contribution of both direct and cascaded processes is comparable. In this way, we can access the ratio χ (3) (−3ω, ω, ω, ω) χ (2) (−3ω, 2ω, ω)χ (2) (−2ω, ω, ω) for THG multistep interaction, and the ratio χ (3) (−ω3 , ω1 , ω1 , −ω2 ) χ (2) (−ω3 , 2ω1 , −ω2 )χ (2) (−2ω1 , ω1 , ω1 ) for FWM multistep cascading. Because more information is available about the χ (2) components, we can determine quite accurately the parameters of the cubic nonlinearity of noncentrosymmetric materials by using this internal calibration procedure which does not require knowledge of the parameters of the laser beam. To illustrate this let us consider phase-matched THG. Under the condition of nondepletion of the fundamental and second-harmonic waves, and neglecting the temporal and spatial walk-off effect, eqs. (3.1) have the following solution for phase matching kTHG = k3 − 3k1 → 0:   σ2 σ5 sin(kTHG L/2) A3 (L) = −i γ + (3.15) |A1 |3 L. kSFG kTHG L/2 The apparent cubic nonlinearity consists of two parts, direct and cascading: (3)

(3)

(3)

χtot = χeff,dir + χeff,casc , with (3)

χeff,casc =

16πdeff,I deff,II . λ1 n2 kSFG

(3.16)

1, § 4]

Phase matching for multistep cascading

39

One way to separate the contributions of the two nonlinearities and to express (3) χeff,dir in terms of the product (deff,I )(deff,II ) is to use the azimuthal (Banks, Feit and Perry [2002]) or input-polarization (Kim and Yoon [2002]) dependence of (3) (3) χeff,dir and χeff,casc . The other way is to compare the TH signal obtained under the condition kTHG → 0 with the TH signal obtained under one of the conditions kSHG → 0 or kSFG → 0, where the signal is only proportional to the fac(3) tor |χeff,casc |2 (Akhmanov, Meisner, Parinov, Saltiel and Tunkin [1977], Chemla, (3)

Begley and Byer [1974]), and calculate χeff,dir ; however this procedure gives two possible values due to an indeterminate sign. Considering all symmetry classes, Feve, Boulanger and Douady [2002] found the crystal directions for which the second-order cascade processes give no contribution and, therefore, are suitable (3) for the measurement of the value of χeff,dir . The parametric interaction that occurs for the degenerated FWM (DFWM) process in noncentrosymmetric media is special because it consists of the steps of optical rectification and linear electro-optic effects (Bosshard, Spreiter, Zgonik and Gunter [1995], Unsbo [1995], Zgonik and Gunter [1996], Biaggio [1999], Biaggio [2001]). Then, the measured χ (3) component can be expressed through the squared electro-optic coefficient of the medium. The cascaded χ (3) contribution in crystals with a large electro-optic effect leads to a very strong cascaded DFWM effect which can exceed by many times the contribution of the inherent direct χ (3) nonlinearity (Bosshard, Biaggio, St, Follonier and Gunter [1999]). Table 7 presents a summary of the experimental works on the measurement of cubic nonlinearities by using cascaded THG or cascaded FWM processes.

§ 4. Phase matching for multistep cascading In general, simultaneous phase matching of several parametric processes is hard to achieve by traditional phase-matching methods, such as those based on the optical birefringence effect, except for some special cases discussed above. However, the situation becomes quite different for nonlinear media with a periodic variation of the sign of the quadratic nonlinearity, as occurs in the fabricated one-dimensional (1D) QPM structures (Fejer, Magel, Jundt and Byer [1992]) or two-dimensional (2D) χ (2) nonlinear photonic crystals (Berger [1998], Broderick, Ross, Offerhaus, Richardson and Hanna [2000], Saltiel and Kivshar [2000a]). In this section we describe the basic principles of simultaneous phase matching of two (or more) parametric processes in different types of 1D and 2D nonlinear optical superlattices.

40

Multistep parametric processes in nonlinear optics

[1, § 4

Table 7 Experimental results for the χ (3) -tensor components measured through the second-order multistep cascading processes Nonlinear crystal

λfund [µm]

PM/NPM/ NCPMa

Cascading scheme

References

ADP GaAs

1.06 10.6

PM NPM

THG FWM

CdGeAs2 KDP

10.6 1.064

PM PM

THG THG

α-quartz β-BBO β-BBO

1.91 1.054 1.053

NPM PM PM

THG THG THG

KTP

1.06

PM

DFWM

β-BBO KD*P d-LAP KTP

1.055

PM

THG

Wang and Baardsen [1969] Yablonovitch, Flytzanis and Bloembergen [1972] Chemla, Begley and Byer [1974] Akhmanov, Meisner, Parinov, Saltiel and Tunkin [1977] Meredith [1981] Qiu and Penzkofer [1988] Tomov, Van Wonterghem and Rentzepis [1992] DeSalvo, Hagan, Sheik-Bahae, Stegeman, Vanstryland and Vanherzeele [1992] Banks, Feit and Perry [1999], Banks, Feit and Perry [2002]

1.62

NCPM

THG

DASTb

1.064

PM

DFWM

α-quartz KNbO3 KTaO3 SF59 BK7 fused silica KNbO3 DAST BaTiO3 AANPb

1.064 1.318 1.907 2.100

NPM

THG

1.06

PM

DFWM

Biaggio [1999, 2001]

1.390 1.402

NCPM

FWM

KDP BBO LiNbO3

1.064 0.532

NPM

DFWM

Taima, Komatsu, Kaino, Franceschina, Tartara, Banfi and Degiorgio [2003] Ganeev, Kulagin, Ryasnyanskii, Tugushev and Usmanov [2003]

Feve, Boulanger and Guillien [2000], Boulanger, Feve, Delarue, Rousseau and Marnier [1999] Bosshard, Biaggio, St, Follonier and Gunter [1999] Bosshard, Gubler, Kaatz, Mazerant and Meier [2000]

a PM – phase matched; NPM – non-phase-matched; NCPM – noncritically phase-matched. b Organic crystals.

4.1. Uniform QPM structures A quadratic nonlinearity in a bulk homogeneous nonlinear crystal is usually homogeneous everywhere. Several methods have been suggested and employed

1, § 4]

Phase matching for multistep cascading

41

Fig. 10. Schematic of the uniform QPM structure: Gm is the reciprocal vector of the mth order; D is the filling factor, and Λ is the period of the QPM structure.

(Fejer [1998]) for creating a periodic change of the sign of the second-order nonlinear susceptibility d (2) , such as in the QPM structure shown in fig. 10. From the mathematical point of view, such a periodic sequence of two domains can be described by a simple periodic function,  gm eiGm z , d(z) = d0 (4.1)  gm =

m=0

2 mπ



sin(πmD),

(4.2)

where Gm = (2πm/Λ) is the reciprocal QPM vector. The uniform QPM structure is characterized by the set of reciprocal vectors ±2π/Λ, ±4π/Λ, ±6π/Λ, ±8π/Λ, . . . , which can be used to achieve the phase-matching conditions when k → 0. The integer number m (which can be both positive and negative) is called the order of the wave vector phase matching. According to eq. (4.1), the smaller the order of the QPM reciprocal wave vector is, the larger is the effective nonlinearity. If the filling factor D = 0.5, the effective quadratic nonlinearities (proportional to the parameter d0 gm ) that correspond to the even-order QPM vectors vanish. Importantly, such uniform QPM structures can be used for simultaneous phase matching of two parametric processes when the interacting waves are collinear or noncollinear to the reciprocal wave vectors of the QPM structure. 4.1.1. Collinear case with two commensurable periods As an example of multistep QPM interaction, we consider the third-harmonic multistep-cascading process under the condition that the interacting waves are collinear to the reciprocal wave vectors of the QPM structure. We denote the mismatches of the nonlinear material without modulation of the quadratic nonlinearity (“bulk mismatches”) by b1 and b2 , where b1 = k2 − 2k1 and

42

Multistep parametric processes in nonlinear optics

[1, § 4

b2 = k3 −k2 −k1 , and choose the period of the QPM structure so as to satisfy the phase-matching conditions Gm1 = −b1 and Gm2 = −b2 . The two parametric processes, SHG and SFG, are characterized by the wave-vector mismatches k1 = k2 − 2k1 + Gm1 and k2 = k3 − k2 − k1 + Gm2 , respectively, and they become simultaneously phase-matched for this particular choice of the QPM period. A drawback of this method is that it can satisfy two phase-matching conditions simultaneously only for discrete values of the optical wavelength. In particular, the values of the fundamental wavelength λ for the double-phase-matching condition can be found from the relation b2 /m2 − b1 /m1 = 0,

(4.3)

where both b1 and b2 are functions of the wavelength. Equation (4.3) is valid for any pair of second-order parametric processes. For the third-harmonic multistep-cascading process, eq. (4.3) is transformed to the following:     m1 3n(3ω) − 2n(2ω) − n(ω) − 2m2 n(2ω) − n(ω) = 0, (4.4) where the arguments shows the wavelength dependence of the refractive index. For a chosen pair of integer numbers (m1 , m2 ), the required QPM period is found from the relation Λ = 2π|m1 /b1 | or Λ = 2π|m2 /b2 |. Such a method was used for double phase matching by many researchers for cascaded single-crystal third- and fourth-harmonic generation (see details and references in Table 2). The highest efficiency achieved with the uniform QPM structure so far is 19.3%, using the combination of SHG (1st-order)/SFG (3rd-order) as reported by He, Liu, Luo, Jia, Du, Guo and Zhu [2002]. 4.1.2. Noncollinear case Collinearity between the optical waves and the reciprocal vectors of the QPM structure is an important requirement for achieving a good overlapping of all the beams and a good conversion efficiency. However, phase matching is possible even when some of the waves propagate under a certain angle with respect to the direction of the reciprocal vectors of the QPM structure, i.e., for the noncollinear case. With this method, double-phase-matching processes can be realized in a broad spectral range (Saltiel and Kivshar [2000b]). This type of noncollinear interaction will be efficient for distances corresponding to the overlap of the interacting beams. As an example, we consider the THG multistep parametric process in the noncollinear geometry as shown in fig. 11. In this case, the simultaneous phase matching of two parametric processes ω + ω = 2ω and ω + 2ω = 3ω is required. We assume a fundamental wave at the input. As shown in fig. 11, for the first process the phase matching can be achieved by the use of the reciprocal vector Gm1 , and

1, § 4]

Phase matching for multistep cascading

43

Fig. 11. Geometry of the noncollinear cascaded THG parametric process in a uniform QPM structure.

the generated SH wave with wave vector k2 is not collinear with the fundamental wave: 2k1 − Gm1 = k2 . The second process can be phase-matched by using the vector Gm2 : k1 + k2 − Gm2 = k3 . From fig. 11, we can derive a result for the period of the QPM structure that allows us to achieve the double phase matching of the processes of cascaded THG in a single QPM structure,   2m1 (k32 − 9k12 ) − 3(m1 + m2 )(k22 − 4k12 ) 1/2 . Λ = 2π (4.5) m1 (m1 + m2 )(2m2 − m1 ) The conditions for double phase matching in a THG multistep-cascading process in LiTaO3 when all waves are polarized along the z-axis of the crystal have been considered by Saltiel and Kivshar [2000b].

4.2. Nonuniform QPM structures Nonuniform QPM structures also permit simultaneous phase matching of two parametric nonlinear processes. We consider three types of such nonuniform QPM structures: (i) phase-reversed QPM structures (Chou, Parameswaran, Fejer and Brener [1999]), (ii) periodically chirped QPM structures (Bang, Clausen, Christiansen and Torner [1999]), and (iii) optical superlattices (Zhu and Ming [1999], Fradkin-Kashi, Arie, Urenski and Rosenman [2002]). 4.2.1. Phase-reversed QPM structures The idea of phase-reversed QPM structures (Chou, Parameswaran, Fejer and Brener [1999]) is illustrated in fig. 12(a). This structure can be explained as a sequence of many equivalent uniform short QPM sub-structures of length 1 2 Λph connected in such a way that at the place of the joint the two end layers have the same sign of the quadratic nonlinearity. Any two neighboring junctions have opposite signs of the χ (2) nonlinearity. In other words, the phase-reversed QPM structure can be thought of as a uniform QPM structure with a change of

44

Multistep parametric processes in nonlinear optics

[1, § 4

Fig. 12. Schematic of nonuniform QPM structures: (a) phase-reversed QPM structure (Chou, Parameswaran, Fejer and Brener [1999]); (b) periodically chirped QPM structure (Bang, Clausen, Christiansen and Torner [1999]).

the domain phase by π characterized by the second (larger) period Λph . The modulation of the quadratic nonlinearity d(z) in a phase-reversed QPM structure with filling factor D = 0.5 can be described by the response function d(z) = d0 (−1)int(2z/ΛQ ) (−1)int(2z/Λph ) ,

(4.6)

and it can be expanded into the Fourier series d(z) = d0

+∞  l=−∞

gl eiGl z

+∞  m=−∞

gm eiFm z = d0

+∞ 

+∞ 

glm eiGlm z ,

(4.7)

l=−∞ m=−∞

where g0 = 0, gl=0 = (2/πl), gm=0 = (2/πm), glm = gl gm , Gl = (2π/ΛQ )l, Fm = (2π/Λph )m, and Glm = (2π/ΛQ )l + (2π/Λph )m. The phase-reversed QPM structure is characterized by a set of reciprocal vectors {Glm } which are collinear to the normal of the periodic sequence, their magnitude depending on all parameters: l, m, ΛQ , Λph . Two of these vectors can

1, § 4]

Phase matching for multistep cascading

45

be chosen to phase-match two parametric processes involved in the multistepcascading, such that Gl1 m1 = −b1 and Gl2 m2 = −b2 . Then, the two QPM periods satisfying the double-phase-matching condition can be found as follows:




2π(l2 m1 − l1 m2 )

2π(l2 m1 − l1 m2 )

,

Λ ΛQ =

(4.8) = ph
l b − l b
. m2 b1 − m1 b2
2 1 1 2 For the THG multistep cascading process, eq. (4.8) are transformed into



λ(l2 m1 − l1 m2 )
, ΛQ =

(4.9) (m1 − 2m2 )n(ω) + 2(m1 + m2 )n(2ω) − 3m1 n(3ω)
Λph





λ(l2 m1 − l1 m2 )
.
=
(l1 − 2l2 )n(ω) + 2(l1 + l2 )n(2ω) − 3l1 n(3ω)


(4.10)

In addition to eqs. (4.9) and (4.10), the design shown in fig. 12(a) imposes the additional condition that the ratio Λph /2ΛQ is an integer number. Nevertheless, the corresponding number of phase-matched wavelengths is larger than that achieved in the uniform QPM structure for the collinear geometry. A phasereverse grating with arbitrary ratio Λph /2ΛQ was considered theoretically by Johansen, Carrasco, Torner and Bang [2002] in their study of spatial parametric solitons. The effect of arbitrary Λph /2ΛQ on the efficiency of different types of frequency-conversion processes has not been investigated yet, to the best of our knowledge. Experimentally, the phase-reversed QPM structure was employed for wavelength conversion (Chou, Parameswaran, Fejer and Brener [1999]) and for the cascaded single-crystal THG process (Liu, Zhu, Zhu, Qin, He, Zhang, Wang, Ming, Liang and Xu [2001], Liu, Du, Liao, Zhu, Zhu, Qin, Wang, He, Zhang and Ming [2002]). 4.2.2. Periodically chirped QPM structures Periodically chirped QPM structures, presented schematically in fig. 12(b), have initially been suggested and designed for increasing the effective (averaged) thirdorder nonlinearity in quadratic media (Bang, Clausen, Christiansen and Torner [1999]). We use the terminology “periodically chirped QPM” in analogy with chirped QPM structures that have a linear growth of the period ΛQ along the structure. This type of structure is very well suited for realizing multiple-phasematching conditions. The properties of periodically chirped QPM structures have been explored experimentally for achieving larger-bandwidth wavelength converters (Asobe, Tadanaga, Miyazawa, Nishida and Suzuki [2003], Gao, Yang and Jin [2004]).

46

Multistep parametric processes in nonlinear optics

[1, § 4

A periodically chirped QPM structure is characterized by a modulated QPM period ΛQ that is itself a periodic function of z (see fig. 12(b)):   2πz Λ = ΛQ + ε0 cos (4.11) . Λch Double-phase-matching conditions written for this type of structure allow one to find two periods of the periodically chirped QPM structure; the results are similar to the formulas describing phase-reversed QPM structures:


2π(l2 m1 − l1 m2 )

,
ΛQ =
(4.12) m2 b1 − m1 b2



2π(l2 m1 − l1 m2 )

. Λch =

(4.13) l2 b1 − l1 b2
The main advantage of the use of the periodically chirped QPM structures is the possibility to satisfy the double-phase-matching conditions for any wavelength in a broad spectral range, i.e. for any pair of b1 and b2 . 4.2.3. Quasi-periodic and aperiodic optical superlattices Another method of double phase matching, studied extensively both theoretically and experimentally, is based on the use of quasi-periodic optical superlattices (QPOS) (Zhu and Ming [1999], Fradkin-Kashi and Arie [1999], Fradkin-Kashi, Arie, Urenski and Rosenman [2002]) and aperiodic optical superlattices (Gu, Zhang and Dong [2000]). In most cases, QPOSs are built with two-component blocks A and B, as shown in fig. 13, which are aligned in a Fibonacci-like or more general quasi-periodic sequence. Importantly, each of the blocks consists of two layers with opposite signs of the quadratic nonlinearity. In order to illustrate the possibility of double phase matching in such structures we take, as an

Fig. 13. Basic blocks of the quasi-periodic and aperiodic QPM superlattices.

1, § 4]

Phase matching for multistep cascading

47

example, the structure consisting of two blocks aligned in a generalized sequence (Fradkin-Kashi and Arie [1999]). The modulation of the quadratic nonlinearity can be described by the following Fourier expansion (Birch, Severin, Wahlstrom, Yamamoto, Radnoczi, Riklund, Sundgren and Wallenberg [1990]):  fm,n eiGm,n z , d(z) = d0 (4.14) m,n

where the reciprocal vectors are defined as Gm,n = 2π(m + nτ )/S, with S = τ LA + L B . For phase matching two parametric processes with mismatch parameters b1 and b2 , we solve the system of equations Gm1 ,n1 = 2π(m1 + n1 τ )/S = −b1 , Gm2 ,n2 = 2π(m2 + n2 τ )/S = −b2 ,

(4.15)

and find the corresponding value of S and τ . The lengths of the blocks LA and LB and the ratio of the sub-layers in each block should be found by maximizing fm1 ,n1 and fm2 ,n2 . The resulting designed structure allows simultaneous phase matching in a broad spectral range without constraints on the ratio b2 /b1 . Equations (4.15) are also valid for Fibonacci-type QPOS lattices, but as the parameter τ is fixed, the double-phase-matching conditions can be satisfied for a limited number of wavelengths (Fradkin-Kashi and Arie [1999]) following from the equation b1 /b2 = (m1 + n1 τ )/(m2 + n2 τ ). In several papers, Zhao, Gu, Zhou and Wang [2003], Gu, Zhang and Dong [2000] and Gu, Dong, Zhang and Yang [1999] studied aperiodic one-dimensional optical superlattices for the simultaneous phase matching of several parametric processes. In this case, the thickness of the layers and their order can be found by solving an inverse problem maximizing the efficiency of both the processes involved in the multistep-cascading parametric interaction. Experimental results employing quasi-periodic and aperiodic optical lattices for realizing multistep parametric interactions are listed in Tables 2, 5, and 6.

4.3. Quadratic 2D nonlinear photonic crystals Nonlinear photonic crystals (NPCs) with a homogeneous change of the linear refraction index and a two-dimensional (2D) periodic variation of the nonlinear quadratic susceptibility have been suggested by Berger [1998, 1999] as a 2D analog of QPM structures. Both theoretical and experimental results published so far show that this kind of structures can effectively be employed as a host medium

48

Multistep parametric processes in nonlinear optics

[1, § 4

Fig. 14. (Left) 2D nonlinear quadratic photonic crystal composed of a periodic lattice of domains (gray) with the reversed sign of χ (2) ; Ga and Gc are the 2D reciprocal vectors. (Right) Fabricated 2D hexagonally poled LiNbO3 structure with period 18.05 µm (Broderick, Ross, Offerhaus, Richardson and Hanna [2000]).

for realizing many different types of multistep-cascading parametric processes (see, e.g., Broderick, Ross, Offerhaus, Richardson and Hanna [2000], Saltiel and Kivshar [2000a], de Sterke, Saltiel and Kivshar [2001], Chowdhury, Hagness and McCaughan [2000]). Below, we discuss some of the possible applications of these 2D structures for multistep parametric interaction and multi-frequency generation. 4.3.1. Phase matching in two-dimensional QPM structures A schematic structure of 2D NPC is shown in fig. 14. A simple way to obtain the phase-matching conditions for 2D NPC is to use a reciprocal lattice formed by the vectors Ga and Gc , defined √ as |Ga | = 2π/Λa and |Gc | = 2π/Λc . For a hexagonal lattice, Λa = Λc = a 3/2, where a is the distance between the centers of two neighboring inverted volumes, the so-called lattice spacing. All reciprocal vectors of the 2D NPC crystal are formed by a simple rule, Gm,n = mGc + nGa . Any two vectors of this set can be used to compensate for the bulk mismatch parameters b1 and b2 , however, the phase-matching conditions require, in most of the cases, noncollinear parametric interactions. The diagrams for calculating the phase-matching conditions for the THG multistep-cascading process are presented in fig. 15 for (a) the process ω + ω = 2ω which is phase-matched by the reciprocal vector Gm1 ,n1 , and (b) the process ω + 2ω = 3ω, which is phase-matched by the reciprocal vector Gm2 ,n2 . Phase matching is achieved by choosing the lattice spacing a and the angle of incidence β. 2D NPC can also be used for simultaneous phase matching of three nonlinear processes, e.g., second-, third-, and fourth-harmonic generation (Saltiel and Kivshar [2000a]), or the generation of a pair of SH waves and thirdharmonic (Karaulanov and Saltiel [2003]) or fourth-harmonic (de Sterke, Saltiel

1, § 4]

Phase matching for multistep cascading

49

Fig. 15. Diagrams of the double-phase-matching conditions for single-crystal THG in 2D NPC structures.

and Kivshar [2001]) generation. Experimentally, simultaneous second-, third-, and fourth-harmonic generation was recently observed in 2D poled bulk LiNbO3 (Broderick, Ross, Offerhaus, Richardson and Hanna [2000] and Broderick, Bratfalean, Monro, Richardson and de Sterke [2002]). The first experiment with phase matching in 2D NPC made in a poled LiNbO3 waveguide slab was reported by Gallo, Bratfalean, Peacock, Broderick, Gawith, Ming, Smith and Richardson [2003]. As shown by He, Tang, Qin, Dong, Zhang, Kang, Sun and Shen [2003], the method of direct electron-beam lithography can also be used for creating 2D nonlinear photonic domain-reversed structures. The shape of the inverted domains is an important property of the 2D NPC structures, and can be employed to effectively optimize parametric conversion processes. Optimization of the domain shapes for the maximum efficiency of THG and FHG processes was reported by Norton and de Sterke [2003b]. For SHG, Lee and Hagness [2003] compared the following three types of domain shapes: hexagonal, circular, and elliptical (aligned to the direction of SHG). The numerical simulation made by the authors revealed that the elliptically poled domain pattern yields the highest frequency-conversion efficiency among these three types of poling structures. The role of the filling factor for the case of SHG was studied both theoretically and experimentally by Wang and Gu [2001], Ni, Ma, Wang, Cheng and Zhang [2003]. 4.3.2. Multi-channel harmonic generation In the first experimental work on harmonic generation in 2D NPC structures (Broderick, Ross, Offerhaus, Richardson and Hanna [2000], Broderick, Brat-

50

Multistep parametric processes in nonlinear optics

[1, § 4

Fig. 16. Multiple second-harmonic (ω2 ), third-harmonic (ω3 ), and fourth-harmonic (ω4 ) generation schemes realized in a 2D NPC structure; orange, SHG (766 nm); green, THG (511 nm); blue, FHG (383 nm) (Broderick, Ross, Offerhaus, Richardson and Hanna [2000]).

falean, Monro, Richardson and de Sterke [2002]), it was noticed that each generated harmonic has multiple outputs at different angles (see fig. 16). The reason for this is that each harmonic can be generated by using several different phase-matching conditions. In several papers, an efficient method was suggested to make use of this property and to combine the multiple outputs in order to generate efficient harmonics with the additional advantage of being collinear to the pump. A phase-matching geometry for THG in 2D NPC, for which the generated TH wave is collinear to the input wave, was suggested by Karaulanov and Saltiel [2003]. This scheme makes use of the fact that the TH wave is generated through two different channels, leading to a factor four improvement in comparison with the conventional collinear scheme. A geometry for multiple phase matching of this THG multichannel parametric interaction is presented in fig. 17(a, b). The process starts with the generation of a pair of SH waves by employing the wave vectors k2 and k2 . The phase-matching conditions are satisfied by two symmetric reciprocal vectors Km,n and Km,−n . Each SH wave interacts again with the fundamental wave (see fig. 17(b)) via a pair of phase-matched interactions with participation of the reciprocal vectors Kp,q and Kp,−q , thus generating a pair of collinear TH waves with one and the same wave vector k3 . The two TH waves interfere constructively in the direction of the input fundamental wave, resulting in a higher overall TH efficiency (in the nondepleted regime). A scheme for a single-crystal FHG process in a 2D QPM structure where the fundamental and FH waves are collinear was studied by de Sterke, Saltiel and Kivshar [2001]. The parametric interaction includes cascading of three phase-matched second-order processes that are simultaneously phase-matched.

1, § 4]

Phase matching for multistep cascading

51

Fig. 17. Multi-channel third- and fourth-harmonic generation in 2D NPC structures: (a) the step ω + ω = 2ω; (b) the step ω + 2ω = 3ω for THG; (c, d) the step 2ω + 2ω = 4ω for FHG; Kp,q , Ki,j , Km,n , and Kk,l are reciprocal wave vectors of the lattice.

de Sterke, Saltiel and Kivshar [2001] demonstrated that the FHG efficiency achieved by this scheme is larger than comparable schemes by a factor that reaches four at low input intensities. The phase-matching geometry for this process is shown in fig. 17(c, d). Two SH waves are generated in the first step (see fig. 17(a)), and they interact noncollinearly generating a phase-matched FH wave, which itself is collinear with the fundamental wave (see fig. 17(c)). As shown by Norton and de Sterke [2003a], there exist two more FHG channels that are simultaneously phase-matched also leading to the generated FH waves collinear with the input fundamental wave (see fig. 17(d)). Thus, in this case the generated FH wave is produced as a result of constructive interference of three FH waves generated at different phase-matching conditions, but propagating in the same direction. 4.3.3. Wave interchange and signal deflection The first experimental demonstration of simultaneous optical wavelength interchange by the use of a 2D NPC structure was reported by Chowdhury, Staus,

52

Multistep parametric processes in nonlinear optics

[1, § 4

Fig. 18. Schematic of simultaneous optical wavelength interchange in a 2D NPC structure (Chowdhury, Staus, Boland, Kuech and McCaughan [2001]).

Boland, Kuech and McCaughan [2001]. A nonlinear 2D lattice fabricated in LiNbO3 was designed to provide an interchange of waves with wavelengths λ1 = 1535 nm and λ2 = 1555 nm. The pump was selected at λp = 777.2 nm. The wavelength-interchange process takes place by means of two concurrent difference-frequency generation processes: ωp −ω1 = ω2 and ωp −ω2 = ω1 . The two difference-frequency processes diffract the converted signals from the unconverted ones, as depicted in fig. 18. The information carried by input beam 1 will be carried by output beam 2 and vice versa. In this experiment, one of the main characteristics of the 2D NPC phase matching, i.e. noncollinearity of the parametric interactions, is used as a real advantage in separating two beams at the output. Another scheme, proposed by Saltiel and Kivshar [2002], is also based on the advantage of noncollinearity of the parametric interactions in the 2D geometry, and it demonstrates how a signal wave can be deflected or split by the pump after the interaction. The phase-matching conditions for this scheme are shown in fig. 19. The collinear pump and the signal are at the same frequency, but they are polarized in orthogonal directions. If the pump carries some information, the signal will be modulated according to this information after the deflection. This interaction belongs to the two-color multistep cascading processes discussed above. It can also be considered as a spatial analog of the wavelength-conversion process discussed in Section 3.2. In concluding this part of the review, we would like to mention that several theoretical studies have predicted that double-phase-matched interactions such as the THG multistep parametric process can be realized in photonic-bandgap (PBG) structures with an efficiency that should be higher by several orders of magnitude than that of conventional nonlinear media of similar length. In particular, this in-

1, § 5]

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Fig. 19. Phase matching for (a) the first and (b) the second cascading step, y1p y1p −z2p , z2p z1s −z1s  . Drawings correspond to n1z < n2z < n1y (Saltiel and Kivshar [2002]).

cludes an infinite PBG system as a host for second-order multistep cascading considered by Konotop and Kuzmiak [1999], cascaded THG in a finite PBG crystal discussed by Centini, D’Aguanno, Scalora, Sibilia, Bertolotti, Bloemer and Bowden [2001], and a photonic-well-type PBG system analyzed by Shi and Wang [2002]. However, no experimental results on the fabrication of PBG structures with strong nonlinear properties were reported yet. This section has presented a brief overview of different techniques for achieving simultaneous phase matching of several nonlinear parametric processes in optical structures with a modulated second-order nonlinear susceptibility. In all of these cases, double-phase-matched interaction becomes possible in a wide range of optical wavelengths, provided the QPM structure used to achieve the phasematching conditions possesses one extra parameter (e.g., the modulation period in chirped QPM structures, or the second dimension in the case of 2D nonlinear photonic crystals). Some possible applications of double-phase-matching processes are the simultaneous generation of several optical frequencies in a single-crystal structure, multi-port frequency conversion, etc. The results presented above look encouraging with respect to the experimental feasibility of the predicted effects in the recently engineered 1D and 2D periodic optical superlattices.

§ 5. Multi-color parametric solitons In the preceding sections, we have discussed the features of multiple parametric processes using the plane-wave and continuous-wave approximations. However, parametric interactions can strongly modify the dynamics of spatial beams or temporal pulses. The parametrically coupled waves in a medium with quadratic

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nonlinearity may experience mutual spatial focusing or temporal compression and lock together into a stationary state, a quadratic soliton (see Sukhorukov [1988], Torner [1998], Kivshar [1997], Etrich, Lederer, Malomed, Peschel and Peschel [2000], Torruellas, Kivshar and Stegeman [2001], Boardman and Sukhorukov [2001], Buryak, Di Trapani, Skryabin and Trillo [2002], and references therein). The first analysis of solitons supported by multistep parametric interactions was reported by Azimov, Sukhorukov and Trukhov [1987] who found that as many as seven waves can be trapped together, and such multi-color solitons may be remarkably stable, particularly in the presence of absorption. The phase mismatch and strength of several parametric interactions can be engineered in nonlinear periodic structures (see Section 4); this provides flexibility in controlling the properties of these solitons, making them useful for potential applications including all-optical switching. In this section, we review the properties of multistep parametric solitons that can form under the conditions of cascaded THG, two-color FWM, frequency conversion, and FHG.

5.1. Third-harmonic parametric solitons The generation of a TH wave through cascaded SHG and SFG processes was discussed in Section 3.1. Komissarova and Sukhorukov [1996] demonstrated that these parametric interactions can result in simultaneous trapping of all the interacting waves and formation of a three-color parametric soliton. Kivshar, Alexander and Saltiel [1999] performed a detailed investigation of these solitons and demonstrated their stability under various conditions. Such solitons can be described by eqs. (3.1) with additional terms accounting for beam diffraction, i 2 dA1 + ∇ A1 = −iσ1 A2 A∗1 e−ikSHG z − iσ3 A3 A∗2 e−ikSFG z , dz 2k1 ⊥ i 2 dA2 + ∇ A2 = −iσ2 A21 eikSHG z − iσ4 A3 A∗1 e−ikSFG z , dz 2k2 ⊥ i 2 dA3 + ∇ A3 = −iσ5 A2 A1 eikSFG z , dz 2k3 ⊥

(5.1)

where the contribution due to the direct THG process is neglected, assuming that 2 acts on the spatial the cascaded χ (2) processes are dominant. The operator ∇⊥ dimensions in the transverse plane (perpendicular to the beam propagation direction z). Similar equations can describe the formation of temporal solitons, where nonlinearity compensates for both the group-velocity mismatch and second-order dispersion effects (Huang [2001]).

1, § 5]

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Fig. 20. Profiles of THG parametric solitons for different values of phase mismatch (normalized units). Labels 1, 2, and 3 indicate the number of harmonic (Kivshar, Alexander and Saltiel [1999]).

Stationary propagation of solitons is possible only when there is no energy exchange between the constituent waves; this requires that the individual phase velocities are synchronized due to nonlinear coupling. Such solutions of eqs. (5.1) have the form Am (r, z) = Bm (r)eimβz , where β is a nonlinear propagation constant and Bm (r) are the transverse soliton profiles. Spatial soliton properties were analyzed in the (1 + 1)-dimensional geometry, where the beam is confined by a planar waveguide and experiences diffraction only in one transverse direction so 2 = ∂ 2 /∂x 2 . There exist two families of single-hump solitons, however that ∇⊥ solutions with higher power are always unstable. Soliton examples from the lowpower branch are shown in fig. 20. It was also found that cascading interactions can lead to radiative decay of self-localized beams, however such quasi-solitons can demonstrate robust propagation for several diffraction lengths. Lobanov and Sukhorukov [2002, 2003] developed a theoretical description of THG solitons in QPM structures, where all the frequency components oscillate along the propagation direction. The spectrum of these oscillations was determined analytically and numerically, and it was found that the solitons can be described by averaged equations with additional terms accounting for induced Kerr effects: self-phase modulation, cross-phase modulation, and third-harmonic generation.

5.2. Two-color parametric solitons As discussed in Section 3.3, two-color multistep cascading can involve up to six steps corresponding to type I and type II interactions between two pairs of orthogonally polarized FF and SH waves. The existence of spatial quadratic solitons under these most general conditions was demonstrated by Boardman, Bontemps and Xie [1998] who found that, by changing the FF polarization at the input, the

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SH output can be precisely controlled. The possibility of achieving beam steering and switching based on collision of two solitons was demonstrated as well. The general case involving four waves coupled by six parametric processes requires three phase-matching conditions to be fulfilled simultaneously, and in addition the respective components of the χ (2) susceptibility tensor should be nonzero. The formation of three-wave solitons which require two phase-matching conditions may be easier to achieve, as indicated by the experimental results in the plane-wave regime (see references in Section 3.3). Kivshar, Sukhorukov and Saltiel [1999] and Sukhorukov [2000] considered the formation of solitons involving two FF components (A and B) that are coupled through a single SH component (S). The soliton formation can be described by the following set of coupled equations that are obtained in the slowly-varying-envelope approximation: i ∂A + ∇ 2 A = −iσAS SA∗ e−ikAS Z , ∂z 2kA ⊥ ∂B i + ∇ 2 B = −iσBS SB ∗ e−ikBS z , ∂z 2kB ⊥ ∂S i 2 ∇ S = −iσAS A2 eikAS Z − iσBS B 2 eikBS Z , + ∂z 2kS ⊥

(5.2)

where σAS and σBS are proportional to the elements of the second-order susceptibility tensor, as discussed in the previous sections, and kAS = kS − 2kA and kBS = kS − 2kB are the wave-vector mismatch parameters for the A–S and B–S parametric interaction processes, respectively. Similar to the case of the THG cascading, multi-component bright solitons are found in the form of stationary waves, of which the phases are locked together. When one of the FF waves is zero, then other two waves can form a type I quadratic soliton (A–S or B–S). However, these solitons may experience an instability associated with the amplification of the orthogonally polarized FF wave through the parametric decay instability of the SH component. Such changes in stability are associated with the bifurcation for two-wave to three-wave solitons, as shown in fig. 21 for a (1 + 1)-dimensional case. We note that the Vakhitov– Kolokolov stability criterion (Vakhitov and Kolokolov [1973], Pelinovsky, Buryak and Kivshar [1995]) is a necessary, but not a sufficient condition: branches with a negative power slope are unstable, but a positive slope does not guarantee stability. The examples in fig. 21 demonstrate the existence of stable solitons that have the same power but different polarizations of the FF waves. Such multistability is a sought-after property of nonlinear systems, since it may permit controlled switching between different states. In fig. 22, we illustrate the development of a

1, § 5]

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Fig. 21. Characteristic dependences of the total soliton power on the propagation constant corresponding to different phase mismatches for two-wave (A–S and B–S) and three-wave (AB–S) soliton families (normalized units). Solid lines, stable solitons; dashed lines, unstable solitons; open circles mark the bifurcation points (figures adapted from Sukhorukov [2000]).

soliton instability which results in a power exchange between the two FF components. Such a type of polarization switching had been predicted for plane waves by Assanto, Torelli and Trillo [1994]. The properties of two-color multistep cascading solitons were analyzed by Towers, Sammut, Buryak and Malomed [1999] and Towers, Buryak, Sammut and Malomed [2000] for the case when two FF waves and the SH component are additionally coupled through a type II parametric process. Due to this interaction, all solitons contain both FF waves, and a transition between states with different polarizations along the soliton family does not involve bifurcations. It was found that soliton multistability can be realized under appropriate conditions, both in planar (one-dimensional) and bulk (two-dimensional) configurations. On the other hand, Towers, Buryak, Sammut and Malomed [2000] reported that no multistability occurs if the BB–S interaction is suppressed and the waves are coupled only through AA–S and AB–S processes.

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Fig. 22. (a) Change of the normalized power in FF (A, solid) and SH (S, dotted) components, which initially constitute a two-wave soliton, and in the guided mode (B, dashed), demonstrating amplification of a guided wave. (b–d) Evolution of the intensity profiles for (b) the effective waveguide (SH), and (c, d) A and B FF components, respectively (Kivshar, Sukhorukov and Saltiel [1999]).

5.3. Solitons due to wavelength conversion The formation of spatial parametric solitons requires strong coupling between the interacting waves, and this can be achieved when the parametric interactions are nearly phase-matched. In the cases of parametric THG and two-color FWM, two (or even more) phase-matching conditions should be satisfied simultaneously. On the other hand, in the wavelength-conversion scheme (see Section 3.2), only a single phase-matching condition has to be implemented, and this greatly simplifies the requirements for experimental observation of spatial beam localization and soliton formation under the conditions of multistep cascading. Indeed, the first experimental study, recently reported by Couderc, Lago, Barthelemy, De An-

1, § 5]

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Fig. 23. Beam profiles at the output of a 2 cm long KTP crystal for low and high input intensities. Horizontal scale: wavelength; vertical scale: transverse dimension on the output pattern. The right-hand part of the figure corresponds to the probe wave (1061 nm), the left-hand part corresponds to the FF component of the soliton (pump at 1064 nm) (Couderc, Lago, Barthelemy, De Angelis and Gringoli [2002]).

gelis and Gringoli [2002], demonstrated that a multi-color soliton composed of a pump and its second harmonic creates an effective waveguide that can trap a weak probe beam of which the frequency is slightly detuned from the pump wave (see fig. 23). This trapping is realized due to parametric coupling of the probe with both the pump and harmonic waves, and additional side-band frequencies are generated in the process according to the principles of wavelength conversion. Couderc, Lago, Barthelemy, De Angelis and Gringoli [2002] demonstrated that the evolution of a weak probe is governed by the equation


∂a 1 2 1 ∂ 2 k

σ2 2 2 i (5.3) + |A| a = 0, ∇⊥ a + δω2 a + − |B|
∂z 2k1 2 ∂ω2 ω1 k where A and B are the profiles of the mutually trapped pump wave and its secondharmonic components, δω is the frequency detuning of the probe, ω1 and k1 are the frequency and the wavenumber of the fundamental-frequency wave, and k is the phase mismatch between the SH and FF components. The last term in eq. (5.3) defines the profile of the effective waveguide experienced by the probe beam, and trapping can be realized when the overall sign is positive. It has been demonstrated in earlier studies that the FF component (|A|2 ) dominates in a quadratic soliton when k > 0, whereas the SH component (|B|2 ) becomes

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larger at negative mismatches, k < 0. Therefore, the probe can be trapped in both cases, however anti-waveguiding effects may occur for particular values of the mismatches and pump intensities. 5.4. Other types of multi-color parametric solitons The formation of spatial parametric solitons due to FHG in a planar waveguide configuration was analyzed theoretically by Sukhorukov, Alexander, Kivshar and Saltiel [2001]. It was found that, similar to the case of two-color parametric solitons (see Section 5.2), there can exist two-wave solitons (coupled second and fourth harmonics) and three-wave solitons containing all three frequency components. Such coexistence of different solitons may give rise to multistability, as illustrated in fig. 24(a). A two-wave soliton can exhibit parametric decay instabil-

(a)

(b) Fig. 24. (a) Thick curve: two-wave (SH + FH) soliton; other curve: three-wave solitons; the curves are solid and dashed for stable and unstable solutions, respectively. The open circle is the bifurcation point. (b) Development of a decay instability of a two-wave soliton corresponding to β = 1 in plot (a), and generation of a three-component soliton. The dotted, solid, and dashed curves in the left-hand plot show the FF, SH, and FH peak normalized intensities vs. distance, respectively (Sukhorukov, Alexander, Kivshar and Saltiel [2001]).

1, § 6]

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61

ity due to the energy transfer into the fundamental-frequency wave, as illustrated in fig. 24(b). Towers and Malomed [2002] performed a theoretical analysis of polychromatic solitons consisting of two low-frequency components ω1 and ω2 and three highfrequency waves, 2ω1 , 2ω2 , and ω1 + ω2 . It was found that the power of a polychromatic soliton can be smaller compared to conventional two-wave parametric solitons, and this can be advantageous for applications. It was also demonstrated that polychromatic solitons can emerge after a collision of two-wave solitons, and soliton interactions may be used to implement a simple all-optical logic XOR gate.

§ 6. Conclusions Parametric interactions and phase matching are key concepts in nonlinear optics. The generation of new waves at frequencies which are not accessible by standard sources and the efficient frequency conversion and manipulation are among the main goals of the current research in nonlinear optics. The QPM technique is becoming one of the leading technologies for optical devices based on parametric wave interaction; it permits the generation of new harmonics and can be made compatible with the operational wavelengths of optical communication systems. In this chapter we have presented, for the first time to our knowledge, a systematic overview of the basic principles of simultaneous phase matching of two (or more) parametric processes in different types of one- and two-dimensional nonlinear quadratic optical lattices, the so-called multistep parametric interactions. In particular, we have discussed different types of multiple phase-matched processes in engineered QPM structures and two-dimensional nonlinear quadratic photonic crystals, as well as the properties of multi-color optical solitons generated by the multistep parametric processes. We have also summarized the most important experimental demonstrations for multi-frequency generation due to multistep parametric processes. We believe that such a comprehensive summary of the achievements to date will become a driving force for future even more active research in this exciting field of nonlinear optics.

Acknowledgements This work was produced with the assistance of the Australian Research Council under the ARC Centers of Excellence Program. The Center for Ultra-high bandwidth Devices for Optical Systems (CUDOS) is an ARC Center of Excellence.

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Solomon Saltiel thanks the Nonlinear Physics Group for hospitality and the Research School of Physical Sciences and Engineering at the Australian National University for a grant of the senior visiting fellowship.

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E. Wolf, Progress in Optics 47 © 2005 Elsevier B.V. All rights reserved

Chapter 2

Modes of wave-chaotic dielectric resonators by

H.E. Türeci Department of Applied Physics, Yale University, New Haven, CT 06520, USA E-mail address: [email protected] (H.E. Türeci)

H.G.L. Schwefel Department of Applied Physics, Yale University, New Haven, CT 06520, USA

Ph. Jacquod Département de Physique Théorique, Université de Genève, CH-1211 Genève 4, Switzerland

and

A. Douglas Stone Department of Applied Physics, Yale University, New Haven, CT 06520, USA

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)47002-X 75

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

§ 2. Failure of eikonal methods for generic dielectric resonators . . . . .

80

§ 3. Ray dynamics for generic dielectric resonators . . . . . . . . . . . .

87

§ 4. Formulation of the resonance problem . . . . . . . . . . . . . . . . .

90

§ 5. Reduction of the Maxwell equations . . . . . . . . . . . . . . . . . .

94

§ 6. Scattering quantization – philosophy and methodology . . . . . . .

97

§ 7. Root-search strategy . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

§ 8. The Husimi–Poincaré projection technique for optical dielectric resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

§ 9. Far-field distributions . . . . . . . . . . . . . . . . . . . . . . . . . .

117

§ 10. Mode classification: theory and experiment . . . . . . . . . . . . . .

119

§ 11. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

Appendix A: Numerical implementation issues . . . . . . . . . . . . . . .

131

Appendix B: Lens transform . . . . . . . . . . . . . . . . . . . . . . . . .

132

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

76

§ 1. Introduction A promising approach to making compact and high-Q optical resonators is to base them on “totally internally reflected” (TIR) modes of dielectric microstructures. Such devices have received considerable attention as versatile components for integrated optics and for low-threshold micron-scale semiconductor lasers (Chang and Campillo [1996], Yamamoto and Slusher [1993]). The interest in such resonators for applications and for fundamental optical physics has motivated the extension of optical resonator theory to describe such systems. All optical resonators are open systems described by modes characterized by both a central frequency and a width (their ratio giving the Q-factor of the mode). In a mirror-based resonator the set of resonant frequencies is determined by the optical path length for one round trip along a path determined by the mirrors within the resonator; the width is determined by the reflectivity of the mirrors, diffraction at the mirror edges and absorption loss within the resonator. Accurate analytic formulas can be found for the resonator frequencies and for the electric field distribution of each mode using the methods of Gaussian optics (Siegman [1986]). The modes are characterized by one longitudinal and two transverse mode indices (in three dimensions). These mode indices play the same role mathematically for the electromagnetic wave equation as do good quantum numbers in characterizing solutions of the wave equation of quantum mechanics. For constructing optical resonators on the micron scale, using total internal reflection from a dielectric interface for optical confinement is convenient as it simplifies the fabrication process. However such dielectric resonators define no specific optical path length; many different and potentially nonclosed ray trajectories can be confined within the resonator. An important point, emphasized in the current work, is that in such resonators there typically exist many narrow resonances characterized by their frequency and width, but such resonances often cannot be characterized by any further modes indices. This is the analog of a quantum system in which there are no good quantum numbers except for the energy. We shall see that the way to determine whether a given mode has additional mode indices (other than the frequency), is to determine whether it corresponds to regular or chaotic ray motion. We will present below an efficient numerical method for cal77

78

Modes of wave-chaotic dielectric resonators

[2, § 1

culating all of the resonances of a large class of dielectric resonators; we will also describe the surface of section and Husimi–Poincaré projection method to determine the ray dynamics corresponding to such a mode. Although both DBR-based and edge-emitting optical resonators rely on reflectivity from a dielectric interface (at normal incidence), we will use the term dielectric resonator (DR) to refer to resonators that rely on the high reflectivity of dielectric bodies to radiation incident from within the dielectric near the critical angle for total internal reflection. This is the only class of resonators we will treat below. We immediately point out that TIR solutions of the wave equation exist only for infinite flat dielectric interfaces; any curvature or finite extent of the dielectric will allow evanescent leakage of propagating radiation from the optically more dense to the less dense medium. As a dielectric resonator is a finite dielectric body embedded in air (or in a lower-index medium) it will of necessity allow some evanescent leakage of all modes, even those which from ray analysis appear to be totally internally reflected. A very large range of shapes for DRs have been studied during the recent years. By far the most widely studied are rotationally symmetric structures such as spheres, cylinders and disks. In this case the wave equation is separable and the solutions can be written in terms of special functions carrying two or three mode indices (neglecting finite size in the axial direction for the cylinders and disks). The narrow (long-lived) resonances correspond to ray trajectories circling around the symmetry axis near the boundary with angle of incidence above total internal reflection; these solutions are often referred to as “whispering-gallery” (WG) modes or morphology-dependent resonances. In this case, owing to the separability of the problem, it is straightforward to evaluate the violation of total internal reflection, which may be interpreted as the tunneling of waves through the angular momentum barrier (Johnson [1993], Nöckel [1997]). Micron-scale, high-Q microlasers were fabricated in the mid-1980s and early 1990s based on such cylindrical or disk-shaped (Q ∼ 104 –105 ) (McCall, Levi, Slusher, Pearton and Logan [1992], Slusher, Levi, Mohideen, McCall, Pearton and Logan [1993], Levi, Slusher, McCall, Tanbunek, Coblentz and Pearton [1992]), and spherical (Q ∼ 108 –1012 ) (Collot, Lefevreseguin, Brune, Raimond and Haroche [1993]) dielectric resonators. However, the very high Q-value makes these resonators unsuitable for microlaser applications, because such lasers invariably provide lowoutput power and furthermore, unless additional guiding elements are used, the lasing output is emitted isotropically. As early as 1994, Nöckel, Stone and Chang proposed to study dielectric resonators based on smooth deformations of cylinders or spheres which were referred to as “asymmetric resonant cavities” (ARCs). The idea was to attempt to combine

2, § 1]

Introduction

79

the high Q provided by near total internal reflection with a breaking of rotational symmetry, leading to directional emission and improved output coupling. General principles of nonlinear dynamics applied to the ray motion (to be reviewed below) suggested that there would be only a gradual degradation of the high-Q modes, and one might be able to obtain directional emission from deformed whisperinggallery modes. Experimental (Nöckel, Stone, Chen, Grossman and Chang [1996], Mekis, Nöckel, Chen, Stone and Chang [1995], Chang, Chang, Stone and Nöckel [2000]) and theoretical (Nöckel and Stone [1997]) work following that initial suggestion has confirmed this idea, although the important modes are not always of the whispering-gallery type (Gmachl, Capasso, Narimanov, Nöckel, Stone, Faist, Sivco and Cho [1998], Gianordoli, Hvozdara, Strasser, Schrenk, Faist and Gornik [2000], Gmachl, Narimanov, Capasso, Baillargeon and Cho [2002], Rex, Türeci, Schwefel, Chang and Stone [2002], Lee, Lee, Chang, Moon, Kim and An [2002]). The calculation of the modal properties of deformed cylindrical and spherical resonators presents a much more challenging theoretical problem. Unless the boundary of the resonator corresponds to a constant coordinate surface of some orthogonal coordinate system, the resulting partial differential equation will not be solvable by separation of variables. The only relevant separable case is an exactly elliptical deformation of the boundary, which turns out to be unrepresentative of generic smooth deformations. Using perturbation theory to evaluate the new modes based on those of the cylindrical or spherical case is also impractical, as for interesting deformations and typical resonator dimensions (tens of microns or larger) the effect of the deformation is too large for the modes of interest to be treated by perturbation theory. The small parameter in the problem for attempting approximate solutions is the ratio of the wavelength to the perimeter, λ/2πR = (kR)−1 . Eikonal methods (Kravtsov and Orlov [1999]) and Gaussian optical methods (Siegman [1986]) both rely on the short-wavelength limit to find approximate solutions. The Gaussian optical method can be used to find a subset of the solutions for generic ARCs, those associated with stable periodic ray orbits, as explained in detail by Türeci, Schwefel, Stone and Narimanov [2002]. The eikonal method can also be used to find a subset of ARC modes, if one has a good approximate expression for a local constant of motion; an example is the adiabatic approximation used by Nöckel and Stone [1997]. However a large fraction of ARC modes cannot be described by either of these methods. The breakdown of the Gaussian optical method is easily understood, as a fully chaotic system will have only unstable periodic orbits, and no consistent solution can be obtained by the Gaussian method based on unstable periodic ray orbits (Türeci, Schwefel, Stone and Narimanov [2002]). The failure of eikonal methods for generic shapes is more subtle, and arises from the possibility of chaotic ray motion in a

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[2, § 2

finite fraction of phase space. Optics textbooks, and even standard research references, often treat the eikonal method as being of completely general applicability; we therefore will devote the next section of this chapter to an explanation of the failure of eikonal methods for resonators with arbitrary smooth boundaries. In Section 3 we describe the phase-space methods which indicate that this failure is generic. In Section 4 we present the formulation of the resonance problem, and Section 5 covers the reduction of the Maxwell equations to the Helmholtz equation for the resonators we study. The failure of all standard short-wavelength approximation methods to describe the solution of wave equations in finite domains with arbitrary smooth boundaries has led to the problem of “quantizing chaos” in the context of the Schrödinger equation and the Helmholtz equation (although our problem is somewhat different due to the dielectric boundary conditions on this equation). Although substantial progress has been made using periodic-orbit methods in obtaining approximations for the density of states of fully chaotic systems, these methods do not yield individual solutions of the wave equation. It is therefore of great importance in this field to develop efficient numerical methods for calculating and interpreting the resonance properties. In Section 6 we present a highly efficient numerical method for ARCs, adapted from the S-matrix methods developed in the field of quantum chaos. In Section 8 we show how to perform the Husimi–Poincaré projection of the real-space numerical solutions so obtained into phase space in order to interpret them in terms of ray dynamics. The calculation of experimental observables relating to emission patterns from microlasers are discussed in Section 9. Finally, in Section 10 we show examples of the main types of modes one encounters in wave-chaotic dielectric resonators generally and in ARCs specifically.

§ 2. Failure of eikonal methods for generic dielectric resonators The use of classical ray theory to describe monochromatic, high-frequency solutions of the wave equation is described in various references (e.g., Keller, Papanicolau and McLaughlin [1995], Babiˇc and Buldyrev [1991], Kravtsov and Orlov [1999]). The connection between rays and waves is derived in a standard way in the context of the Helmholtz equation

2 ∇ + n2 (x)k 2 ψ(x) = 0;

(2.1)

2, § 2]

Failure of eikonal methods for generic dielectric resonators

81

the wave equation for the resonator problem will be reduced to this equation in Section 5. The eikonal approach uses the asymptotic ansatz ψ(x) ∼ eikS(x)

∞  Aν (x) ν=0

kν

(2.2)

in the limit k → ∞. Inserting eq. (2.2) into eq. (2.1) one finds, to lowest order in the asymptotic parameter k1 , the eikonal equation (∇S)2 = n2 (x)

(2.3)

and the transport equation 2∇S · ∇A0 + A0 ∇ 2 S = 0.

(2.4)

Note that at this point we only assume one eikonal S(x) and one amplitude A(x) at each order in the expansion. We will also specialize to a uniform medium of dielectric constant n. In this framework, each wave solution ψ(x) corresponds to a family of rays defined by the vector field p(x) = ∇S(x),

(2.5)

where the field has a fixed magnitude, |∇S| = n. The solution for the function S(x) can be found by the specification of initial conditions on an open curve C: x = x(s) and propagating the curve using the eikonal equation. Such an initial value solution can thus be extended until it encounters a point at which two or more distinct rays of the wavefront converge; at or nearby such a point will occur a focus or caustic at which the amplitude A will diverge, and in the neighborhood of which the asymptotic representation becomes ill-defined. (A caustic is a curve to which all the rays of a wavefront are tangent; if the curve degenerates to a point it is a focus (Kravtsov and Orlov [1999]).) This causes only a local breakdown of the method and can be handled by a number of methods (Berry [1976]). At a distance much greater than a wavelength away from the caustic the solutions are still a good approximation to the true solution of the initial value problem. In contrast, to find asymptotic solutions on a bounded domain D with boundary conditions, one must introduce more than one eikonal at each order in the asymptotic expansion. While we will illustrate the important points here with Dirichlet boundary conditions on the boundary ∂D, the basic argument holds for any linear homogeneous boundary conditions and, with minor modifications, for the matching conditions relevant for uniform dielectric resonators of index of refraction n with boundary shape ∂D, within an infinite medium of index n = 1. For the discussion of Dirichlet boundary conditions we will set the index n = 1 for convenience within the domain D. The leading order in the asymptotic expansion of

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[2, § 2

the solution takes the form ψ(x) =

N 

Am (x)eikSm (x)

(2.6)

m=1

with N
2. It is easily checked that there must be more than one term (eikonal) in the solution in order to have a nontrivial solution; if there were only one term in the expression for ψ(x) then any solution that vanished on the boundary would vanish identically in D due to the form of the transport equation. The question we now address is the following. For what boundary shapes ∂D in two dimensions do there exist approximate solutions of the form eq. (2.6) that are valid everywhere in D except in the neighborhood of caustics (which are a set of measure zero)? First we note that with Dirichlet boundary conditions we have a hermitian eigenvalue problem and so we know that solutions will exist only at a discrete set of real wavevectors k. In the eikonal approximation the quantization condition for k arises from the requirement of single-valuedness of ψ(x) as will be reviewed briefly below. Here our primary goal is to show that the existence of eikonal solutions to the boundary-value problem is intimately tied to the nature of the ray dynamics within the region D. Moreover, for the case of fully chaotic ray dynamics this connection shows that eikonal solutions do not exist. We will prove this latter statement by showing that a contradiction follows from assuming the existence of eikonal solutions in the chaotic case. This argument will be a “physicist’s proof” without excessive attention to full mathematical rigor. The proposed solution for ψ(x) posits the existence of N scalar functions Sm (x) each of which satisfies the eikonal equation, (∇Sm (x))2 = 1, and which, while themselves not being single-valued on the domain D, allow the construction of single-valued functions ψ(x) and ∇ψ(x). Moreover, for the asymptotic expansion to be well-defined, the “rapid variation” in ψ(x) must come from the largeness of k; i.e., to define a meaningful asymptotic expansion in which terms are balanced at each order in k the functions Sn cannot vary too rapidly in space. From the eikonal equation itself we know that |∇Sm | = 1, but we must also have that the curvature ∇ 2 Sm k for the asymptotic solution to be accurate. This condition fails within a wavelength of a caustic, as one can check explicitly, e.g., for the case of a circular domain D; but for a solvable case like the circle it holds everywhere else in D. It is convenient for our current argument to focus on ∇ψ instead of ψ itself. Consider an arbitrary point x0 in D where ∇ψ(x 0 ) = 0; to leading order in k and

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away from caustics we have ∇ψ(x 0 ) = ik

N 

Am (x 0 )∇Sm (x 0 )eikSm (x 0 ) .

(2.7)

m

The N unit vectors ∇Sm (x 0 ) ≡ pˆ m define N directions at x0 ; these are the directions of rays passing through x0 in the stationary solution. An important point is that due to the condition on the curvature just noted, these directions are constant at least within a neighborhood of linear dimension λ = 2π/k around x0 . Choose one of the ray directions, call it pˆ 1 and follow the gradient field ∇S1 to the boundary D. For a medium of uniform index (as we have assumed) the vector ∇S1 is strictly constant in both direction and magnitude along a ray. Thus one can find the direction of ∇S1 at the boundary and calculate its “angle of incidence”, nˆ · ∇S1 , where nˆ is the normal to the boundary at the point of intersection. The condition ψ = 0 on the boundary implies that there is a second term with the eikonal S2 in the sum, which satisfies S1 = S2 and A1 = −A2 on the boundary. As a result, tangent derivatives of S1,2 on the boundary are also equal, and together with eq. (2.3) this implies that for a nontrivial solution nˆ · ∇S2 = −nˆ · ∇S1 . In other words, a ray of the eikonal S1 must specularly reflect at the boundary into a ray of another eikonal in the sum, which we label S2 . Hence we know the direction of ∇S2 at the boundary and can follow it until the next “reflection” from the boundary. Thus each segment of a ray trajectory corresponds to a direction of ∇Sm for some m in eq. (2.7). A ray moving linearly in a domain D and specularly reflecting from the boundary describes exactly the same dynamics as a point mass moving on a frictionless “billiard” table with boundary walls of shape ∂D. Such dynamical billiards have been studied since Birkhoff [1927] as simple dynamical systems which can, and typically do, exhibit chaotic motion. Thus the problem of predicting the properties of the vector fields Sm is identical to the problem of the long-time behavior of dynamical billiards. One property of any such bounded dynamical system (independent of whether it displays chaos) is that any trajectory starting from a point x0 will return to a neighborhood of that point an infinite number of times as t → ∞ (the Poincaré recurrence theorem (Poincaré [1890])). Therefore we are guaranteed that the ray we followed from x0 in the direction pˆ 1 will eventually re-enter the neighborhood of size λ around x0 . By our previous argument, each linear segment of the ray trajectory corresponds to one of the directions ∇Sm , and thus when the ray re-enters the neighborhood of x0 for ∇ψ to be single valued it is necessary that the ray travel in one of the directions ∇Sm (x 0 ) = pˆ m . There can be two categories of ray dynamics: (1) Although the ray enters the neighborhood of x0 an infinite number of times it only does so in a finite number N of ray directions. (2) The number

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of ray directions grows monotonically with time and tends to infinity as t → ∞. We will now show that the general applicability of the eikonal method depends on which category of ray motion occurs. Let us first consider a billiard ∂D with fully chaotic dynamics. In the current context “fully chaotic” means that for arbitrary choice of x0 and the direction pˆ 1 (except for sets of measure zero, such as unstable periodic orbits) the distribution of return directions (momenta) is continuous and isotropic as t → ∞. Therefore the number of terms in an eikonal solution of the form eq. (2.6) would have to be infinite, contradicting our initial assumption of finite N . Thus there do not exist eikonal solutions with finite N for wave equations on domains with fully chaotic ray dynamics. A very closely related point was made by Einstein as early as 1917 (Einstein [1917]) (he phrased it as the nonexistence of a multi-valued vector field defined by the N “sheets” of a function S(x)). One may ask whether an eikonal solution with an infinite number of terms could be defined; this appears unlikely as the amplitudes for the wavefronts are bounded below by (λ/L)1/2 , where L is the typical linear dimension of D, so that only a very special phase relationship between terms would allow such a sum to converge. The essential physics of the breakdown of the eikonal method in a chaotic system is the following. Hermiticity of the eigenvalue problem guarantees that wave solutions exist and the Weyl expansion shows that their average number per unit wavevector is the same as in a regular system with the same volume. However the solutions in the chaotic case require wavefronts which are not straight on a scale much larger than a wavelength, but in fact change direction “turbulently” on the wavelength scale. Therefore the assumption that one can remove the rapid variation in the solution by an overall scale factor k, leaving a set of smooth scalar functions Sm and smoothly varying ray-direction fields ∇Sm simply fails, and no well-ordered asymptotic expansion is possible. Returning now to the case of a boundary ∂D for which the distribution of return momenta is always discrete, this means that there exist exactly N ray directions for each point x0 and any choice of pˆ 1 . In this case the entire spectrum of the wave equation on ∂D can be obtained by an eikonal approximation with N terms of the form eq. (2.6). The quantized values of k are determined by the conditions that the eikonal only advance in phase by an integer multiple of 2π upon each return to x0 and hence the solution is single-valued. The correct quantization condition must take into account phase shifts which occurs for rays as they pass caustics. The details of implementing this condition have become know as Einstein– Brillouin–Keller quantization, since Einstein first proposed the relevant quantization condition within the context of the old quantum theory (Einstein [1917])

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Fig. 1. (a) Typical quasi-periodic ray motion in a circular billiard. The two possible ray return directions for a specific point x0 and initial direction pˆ 1 are shown in gray. (b) The Bunimovich stadium, consisting of two semi-circles connected by straight segments, for which the ray motion is completely chaotic. The ray simulation is shown for 200 bounces, with initial conditions along the gray arrow. In the infinite-time limit, the ray return directions are infinite, making an eikonal solution impossible. (c, d) A quadrupole with deformation parameter ε = 0.16, which has a mixed ray dynamics. (c) An initial condition in the “bouncing-ball” region leads to a quasi-periodic motion with four possible ray directions at a specific point on the trajectory. An eikonal solution corresponding to ray motion in the bouncing-ball regime is possible and includes four terms (N = 4 in eq. (2.6)). (d) Simulation of an initial condition with the same position in real space as in (c) but a different ray direction, which can lead to a chaotic motion with infinite return directions in the infinite-time limit. The inset next to each simulation zooms in to the neighborhood of the initial conditions.

but the application to eikonal approximations and the correct phase shifts were determined by Keller and Rubinow [1960]. From modern studies of billiard dynamics we know that both of the cases we have just considered are exceptional. Billiards for which eikonal solutions for the entire spectrum exist are called integrable, and their ray dynamics has one global constant of motion for each degree of freedom. For example, in the circular billiard both angular momentum and energy are conserved and for each choice of x0 and direction pˆ 1 there are exactly two return directions (Keller and Rubinow [1960]) (see fig. 1). While the circle is a good and relevant example here, there are other shapes, such as rectangles and equilateral triangles, for which the method also works; obviously these are shapes of very high symmetry. It is also known that an elliptical billiard of any eccentricity is integrable; however this is believed to be the only integrable smooth deformation of a circle (Poritsky [1950], Amiran [1997]). Thus there is a relatively small class of boundaries for which eikonal methods work globally; this point does not seem to be widely appreciated in the optics community.

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As already noted, the types of boundary shape which generate continuous return distributions for each choice of x0 and direction pˆ 1 correspond to completely chaotic billiards, and such shapes are quite rare. No smooth boundary (i.e. ∂D for which all derivatives exist) is known to be of this type. A well-known example of relevance to us is the stadium billiard, consisting of two semi-circular “endcaps” connected by straight sides. Note that the generation of continuous return distributions would fail for a point x0 between the two straight walls if we chose pˆ 1 perpendicular to the walls, generating a (marginally stable) two-bounce periodic orbit passing through x0 . However this choice represents a set of measure zero of the initial conditions in the phase space. It follows from our above arguments that eikonal methods would fail for the entire spectrum in such a billiard (except a set of measure zero in the short-wavelength limit). The generic dynamics of billiards arises when the boundary is smooth but there is not a second global constant of motion; this is exemplified by the quadrupole billiard we will study extensively below (see definition in eq. (3.1)). Such a billiard has “mixed” dynamics; we shall explain what this means and how it is studied in more detail below. For such a billiard, depending on the choice of the initial phase-space point (x0 , pˆ 1 ), one may get either a finite number N of return directions or an infinite number as t → ∞. It is plausible that in such a case a finite fraction of the spectrum can be found by eikonal methods but the remainder cannot; the eikonal solutions correspond to regular types of motion in the mixed phase space. If one can obtain a relatively tractable expression for the locallyconserved quantity which leads to a finite number of return directions N , and hence to regular motion, then analytic approximate solutions can be found for this subset of states. Examples of this are solutions near stable periodic orbits (Keller and Rubinow [1960], Ralston [1976]) or near the boundary of convex billiards (Keller and Rubinow [1960], Lazutkin [1993]); closely related are solutions based on the adiabatic approximation of Robnik and Berry [1985] as found by Nöckel and Stone [1997]. Eikonal solutions obtained in this way typically diverge on the caustics. Uniformized solutions near isolated stable periodic orbits can be found by the methods of Gaussian optics; this is worked out in detail for dielectric billiards with mixed dynamics by Türeci, Schwefel, Stone and Narimanov [2002]. The theory of the Kolmogorov–Arnold–Moser (KAM) transition to chaotic dynamics implies that for weak nonintegrable perturbations a large fraction of phase space is filled with “irrational KAM tori” corresponding to regular quasi-periodic motion (see below), but we are not aware of any tractable semiclassical technique to quantize this torus motion in the mixed dynamical regime. Moreover, both the eikonal and Gaussian methods fail for a fraction of the spectrum which approaches unity as the chaotic fraction of phase space approaches unity.

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Since the traditional analytic methods of optics fail for these systems, what other short-wavelength approaches exist? The development of short-wavelength approximations for mixed and chaotic systems is precisely the problem of quantum chaos which has been studied widely in atomic, nuclear, solid-state and mathematical physics over the past two decades (Gutzwiller [1990], Haake [2000], Giannoni, Voros and Zinn-Justin [1989], Berggren and Åberg [2000]). Powerful analytic methods have been developed which are essentially different in character from eikonal or Gaussian methods (these latter techniques are typically referred to as semiclassical methods in the quantum chaos literature). The analytic methods in quantum chaos theory are all of a statistical character and do not allow one to calculate individual modes. Instead, the methods focus on the fluctuating part of the density of modes and the statistical properties of the spectrum (e.g., levelspacing distributions). The results are useful in many contexts, but less useful in the context of optical resonators and microlasers for which a single mode or a small set of modes will be selected and one is interested in their emission patterns and Q-values. Therefore it is particularly important to develop efficient numerical methods for calculating the spectrum and modes of such dielectric resonators, and we will discuss our method for doing this in Section 6 below. We are primarily interested in ARC resonators with mixed billiard dynamics as these shapes lead to resonances with high Q and directional emission. For such resonators, the ray phase space is not fully chaotic, but is highly structured. Moreover, the possibility of ray escape decreases the randomizing effect of chaotic motion at t → ∞. Therefore using methods which maintain a connection between the wave solutions and the ray phase space is very helpful. We shall describe such a method, known as Husimi–Poincaré projection, in Section 8 below.

§ 3. Ray dynamics for generic dielectric resonators Before introducing our numerical method and the Husimi–Poincaré projection method, we review the properties of mixed phase space via the surface of section method in the context of the quadrupole billiard/ARC. This billiard is described by the boundary shape R(φ) = R0 (1 + ε cos 2φ)

(3.1)

which in the zero deformation limit ε = 0 reduces to a circular billiard that, as we have already noted, is integrable. Therefore the variation of the parameter ε starting from zero induces a transition to chaos. As the perturbation is smooth, various results in dynamical systems (collectively known as Kolmogorov–Arnold–Moser

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Fig. 2. Construction of the surface of section plot. Each reflection from the boundary is represented by a point in the SOS recording the angular position of the bounce on the boundary (φ) and the angle of incidence with respect to the local outward pointing normal (sin χ). For a standard dynamical billiard there is perfect specular reflection and no escape. For “dielectric billiards” if sin χ > sin χc = 1/n, total internal reflection takes place, but both refraction and reflection according to Snell’s law result when a bounce point (bounce #4 in the figure) falls below the “critical line” sin χ = sin χc . Note that sin χ < 0 corresponds to clockwise sense of circulation. We do not plot the sin χ < 0 region as the SOS has reflection symmetry. Below we will plot the SOS for ideal billiards without escape unless we specify otherwise.

theory) imply that the transition to chaos is gradual (Arnold [1989], Lazutkin [1993]). The quadrupole billiard displays the typical behavior characteristic of this transition. In our initial discussion here we treat the ideal perfectly reflecting billiard; later we will discuss the role of ray escape in ARCs. When the shape is gradually deformed, it quickly becomes unfeasible to capture the types of ensuing ray motion by standard ray-tracing methods in real space. A standard tool of nonlinear dynamics, which proves to be very useful in disentangling the dynamical information, is the Poincaré surface of section (SOS) (Lichtenberg and Lieberman [1992], Reichl [1992]). In this two-dimensional phase-space representation, the internal ray motion is conveniently parametrized by recording the pair of numbers (φi , sin χi ) at each reflection i, where φi is the polar angle denoting the position of the ith reflection on the boundary, and sin χi is the corresponding angle of incidence of the ray at that position (see fig. 2). Each initial point is then evolved in time through the iteration of the SOS map i → i+1, resulting in basically two general classes of distributions. If the iteration results in a one-dimensional distribution (an invariant curve), the motion represented is regular. On the other hand, exploration of a two-dimensional region is the signature of chaotic motion.

2, § 3]

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Modes of wave-chaotic dielectric resonators

[2, § 4

Fig. 3. SOS of a closed quadrupole billiard at fractional deformations ε = 0, 0.05, 0.11, 0.18. The closed curves and the curves crossing the SOS represent two types of regular quasi-periodic motion: librational motion near a stable periodic orbit and marginally stable rotational motion, respectively. The regions of scattered points represent chaotic portions of phase space. A single trajectory in this “chaotic component” will explore the entire chaotic region. With increasing deformation the chaotic component of the SOS (scattered points) grows with respect to regular components and is already dominant at 11% deformation. Note in (b) the separatrix region associated with the two-bounce unstable orbit along the major axis where the transition to chaotic motion sets in first.

The transition to ray chaos in the quadrupole billiard is illustrated in fig. 3. At zero deformation the conservation of sin χ results in straight-line trajectories throughout the SOS and we have globally regular motion. These are the wellknown whispering-gallery (WG) orbits for sin χ > n1 . As the deformation is increased (see fig. 3) chaotic motion appears (the areas of scattered points in fig. 3) and a given initial condition explores a larger range of values of sin χ. Simultaneously, islands of stable motion emerge (closed curves in fig. 3), but there also exist extended “KAM curves” (Lazutkin [1993]) (open curves in fig. 3), which describe a deformed WG-like motion close to the perimeter of the boundary (these are examples of the regular quasi-periodic motion mentioned above). These islands and KAM curves cannot be crossed by chaotic trajectories in the SOS. As the transition to chaos ensues, a crucial role is played by the periodic orbits (POs), which appear as fixed points of the SOS map. The local structure of the islands and chaotic layers can be understood through the periodic orbits which they contain. Thus, the center of each island contains a stable fixed point, and close to each stable fixed point the invariant curves form a family of rotated ellipses. The Birkhoff fixed point theorem (Lichtenberg and Lieberman [1992]) guarantees that each stable fixed point has an unstable partner residing on the intersection of separatrix curves surrounding the elliptic manifolds. Chaotic motion sets in at separatrix regions first, and with increasing deformation pervades larger and larger regions of the SOS. Already at ε = 0.1, much of the phase space is chaotic and a typical initial condition in the chaotic sea explores a large range of sin χ, eventually reversing its sense of rotation.

§ 4. Formulation of the resonance problem A dielectric resonator is significantly different from the closed (Dirichlet) problem due to its openness. In contrast to ideal metallic cavities which possess normal modes at discrete real frequencies, dielectric resonators are characterized by a

2, § 4]

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discrete set of quasi-bound modes (Leung, Liu and Young [1994], Ching, Leung, van den Brink, Suen, Tong and Young [1998]), or resonances. As a result, the quasi-bound modes of a resonator are characterized by a frequency ω = ck and a lifetime τ , where c is the speed of light and k = 2π/λ is the wavevector in vacuum. Experiments on resonators fall into two broad categories, and the presence of quasi-bound modes is manifested differently in these two situations. In scattering experiments, an incoming field produced by a source in the far field (spatial infinity) gives rise to an outgoing field which represents the response of the resonator, as measured by an ideal detector in the far field. In the ideal case, where absorption is absent, this corresponds to a situation where energy is conserved and hence in this situation the EM field has a real frequency, ω, which is arbitrary and set by the source. In emission experiments, on the other hand, there is no incoming field, but only an outgoing field. As a result, energy is depleted from the system, and this process is characterized by decay. The simplest mathematical description of these two experiments corresponds to the solution of the wave equation (which is derived from the Maxwell equations as described in Section 5)   n2 (x) ∂ 2 ∇2 − (4.1) Ψ (x, t) = 0, c2 ∂t 2 where the solutions have the separable, time-harmonic dependence Ψ (x, t) = ψ(x)eiωt

(4.2)

so that ψ(x) obeys the Helmholtz equation

2 ∇ + n2 (x)k 2 ψ(x) = 0.

(4.3)

Here, n(x) represents the index of refraction. In general, one can define a com(−) (+) plete set of incoming {ψµ (k; x)} and outgoing {ψµ (k; x)} modes at a given k, in the absence of the resonator. The exact form of these sets is dictated by convenience, and in the present discussion we will employ cylindrical harmonics. The two experimental situations at this point are distinguished by two different boundary conditions in the far field. The scattering experiment corresponds to the boundary condition  ψ(x) ∼ ψµ(−) (k; x) + (4.4) Sµν (k)ψν(+) (k; x), |x| → ∞, ν

and experimentally, it is the scattering matrix Sµν (k), which contains the information measured by the far-field detector. In the typical case, Sµν (k) (as well as IFF (θ ) = |ψ(r → ∞, θ )|2 ) will display sharp peaks at a discrete set of real wavevectors ki (in case of isolated resonances; see the back panel (Re[kR]–I

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Fig. 4. Comparison of scattering and emission pictures for quasi-bound modes. Variation of the intensity scattered off a dielectric circular cylinder with wavenumber k of an incoming plane wave is plotted on the back panel (I –Re[kR] plane). The intensity is observed at 170◦ with respect to the incoming wave direction; it would look significantly different in another direction due to interference with the incident beam. The complex quasi-bound mode frequencies are plotted on the Re[kR]–Im[kR] plane. Notice that the most prominent peaks in scattering intensity are found at the values of k where a quasi-bound mode frequency is closest to the real-axis. These are the long-lived resonances of the cavity. Also visible is the contribution of resonances with shorter lifetimes (higher values of Im[kR]) to broader peaks and the scattering background.

plane) on fig. 4). This is the signature of long-lived quasi-bound modes with frequency ω = cki ; their lifetimes τ are encoded in the functional form of the peaks, which in general is of Fano shape, with direction-dependent parameters. This makes scattering boundary conditions less convenient for the extraction of the quasi-bound mode structure. Emission experiments are modeled by the outgoing wave boundary conditions at infinity  γν (ki )ψν(+) (ki ; x), |x| → ∞. ψ (i) (x) ∼ (4.5) ν

This form at infinity does not permit solution for any real k as it manifestly violates current conservation. Instead, the solutions of eq. (4.3), indexed by i, exist only at discrete complex wavevectors ki = κi + iΓi . The connection to quasibound modes is then direct; the real part gives the quasi-bound mode frequency

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ωi = cκi , and the imaginary part represents the lifetime of the mode, τi = 1/cΓi . Here, we will use the radiation boundary conditions exclusively, and quote the dimensionless complex variable kR instead, where R is the mean radius of the resonator. In an active medium, one may think of these resonances as being pulled up to real wavevectors by the gain. Although the description of actual (stationary) laser modes requires the solution of a nonlinear wave equation (Türeci [2003]), we will focus here on the problem of linear resonances, an approximation which is often used in laser theory (Siegman [1986]). The relation between the linear emission and scattering pictures is easily visualized in the extended complex wavevector space of the scattering matrix Sµν (k), depicted in fig. 4. The discrete quasi-bound wavevectors ki are the poles of Sµν (k). As can be seen from the figure (where we plot IFF (θ ), the actual experimentally accessible quantity), in general there are multiple quasi-bound modes contributing to a given resonance peak, but the quasi-bound modes which are closest to the real axis lead to the sharpest peaks (some of which might not even be resolved in the scattering profile). Note that via eq. (4.2), the quasi-bound mode solutions damp in time. An important experimental value often quoted is the Q-value of a resonator, which is defined by the number of cycles of the optical field at frequency ω it takes to decay to half of its value, and thus can be related to quasi-bound mode parameters by the relation Q = 2ωτ = 2 Re[kR]/| Im[kR]|. It is possible to generalize eikonal theory to calculate the quasi-bound states of complex k of dielectric cavities of integrable shape (Türeci [2003]). The dielectric boundary conditions then reduce to ray trajectories which still propagate on straight lines and undergo specular reflection, much like in the case of billiards. The additional feature is that dielectric billiards exhibit ray splitting at the boundary, and give rise to both a refracted and a reflected ray, with amplitude and direction obtained from the application of the laws of Snell and Fresnel for a flat dielectric boundary. The transport equations for the amplitude have to be supplemented by an additional complex multiplicative factor at each encounter with the boundary. The practical implication is that the ray motion as displayed on the SOS can be used for the dielectric problem, when augmented with an escape condition. The escape probability will be exponentially small in wavenumber k for angles of incidence above the critical angle χc = sin−1 1/n, since it is due to a tunnelinglike process (Nöckel [1997], Türeci [2003]). This condition is demarcated by the line sin χ = sin χc in the SOS; any ray falling below this line refracts out with the probability given by the local Fresnel law of refraction (assuming a TM mode):  2 1 − sin2 χ  T (sin χ) =  (4.6) 1 − sin2 χ + sin2 χc − sin2 χ

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providing a classical loss mechanism and leading to finite lifetimes of the corresponding modes. Thus the openness of dielectric resonators is not the cause of the failure of standard methods; these methods fail for chaotic resonator shapes for the same reasons of dynamical complexity as discussed in Section 2 for the Dirichlet case. We now discuss the reduction of the wave equation for dielectric resonators to the Helmholtz equation to lay the groundwork for the numerical method of Section 6, which allows us to solve for the resonances of chaotic shapes.

§ 5. Reduction of the Maxwell equations Consider the problem of the excitation of electromagnetic waves in an infinite dielectric rod of arbitrary cross-section (see schematics in fig. 5), which is extended along the z-axis. In practical situations, the structure is of finite extent and there are planar endcaps which makes it truly a resonator. In other cases, it is a fiber-optic cable of practically infinite extent. In any case, we will for now assume translational symmetry along the z-axis, and we will show later that this is a perfectly valid assumption for the modes of relevance to us. We will assume a harmonic variation in time for the electromagnetic fields of the resonator: E E (x, t) = (x)e−iωt , (5.1) B B where ω is the frequency of the field, and in general is a complex number. The spatial distribution of these fields is governed by the reduced Maxwell equations ∇ × E = ikB, ∇ · E = 0, ∇ × B = −in2 kE, ∇ · B = 0,

(5.2)

Fig. 5. Illustration of the reduction of the Maxwell equation for an infinite dielectric rod of general cross-section to the 2D Helmholtz equation for the TM case (electric field parallel to axis) and k = kz = 0.

2, § 5]

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95

√ where k = 2π/λ = ω/c is the wavevector in vacuum; n = µε is the index of refraction, µ is the permeability and ε is the dielectric constant of the medium, which we assume to be piecewise continuous functions of position. It follows that both vector fields satisfy the Helmholtz equation

E 2 2 2 ∇ + n(x) k (5.3) = 0. B The translational symmetry along the z-axis allows us to express the z-variation of the fields as E E (5.4) (x) = (x, y)e−ikz z . B B Following Jackson [1998], we separate the fields and operators into components parallel and transverse to the z-axis:

E E⊥ Ez ∂ (x, y) + (x, y), ∇ = ∇ ⊥ + zˆ . (x, y) = zˆ ∂z Bz B⊥ B (5.5) Writing out the transverse and longitudinal projections of eq. (5.2), we obtain after a few manipulations: i (k zˆ × ∇ ⊥ Bz + kz ∇ ⊥ Ez ), γ2

i B ⊥ = 2 n2 k zˆ × ∇ ⊥ Ez − kz ∇ ⊥ Bz , γ E⊥ = −

(5.6) (5.7)

where γ 2 = n(x)2 k 2 − kz2 . It is evident from these four (scalar) equations that Ez and Bz are the fundamental fields we should be after, and that once they are determined we can solve for E ⊥ and B ⊥ . Thus, the Maxwell equations themselves decouple completely, which was already obvious from eq. (5.3). The actual complication of solving the vector Helmholtz equation stems from the fact that the boundary conditions are coupled. The Maxwell boundary conditions are

nˆ · n21 E 1 − n22 E 2 = 0, nˆ × (E 1 − E 2 ) = 0, (5.8) nˆ × (B 1 − B 2 ) = 0,

nˆ · (B 1 − B 2 ) = 0,

(5.9)

in the absence of surface currents and charges and for a linear, isotropic medium. We will assume n1 = n > n2 = 1. Note that these are six conditions altogether. The subscripts denote the media on respective sides of the interface, and nˆ is the unit normal on the interface, pointing out from the cylinder. We will also define ˆ yielding the triad (ˆz, tˆ , n) ˆ to the unit tangent vector at the interface, tˆ = zˆ × n,

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express any given vector field at the interface. The boundary conditions (5.8)– (5.9) can be re-expressed in terms of the primary scalar fields Ez , Bz using i (k∂n Bz + kz ∂t Ez ), γ2 i En = 2 (k∂t Bz − kz ∂n Ez ), γ

i Bt = 2 n2 k∂n Ez − kz ∂t Bz , γ

i Bn = − 2 n2 k∂t Ez + kz ∂n Bz , γ Et = −

(5.10) (5.11) (5.12) (5.13)

the projections of eqs. (5.6)–(5.7) at the interface. It is possible to show that the resulting six scalar boundary conditions in terms of Ez1 , Ez2 , Bz1 , Bz2 are not all linearly independent. One convenient choice of a linearly independent set of four boundary conditions comprises the continuity of the fields, Ez1 = Ez2 ,

Bz1 = Bz2 ,

(5.14)

and the following conditions on the tangent and normal derivatives of the fields:   kz k k kz (5.15) ∂n Bz1 − 2 ∂n Bz2 = − 2 − 2 ∂t Ez1 , γ12 γ2 γ1 γ2   n22 k n21 k kz kz ∂n Ez1 − 2 ∂n Ez2 = + 2 − 2 ∂t Bz1 . (5.16) γ12 γ2 γ1 γ2 Thus, a compact expression of the boundary-value problem to be solved is given by the two-dimensional Helmholtz problem   2 ∇⊥ + γi2 Ψ i (x, y) = 0 (5.17) supplemented with the boundary conditions Ψ 1 |∂D = Ψ 2 |∂D , A1 Ψ 1 |∂D = A2 Ψ 2 |∂D , Ez

where Ψ = Bz , and A1 , A2 are matrices given by   kkz (n22 − n21 )∂t γ22 ∂n A1 = , n21 γ22 ∂n −kkz (n22 − n21 )∂t   0 γ12 ∂n A2 = n22 γ12 ∂n 0

(5.18)

(5.19)

to be evaluated on ∂D. It is possible to solve this vector-partial differential equation with the method outlined in the following section. We are however interested

2, § 6]

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97

only in the long-lived modes of the resonator. Modes with a finite kz correspond in the semiclassical limit to rays which spiral up and down along the cylinder walls and escape through the endcaps by refracting out. Thus under most circumstances the longest-lived modes have kz ≈ 0, and correspond to modes which are effectively two-dimensional, i.e. can be expressed by the dynamics of rays on the cross-sectional plane. In that case, the boundary conditions eq. (5.18) also become diagonal and we have complete decoupling. We choose to work with ψi (x, y) = Ezi (x, y) and Bzi (x, y) = 0, corresponding to TM polarized resonant modes, for which the problem reduces to the two-dimensional Helmholtz equation for the scalar field ψ with continuity conditions 2

∇⊥ + n2i k 2 ψi (x, y) = 0, (5.20) ψ1 |∂D = ψ2 |∂D ,

∂n ψ1 |∂D = ∂n ψ2 |∂D .

(5.21)

Note that this boundary-value problem is equivalent to that of the stationary Schrödinger equation of quantum mechanics. Hereafter, we will drop all references to the original three-dimensional and vector character of the problem and work with eqs. (5.20)–(5.21).

§ 6. Scattering quantization – philosophy and methodology In this section we will describe a numerical method for solving eq. (5.21) that is both efficient and physically appealing. Our approach is a generalization to open systems (specifically, dielectric resonators) of the scattering quantization approach to quantum billiards (Doron and Smilansky [1992], Dietz, Eckmann, Pillet, Smilansky and Ussishkin [1995]). This approach is based on the observation that every quantum billiard interior problem (Helmholtz equation for a bounded region with Dirichlet/Neumann boundary conditions) can be viewed as a scattering problem, and the spectrum can be uniquely deduced from the knowledge of the corresponding scattering operator. In the case of closed systems, the internal scattering problem can be mapped rigorously to an external scattering problem (Eckmann and Pillet [1995]), and the resulting (exact) scattering matrix is unitary. For the dielectric resonator problem with radiation boundary conditions, we will see that the corresponding scattering operator is inherently nonunitary, reflecting the physical fact that we are dealing with a leaky system. Thus we will define below a new “S-matrix” which is nonunitary and distinct from the true S-matrix describing external scattering from the system. We retain the terminology “S-matrix” nonetheless because of the conceptual similarity to the quantum

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Modes of wave-chaotic dielectric resonators

[2, § 6

billiard method of (Doron and Smilansky [1992], Dietz, Eckmann, Pillet, Smilansky and Ussishkin [1995]). The generalization of this approach to dielectric billiards was first made by Narimanov, Hackenbroich, Jacquod and Stone [1999], however without the efficient algorithm presented below. We assume that the resonator is bounded by an interface ∂D of the form r = R(φ), where R(φ) is some smooth deformation of the boundary such that there exists only one point of the boundary for each angle φ. We decompose the internal and external fields into cylindrical harmonics with a constant k: ψ1 (r, φ) = ψ2 (r, φ) =

∞ 

imφ − αm H+ , m (nkr) + βm Hm (nkr) e m=−∞ ∞ 



imφ − γm H + , m (kr) + δm Hm (kr) e

r < R(φ),

r > R(φ).

(6.1)

(6.2)

m=−∞

Each of the terms ± imφ ψm (r, φ) = H+ m (nkr)e

(6.3)

in the sum is a solution of the appropriate (interior or exterior) Helmholtz equa± } forms tion, but does not satisfy the matching conditions by itself. Note that {ψm a normal basis in the infinite space. Owing to the completeness of this basis, the expansion is exact for r < Rmin and r > Rmax as long as the sum runs over an infinite number of terms, where Rmin and Rmax are the lower and upper bounds of R(φ), respectively. The assumption that the expansions can be continued analytically to the region Rmin < r < Rmax is known as the Rayleigh hypothesis (Rayleigh [1907]). It has been shown (van den Berg and Fokkema [1980]) that for a family of deformations parametrized by ε, there is typically a critical deformation εc beyond which the hypothesis breaks down because the expansion ceases to be analytic in the region Rmin < r < Rmax . For the deformations eq. (3.1), this happens long after the shape becomes concave; we are not interested in this regime. Although this issue seems thus to be resolved, we shall see that precursors of the nonconvergence emerge in the form of numerical instabilities for ε < εc . We will assume that δm = 0 (no incoming waves), thus confining our attention to quasi-bound modes. Turning to the interior expansion, the regularity of the solution at the origin requires that we take αm = βm , but we will not implement this condition at this stage. The continuity conditions eq. (5.21) give us further relations among the remaining coefficients:



ψ1 φ, R(φ) = ψ2 φ, R(φ) , (6.4)

∂ψ1

∂ψ2

(6.5) = . ∂r
∂r
φ,R(φ)

φ,R(φ)

2, § 6]

Scattering quantization – philosophy and methodology

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In eq. (6.5), we have replaced the normal derivative condition by the radial derivative condition, because eq. (6.4) shows that the tangential derivatives are also continuous. Note that this latter set of equations containing radial derivatives is equivalent to the set of eq. (5.21) using normal derivatives. These conditions can be written out as ∞ 



imφ − αm H+ m nkR(φ) + βm Hm nkR(φ) e m=−∞

= n

∞ 



imφ γm H + , m kR(φ) e

(6.6)

m=−∞ ∞ 





imφ  − αm H+ m nkR(φ) + βm Hm nkR(φ) e

m=−∞ ∞ 

=



imφ  γm H + . m kR(φ) e

(6.7)

m=−∞

We multiply both sides by wn (φ)e−inφ and integrate with respect to φ to get a matrix equation for the coefficient vectors |α, |β and |γ  H1+ |α + H1− |β = H2+ |γ , (6.8) 1 − + DH+ (6.9) 1 |α + DH1 |β = n DH2 |γ . Various choices of the weight function w(φ) are possible (Narimanov, Hackenbroich, Jacquod and Stone [1999]); here we choose w(φ) = 1. The matrices in eqs. (6.8) and (6.9) are defined by  2π  ±

i(m−l)φ Hj lm = (6.10) dφ H+ , m nj kR(φ) e 0

  DH± j lm =



2π 0



i(m−l)φ  dφ H± . m nj kR(φ) e

(6.11)

Eliminating |γ  between eq. (6.8) and eq. (6.9), we obtain S(k)|α = |β, where the matrix S(k) is given by 

−1 + −1 − −1 S(k) = n DH+ DH− H1 2 1 − H2   + −1 + + −1 × H2 H1 − n DH2 DH+ 1 .

(6.12)

(6.13)

As noted earlier, this S-matrix is different from the standard external scattering matrix introduced in eq. (4.4). It is straightforward to check that, for real k, S(k) is

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[2, § 6

nonunitary. Consider now the eigenvalue problem of S(k), S(k)|α = eiϕ |α,

(6.14)

where the phase ϕ is in general complex. Once we find a complex kq where one (or several) of the ϕ is a multiple of 2π, we have |α = |β, which is exactly the condition of regularity at the origin. This is the quantization condition which will provide us with the quantized eigenvalues and eigenvectors (kq , |α (q) ) that allow us to construct the resonant solutions of the interior and exterior problem we set out to find. This condition is often expressed in terms of the secular function ζ (k) (Doron and Smilansky [1992]) given by   ζ (k) = det 1 − S(k) . (6.15) The spectrum is obtained as the zeros of the secular equation ζ (k) = 0. As noted, the values kq for which we obtain a unit eigenvalue and the secular function eq. (6.15) has a root, are always complex, and the eigenvalues of S(k) are not pure phases. The practical upshot of this is that this requires a two-dimensional root search for the equation ζ (k) = 0. An often-employed numerical procedure involves a sweep in the complex k-plane of the singular values of the operator T (k) = 1 − S(k), with proper care of the numerical null-space of T (k) (Barnett [2000]). This requires several calculations of the entries of T (k) and its singular value decomposition per quantized state. In the next section, we will present an efficient root-finding method which ideally requires two diagonalizations per nkR quantized states. Before doing that however, it is worthwhile to investigate the structure of S(k) based on simple physical considerations. A physical interpretation of the internal scattering operator S(k) and its eigenvectors can be given even off-quantization (ϕ(k) = 2π) (Klakow and Smilansky [1996], Frischat and Doron [1997]). We can visualize this approach in our case by dividing the interior of the resonator into two subdomains joined along the curve C, which we take to be a circle of radius RC  Rmin , and considering it as a boundary at the junction of two back-to-back scattering systems. We furthermore introduce a tiny metallic inclusion of radius δ at the origin (this is introduced for the sake of the argument and could be omitted). This is our first scattering system, which scatters an incoming wave |β into |α via the scattering operator Sδ : |β = Sδ (k)|α.

(6.16)

Sδ (k) is exactly the exterior scattering operator for a metallic circle (immersed in a medium with index of refraction n):

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101

Fig. 6. Schematics of the quantum Poincaré mapping induced by the internal scattering operator; see discussion in text.

  H− (nkδ) Sδ (k) mm = − m δmm . H+ m (nkδ)

(6.17)

The second scattering system is the boundary itself, scattering an incoming wave (with respect to the boundary) |α into |β, and the scattering operator for this system is simply S(k) whose form is given in eq. (6.13). Consider now a whole cycle, starting with the state |α on C, being first scattered off the boundary, then from the tiny circle returning to C again (see fig. 6). The resulting scattered vector is Sδ S|α. Now, as kδ → 0, we have Sδ → 1, and the re± in sulting scattered vector is S|α. Because the individual normal modes ψm our expansion correspond to ray trajectories with well-defined angular momenta m sin χ = nkR , the mapping S|α can be interpreted as a wave analogue of the C Poincaré SOS mapping on the section C, parametrized by (φ, sin χ). This link has been used fruitfully to obtain short-wavelength forms of the scattering operator S(k), for various closed systems (Klakow and Smilansky [1996]). We will not pursue this approach here, but will make use of this visualization to develop a meaningful truncation scheme for a numerical implementation of our method. ± , for which First of all, at a given k, an angular momentum eigenstate ψm m > nkRmax is a closed channel for the internal scattering system, because it

102

Modes of wave-chaotic dielectric resonators

[2, § 6

Fig. 7. Gray-scale representation of the scattering matrix eq. (6.13), calculated for a quadrupolar resonator at ε = 0.1 deformation, n = 2.5 and nkR = 40. The number of evanescent channels used in the calculation is Λev = 15. Note the strong diagonal form for |m| > nkR. The spread around the diagonal is proportional to the deformation. Here the internal scattering couples approximately 20 angular momentum modes.

corresponds to classical motion with a circular caustic of radius larger than Rmax . Such channels are called evanescent, and are not expected to be scattered significantly. In fact, a plot of the matrix S(k) (fig. 7) reveals that as m grows beyond a critical value mc ≈ nkRmax , the scattering matrix becomes strongly diagonal, i.e. [S(k)]mm ≈ δmm for |m|, |m | > mc . Furthermore, there is a transition region nkRmin < |m|, |m | < nkRmax , where the matrix is heading towards diagonality, and this region corresponds to evanescent components which undergo enhanced scattering because they overlap significantly with only certain regions of the resonator. This region grows with the deformation of the resonator, and consequently, Λev evanescent channels have to be included in the number Λ of (positive) channels contributing to a given internal scattering matrix. Denoting the critical matrix size at the evanescent channel boundary by Λsc = JnkRmin K (where J·K stands for the integer part), the size of the S-matrix is then Ntrunc = 2Λ + 1, with Λ = Λsc + Λev .

2, § 7]

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103

Fig. 8. Distribution of scattering eigenvalues (circles) in the complex plane for nkR = 106, ε = 0.12, n = 2.65. The dashed line is the unit circle |z| = 1. Long-lived states have the modulus of their eigenvalue very close to unity, i.e. the eigenphase has only a small imaginary part η.

§ 7. Root-search strategy A typical run at nkR0 = 106 for ε = 0.1 produces the eigenvalue distribution {eiϕk } plotted in fig. 8 in the complex z = eiϕ plane. We put ϕ = θ + iη, where θ and η are real numbers, so that |z| = exp(−η). Note that Λsc = JnkR0 × (1 − ε)K = 93 and we have included Λev = 55 evanescent channels. The handling of numerical stability issues relating to the inclusion of such a large number of evanescent channels is outlined in Appendix A. Our first observation is that all the eigenvalues are strictly distributed within the unit circle |z| = 1, i.e. Im[ϕ] < 0. This is because of the restriction of solutions to outgoing waves only. Furthermore, there is an accumulation of eigenvalues on the boundary of the circle, particularly at θ = 2π + . As we have established, an eigenvalue for which ϕ (l) (k) = 2π within a given numerical precision yields a quantized mode of the resonator. However, we should resist the temptation to simply take all the scattering eigenstates whose eigenphases are ϕ ≈ 2π to be quantized.

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Modes of wave-chaotic dielectric resonators

[2, § 7

As was pointed out by Dietz, Eckmann, Pillet, Smilansky and Ussishkin [1995] in the case of a closed system, there is an accumulation of scattering eigenphases at ϕ ≈ 2π + , which do not correspond to proper physical eigenmodes of the resonator. These modes are primarily composed of evanescent channels, and can easily be distinguished from regular modes by their lack of k-dependence, as we shall see below.

7.1. Zero deformation – case of the rotationally symmetric dielectric A lot can be learned by way of a simple example. We will consider a case where we know the exact solutions, namely the dielectric circle. The exact eigenstates of the scattering matrix for the circle can be given a precise physical meaning in terms of classical processes in the short-wavelength limit. They correspond to motion with a conserved angular momentum, or in terms of our notation in Section 3, a given impact angle sin χ on the dielectric interface. The resulting scattering matrix is diagonal in the angular momentum representation. This signifies the fact that a “channel” with a given m upon encountering the boundary will be scattered to the same channel m, corresponding to specular reflection. The scattering matrix can be written as   H+ (nkR) S(k) mm = −δmm m (7.1) fm (k), H− m (nkR) where the function fm (k) is given by    −1  +  + H+ H− m (nkR) Hm (kR) m (nkR) Hm (kR) . fm (k) = 1 − n + × 1−n −   Hm (nkR) H+ Hm (nkR) H+ m (kR) m (kR) (7.2) This form in terms of the particular ratios of Hankel functions will help us simplify the expressions considerably in the asymptotic limit nkR → ∞. Notice that when fm (k) = 1,   H− (nkR) Sc (k) mm = − m (7.3) δmm H+ m (nkR) is the (inverse of the) external scattering matrix for the closed circular cavity, which is unitary. Then our quantization condition [Sc (k)]mm = 1 yields Jm (nkR) = 0,

(7.4)

which is the exact quantization condition for wavevectors nk of a metallic cavity. Hence in the form (7.1), the corrections due to the openness of the system are lumped into the factor fm (k).

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105

Let us first consider the diagonal elements of eq. (7.1) for m > nkR. We will use the notation α = cosh−1 (m/nkR), α  = cosh−1 (m/kR), β = cos−1 (m/nkR) and β  = cos−1 (m/kR). Note that α  > α  1. Using the large-order asymptotic representations for Bessel functions (Abramovitz and Stegun [1972]) and with proper attention for exponentially small terms, it can be shown that (Türeci [2003])   S(k) mm ∼ 1 + i(1 + 2n)e−2mα (7.5) for m  nkR. As noted, these entries correspond to scattering of evanescent channels and result in eigenphases exponentially close to zero, ϕ ∼ (1 + 2n) × e−2mα . Thus, the accumulation of eigenphases on the unit circle close to the quantization point ϕ = 2π in fig. 8 can be linked to such extremely evanescent channels, which are not the physical modes of the cavity. These modes can be interpreted as creeping waves, evanescent modes which cling to the surface of the resonator (Nussenzveig [1992]). Note that the number of such scattering eigenstates depends strongly on the choice of Λev in our numerical implementation. Next, we will look at the internally reflected channels. These are obtained for the entries kR < m < nkR. The asymptotic form of the corresponding matrix elements are (Türeci [2003])       e−2mα  , S(k) mm ∼ eiΘ 1 − i 2n sin βe−α − i (7.6) n sin β where Θ, which is identical to the closed-circle eigenphase viz. eq. (7.3), is real and is given by π Θ(k) = −2m(β − tan β) − . (7.7) 2 These channels yield eigenvalues which accumulate exponentially close to the unit circle |z| = 1 but, unlike the evanescent modes (7.5), with arbitrary phases. Note that the exponentially small difference from |z| = 1 represents the evanescent leakage which vanishes in the short-wavelength limit. It is possible to assign a velocity to these eigenphases in k-space:   1 dΘ = 2 sin β + O > 0. (7.8) d(nkR) nkR A useful observation at this point is that this velocity is twice the cosine of the conserved ray impact angle χ = π2 − β in the circular billiard corresponding to the motion with angular momentum m (see fig. 9). The picture this entails is the following: When we slowly increase k, the individual eigenphases move with an approximately constant but mode-dependent

106

Modes of wave-chaotic dielectric resonators

[2, § 7

Fig. 9. Geometric representation of the angle β; the velocity of the eigenvalues in the complex plane is 2 sin β, which is also the chord length of the corresponding ray. We note that for the diametral two-bounce orbit the speed is maximal (corresponding to the minimum free spectral range) while for whispering-gallery modes the chord length is minimal and the free spectral range is the largest.

speed given by eq. (7.8) counter-clockwise around the unit circle. Each time one of the eigenphases passes through ϕ = 2π, the quantization condition is fulfilled and the resulting eigenvector is a quantized mode of the resonator. Hence, the eigenvectors of S(k) can be assigned a physical meaning and identity even when k is not tuned to resonance ϕ(k) = 2π. In the present case, they correspond to TIR whispering-gallery modes. Finally, we investigate the classically open channels, which corresponds to rays which are refracted out. In this regime m < kR and   sin β  − n sin β iΘ S(k) mm ∼ e . sin β  + n sin β

(7.9)

Note that the algebraic prefactor is the Fresnel reflection factor for a ray coming in at an angle χi = π2 − β. Thus, the proximity of the scattering eigenphase to the unit circle is a measure of the lifetime. The smaller the radius of the eigenphase, the smaller is the associated lifetime. As we change k, the variation of the eigenphase of a given solution will be dominated by the phase factor eiΘ . The path to quantization goes thus by first increasing Re[k] until Re[Θ] = 2π, and then adding a small imaginary part ik so that
−iΘ(k+ik)
sin β  − n sin β
e
= , sin β  + n sin β

(7.10)

driving the eigenphase right to the quantization point. From this condition, we can extract an approximate value for the imaginary part of the resulting quasi-bound

2, § 7]

Root-search strategy

mode: Im[nkR] = −


 
sin β  − n sin β

1

log . 2 sin β
sin β  + n sin β


107

(7.11)

This is precisely the lifetime of refractive WG modes due to Fresnel scattering, which can be obtained using different methods (Nöckel [1997], Türeci [2003]). The crucial point here is that these statements are only valid for an interval of (nkR) ∼ O(1), so that β is approximately constant:   dβ 1 (7.12) =O . d(nkR) nkR Furthermore, the assumption that Im[nkR] Re[nkR] is also implicit in these derivations. These procedures have to be implemented carefully because of the Stokes phenomenon (Bender and Orszag [1999], Bleistein and Handelsman [1986]) in the asymptotic expansion of the Hankel functions with complex argument. However, as long as the latter condition is satisfied, these estimates are valid.

7.2. Deformed dielectric resonators In the light of our findings for the undeformed case, it is possible to develop a powerful search strategy for the general, deformed case. The reason for our ability to “track” the scattering eigenphases through quantization in the case of the circular resonator was the fact that the angular momentum channels did not mix when we changed k, owing to the diagonality of the scattering matrix over all k, i.e., there we had a good label m which was conserved. This will not be the case when we deform the resonator. For small deformations, the internal scattering matrix S(k) will remain approximately diagonal, with fluctuations due to inter-channel scattering. The resulting eigenstates will show a broadening in their angular momentum distributions. In that case, one can still define an average phase velocity given by dΘ¯ = 2 sin β¯ ¯ d(nk R)

(7.13)

defined by the average angular momentum m ¯ β¯ = cos−1

m ¯ , nkR

 1 m|αm |2 . 2Λ + 1 Λ

m ¯ =

−Λ

(7.14)

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Modes of wave-chaotic dielectric resonators

[2, § 7

At first sight, there is no reason for such a solution to persist over a given interval k. Following Frischat and Doron [1997], we suggest that the scattering eigenvectors have an identity beyond a given k-value, and more importantly, that the resonances, the quantized modes, have an identity even when they do not fully satisfy the boundary conditions. We can quantify this statement by defining a simple scalar product between eigenvectors of the internal S-matrix at different k:
   ∗ α(k)
α(k + k) = (7.15) αm (k)αm (k + k). m

Then our claim is tantamount to the adiabaticity of α(k)|α(k + k). The reason that this is possible lies in the subtle correlations among the matrix elements induced by the underlying classical motion in the short-wavelength limit. We have already emphasized the connection between the scattering matrix in the shortwavelength limit and the classical SOS map. As long as there are invariant curves in the SOS – which we have seen is guaranteed by the KAM scenario for nearintegrable deformations in Section 3 – there will be eigenstates of the scattering matrix which will display the aforementioned adiabatic behavior. In fig. 10 we trace the overlap (7.15) of a set of eigenvectors in an interval of (nkR) ∼ O(1). First, a diagonalization of S(k) is performed at some k0 , the eigenvectors are determined, and then further diagonalizations are performed at regular intervals k = k0 + j k, where nkR = 0.03. At each step, there is in general a single state having markedly higher overlap with the respective original state at k0 than the others, and that value is plotted. The result shows that an adiabatic identity can in fact be defined for certain states. This procedure allows the tracking of the majority of states, as long as the deformation is not too large. In fact, it is possible to show that  
  1 α(k)
α(k + j k) = 1 − j nkR · O (7.16) . nkR At this point it may be helpful to clarify what we mean by the “identity” of a state in the chaotic case in which the state is not associated with a stable periodic orbit or a family of quasi-periodic orbits. In fig. 11 we show both a real-space solution for the electric field of a TM resonance and its projection onto the surface of section using the Husimi–Poincaré projection technique described in Section 8. This method allows one to associate any solution, even a nonquantized one, with a region in the SOS and hence with an approximate ray-dynamical (classical) meaning. Furthermore, at the values of nkR at which we work, these chaotic states are often associated with unstable periodic orbits or their unstable manifolds (this is the case in figs. 10 and 11); in fig. 11 the top states are associated with the unstable fish orbit and the bottom states are associated with the unstable manifolds of

2, § 7]

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109

Fig. 10. Overlap calculated for a set of states in the interval nkR = 106–107.5, for ε = 0.12 and n = 2.65. The associated classical structures are found from the Husimi–Poincaré projections of the respective states. The schematics identify the ray orbits with which they are associated (see discussion in text). At this deformation the short two-bounce orbit is stable, the long one unstable; the diamond orbit is stable and the fish and triangle orbits are unstable.

the unstable two-bounce orbit along the major axis of the resonator. States localized on unstable periodic orbits have been termed “scars” in the quantum chaos literature and will be discussed further in Section 10. It turns out that one can extend this strategy to higher deformations, where the SOS displays large chaotic components, with proper attention for eigenstates which have an appreciable overlap with chaotic regions. A typical scenario encountered is the avoided crossing of two scattering eigenvectors. This is captured in fig. 11, where two eigenvectors are traced over a mean level spacing. Originally, the two states are well-distinguished; they have approximately zero overlap with each other. They have different classical meaning as well, as shown by their Husimi–Poincaré projections on the SOS (see Section 8 for definition). One state is associated with the border of the stable bouncing-ball region of the SOS and has no intensity near φ = 0, π; the other is concentrated in the separatrix region associated with the unstable period-two orbit along the major axis (we have plotted its unstable manifold for reference). At the crossing they perturb each other strongly, and an approximate superposition state results. However, if we continue

110 Modes of wave-chaotic dielectric resonators

Fig. 11. Two eigenvectors are traced by the criterion that the overlap is largest in two consecutive iterations. The figure shows the overlap of the resulting two sets of states with one of the initial states, |α0 . Away from the avoided crossing the states have distinct classical meaning as discussed in the text; they exchange “identity” at the avoided crossing. Shown are both the real-space electric field intensities (false color scale) and the Husimi–Poincaré projections (to be discussed in Section 8) of the states before and after the avoided crossing. Note that both modes are localized on the unstable manifolds of the unstable bouncing-ball orbit, superposed on the surface of sections.

[2, § 7

2, § 7]

Root-search strategy

111

Fig. 12. Several representative eigenvalues z (corresponding to states associated with the ray orbits above and below via color code) traced in the complex plane as one changes the real and imaginary values of k. First Re[k] is varied, resulting in the circular arcs of fixed radius (Im[z]); subsequently Im[k] is varied, resulting in a radial motion and fixed Re[k]. On the right we show the constancy of the derivative of the phase angle θ with respect to Re[nkR] and of the derivative of the logarithm of the radius η(k) with respect to Im[nkR], implying constant speed of the eigenphases as a function of k in the complex plane. The simulations are performed at ε = 0.12 quadrupolar deformation, n = 2.65.

to change k, the states emerging from the avoided crossing will still have a pronounced overlap with the states before the crossing. Notice that the overlaps are calculated with reference to one of the original states |α0 . This example represents a case where a numerical tracing algorithm has to be properly conditioned. After having established that we can assign an identity to the scattering eigenvectors as k varies, we next investigate how precisely the corresponding eigenvalues move within the complex unit circle as we vary k, both through real and imaginary values. Figure 12(a) shows such a tracing of several representative states. First, the initial eigenvalues are followed while varying the real part of k; each of the eigenvalues follows approximately a circular trajectory, followed by a pure imaginary change in k resulting in the eigenvalues following an almost precisely radial path. We write the radius of the complex eigenvalue as |eiϕ(k) | = eη(k) and call θ the angle in the complex plane for the eigenvalue eiθ(k) .

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Modes of wave-chaotic dielectric resonators

[2, § 7

This simple behavior can be understood from the fact that the classical channels (of angular momentum in our case) in the expansion preserve their identity over a range of (nkR) ∼ O(1), and the weights of these channels, embodied in 1 ). In conclusion, the radial and the expansion coefficients αm , change only O( nkR angular speeds of the eigenvalue are approximately “decoupled”. This speed is to high accuracy constant for the eigenphases, i.e. for the log of the eigenvalues as shown in fig. 12(b, c). We have developed an efficient numerical algorithm to determine the quasinormal modes of an smoothly deformed dielectric resonator based on all of these observations: (1) A diagonalization of S(k) is performed at a given k, and Ntrunc eigenphases (i) and eigenvectors are determined, denoted by |α0 , i = 1, . . . , Ntrunc ; (i) (i) m|α0  = αm . (2) A second diagonalization is performed at k + k, where k is a small complex number so that |k| k. (3) Approximate radial and angular eigenphase speeds are determined. (4) Assuming the constancy of the individual speeds, an approximate quanti(i) zation wavevector kq is determined for each of the initial eigenvectors. (5) Finally, the quasi-bound modes are constructed by ψq(i) (r, φ) =

Λ 



(i) αm Jm nkq(i) r eimφ .

(7.17)

m=−Λ (i)

Due to the small change in {αm } with k, we simply use their nonquantized values (i) in the expansion with the extrapolated k-value kq . We have checked that it is (i) important to use kq instead of the original k. In the ideal case this means that Ntrunc ∼ nkR quasi-bound modes are found in only two diagonalizations. In practice, this ideal limit is not fully attained. However, depending on the deformation and the value of nkR, a large fraction of the quasi-bound modes can be calculated approximately in this manner. Table 1 shows a typical run and the quality of the results compared to “exact” solutions. For increased numerical stability we have found it convenient to use the same algorithm in the context of solving the equivalent generalized eigenvalue problem for the system; this is discussed briefly in Appendix A. The implementation of the algorithm can be adapted to the particular result of interest. In fact, when the quantization of a single state is desired, a more exact eigenphase quantization can be performed by multiple scans with update of the speeds, quite like Newton’s root-search method.

2, § 8]

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113

Table 1 Typical run of the numerical algorithm at kR0 = 40, ε = 0.12 and n = 2.65. The first column represents the predicted value assuming a constant speed as determined from two successive diagonalizations separated by kR = 10−4 + i10−4 . The second column is the error of this prediction, obtained by a full root search, measured by the distance in the complex plane between the eigenphase and the quantization point. The last two columns are the overlaps of the original eigenvectors (internal and external) with the actual quantized ones kRq 40.530139923096 − i0.128341300297e−03 40.354640960693 − i0.346842617728e−02 40.362663269043 − i0.262046288699e−01 40.597846984863 − i0.617872737348e−02 40.760002136230 − i0.517168489750e−03 39.372772216797 − i0.782183464617e−02 39.384689331055 − i0.253524887376e−02 39.524833679199 − i0.416035996750e−03 40.427906036377 − i0.654014274478e−01 40.367130279541 − i0.640191137791e−01 40.508068084717 − i0.814560204744e−01 40.537075042725 − i0.717425644398e−01 40.627620697021 − i0.913884192705e−01 39.421646118164 − i0.850722268224e−01

|eiϕ − 1|

αq |α0 

γq |γ0 

0.1049356E−01 0.2512625E−01 0.3483059E−01 0.4986330E−01 0.2216994E−02 0.3692186E−01 0.1560470E−01 0.6675268E−02 0.2790542E−02 0.2049567E−01 0.1035770E−02 0.6688557E−01 0.8095847E−02 0.5916540E−01

0.935212779 0.800668216 0.875476530 0.885584168 0.900596963 0.670832741 0.644713713 0.918124783 0.989788008 0.949528775 0.979746925 0.910022901 0.858722180 0.906494113

0.815777069 0.888131129 0.959173141 0.902043947 0.598602624 0.857146071 0.768504668 0.571120689 0.989591927 0.983221172 0.983924594 0.939244195 0.846152346 0.943557886

As already hinted, the method is most powerful when applied to calculating classically meaningful quantities (such as the Husimi–Poincaré projection, see Section 8 below) for which finding the quantized k values is not important. This is most evident in considering the physical observables which are unique to open systems: far-field emission patterns and boundary image-fields. The calculation of these observables will be described below.

§ 8. The Husimi–Poincaré projection technique for optical dielectric resonators In this section, we will describe the Husimi–Poincaré projection technique, which allows us to relate a given mode to the phase-space structures in the SOS. Just as for quantum wavefunctions, for these two-dimensional electromagnetic fields we can represent the solutions in real space (the solutions we have been calculating) or, by Fourier-transforming them, in momentum space. However, we are interested in representing the solutions in the phase space of the problem so that we can understand their ray-dynamical meaning, and ultimately in projecting such phase-space densities onto the SOS which is our standard interpretive tool. Just as in quantum mechanics, we cannot have full information about real space and

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[2, § 8

momentum space at the same time due to the analog of the uncertainty principle, which here is reflected by the property of Fourier transforms: x · p
1,

(8.1)

˜ where x and p are the widths of ψ(x) and ψ(p) in real and momentum space. Here,  ˜ ψ(p) = d2 x eip·x ψ(x). (8.2) Thus our goal is to take the solution ψ(x) and associate with it a momentum content in some region around each point x, recognizing that our resolution in real space is limited by the uncertainty relation. One familiar method for doing this in quantum mechanics is the Wigner distribution function (Wigner [1932]), which preserves exactly certain moments of the wavefunction, but has the interpretive problem that it can be negative in some regions of phase space, and the practical problem that it is typically subject to rapid oscillations. An alternative approach, which is often more useful, was introduced by Husimi [1940] and can be thought of as a Gaussian smoothing of the Wigner distribution. In our context however one can think of the Husimi projection as a “windowed” two-dimensional Fourier transform, which involves integrating the real-space solution ψ(x) against the coherent states     1 1/4 1 2 ¯ , exp(ik p¯ · x) exp − 2 |x − x| Zx¯ p¯ (x) = (8.3) πη2 2η which are approximate solutions of the Helmholtz equation optimally localized in both momentum and configuration space while respecting the uncertainty rela¯ while its Fourier transform tion. The coherent state Zx¯ p¯ (x) is peaked around x,   2 2

k η ¯ 2 ¯ · x¯ exp − |p − p| (8.4) exp −ik(p − p) 2 √ ¯ The width parameter is η = /k, and  is a parameter with is peaked around p. the dimension of a length, which has to be determined based on the domains of variation of the conjugate pair (p, x). Note that the momentum vector has been ¯ so that p¯ is a unit vector denoting the direction of the wavevecfactorized as k p, tor. In these scaled variables, the functions Zx¯ p¯ (x) and Z˜ x¯ p¯ (p) have standard deviations  1 η  , p = √ x = √ = (8.5) 2k 2k 2 Z˜ x¯ p¯ (p) =



η2 π

1/4

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115

so that they saturate the lower bound of the uncertainty relation which takes the form 1 , x · p
(8.6) 2k and represent a “minimum uncertainty basis” for projecting the solutions onto the phase space. Note that both scales x and p are sharpened as k → ∞. The Husimi distribution in phase space is then defined as

2
2


¯ p) ¯ =
z|ψ
=

d2 x Zx∗¯p¯ (x)ψ(x)

. ρψ (x, (8.7) Note that this distribution, unlike the Wigner distribution, is positive definite in the ¯ p). ¯ As defined, the Husimi distribution is on a four-dimensional phase space (x, phase space of the billiard (restricted to the three-dimensional “constant energy surface” because the momentum p¯ is normalized to unity). One can then visualize this distribution as a vector field on a grid of size η2 in real space (Heller [1989]); however, based on our earlier discussion we find it more illuminating to define a projection of the Husimi distribution onto the surface of section of the billiard, the Husimi–Poincaré projection. For billiard systems, this idea was first carried out by Crespi, Perez and Chang [1993], but with a different choice of section than we are using here. We instead follow an approach similar to that described in Hackenbroich and Nöckel [1997]. The problem is that the boundary of the billiard is not a constant coordinate surface of a separable coordinate system. We therefore introduce the windowing functions in cylindrical coordinates and calculate the Husimi distribution at a fixed radius r = Rc . The coherent states in cylindrical coordinates take the form Zr¯ φ¯ p¯r p¯φ (r, φ) = Zr¯ p¯r (r)Zφ¯ p¯φ (φ), where

   1 1/4 1 2 exp[ip¯ r r] exp − 2 (r − r¯ ) , Zr¯ p¯r (r) = πηr2 2ηr 1/4   ∞   1 Zφ¯ p¯φ (φ) = exp ip¯ φ (φ − 2πl) 2 πηφ l=−∞   1 × exp − 2 (φ − φ¯ − 2πl)2 . 2ηφ

(8.8)



(8.9)

(8.10)

The sum on l is necessary to insure periodicity in the φ variable. We define the projection of the full four-dimensional Husimi function onto the SOS at constant

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radius r = Rc by ¯ p¯ φ ) = lim H (φ,



η→0

∞ 0



dp¯ r





π −π

∞

dφ

dr 0

[2, § 8


2


ZR∗ φ¯ p¯ p¯ (r, φ)ψ(φ, r)

c r φ

.

(8.11) Note that the integration only extends over positive radial momenta to accord with the definition of the SOS which only counts trajectories which encounter the boundary in the outgoing direction. We have included in the definition (8.11) the limit ηr → 0 to sharpen the resolution around r = Rc and evaluate the integral in the short-wavelength limit where k → ∞. However, care must be exercised in the order of limits here. For example, a closed circular billiard would have ψ(r = Rc ) = 0 and the integrand of eq. (8.11) would vanish if ηr were taken to zero before k → ∞ due to the concentration of the coherent state to a region less than a wavelength from the boundary; hence the Husimi–Poincaré projection would vanish. Letting η → 0 and k1 → 0 while kη > 1, it can be shown that
 π
2  ∞  ∞

∗
lim dp¯ r
dφ dr ZR φ¯ p¯ p¯ (r, φ)ψ(φ, r)

η→0

−π

0



≈



π −π

0

c

r φ


2


dφ Zφ∗¯ p¯ (φ)ψ (+) (Rc , φ)

φ

,

(8.12)

where ψ (+) (Rc , φ) contains only the wavefunction components with Hankel functions of the first kind: ∞ 

ψ (+) (Rc , φ) =

imφ αm H+ . m (nkRc )e

(8.13)

m=−∞

The presence of Hankel functions of only one kind is in accordance with the short-wavelength interpretation that H+ m (nkr) represent incoming and outgoing waves, respectively. This interpretation is quickly obscured for components m > nkR, which however have vanishing weights in the expansion (8.13) through the physical considerations laid out in Section 6. Using eq. (8.12) and the short-wavelength correspondence pφ ↔ m = nkRc sin χm in eq. (8.11) we obtain ¯ sin χ¯ ) = Hψ (φ,



1 πηφ2

1/4  ∞  l=−∞

π −π

  dφ exp −inkRc sin χ(φ ¯ − 2πl)

  1 × exp − 2 (φ − φ¯ − 2πl)2 Ψ + (φ). 2ηφ

(8.14)

2, § 9]

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117

Noting that the integrand in eq. (8.14) is 2π-periodic, the integration limits can be extended to infinity and the resulting Gaussian integral over φ can be evaluated analytically, yielding
∞
 −inkRc (sin χm −sin χ¯ )φ¯ ¯ sin χ¯ ) =

αm H+ Hψ (φ, m (nkRc )e
−∞
2
− 12 ηφ2 (nkRc )2 (sin χm −sin χ¯ )2
×e (8.15)
.
√ For optimal resolution in both SOS coordinates, we choose ηφ ∼ 1/ nkRc (Rc can be chosen at any convenient value). Equation (8.15) is a perfectly good Husimi–Poincaré distribution, but it does not correspond to our conventional ¯ sin χ¯ ) on the choice of the SOS at the boundary. However, for each value of (φ, circle r = Rc we can simply calculate the values of (φ, sin χ) that would result from following this ray to the boundary r = R(φ) and assign to the corresponding point on the boundary the values of the circle Husimi–Poincaré projection at ¯ sin χ¯ ) corrected by a Jacobian factor for the Gaussian propagation between (φ, the two sections:

¯ sin χ¯ ), sin χ(φ, ¯ sin χ) ¯ sin χ¯ )Hψ (φ, ¯ sin χ¯ ). ¯ = J (φ, sin χ; φ, Hψ φ(φ, (8.16) This is the quantity we use to compare and interpret wave solutions in the classical SOS of the problem. Hence EM wave solutions only resolve the classical structures in the SOS on a scale of area (2nkR)−1 ; this is the EM analog of the statement in quantum chaos theory that wavefunctions are only sensitive to classical structures of order h1¯ . § 9. Far-field distributions In typical microlaser experiments there are two basic data-acquisition modes. In the far-field acquisition mode, the CCD camera is used without a lens and aperture as a simple photodiode, and at each far-field angle θ the far-field emission intensity IFF (θ ) is recorded. The far-field emission pattern is one of the few clues we have as to the lasing mode of the resonator. Our algorithm is very efficient in calculating all possible far fields achievable with a given resonator shape and index of refraction. At a given k which is chosen close to the lasing frequency ω = ck, we solve the scattering problem without reference to any quantization condition. The far fields are computed from the external wavefunction (6.2) (with

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Modes of wave-chaotic dielectric resonators

[2, § 9

Fig. 13. Far-field emission intensity pattern for a “fish” mode (see insert of fig. 10), quantized at kR = 19.7392 − 0.06240i (solid black line) and at kR = 21.0210 − 0.06240i (dashed line). The dash-dotted and dotted lines are the far-field of the unquantized states in-between with eigenvalues z = −0.4664 + 0.5692i and z = 0.3407 − 0.6472i. Clearly the essential features of the emission patterns are calculable with nonquantized modes. In the inset we show the eigenvalues z in the unit circle.

δm = 0) by using the large-argument asymptotic form of the Hankel function (Abramovitz and Stegun [1972]),
2


 im(φ− π2 )

γm e I (φ) ∝
(9.1)
, m

where we have extracted all the quantities independent of m and φ. The far-field intensity distributions computed in this way are very well-behaved and insensitive to the k-value used, because the strongly varying Hankel functions drop out of this quantity. As seen in fig. 13, the far fields computed are virtually identical to those obtained from the quantized modes as k is varied over two level-spacings. Using this simplification the method here can be used to very rapidly evaluate all possible emission patterns for a wide range of dielectric resonators. Several recent experiments have studied dielectric microlasers using an imaging technique for data acquisition (Rex, Chang and Guido [2001], Rex, Türeci,

2, § 10]

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119

Schwefel, Chang and Stone [2002], Schwefel, Rex, Türeci, Chang, Stone and Zyss [2003], Chern, Türeci, Stone, Chang, Kneissl and Johnson [2003]); the CCD camera records a magnified image of the intensity profile on the sidewall viewed from angle θ in the far field. The pixels can then be mapped to sidewall angle φ via the solution of a simple transcendental equation. This yields a two-dimensional plot, called the boundary image-field, where a given data point I (φ, θ ) denotes the intensity emitted from sidewall position φ towards the far-field angle θ . Given φ and the boundary shape, each far-field angle θ can be associated via Snell’s Law with an internal incidence angle sin χ for the emitted radiation, allowing one to project the emitted intensity back onto the surface of section. In making this projection, the aperture has the important role of defining a window ( sin χ), so that a given pixel on the camera can be identified up to a diffraction-limited resolution with a pair (φ, sin χ). Mathematically, the effect of the lens–aperture combination is equivalent to a windowed Fourier transform of the incident field on the lens (Goodman [1996]); thus it is simply connected to the Husimi–Poincaré distributions we have just discussed (Türeci [2003]). It has to be emphasized that image data only probe the far field, and do not contain the “near-field” details we would see in a typical numerical solution, nor does it contain information about the internally reflected components of the cavity field (see Appendix B for further details). On the other hand, it contains (with some finite resolution) the same information as the Husimi–Poincaré distribution of the emitting components of the field and does allow us therefore to put forward a ray interpretation of the emission and the internal resonance. This was done to compare experiment and a ray-based model for emission by Schwefel, Rex, Türeci, Chang, Stone and Zyss [2003]. If one wishes to compare image data to numerically calculated resonance solutions of the type we have just discussed it is possible to calculate the boundary image-field of a numerically obtained resonance in the manner described in Appendix B. In fig. 14(b) we plot an example of such a numerically generated boundary image-field for the case of a circular resonator; further examples with comparison to experiment will be given in the next section.

§ 10. Mode classification: theory and experiment As noted initially, for dielectric resonators with nonintegrable ray dynamics typically the modes cannot be classified by a complete set of mode indices, the main exception being modes associated with stable periodic orbits. Nonetheless there

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Modes of wave-chaotic dielectric resonators

[2, § 10

Fig. 14. (a) Experimental setup for simultaneously measuring far-field intensity patterns and boundary images as implemented by Rex, Türeci, Schwefel, Chang and Stone [2002]; the lens and aperture are in the far field and the schematic is not to scale. (b) Theoretically calculated boundary image-field for a circular dielectric resonator. This image field corresponds to a whispering-gallery mode close to critical incidence.

is a strong correspondence between the ray and wave behavior which can be used both to provide an intuitive understanding of the type of mode involved and as a rough classification scheme based on classical phase-space properties or structures. Such an approach has been employed to interpret a large number of microlaser experiments using ARC resonators. In this section we will illustrate this classification scheme using the tools of the quasi-bound mode algorithm and Husimi–Poincaré projection discussed above, and in several cases briefly describe lasing experiments in which a certain type of mode was found to be relevant to the lasing properties. We note at the beginning that the real-space plots and the Husimi–Poincaré distributions shown below are constructed using the nonquantized scattering eigenvectors, as we described in the previous section. Consider first the near-integrable regime in the quadrupolar deformation. At deformations of ε ≈ 0–0.06, the SOS is dominated by unbroken invariant curves corresponding to quasi-periodic whispering-gallery type of motion, interrupted by √ tiny islands of stability of size ∼ ε and narrow regions of stochastic motion on the separatrices surrounding these islands. For TIR modes, the emission pattern then becomes slightly anisotropic, but light still leaks out by evanescent escape as for a circular whispering-gallery mode (see fig. 15). Note that the emission process here does not have a ray analog, because the motion is highly confined by unbroken invariant curves, so that light has to tunnel through these dynamical bar-

2, § 10]

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121

Fig. 15. Real-space false color plots and Husimi–Poincaré projections of regular quasi-periodic solutions for n = 2.0 at (a, b) ε = 0, kR = 90.5591 − 0.00149i (Q ≈ 12156) and (c, d) ε = 0.03, kR = 89.09214 − 0.0032i (Q ≈ 5568).

riers. This behavior is very similar to the emission mechanism of low-eccentricity elliptic resonators. Low-index materials, such as polymers, fused silica and liquid microdroplets, are found to display “universal behavior” (Nöckel and Stone [1997]) at intermediate deformations, ε ≈ 0.08–0.20. The lower limit represents the deformation at which the last irrational torus (the so-called “golden mean torus”) is broken and there are no more dynamical barriers left to confine the motion in the sin χ direction. Note that there are still regular whispering-gallery orbits in the Lazutkin region sin χ ∼ 1, however the size of this region in the SOS becomes vanishingly small for higher deformations. The emission is then dominated by chaotic whispering-gallery modes. These are modes which are localized predominantly

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Modes of wave-chaotic dielectric resonators

[2, § 10

in the region of the SOS which is trapped by total internal reflection and hence have high enough Q-values to lase, but emit by the “classical” mechanism of ray escape (Fresnel refraction). The basis of this interpretation is the observation that the high-emission directions are obtained by ray simulations in which a randomly chosen bundle of trapped rays is propagated forward, allowed to escape and binned in the far field to generate a “classical” emission pattern (Nöckel and Stone [1997], Schwefel, Rex, Türeci, Chang, Stone and Zyss [2003]). Particularly interesting is that such chaotic whispering-gallery modes can give highly directional emission despite their association with the chaotic region of phase space, as first suggested by Nöckel and Stone [1997] and later confirmed by several experiments (Chang, Chang, Stone and Nöckel [2000], Rex [2001], Schwefel, Rex, Türeci, Chang, Stone and Zyss [2003]). The high-emission directions were also predicted to be highly sensitive to the shape of the resonator, a result which was confirmed by subsequent experiments (Schwefel, Rex, Türeci, Chang, Stone and Zyss [2003]). An example of such a chaotic whispering-gallery mode for an index n = 1.49 is given in fig. 16(a, b). Note that it emits primarily at an angle of roughly 35◦ to the major axis of the quadrupole and that its main intensity in the Husimi–Poincaré plot is in the chaotic region of phase space above the line denoting the critical angle for total internal reflection. The emission directionality for modes of this type is similar over deformations of the quadrupole in the range of ε = 0.08–0.20, for low-index resonators. The directionality of the emission patterns from such quadrupolar resonators is dictated by the geometry of the unstable manifolds of the short unstable periodic orbits, as explained in detail by Schwefel, Rex, Türeci, Chang, Stone and Zyss [2003], and is completely captured by a ray-escape model. Detailed experiments contrasting resonator shapes in dye-doped (DCM) polymer (PMMA) microlasers have recently confirmed the existence of such modes and their ray interpretation. In fig. 16(c, d), an experimental boundary image-field is compared to the theoretical image field generated by the numerical resonance of fig. 16(a, b) using the method discussed in Appendix B, and good agreement is found. Earlier experiments on lasing droplets were also interpreted as being due to similar chaotic whispering-gallery modes (Chang, Chang, Stone and Nöckel [2000]). Large islands in the SOS (corresponding to stable periodic orbits) support multiple quasi-bound modes, which can be calculated approximately by the methods of Gaussian optics (Türeci, Schwefel, Stone and Narimanov [2002]). It is possible to find such a series of modes corresponding to any stable island of sizable extent in the SOS, with two characteristic spacings, the free spectral range

2, § 10]

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123

Fig. 16. (a) Real-space false color plot and (b) Husimi–Poincaré projection of a chaotic whispering-gallery mode localized near the unstable four-bounce rectangular periodic orbit in the quadrupole. This mode was found to describe the emission pattern of dye-doped polymer microcylinder lasers with n = 1.49 and ε = 0.12 by Schwefel, Rex, Türeci, Chang, Stone and Zyss [2003]. The quasi-bound mode plotted is quantized at kR = 49.0847 − 0.0379i. (c) Experimental and (d) theoretically calculated image field corresponding to the quasi-bound mode in (a).

(FSR) k = 2π/L (where L is the length of the periodic orbit), and the transverse mode spacing, which depends on the eigenvalues of the stability matrix (Türeci, Schwefel, Stone and Narimanov [2002]). Such modes are very similar to the familiar Gaussian modes of mirror-based resonators, except that their Q-values are controlled by different factors (Fresnel scattering, or in the case of TIR modes an effect known as “chaos-assisted tunneling” (Türeci, Schwefel, Stone and Narimanov [2002])). The modes which were observed to be lasing in InGaAs/InAlAs quantum-cascade (QC) lasers (Gmachl, Capasso, Narimanov, Nöckel, Stone, Faist, Sivco and Cho [1998]) for higher deformations were exactly of this character. In this system with an effective index of refraction of n = 3.3, relatively weak directional emission was observed up until a deformation of ε = 0.16, whereupon a large directional peak emerged in the far field at θ ≈ 45◦ .

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Modes of wave-chaotic dielectric resonators

[2, § 10

Fig. 17. (a) Scanning electron micrographs of the top and side views of a deformed cylindrical quantum cascade microlaser. The shape is close to but not exactly quadrupolar. (b) Angular dependence of the emission intensity for deformations ε = 0 (triangles), ε = 0.14 (open circles) and ε = 0.16 (solid circles), where ε denotes the ratio of the minor and major axes. The right-hand inset shows the bowtie periodic orbit, the left-hand inset shows the logarithmic plot of the measured power spectrum. The FSR of the peaks is found to agree with the calculated bowtie FSR (after Gmachl, Capasso, Narimanov, Nöckel, Stone, Faist, Sivco and Cho [1998]) and the far-field directionality for ε = 0.16 conforms with the far-field emission pattern of a second excited transverse bowtie mode, fig. 18(c), plotted as a dashed line.

The sudden onset with deformation was markedly different from the low-index chaotic whispering-gallery modes which already were highly anisotropic at lower deformations; moreover the directionality found was not consistent with randomray simulations. On the other hand, in this experiment spectral data were available, and the observation of a set of equally-spaced lasing modes (see inset, fig. 17(b)) with a free spectral range corresponding to the inverse length of a stable periodic orbit with the geometry of a bow-tie suggested strongly that the lasing emission was due to such modes. Numerical simulations using the methods we have developed in previous sections find good agreement between the directional emission pattern from the appropriate bow-tie mode and the experimental observations. An interesting feature of the analysis was the discovery that the far-field emission pattern was not consistent with a “ground-state” bow-tie mode (i.e. a state of zero transverse excitation) but agreed well with the second excited transverse mode. A typical sequence of such modes based on the bow-tie periodic orbit is shown in fig. 18. The nature of the mode selection and mode competition in such lasers is not yet understood in any detail. These experiments generated particular interest because by measuring a series of increasingly deformed cylinder lasers it was found that the peak output

2, § 10]

Mode classification: theory and experiment

125

126

Modes of wave-chaotic dielectric resonators

[2, § 10

Fig. 18. Numerically calculated transverse series of modes based on the bow-tie periodic orbit. Such modes were observed experimentally by Gmachl, Capasso, Narimanov, Nöckel, Stone, Faist, Sivco and Cho [1998]. The calculations are performed at nkR ≈ 128, n = 3.3, ε = 0.16. With these parameters the bowtie periodic orbit hits the boundary exactly at the critical angle. (a) Fundamental mode, (b) first excited mode, (c) second excited mode. Note the strong “near-field” fluctuations, particularly in the case of the second-order mode in (c). The middle row shows the Husimi–Poincaré distributions. The bottom row presents for each of the modes a calculated boundary image, showing that one can clearly distinguish the three different modes from their boundary image-fields.

power increased by nearly three orders of magnitude between a circular cylinder and a deformed cylinder with approximately 2:1 aspect ratio. This indicated that ARC-based microlasers are potentially much more efficient semiconductor light sources than disk lasers. A full understanding of the power increase in these lasers is an outstanding challenge for the theory of these systems. The importance of periodic orbits for chaotic resonators is not limited to stable orbits. Short unstable periodic orbits, especially those which are least unstable, can make their presence felt in the mode structure of resonators, despite the fact that the methods of Gaussian optics fail for such modes (Türeci, Schwefel, Stone and Narimanov [2002]). These modes are referred to as “scars” in the quantum chaos literature; they display an enhanced intensity along an unstable periodic orbit and have been studied widely (Heller [1984], Kaplan [1999]). Evidence for lasing action on scarred states was obtained in experiments on GaN diode lasers (Rex, Türeci, Schwefel, Chang and Stone [2002]) with index of refraction n = 2.65 as well as in GaInAs/GaAs/GaInP diode lasers (Gmachl, Narimanov, Capasso, Baillargeon and Cho [2002]) with index of refraction n = 3.4. In the work of Rex, Türeci, Schwefel, Chang and Stone [2002], the scarred states which account well for the observed far-field images were based on two symmetryrelated triangular orbits with a Lyapunov exponent of λ ∼ 1.62 (see fig. 19). Again we were able to confirm the nature of the lasing mode by comparing the experimental and theoretical boundary image-fields as shown in fig. 20. Finally, we would like to note that lasing on scarred states was also observed in mirrorbased diode lasers with a “quasi-stadium” shape (Fukushima, Biellak, Sun and Siegman [1998]). Modes can localize not only on unstable fixed points but also on their associated stable and unstable manifolds. In fig. 21(a–f) we show a series of three states associated with the shortest unstable periodic orbit of the system, the unstable bouncing-ball orbit. Note that the first mode represents the “fundamental” mode which localizes on the fixed point itself (see fig. 21(b)). The real-space plots in fig. 21(c, e) do not show any distinct structure. Their respective Husimi–

2, § 10]

Mode classification: theory and experiment

127

Fig. 19. (a) Real-space false color plot of the quasi-bound state at kR0 = 48.437 − 0.0204i and ε = 0.12 which is scarred by the triangular periodic orbits shown in the inset. The four points of low incidence angle which should emit strongly are indicated. Note that there is no emission from the upper and lower vertices of the triangular periodic orbits at φ = ± π2 , which are incident well above the critical angle for total internal reflection. (b) Husimi–Poincaré projection for the same mode superimposed onto the surface of section of the resonator. The surface of section for the corresponding ray dynamics is shown in black, indicating that there are no stable islands (orbits) near the high-intensity points for this mode. Instead, the high-intensity points coincide well with the bounce points of the unstable triangular orbits (triangles). The black line denotes sin χc = n1 for GaN (n = 2.65); the triangle orbits are just above this line and would be strongly confined, whereas the stable bowtie orbits (bowtie symbols) are well below and would not be favored under uniform pumping conditions.

Fig. 20. (a) Experimental data showing in color scale the CCD images as a function of camera angle θ. Three bright spots are observed on the boundary for camera angles in the 1st quadrant, at φ ≈ 17◦ , 162◦ , −5◦ . (b) Calculated image field corresponding to the scarred mode shown in fig. 19.

Poincaré plots, fig. 21(d, f), however, reveal that the modes localize on the heteroclinic intersections of the stable and unstable manifolds emanating from the unstable bouncing-ball fixed points. A similar behavior was reported by Frischat and Doron [1997] in the context of a quantum billiard.

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Modes of wave-chaotic dielectric resonators

[2, § 10

Fig. 21. Real-space plots and Husimi–Poincaré distributions of modes which are related to the unstable bouncing-ball orbit (scars, see text). Superimposed on the SOS are the stable and unstable manifolds of this orbit which seem to play a role in localizing the solutions in certain regions of the chaotic component of phase space. (a, b) simple scar of the two-bounce orbit; (c–f) states localized on intersections of the manifolds and appearing more chaotic in real space. These solutions are found at nkR = 106, ε = 0.12 and n = 2.65.

2, § 11]

Conclusion

129

Fig. 22. Real-space false-color plot and Husimi–Poincaré distribution of a chaotic mode at a quadrupolar deformation of ε = 0.18 and n = 2.65, quantized at kR = 32.6638 − 0.06964i.

Finally, at large deformations the spectrum contains chaotic modes which cannot easily be associated with any particular classical phase-space structure. One such mode is plotted in fig. 22. Note that the support of the mode is entirely in the chaotic portion of the SOS. Recalling our arguments in Section 2 as to the failure of eikonal methods, it is instructive to note here the complexity of the wavefronts and the large portions of the resonator in which no series of parallel wavefronts is discernible. It should also be noted however that these states do not fill the chaotic region of the SOS uniformly, suggesting that they cannot be considered completely random.

§ 11. Conclusion We have considered a general class of optical resonators which are based on deformations of cylindrical dielectric resonators. Such resonators are being studied for applications in integrated optics and optoelectronics, and for their intrinsic interest as wave-chaotic systems. The basic physics of such resonators can be best understood by the methods of classical and quantum chaos, applicable in the short-wavelength limit. We have pointed out the characteristic global breakdown of conventional geometric-optics approaches due to the transition to chaos in the associated ray dynamics. While real-space ray-tracing methods quickly become powerless with increasing deformation (degree of chaos), the essential structures are uncovered effectively in the Poincaré surface of section. Information about important

130

Modes of wave-chaotic dielectric resonators

[2, § 11

physical properties of deformed dielectric resonators and lasers, such as emission characteristics, internal modal distributions, spectra and lifetimes can be extracted from the types of ray motion in the equivalent refractive billiard system. However no general analytic technique exists for calculating approximately all of the modes of a generic resonator of this type. We have presented an efficient numerical method for the calculation of quasibound modes of dielectric cavities. The method we are proposing is a hybrid between a point-matching technique (Manenkov [1994]) and scattering approach to quantization (Doron and Smilansky [1992], Frischat and Doron [1997]). In contrast to existing methods (Manenkov [1994], Nöckel [1997], Hentschel and Richter [2002]) which employ the external scattering matrix to extract the quasibound modes of a dielectric resonator, we consider the internal scattering operator (Doron and Smilansky [1992], Dietz, Eckmann, Pillet, Smilansky and Ussishkin [1995], Eckmann and Pillet [1995]). An important conceptual difference with respect to wave-function matching methods is that the internal scattering approach permits the identification of a discrete set of internal scattering states at each value of k. This is realized by writing the matching conditions in the form of an off-shell eigenvalue problem instead of a linear inhomogeneous equation. Many of the physical properties of the modes within a given linewidth of the order of 1/nk0 R around nk0 R are contained in the eigenvectors of the internal S-matrix at k0 , and can be extracted without quantizing the mode (i.e. without tuning k to satisfy the regularity condition at the origin). The quantized spectrum and the exact quasi-bound modes can be easily accessed by an extrapolation technique which requires only two diagonalizations of this S-matrix. In principle, variants of this technique can be extended to very high wavevectors k since it scales only as the ratio of the perimeter of the resonator to the wavelength. Finally, we have presented a version of the Husimi projection technique that is well-suited to dielectric resonators, and showed that the quasi-bound modes can be associated to classical phase-space structures. In conclusion, the methods and tools presented in this work provide a unified conceptual framework for treating dielectric resonators beyond the standard numerical and analytic approaches to such electromagnetic problems.

Acknowledgements We would like to thank Evgenii Narimanov and Gregor Hackenbroich for their contributions to the early phase of the development of the numerical method. We thank Martin Gutzwiller for making us aware of Einstein’s 1917 paper and

2, § 11]

Acknowledgements

131

for helpful discussions of its main point. We would further like to acknowledge helpful discussions with Jens Nöckel. Special thanks are due to our experimental collaborators Nathan Rex, Richard Chang, Claire Gmachl and Federico Capasso. A.D.S. would like to acknowledge support from NSF grant DMR-0084501, and P.H.J. acknowledges support from the Swiss National Science Foundation.

Appendix A: Numerical implementation issues Although the development in the text using the eigenvalue problem for the truncated S-matrix S(k) is perfectly sound, its direct implementation is not very efficient. The numerical problems already surface at the stage of computing the truncated scattering matrix S(k) itself. The expression given in eq. (6.13) requires numerical inversion of 5 matrices. At a given k, as the number of channels Λ of the interior matrices designated by index 1 grows beyond Λsc , they become increasingly more singular. As evident from our previous discussion, this ill-conditioning is caused by blindly including evanescent channels in the scattering problem. A quick solution is to truncate the matrices at the singularity boundary suggested by our ray interpretation and include only Λsc = JnkRmin K channels of |m|. But this turns out to produce some states which do not satisfy the boundary conditions well enough. As noted by Doron and Smilansky [1992], some of the evanescent channels Λev have to be kept, enough to be able to proceed with our numerical computation and provide the missing (evanescent) components of those states which require it. Thus, the properly truncated scattering matrix will have a size of Ntrunc = 2(Λsc + Λev ) + 1. The conditioning of S(k) is highly sensitive to the choice of Λev . Singular value decomposition can be employed to calculate the inverses and build up S(k). However, one way to sidestep this issue in favor of a more robust method is to trade the eigenvalue problem eq. (6.14) for a generalized eigenvalue problem. We rewrite eqs. (6.8)–(6.9) in the form A|Υ  = eiϕ B|Υ , where the 2Ntrunc × 2Ntrunc matrices are given by  +   H1 −H2+ −H1− A= , B = DH+ − n1 DH+ −DH− 1 2 1 and   |α |Υ  = . |γ 

(A.1)

0 0

 (A.2)

(A.3)

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[2, § 11

This way, the common null-space of the matrices A and B can be removed by existing powerful generalized eigenvalue solvers, such as ZGGEV of the LAPACK library. As a byproduct we get both the inside and the outside vectors at one shot. This method turns out to be more stable than the one based on inversion, and yields good results with arbitrarily large Λev . The numerical problems associated with the regions of evanescent behavior (Rmin < r < Rmax ) remain, but are tractable for nkR  200–300 (this range is larger for modes without evanescent components).

Appendix B: Lens transform In the experimental imaging system, radiation emanating from the resonator is collected through an aperture and after passing through a lens, and an image is recorded for a discrete number of angles in the far field. The resonator is placed at the focal plane of the lens, so that the image is effectively formed at infinity. Just in front of the lens the field distribution is given by the resonance wavefunction Ψ (x) which, at the observation (FF) angle θ , can be expressed as     im(φ+θ) 2 2 γm H + , Ψ (x) ∼ (B.1) m k x + z1 e m

where φ = tan−1 zx1 ≈ zx1 (see fig. 23). The lens effectively adds a quadratic phase, so that the field immediately behind the lens is given by   k Ψ  (x) = Ψ (x)P (x) exp −i x 2 . (B.2) 2f Here P (x) is the pupil function, which takes care of the effect of the aperture, f is the focal length of the lens, and x is the position on the lens. The field at the camera is given by propagating this field with the Fresnel propagator (Goodman [1996]), which is well-justified as the lens–camera distance is much larger than the wavelength:    k ∞ k dx Ψ  (x) exp i (u − x)2 . Ψθ (u) = (B.3) iz2 −∞ 2z2 Using the expression eq. (B.1) for the wavefunction, k i 2zk u2  Ψθ (u) = e 2 γm iz2 m  ∞    2 + x2 k × dx P (x)H+ z m 1 −∞

(B.4)

2, § 11]

Acknowledgements

133

Fig. 23. Variables used in the lens transform to generate the boundary image-field as measured experimentally.

×e

im(θ+ zx ) i k2 ( z1 − f1 )x 2 −i zk ux 1

e

2

e

2

.

(B.5)

Using the large-argument asymptotic expansion of the Bessel functions:        2 π π + 2 2 2 + z2 − im exp ik x − i . Hm k x + z1 ∼ 1 2 4 k(x 2 + z12 )1/2 (B.6) Expanding the square root in the exponential to O( zx1 )4 , i.e. to the same order as the Fresnel approximation, and rearranging the terms, we have 1 ikz1 +imθ−im π −i π k i 2zk u2  2 4 e 2 γm e Ψθ (u) = (B.7) iz2 kz1 m  ∞ im x − k ux i k ( 1 + 1 − 1 )x 2 × dx P (x)e z1 z2 e 2 z1 z2 f . (B.8) −∞

The second exponent in the integral is exactly the lens law, so it vanishes. Setting m k = R0 sin χm , the intensity recorded at pixel u of the camera at the far-field angle θ can asymptotically be written as


2
 imθ
Ψ (u)
∼
γm H + m (kz1 )e θ
m



×

 
2
k dx P (x) exp i (MR0 sin χm − u)x

, z2 −∞ ∞

(B.9)

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[2, § 11

where M = z2 /z1 is the magnification of the lens. For a simple aperture, P (x) is given by  1, |x| < A2 , P (x) = (B.10) 0, |x| > A2 , so that the integral can be performed exactly to yield

2  


2
 1 u + imθ
,
Ψ (u)
∼
A sin χm − γm Hm (kz1 )e sinc θ

∆ MR 0 m

(B.11)

2z1 AkR0 and sinc(x) = sin x/x. Note that in the short-wavelength limit x and π1 sinc( ∆ ) → δ(x). This expression allows us to make predictions

where ∆ =

∆→0 based on short-wavelength limit and geometric ray optics, which includes effects of diffraction as well. For instance, for a circular cylindrical resonator, the resonances are composed of a single angular momentum component m (and its degenerate partner −m). In that case, according to the expression (B.11),
2
   


2
u imθ −imθ

Ψ (u)
∝
δ sin χm − u ± δ sin χ + e e m θ
.
MR0 MR0 (B.12) Note that the image field contains only information captured from the far-field distribution. The actual details of the resonance in the “near field” can be quite different, due to evanescent contributions close to critical incidence. For instance, the points of brightest emission inferred from the image field might be shifted due to an “optical mirage”-like effect (see fig. 18(c)). The mirage is formed not because of a continuously varying index of refraction but because of a discontinuous interface. The image field has an interesting connection to the (SOS-projected) Husimi– Poincaré distribution. The Husimi distribution of the field projected onto the SOS at a distance R → ∞ is given by
2


imθ − 12 η2 (m−pθ )2
γm H + (kR)e e HΨ (θ, pθ ) =

(B.13) m
. m

Comparing with eq. (B.11), we see that the two functions contain almost the same information. In fact, by choosing an aperture with a Gaussian transmittance P (x), one would obtain exactly the same form as eq. (B.11). Note that the freedom of smoothing to obtain various phase-space distributions which represent the same physical system gains a physical meaning here, namely it translates to the choice of optical apparatus (lens, aperture, etc.) to observe the resonator. This connection

2]

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was used by Schwefel, Rex, Türeci, Chang, Stone and Zyss [2003] to reconstruct from the boundary image-field a Husimi–Poincaré projection for the emitted radiation which was found to agree well with ray-escape simulations.

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E. Wolf, Progress in Optics 47 © 2005 Elsevier B.V. All rights reserved

Chapter 3

Nonlinear and quantum optics of atomic and molecular fields by

Chris P. Search Department of Physics and Engineering Physics, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA

and

Pierre Meystre Optical Sciences Center, The University of Arizona, Tucson, AZ 85721, USA

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)47003-1 139

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

§ 2. Field quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

§ 3. Quantum-degenerate atomic systems . . . . . . . . . . . . . . . . .

149

§ 4. Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

§ 5. Mean-field theory of Bose–Einstein condensation . . . . . . . . . .

164

§ 6. Degenerate Fermi gases . . . . . . . . . . . . . . . . . . . . . . . . .

172

§ 7. Atomic solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

§ 8. Four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . .

182

§ 9. Three-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . .

194

§ 10. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207

Appendix A: Feshbach resonances . . . . . . . . . . . . . . . . . . . . . .

207

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Introduction The idea that light carries a mechanical momentum and can modify the trajectories of massive objects can be traced back to Kepler, who offered it as an explanation for the direction of the tail of comets away from the Sun. More rigorously, the force exerted by light on atoms is implicit in Maxwell’s equations. For example, it is readily derived from the classical Lorentz model of atom–radiation interaction, where the force of light on atoms is found to be F(r, t) = −∇r V (x, r, t). Here r is the center-of-mass coordinate of the atom, x is the position of the electron relative to the nucleus, V (x, r, t) = −qx · E(r, t), is the dipole potential due to the light, E(r, t) is the electric field at the center-ofmass location of the atom and q = −e is the electron charge. The force F(r, t) is often called the dipole force, or the gradient force. It indicates that it is possible to use light to manipulate atomic trajectories, even when considered at the classical level. An additional key element in understanding the motion of atoms in light fields derives from basic quantum mechanics: since the 1923 work of Louis de Broglie we know that any massive particle of mass M possesses wave-like properties, characterized by a de Broglie wavelength h , Mv where h is Planck’s constant and v the particle velocity. Combining de Broglie’s matter wave hypothesis with the idea that light can exert a mechanical action on atoms, it is easy to see that in addition to conventional optics, where the trajectory of light is modified by material elements such as lenses, prisms, mirrors, and diffraction gratings, it is also possible to manipulate matter waves with light, resulting in atom optics. Indeed, atom optics (Meystre [2001]) often (but not always) proceeds by reversing the roles of light and matter, so that light provides the “optical” elements for matter waves. λdB =

141

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Nonlinear and quantum optics of atomic and molecular fields

[3, § 1

Very much like conventional optics can be organized into ray, wave, nonlinear, and quantum optics, matter-wave optics has recently witnessed parallel developments. Ray atom optics is concerned with those aspects of atom optics where the wave nature of the atoms does not play a central role, and the atoms can be treated as point particles. Wave atom optics deals with topics such as matter-wave diffraction and interference. Nonlinear atom optics considers the mixing of matter-wave fields, such as in atomic four-wave mixing, and the photoassociation of ultracold atoms – a matter-wave analog of second-harmonic generation. In such cases, the nonlinear medium appears to be the atoms themselves, but in a more proper treatment it turns out to be the electromagnetic vacuum, as we discuss in some detail later on. Finally, quantum atom optics deals with topics where the quantum statistics of the matter-wave field are of central interest. Examples include the generation of entangled and squeezed matter waves. In contrast to photons, which obey bosonic statistics, atoms can be either composite bosons or fermions. Hence, in addition to the atom optics of bosonic matter waves, which finds much inspiration in its electromagnetic counterpart, the atom optics of fermionic matter waves is now actively studied by a number of groups. This emerging line of investigations is likely to lead to the discovery of novel phenomena completely absent from bosonic atom optics. This chapter reviews some of the key recent developments in nonlinear and quantum atom optics that result from the availability of Bose–Einstein condensates and quantum-degenerate Fermi systems. After an elementary review of the formalism of second quantization, which describes atoms as a quantum field and leads to a simple understanding of much of atom optics in direct analogy to the optical case, we recall some important features of Bose–Einstein condensation and of quantum-degenerate Fermi systems. One important distinction between optical and matter-wave fields is that the latter ones are self-interacting, a result of atomic collisions. As it turns out, collisions play for atoms a role analogous to that of a nonlinear medium for light; hence it is important to introduce their main characteristics in the context of ultracold atoms. We show that attractive two-body interactions are the de Broglie waves analog of a self-focusing medium in optics, while repulsive interactions correspond to defocusing. We also discuss at some length the physics of Feshbach resonances, which provide us with an exquisite tool to change two-body collisions from being attractive to repulsive, with important implications in nonlinear atom optics. Indeed, much of the recent work in that field relies heavily on these resonances, as we shall see. After having understood the source of nonlinearities in de Broglie optics in this way, we turn our attention to the mean-field description of bosonic matter-

3, § 2]

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wave fields, the analog of the semiclassical approximation in optics. We introduce the Gross–Pitaevskii equation, and study departures from its predictions in a linearized approach that introduces the concept of quasiparticles. We also introduce particle–hole operators that are of particular use in the description of fermionic fields. This formalism being established, we turn to nonlinear atom optics per se. Concentrating first on bosonic atoms, we discuss the focusing and defocusing of coherent atomic matter waves and the generation of dark and bright solitons. For lack of space, we omit several important topics including much of the fascinating work on optical lattices and the generation of vortices and vortex lattices. We also omit the nonlinear mixing of optical and matter waves, where the progress has been somewhat slower in the last three years than in the topics that we cover. The reader is referred to Chapter 13 of the monograph by Meystre [2001] for a discussion of this topic that is still reasonably current. We then turn to four-wave mixing, starting with bosonic atoms, which was one of the first nonlinear atom-optical effects demonstrated experimentally. We then extend our considerations to the four-wave mixing and phase conjugation of fermionic matter waves, drawing an analogy between this process and Dicke superradiance. Further extending the analogy with optics, the following section discusses three-wave mixing. We first return to quasiparticles to interpret Baliev and Landau damping in terms of nonlinear wave mixing, and then proceed with discussion of the mixing between atomic and molecular (dimer) matter-wave fields. This allows us to make some comments of a general nature on the so-called BEC–BCS cross-over and the potential use of Feshbach resonances to achieve resonant superfluidity in ultracold bosonic atomic samples. We conclude this section with the discussion of a molecular analog of the cavity QED micromaser.

§ 2. Field quantization It is well known in optics that the easiest way to describe wave and interference phenomena is to describe light directly in terms of the electric field operator (or its classical version). While it is certainly possible to understand these properties in terms of ‘photons’, this is much more cumbersome. The situation is similar in the case of atom optics, where it is more convenient to describe the dynamics of the system in terms of a field – the Schrödinger field – than to concentrate on the corpuscular nature of the atoms. Even though the treatment of N interacting particles may be described fully using the N -particle wave function, Ψ (r1 , r2 , . . . , rN ; t), for the system, this

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approach is very cumbersome when N  1 due to the requirement that the wave function be fully symmetric (bosons) or antisymmetric (fermions) with respect to the interchange of any two particle coordinates ri and rj . In second quantization, we work not with the wave function for the particles, but with a field operator, Ψˆ (r), and its adjoint, Ψˆ † (r), that annihilate and create particles in an abstract Hilbert space that describes the number of atoms in each possible quantum state. This is analogous to the way the positive- and negative-frequency components of the electric field operator annihilate and create photons in each of the normal modes of a system. Calculations performed using this number state basis for the particles along with the appropriate second-quantized operators are equivalent to working directly with Ψ (r1 , r2 , . . . , rN ; t). Here we only provide a brief summary of the main results of second quantization. A more detailed discussion can be found in a number of texts (Merzbacher [1998], Pathria [1996], Huang [1987], Fetter and Walecka [1971], Cohen-Tannoudji, Dupont-Roc and Grynberg [1989]). The great utility of second quantization, or field quantization, is that the complicated symmetrization requirements for Ψ (r1 , r2 , . . . , rN ; t) needed to insure that the particles obey Bose–Einstein or Fermi–Dirac quantum statistics are embodied completely by the commutation relations for the field operator. For bosons one has   Ψˆ (r), Ψˆ † (r ) = δ(r − r ), (2.1)   Ψˆ (r), Ψˆ (r ) = 0, while for fermions the commutators are replaced with anti-commutators:   Ψˆ (r), Ψˆ † (r ) + = δ(r − r ),   Ψˆ (r), Ψˆ (r ) + = 0.

(2.2)

In this last expression, [· · ·]+ is an anti-commutator, [A, B]+ = AB + BA. The operators Ψˆ † (r) and Ψˆ (r) act on a Hilbert space with basis vectors given by | . . . , nr1 , . . . , nr2  where nr is the number of particles in the infinitesimal volume d3 r around r. Each basis vector then corresponds to a certain total number of particles, N, with a particular distribution in position space. (For simplicity we ignore the mathematical difficulties associated with the continuum of spatial coordinates and simply treat space as being discretized into elements of volume d3 r.) For example, |0 is the “vacuum state”, which contains no particles while |1r  is a state with a single particle at r. The operators Ψˆ † (r) and Ψˆ (r) can be interpreted as creating and annihilating a particle at the position r, respectively. To see the action of the field operators more clearly, consider the second-quantized version

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of the particle density operator, ρ(r) ˆ = Ψˆ † (r)Ψˆ (r)

(2.3)

and a vector, |Φ in our Fock space which has nr particles in the volume d3 r centered at r, ρ(r) ˆ d3 r |Φ = nr |Φ. The action of Ψˆ † (r) and Ψˆ (r) on |Φ can be found using eqs. (2.1) or (2.2):

ρ(r) ˆ d3 r Ψˆ † (r )|Φ = nr + δ(r − r ) d3 r Ψˆ † (r )|Φ, (2.4)

ρ(r) ˆ d3 r Ψˆ (r )|Φ d3 r = nr − δ(r − r ) d3 r Ψˆ (r )|Φ. (2.5) In the limit that space is discretized into volume elements d3 r, then δ(r−r ) d3 r ≈ δr,r , where δi,j is the Kronecker delta, and one can easily see from eqs. (2.4)– (2.5) that the effect of Ψˆ † (r) and Ψˆ (r) is simply to raise and lower the number of particles at r by one, respectively. The dynamics of Ψˆ (r) are determined by the second-quantized Hamiltonian, which in the absence of any interactions between the particles has the form    h¯ 2 2 ∇ + V (r) Ψˆ (r), H = d3 r Ψˆ † (r) − (2.6) 2M where the term in parentheses can be identified as the single-particle Hamiltonian, H , with the external potential, V (r). From the second-quantized Hamiltonian, it is easy to find the Heisenberg equation of motion for the field operator Ψˆ (r), ih¯

 dΨˆ (r, t)  = Ψˆ (r, t), H dt      h¯ 2 2 = d3 r  Ψˆ (r, t), Ψˆ † (r , t) − ∇ + V (r ) Ψˆ (r , t) 2M   2 h¯ ∇ 2 + V (r) Ψˆ (r, t), = − (2.7) 2M

an equation readily shown to be valid both for bosons and for fermions. The similarity of eq. (2.7) to the “first-quantized” Schrödinger equation allows one to think of Ψˆ (r, t) as the quantized form of the single-particle (or very loosely speaking “classical”) Schrödinger field ψ(r, t). Hence the name “second quantization” follows from the notion that one can quantize the Schrödinger wave function in much the same way that one quantizes the electromagnetic field.

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We will shortly see the importance of interactions between particles, in particular two-body collisions, in nonlinear atom optics. In the framework of second quantization, two-body interactions characterized by the central potential V2 (ri − rj ) between two particles at ri and rj are described by the interaction Hamiltonian  1 V= (2.8) d3 r d3 r  Ψˆ † (r)Ψˆ † (r )V2 (r − r )Ψˆ (r )Ψˆ (r), 2 a result used repeatedly in this review. By adding eq. (2.8) to eq. (2.6), one can show that the Heisenberg equation of motion for Ψˆ (r) is now given by    dΨˆ (r, t) h¯ 2 2 3  ˆ†   ˆ  ih¯ = − ∇ + V (r) + d r Ψ (r )V2 (r − r )Ψ (r ) dt 2M × Ψˆ (r, t). (2.9) Equation (2.9) is analogous to the equations of motion for the electric field in a medium with a nonlocal χ (3) susceptibility. The relationship between the second-quantized N -particle state, |ΦN , and the Schrödinger wave function, Ψ (r1 , r2 , . . . , rN ; t), is furnished by the following result (Merzbacher [1998]): 1 Ψ (r1 , . . . , rN ; t) = √ 0|Ψˆ (r1 , t)Ψˆ (r2 , t) · · · Ψˆ (rN , t)|ΦN . N!

(2.10)

Equation (2.10) may be inverted to give an expression for |ΦN  in terms of the wave function,  1 d3 r1 d3 r2 · · · d3 rN Ψˆ † (rN , t) · · · Ψˆ † (r2 , t)Ψˆ † (r1 , t)|0 |ΦN  = √ N! (2.11) × Ψ (r1 , . . . , rN ; t). It is easy to see using the commutation relations (2.1) or (2.2) that Ψ (r1 , r2 , . . . , rN ; t) is either symmetric (bosons) or antisymmetric (fermions) with respect to the interchange of the particle coordinates ri . Even though |ΦN  and Ψ (r1 , r2 , . . . , rN ; t) are completely equivalent descriptions of an N -particle system, the occupation-number Hilbert space of second quantization is much more general since it can describe systems that are in a superposition of states with different  total numbers of particles, N cN |ΦN , as well as systems in which one type of particle is converted to a different kind of particle. In the later case, each type of particle is described with its own independent field operator Ψˆ  (r), where  labels the type of particle. This will prove useful when we treat the conversion of pairs of atoms into molecules.

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Just as it is useful in quantum optics to introduce a mode expansion of the elecˆ t), we can introduce a mode expansion of the Schrödinger tric field operator, E(r, field operator. To illustrate this procedure, we expand Ψˆ (r, t) on the complete set of orthonormal eigenfunctions, ϕn (r), of the time-independent Schrödinger equation   h¯ 2 2 H ϕn (r) = − ∇ + V (r) ϕn (r) = En ϕn (r) 2M as  Ψˆ (r, t) = (2.12) ϕn (r)cˆn (t), n

where the mode label n stands for the complete set of quantum numbers necessary to characterize that mode. The operators cˆn and cˆn† will soon be interpreted as annihilation and creation operators for a particle in mode n, in complete analogy with the familiar optical situation. Inserting the expansion (2.12) and its hermitian conjugate into the second-quantized Hamiltonian (2.6) gives  H= (2.13) En cˆn† cˆn , n

where we have made use of the orthonormality relation  d3 r ϕn (r)ϕm (r) = δnm

(2.14)

and we have assumed a discrete energy spectrum for simplicity. With eq. (2.14) we can express the operators cˆn from eq. (2.12) as  cˆn (t) = d3 r ϕn (r)Ψˆ (r, t). (2.15) For the bosonic commutation relations (2.1), this gives   † cˆn , cˆm = δnm , [cˆn , cˆm ] = 0,

(2.16)

which we recognize from the quantization of the electromagnetic field, while for fermionic particles we find   † cˆn , cˆm = δnm , + (2.17) [cˆn , cˆm ]+ = 0. The form of the second-quantized Hamiltonian (2.13) together with the bosonic commutation relations (2.16) shows that the problem has been reduced to that of a set of independent harmonic oscillators with energies En . We can therefore

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immediately interpret cˆn as an annihilation operator and cˆn† as a creation operator for mode n, with  cˆn |Nn  = Nn |Nn − 1,  cˆn† |Nn  = Nn + 1|Nn + 1, (2.18) cˆn† cˆn |Nn  = Nn |Nn , whereby Nn is the number of particles in the mode. This shows that in second quantization, the state of the system can be easily expressed in terms of the distribution of excitations of each single-particle mode. The total number of particles in the system is clearly    † ˆ ˆ N= (2.19) cˆn cˆn = d3 r Ψˆ † (r)Ψˆ (r). Nn = n

n

As is the case for the simple harmonic oscillator, nothing prevents us from putting as many bosons in mode n as we wish. This is not the case for fermions, however, as is apparent from the anti-commutation relation [cˆn , cˆm ]+ = 0, which for m = n yields † † cˆm cˆm = cˆm cˆm = 0.

These remarkable identities indicate that a given mode can never be populated with more than a single particle, a property further evidenced by the fact that the number operator Nˆ n and its square Nˆ n2 are easily shown to be equal, Nˆ n2 = Nˆ n . Hence the population of a given mode must be either zero or one. In addition, one finds that Nˆ n cˆn† |0 = cˆn† |0,

(2.20) cˆn† |0

where |0 is the vacuum state (absence of particle), indicating that is an eigenstate of mode n with value 1. These results are nothing but Pauli’s Exclusion Principle, expressed in the formalism of second quantization in terms of anticommutation relations. Another consequence of the fermionic anti-commutation relations is that, unlike for bosons, the construction of the N -particle state vector |ΦN  starting from the vacuum depends on the ordering of the creation operators, with different or† derings differing by an overall sign. For example, the two-particle state cˆn† cˆm |0 † † differs from the physically identical state cˆm cˆn |0 by a minus sign. In general when dealing with fermions one must choose a particular ordering for the creation operators when constructing |ΦN  and stick with it in order to avoid any sign ambiguities.

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§ 3. Quantum-degenerate atomic systems 3.1. Bose–Einstein condensates It has been known since the seminal work of Einstein more than 75 years ago (Einstein [1925a, 1925b]) that when a sample of massive bosonic particles is cooled to a point where the atomic separation becomes comparable to their thermal de Broglie wavelength, a remarkable process takes place: a macroscopic fraction of the particles “condenses” into the lowest-energy single-particle quantum state of the system, while all other levels remain microscopically populated. More specifically, the onset of the condensation occurs in a noninteracting uniform gas of bosons when nλ3T
2.612 . . . ,

(3.1) √ where n is the particle density and λT = h¯ 2π/mkB T is the thermal de Broglie wave length at the temperature T for particles of mass m. For particles in free space the condensate forms in the state with zero momentum, while in the presence of an external trapping potential the condensate forms in the quantummechanical ground state of the potential. As T → 0, the fraction of atoms in the condensate state approaches 100% for noninteracting particles. This effect, known as Bose–Einstein condensation (BEC), was put forward as an explanation for the unique properties observed in 4 He at a few degrees above absolute zero (London [1938a, 1938b]). However, due to the strong interaction between particles in that system, the condensate fraction is only about 10% of the total particle number at T = 0 (Huang [1987], Penrose and Onsager [1956]). It is also believed that the core of neutron stars is a BEC, and there is some evidence for the existence of an excitonic condensate in semiconductors (Griffin, Snoke and Stringari [1995]). It is worth mentioning that while photons are bosons they cannot undergo Bose–Einstein condensation because the number of photons is not a conserved quantity since they can be absorbed or emitted by the surrounding matter. As a result, the lowest-energy state for a photon gas is simply the vacuum state. A major breakthrough in the study of Bose–Einstein condensation occurred in 1995 when it became possible to achieve it in weakly interacting, low-density atomic systems (Anderson, Ensher, Matthews, Wieman and Cornell [1995], Davis, Mewes, Andrews, van Druten, Durfee, Kurn and Ketterle [1995], Bradley, Sackett, Tollett and Hulet [1995], Bradley, Sackett and Hulet [1997]). Compared to other condensates, atomic systems present a number of important advantages, the most significant perhaps being that a very large fraction of the atoms in the

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sample, close to 100%, are in the condensate state since the interactions between atoms are relatively weak compared to 4 He. However, as we further discuss later on, the strength of the inter-atomic interaction can be varied almost at will, for instance using Feshbach resonances (Feshbach [1962], Timmermans, Tommasini, Hussein and Kerman [1999]). This permits one to study condensates of interacting atoms in a precisely controlled manner in various regimes, from attractive to repulsive interactions and from weakly to strongly interacting systems. Furthermore, the diluteness of these condensates allows one to model the two-body interactions using a very simple form for V2 (r − r ) that is independent of the precise form of the interatomic potential. To produce a BEC, an isolated sample of bosonic alkali atoms are usually cooled initially with laser cooling followed by forced evaporation in an external confining potential that is created by either the Zeeman interaction of the atoms with an external magnetic field or by the AC Stark shift of an off-resonant laser. These confining potentials are well described as being harmonic potentials of the form

1 V (r) = m ω1 x 2 + ω2 y 2 + ω3 z2 . (3.2) 2 The cooling process results in temperatures well below the critical temperature for Bose–Einstein condensation, TC , which for ideal bosons is given by 1/3 1/3 ≈ 0.94h¯ (ω1 ω2 ω3 )1/3 N 1/3 , kB TC = ζ (3)−1/3 h(ω ¯ 1 ω2 ω3 ) N

(3.3)

where N is the number of atoms and ζ is the Riemann zeta function (Pethick and Smith [2002]). TC is typically on the order of 100 nK–1 µK with corresponding atomic densities that are typically 1013 –1015 atoms per cubic centimeter, making these gases more than ten thousand times more dilute than the air we breathe. At the point during the cooling process when the temperature reaches TC , a macroscopic fraction of the atoms fall into a state that is essentially the ground state of the confining potential. As the temperature is lowered further, the fraction of the atoms in the condensate, N0 , increases as   α  T , N0 = N 1 − (3.4) TC where α = 3 for a harmonic potential and 32 for free-space. The point in the cooling process characterized by a sudden macroscopic occupation of a single quantum state represents the onset of Bose–Einstein condensation. Atomic Bose– Einstein condensates have now been achieved for many alkali-atom isotopes, including 7 Li (Bradley, Sackett, Tollett and Hulet [1995], Bradley, Sackett and Hulet [1997]), 23 Na (Davis, Mewes, Andrews, van Druten, Durfee, Kurn and Ketterle

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[1995]), 41 K (Modugno, Ferrari, Roati, Brecha, Simoni and Inguscio [2001]), 85 Rb (Cornish, Claussen, Roberts, Cornell and Wieman [2000]), 87 Rb (Anderson, Ensher, Matthews, Wieman and Cornell [1995]), 133 Cs (Weber, Herbig, Mark, Nägerl and Grimm [2003]). In addition to these alkali metals, Bose–Einstein condensation has also been achieved in the rare-earth metal ytterbium (Takasu, Maki, Komori, Takano, Honda, Kumakura, Yabuzaki and Takahashi [2003]), as well as in hydrogen (Fried, Killian, Willmann, Landhuis, Moss, Kleppner and Greytak [1998]) and metastable helium (Robert, Sirjean, Browaeys, Poupard, Nowak, Boiron, Westbrook and Aspect [2001], Pereira Dos Santos, Leonard, Wang, Barrelet, Perales, Rasel, Unnikrishnan, Leduc and Cohen-Tannoudji [2001]), providing fascinating systems whose quantum-mechanical behavior is observable on a macroscopic scale. In addition to this, the exquisite isolation of these atomic gases from their surrounding environment combined with the relative ease with which they can be manipulated using external optical or magnetic fields has made them a test bed for many-body physics theories. One remarkable property of BECs is that they are characterized by coherence properties similar to those of laser light, leading to the realization of “atom lasers”. The coherence properties of BECs were first demonstrated with the observation of interference fringes produced by two overlapping condensates (Andrews, Townsend, Miesner, Durfee, Kurn and Ketterle [1997]). Further work showed that the higher-order correlation functions g (2) (0) and g (3) (0) could be inferred from the interaction energy between atoms (Ketterle and Miesner [1997]) and the inelastic losses from the condensate (Burt, Ghrist, Myatt, Holland, Cornell and Wieman [1997]). The experimental results were consistent with a coherent state such as is produced by a laser. There have been numerous demonstrations of primitive atom laser beams created by using a radio-frequency transition (Mewes, Andrews, Kurn, Durfee, Townsend and Ketterle [1997]) or a stimulated two-photon Raman transition (Hagley, Deng, Kozuma, Wen, Helmerson, Rolston and Phillips [1999]) to output-couple the atoms from their confining potential. Thus a BEC is one of the key ingredients for experiments on nonlinear and quantum atom optics, much like the laser is central to the fields of nonlinear and quantum optics.

3.2. Quantum-degenerate Fermi systems While ultracold atomic bosons can form a Bose–Einstein condensate, fermions can of course not condense, since the Pauli Exclusion Principle prohibits more than one atom occupying each quantum state. The best one can achieve with a sample of N atoms is to occupy the N lowest-energy eigenstates, up to the

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Fermi energy EF , with no occupation above that energy. For atoms in a trap characterized by the potential (3.2), the Fermi energy is given by (Butts and Rokshar [1997]) EF = h¯ (ω1 ω2 ω3 )1/3 (6N )1/3 .

(3.5)

Despite the lack of a sharp phase transition, quantum-degenerate Fermi gases also exhibit a number of nonclassical features that can again be traced back to the Pauli principle, which limits their dynamical behavior to a relatively small volume of phase space centered around the Fermi energy. One important aspect of ultracold identical fermions is that they basically cease to collide, since the Pauli principle prohibits them from getting close enough to undergo a collision. This has important consequences for atom interferometry since recent results indicate that the lack of collisions makes ultracold fermions much more sensitive than their bosonic counterparts for some kinds of precision interferometry (Search and Meystre [2003a], Roati, de Mirandes, Ferlaino, Ott, Modugno and Inguscio [2004]). However, the cooling of fermions presents major challenges: In the experimental realization of atomic Bose–Einstein condensation, the last cooling stage is achieved by evaporative cooling. In this process, the depth of the trapping potential is slowly lowered, allowing the hottest atoms to escape. Interatomic collisions between the remaining atoms permits them to rethermalize to a lower temperature, at which point the trap potential is further lowered, etc. The absence of collisions between identical fermions at low energies would imply that evaporative cooling does not work in that case, seriously limiting the achievable temperatures. One way around this difficulty is to use a mixture of fermions in two different hyperfine spin states so that atoms in different states can undergo the necessary rethermalizing collisions (DeMarco and Jin [1999], DeMarco, Bapp and Jin [2001]). The other technique that has been employed is sympathetic cooling of the fermions using a gas of bosons (Truscott, Strecker, McAlexander, Partridge and Hulet [2001], Schreck, Ferrari, Corwin, Cubizolles, Khaykovich, Mewes and Salomon [2001], Schreck, Khaykovich, Corwin, Ferrari, Bourdel, Cubizolles and Salomon [2001]). In this case the bosons are evaporatively cooled and the collisions between bosons and fermions result in the cooling of the fermions to the same equilibrium temperature as the bosons. These techniques have proven successful, and it is now possible to realize quantum-degenerate Fermi systems of either 6 Li or 40 K with temperatures as low as 0.05TF (Hadzibabic, Gupta, Stan, Schunck, Zwierlein, Dieckmann and Ketterle [2003]), where TF = EF /kB is the Fermi temperature, representing the point at which quantum degeneracy starts to become pronounced. Recently, an al-

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ternative cooling technique has been proposed that consists of first forming molecules from fermionic atoms using a Feshbach resonance and then evaporatively cooling the bosonic molecules. By reversing the sweep through the resonance adiabatically one reproduces the original Fermi gas but at a temperature that is much lower than that of the molecules (Carr, Shlyapnikov and Castin [2004]). As in the case of BEC, the possibility of varying the strength of collisions between fermions in different hyperfine states using a Feshbach resonance allows one to study these systems in both weakly and strongly interacting regimes. In addition, it is well known from condensed-matter physics that if ultracold fermionic atoms in different spin states interact via an attractive two-body interaction, this interaction leads to an instability in the normal state of the gas that results in the Bardeen–Cooper–Schrieffer (BCS) transition to a superfluid state characterized by the appearance of strong correlations between atoms with opposite momentum and spin (Bardeen, Cooper and Schrieffer [1957]). This mechanism, which was originally discussed with electrons instead of atoms, is the source of superconductivity in metals and is also responsible for the onset of superfluidity in 3 He. While smoking-gun evidence of the superfluid transition has not yet been observed in these dilute Fermi gases, there is already tantalizing evidence of its existence (Kinast, Hemmer, Gehm, Turlapov and Thomas [2004]). The ability to experimentally control in a continuous manner the crossover from molecular condensates to quantum-degenerate superfluid Fermi systems affords a remarkable opportunity to investigate detailed aspects of strongly correlated Fermi systems under well-controlled conditions, with clear implications well past the confines of Atomic, Molecular and Optical science.

§ 4. Collisions 4.1. s-wave scattering Early atom-optics experiments considered low-density atomic samples, where atom–atom interactions are negligible. However, when the sample densities or atomic beam fluxes become large enough for collisions to become important, the dynamics of a given atom are modified by the presence of other atoms. The point of view generally adopted in atom optics is that these interactions allow matter waves to interact and mix, very much like optical waves mix in a nonlinear medium. As such, collisions represent an essential element of nonlinear atom optics and are the matter-wave analog of a nonlinear polarization in optics. In fact, nonlinear atom optics can be thought of as a reversal of the roles

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played by the atoms and the electromagnetic field. In nonlinear optics the nonlinear self-interaction of the electromagnetic field originates from the interaction of the photons with the atomic medium, while in nonlinear atom optics the interaction between atoms is a result of electromagnetic forces between atoms, more precisely the exchange of scalar photons that give rise to the electrostatic potential between atoms (Cohen-Tannoudji, Dupont-Roc and Grynberg [1989]). For ultracold atoms, the dominant interaction between atoms is normally due to ground-state collisions. Trapped atomic BECs and degenerate Fermi gases are considered to be dilute in the sense that the average distance between atoms is much larger than the range of the interatomic potential V2 (r), n1/3 r0 1, where n is the atomic density and r0 ∼ 102 a0 is the range of the interatomic potential, with a0 = 5.29 × 10−11 m the Bohr radius. This diluteness, combined with the low collision energies of the atoms, allows us to develop a very simple model for the two-body interactions that depends only on the scattering length, a. In practice, the theoretical prediction of the scattering length a is a very difficult challenge: it is related to the actual phase shift of the scattered wave function produced by the whole potential. As such it depends sensitively on the detailed shape of the interatomic potential energy V2 (r), including its short-range form, and can vary between −∞ and +∞. One simple way to think about it is via the semiclassical WKB quantization condition, which puts bound states whenever the accumulated phase of the wave-function across the potential equals (n + 12 )π, with n an integer (Merzbacher [1998]). The total phase accumulated by the scattered wave function across the full range of the potential modulo π determines the phase shift relative to the unscattered wave function. Roughly speaking, the scattering length is the displacement between the nodes of the scattered and unscattered wave functions for r  r0 that result from that shift. Thus, one would have to have detailed knowledge of the potentials from short to long range with sufficient accuracy to calculate the remaining phase shift of the scattered wave function after subtracting off the nπ phase shifts associated with each of the bound states of the potential. Since one normally does not know the potential with this type of accuracy, what is done in practice is to get a set of model potentials, and adjust them to fit experimental observables – such as photoassociation spectra – that are sensitive to the scattering length (Weiner, Bagnato, Zilio and Julienne [1999]). To be more specific, we recall that in order to describe the scattering between two distinguishable particles with masses m1 and m2 and reduced mass µ = m1 m2 /(m1 + m2 ), one expresses the wave function of relative motion, ψ(r), as the sum of an incoming plane wave along the z-direction and a scattered wave

3, § 4]

Collisions

that can be taken to be spherical for large enough r,   eikr , ψ(r) = ℵ eikz + f (k) r

155

(4.1)

where ℵ is a normalization constant and E = h¯ k 2 /2µ is the collision energy. If the interaction between atoms is spherically symmetric, the scattering amplitude f (k) is only a function of the scattering angle, θ , between k and the z-axis, f (k) → f (θ ), and f (θ ) approaches a constant −a in the zero-energy limit, k → 0. In this limit, eq. (4.1) reduces to   a , ψ ℵ 1− (4.2) r which shows that the scattering length a can be interpreted as the intercept of the asymptotic wave function ψ with the r-axis. We recall that the differential scattering cross-section is related to the scattering amplitude by
2
dσ =
f (θ )
, dΩ

(4.3)

so that the total cross-section is 4πa 2 in the zero-energy limit. For spherically symmetric potentials V2 (r) the scattering amplitude can be calculated directly by solving the radial Schrödinger equation for atoms colliding with angular momentum h¯ 2 (+1), which gives rise to a centrifugal potential barrier of h¯ 2 ( + 1)/2µr 2 in the radial Schrödinger equation. If h¯ 2 ( + 1)/2µr02  E, then there is a negligible portion of the radial wave function that penetrates the centrifugal barrier into the region where V2 (r) is nonzero and as result there is no scattering for those angular momenta. In the limit E → 0, then, only s-wave ( = 0) scattering results in a nonzero scattering amplitude. The s-wave solution of the Schrödinger equation for r → ∞ is

ℵeiδ0 (k) sin kr + δ0 (k) , (4.4) r→∞ kr where δ0 (k) is the phase shift acquired in the region where V2 (r) = 0. For s-wave scattering, the scattering amplitude is independent of θ . By expanding the plane wave in eq. (4.1) in terms of spherical waves for large r, lim ψ(r) =

lim eikz =

r→∞

 sin(kr − π/2) (2 + 1)i P (cos θ ), kr 

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where the P are Legendre polynomials, and comparing to eq. (4.4), one finds that the  = 0 component of the scattering amplitude is given by ei2δ0 (k) − 1 , 2ik which allows one to relate the scattering length to the phase shift by f0 (θ ) =

a = − lim

k→0

δ0 (k) . k

(4.5)

(4.6)

For identical particles ψ must be properly symmetrized with respect to the interchange of the two particles. This implies that ψ(−r) = ±ψ(r), where the upper sign is for bosons and the lower sign for fermions. This leads to a scattering cross-section of
2
dσ =
f (θ ) ± f (π − θ )
, (4.7) dΩ which shows that the cross-section vanishes for s-wave scattering between identical fermions while for identical bosons the total cross-section is twice that of distinguishable particles, 8πa 2 . In the case of ultra-cold atomic gases, the thermal de Broglie wavelength of the atoms is much larger than the range of the interatomic potential, λT  r0 , while at the same time the average distance between particles is also much larger than r0 , n1/3 r0 1. Since a particle cannot be localized to a region less than its de Broglie wavelength, the atoms only feel an interatomic potential that has been averaged over a volume ∼ λ3T . As a result, the precise form of the interatomic potential should be unimportant for treating the interactions between atoms. In fact, we have just seen that at low energies the scattering amplitude is independent of V2 and is given by −a, which is the same as that produced by scattering from a hard sphere of radius a. We can therefore treat the atoms as if they were interacting via an effective hard-sphere potential. At the same time, the diluteness of the gases implies that if we are only interested in the dynamics on length scales much larger than r0 , then we should be able to replace the interatomic potential in eq. (2.8) with an effective contact interaction. This can be accomplished by the method of pseudopotentials, which allows one to replace the hard sphere with a delta-function interaction at r = 0 that reproduces the correct wave function for r > a ∼ r0 (Huang [1987]). The stationary s-wave Schrödinger equation describing the relative motion of the two particles for a hard-sphere interaction between the particles is   1 d 2 d 2 ψ(r) = 0 (r > a), r + k r 2 dr dr

3, § 4]

ψ(r) = 0 with the solution 1 sin(kr + δ0 ), ψ(r) ∝ kr 0,

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157

(r  a),

(4.8)

r > a, r  a,

(4.9)

where δ0 = −ka is the s-wave phase shift. The idea of the pseudopotential method is to replace the hard-sphere boundary condition by a collection of sources at the origin (Huang [1987]). This is achieved by introducing an extended wave function, ψex (r), such that

2 ∇ + k 2 ψex (r) = 0 (4.10) everywhere except at r = 0, and satisfying the boundary condition ψex (a) = 0.

(4.11)

The solution is given by 1 sin(kr + δ0 ), r = 0, (4.12) kr where χ is given by the boundary conditions at r → ∞. The explicit value of this boundary condition is not required, however, because we can invert eq. (4.12) to obtain an expression for χ in terms of the ψex (r) that is assumed to be wellbehaved at large r,   ∂ 1 (rψex ) . χ= (4.13) cos δ0 ∂r r=0 ψex (r) = χ

By using the Green function for the Helmholtz equation,

eikr 2 = −4πδ(r), ∇ + k2 r we can extend eq. (4.10) to include the origin, 2

sin δ0 ∇ + k 2 ψex (r) = −4π χδ(r) k

−4π ∂ rψex (r) . = δ(r) k cot δ0 ∂r The operator   ∂ 2π h¯ 2 δ(r) r, µk cot ka ∂r

(4.14)

(4.15)

(4.16)

where we have used δ0 = −ka, is the pseudopotential. It reproduces the hardsphere wave function for r > a. Its s-wave scattering solutions for k → 0 are

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[3, § 4

identical to those corresponding to the exact interatomic potential. The regularized δ-function, δ(r)∂r r, removes terms in the wave function diverging as 1r . If the effect of the pseudopotential is a weak perturbation to be treated to first order in perturbation theory and the operator ∂r r acts on unperturbed wave functions wellbehaved at the origin, it can be set equal to unity. For ka 1 (or equivalently λT  r0 ), the two-body pseudopotential then reduces to 4πa h¯ 2 2πa h¯ 2 δ(r) = δ(r), V˜2 (r) = µ m

(4.17)

where in the second expression we have assumed equal masses for the particles, m1 = m2 = m. V˜2 (r) can be used in place of the exact two-body potential in eq. (2.8) in the second-quantized Hamiltonian. As a result, the nonlinear self-interactions between the matter-wave fields are fully described by a local contact interaction whose strength is given by the scattering length. We note that a can be positive or negative, which causes either a repulsion (a > 0) or attraction (a < 0) between atoms. From eq. (2.8), the local interaction energy density is

1 4π h¯ 2 a ρ(r) ˆ 2 − ρ(r) ˆ , 2 m which shows that for a > 0 the energy is minimized by lowering the density while for a < 0 the energy is minimized by maximizing the density.

4.2. Feshbach resonances We mentioned in the introduction that Feshbach resonances provide us with an exquisite tool to vary the s-wave scattering length, thereby changing the strength of the nonlinear wave mixing, and even modifying its nature from self-focusing to self-defocusing. In addition, Feshbach resonances have recently played a central role in the generation of ultracold molecular samples, the generation of molecular condensates, resonance superfluidity, and the study of the BEC–BCS cross-over. We return to all of these points later in the review, but in order to do so, we need to understand better the basic properties of Feshbach resonances. Up to this point, we have ignored the internal states of the atoms undergoing a collision. These states, which for ground-state alkali atoms correspond to the electronic and nuclear spin of each atom, can be used to define a collision channel labeled by the eigenstates of the unperturbed Hamiltonian, |α1 , α2 , where αj labels the spin quantum numbers of the two atoms (j = 1, 2). Different collision channels correspond to different combinations of electronic and nuclear spins for

3, § 4]

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the pair of colliding atoms. The unperturbed Hamiltonian is given by H0 =

 pˆ 2j j =1,2

where Hspin =

2mj

+ Hspin ,

  ahf j =1,2

h¯

(4.18)



s · i + B · [gs µB sj − gi µN ij ]/h¯ 2 j j

(4.19)

and Hspin |α1 , α2  = (εα1 + εα2 )|α1 , α2 .

(4.20)

Here ahf is the strength of the hyperfine coupling, B is an external magnetic field, µB is the Bohr magneton, µN = µB /1836 is the nuclear magneton, gs = 2 is the electronic g-factor, and gi is the nuclear g-factor. For r < r0 , the interatomic potential, which depends on the charge density of the valence electrons, starts to dominate over the unperturbed energies. For alkali atoms having a single valence electron, V2 (r) can be decomposed into the sum of a singlet and a triplet potential corresponding respectively to symmetric and antisymmetric spatial wave functions for the two electrons of the atoms,  V2 (r) = (4.21) VS (r)ΠS , S=0,1

where S is total electronic spin quantum number and ΠS are projection operators onto the singlet (S = 0) and triplet (S = 1) electronic states. Since V2 is diagonal in the total electronic spin, it is convenient to rewrite the Hamiltonian in terms of the total electronic spin S = s1 + s2 and total nuclear spin I = i1 + i2 , for r r0 , H = H+ + H− ,

(4.22)

where H+ =

 pˆ 2j j =1,2

H− =

ahf 2h¯ 2

2mi

+

ahf 2h¯ 2

S · I + B · [gs µB S − gi µN I]/h¯ + V2 (r),

(s1 − s2 ) · (ii − i2 ).

(4.23) (4.24)

The Hamiltonian H+ conserves the total electronic and nuclear spin quantum numbers S and I . The collision channels associated with this part of the total

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Hamiltonian H correspond to the eigenstates of H+ ,  (j ) CMS ,MI |S, I, MS , MI , |ψj  = MS ,MI

where MS and MI are the components of the total electronic and nuclear spin in the direction of the external B field, taken to be along zˆ . In contrast, H− is odd with respect to s1 ↔ s2 and therefore only has nonvanishing matrix elements between singlet and triplet spin states. Hence the hyperfine interaction leads to a coupling between singlet and triplet collision channels (Tiesinga, Verhaar and Stoof [1993], Moerdijk, Verhaar and Axelsson [1995], Timmermans, Tommasini, Hussein and Kerman [1999]). For two atoms in the spin configuration |α1 , α2  colliding with energy E = h¯ 2 k 2 /2µ + εα1 + εα2 we can express the scattering wave function for large r as   exp[ikα  ,α  r]
 α  ,α   ikz   1 2 1 2
α1 , α2 . fα1 ,α2 (k) ψ(r) = ℵ e |α1 , α2  + (4.25) r   α1 ,α2

Because the total energy is conserved in a collision, we can distinguish between “open channels” that satisfy εα1 + εα2  E and therefore contribute to the sum in eq. (4.25), and “closed channels” satisfying εα1 + εα2 > E, which do not contribute to the sum since there is no kα  ,α  that can satisfy energy conservation. 1 2 Feshbach resonances are scattering resonances that occur whenever the total energy of two colliding atoms is close to the energy of a bound molecular state in a closed channel. Virtual transitions from the open-channel continuum wave function to the bound-state wave function then result in a dramatic increase in the scattering phase shift in much the same way that the off-resonant interaction of light with atoms leads to a modification of the index of refraction due to the virtual absorption and emission of photons. From eq. (4.19), it is possible to magnetically shift the energy difference between different collision channels via the Zeeman effect. The bound state of the closed channel can therefore be tuned into resonance with the collision energy of the atoms. The coupling between the open-channel scattering wave function, ψ(r), of the atoms and the closed-channel bound state, φm (r), is given by (Tiesinga, Verhaar and Stoof [1993], Moerdijk, Verhaar and Axelsson [1995], Timmermans, Tommasini, Hussein and Kerman [1999]) 
 χ = α1 , α2
H− |α1 α2  d3 r φm (r)ψ(r), (4.26) where |α1 α2  is the spin configuration of the atoms in the entrance channel and |α1 α2  is the configuration in the closed channel. All of the Feshbach resonances that have been observed experimentally in the alkali atoms occur at magnetic

3, § 4]

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field strengths of 102 –103 G, at which the hyperfine coupling and Zeeman shift in eq. (4.18) are of comparable magnitudes. At these field strengths the eigenstates for the individual atoms are c−1/2 |s, ms = −1/2; i, mi = m + 1/2 + c+1/2 |s, ms = 1/2; i, mi = m − 1/2, where s = 12 , i is the nuclear spin quantum number of each atom, and ms and mi are the components of the angular momentum in the direction of the B field. Thus the colliding atoms will, in general, be in a combination of several singlet and triplet states collision channels, |ψj . Appendix A gives the formal derivation of the Feshbach resonance and the effect on the collisional phase shift. In particular, eq. (A.16) expresses the enhancement of the scattering length in terms of the collision energy, E, the bound-state energy, EM , and the coupling strength between the open and closed channels, γ ∼ |χ|2 , as a = a0 −

γ , EM − E

(4.27)

where a0 is the unperturbed scattering length. Equation (4.27) can be used in the pseudopotential (4.17) to describe the resonantly enhanced interactions between the atoms, which can be varied from attractive to repulsive. For zero energy collisions, EM − E is given by the difference in the Zeeman shifts between the two channels, ε = EM − E ≈ (∂/∂B)[B − B0 ],

(4.28)

where ∂/∂B is the difference in magnetic moments between the open and closed channels and B0 is the magnetic field strength at which the bound-state energy equals the open-channel collision threshold. Equation (4.27) can then be expressed in terms of experimental observables as   B a(B) = a0 1 − (4.29) , B − B0 where B = γ /[a0 (∂/∂B)] is the width of the resonance expressed in terms of the applied magnetic field. Very close to resonance eq. (4.29) breaks down, and the effect of the molecular state on the interaction between the atoms can no longer be described in terms of a modified scattering length. In that case we must explicitly include the dynamics of the molecules in our many-body theory. The coupling between atom pairs in

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[3, § 4

the open-channel and closed-channel molecules can be described using a secondquantized Hamiltonian,   2 2   h¯ ∇ 3 † ˆ H = d r Ψ (r) − + V (r) Ψˆ (r) 2m  2 2  1 h¯ ∇ ˆ + ga Ψˆ † (r)Ψˆ † (r)Ψˆ (r)Ψˆ (r) + Φˆ † (r) − + V (r) + ε Φ(r) 4m 2 1 ˆ Φ(r) ˆ ˆ + gm Φˆ † (r)Φˆ † (r)Φ(r) + gam Φˆ † (r)Ψˆ † (r)Ψˆ (r)Φ(r) 2 

1 ˆ + H.c. , + √ χ Ψˆ † (r)Ψˆ † (r)Φ(r) (4.30) 2 ˆ where Φ(r) is a bosonic annihilation operator for the molecules. The coupling between atom pairs and molecules is taken to be local in space since the size of the molecules, of the order of r0 , is much smaller than the inter-particle separation. In addition to the atom–atom pseudopotential with coupling constant ga = 4π h¯ 2 a0 /m, we have also included atom–molecule and molecule–molecule interactions with coupling constants gam = 6π h¯ 2 aam /2m and gm = 2π h¯ 2 am /m, respectively. In the limit that |ε| is much larger than the kinetic energies of the atoms and molecules and the interaction energies, the molecular field adiabatically follows the atomic field, χ ˆ Φ(r) ≈ − √ Ψˆ (r)Ψˆ (r). 2ε Substituting this expression back into the Hamiltonian, one finds that the atom– atom interactions are given by 1 4π h¯ 2 a(B) † Ψˆ (r)Ψˆ † (r)Ψˆ (r)Ψˆ (r), (4.31) 2 m thereby recovering the result from the two-body scattering calculation. An important implication of eq. (4.30) is that it shows that matter-wave optics is not confined to just atoms any longer. In many ways, the creation of molecular dimers out of free atoms can be thought of as an atom-optical analog of threewave mixing in optics, as will be discussed in more detail later on.

4.3. Photoassociation The coherent coupling between pairs of atoms and molecular dimers in eq. (4.30) can be realized in a completely different manner by using optical fields. The

3, § 4]

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163

process by which two colliding atoms absorb a single photon and form a molecule in an excited electronic state is known as photoassociation. Photoassociation has proven to be a sensitive probe of interatomic collision potentials since the transition rates depend on the overlap between the collisional wave function and the bound-state vibrational wave function as well as the detuning of the lasers from the corresponding binding energies of the vibrational states of the molecules (Weiner, Bagnato, Zilio and Julienne [1999]). When the photoassociation laser is detuned from the excited molecular state, virtual transitions to that state lead to modified atom–atom interactions with the same form as eq. (4.31) but with the a(B) determined by the detuning from the excited state and the laser Rabi frequency (Fedichev, Kagan, Shlyapnikov and Walraven [1996]). These optically induced Feshbach resonances have recently been observed experimentally (Theis, Thalhammer, Winkler, Hellwig, Ruff, Grimm and Hecker Denschlag [2004]). However, single-photon photoassociation is impractical for studying the resonant coupling between quantum-degenerate atomic and molecular gases because the excited molecules decay quickly due to spontaneous emission, with the resulting atomic recoil leading to rapid heating of the gas. Instead, two-photon stimulated Raman transitions are used to transfer the unbound atom pairs into molecules in their electronic ground state as first proposed by Julienne, Burnett, Band and Stwalley [1998]. The first experiments using stimulated Raman transitions to form stable molecules starting from a BEC were performed at the University of Texas by Wynar, Freeland, Han, Ryu and Heinzen [2000]. Further theoretical work showed that two-photon Raman photoassociation could be described in terms of coherent coupling of atomic and molecular fields, with the resulting Hamiltonian having the same form as eq. (4.30) (Heinzen, Wynar, Drummond and Kheruntsyan [2000]). However, the meaning of ε and χ is quite different in this case, with ε = −εb − h¯ (ω2 − ω1 ),

(4.32)

where εb < 0 is the binding energy of the molecule relative to the collision threshold, ω1 is the frequency of the laser that excites the unbound atoms to the molecular excited state, and ω2 is the frequency of the laser that transfers the excited molecule to its electronic ground state. Finally, χ is now a two-photon Rabi frequency, which in the case that the intermediate excited state can be adiabatically eliminated, has the form (Heinzen, Wynar, Drummond and Kheruntsyan [2000]) ∗

 I1,v I2,v 1 χ/h¯ = − Ω1 Ω2∗ . 2 ∆ν ν

(4.33)

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[3, § 5

In this expression the frequencies Ωi are the Rabi frequencies of the two lasers, ∆ν = Eν /h¯ − ω2 is the detuning from the vibrational level ν of the electronically excited molecule with energy Eν , and  I1,ν = φe,ν (r)ψ ∗ (r)r 2 dr is the Franck–Condon factor representing the overlap between the wave function of the colliding atoms and the excited vibrational wave function, φe,ν . Similarly,  I2,ν = φe,ν (r)φ2∗ (r)r 2 dr is the overlap between the excited-state molecular wave function and that of the final ground-state molecule, φ2 . Even though the lasers are far detuned from the excited state, some population is still transferred to this state where it is subsequently lost due to spontaneous emission. These losses limit the efficiency of the atom–molecule conversion. To overcome these difficulties, Drummond, Kheruntsyan, Heinzen and Wynar [2002] have theoretically explored the use of stimulated Raman adiabatic passage (STIRAP) to transfer population from the atoms to the molecules via a dark state that does not produce any population in the molecular excited state.

§ 5. Mean-field theory of Bose–Einstein condensation 5.1. The Gross–Pitaevskii equation We have seen in Section 2 that the many-body Hamiltonian describing a lowdensity sample of atoms trapped in an external potential V (r) and subject to twobody collisions through the potential V2 (r − r ) is    h¯ 2 2 ∇ + V (r) Ψˆ (r) H = d3 r Ψˆ † (r) − 2M  1 + (5.1) d3 r d3 r  Ψˆ † (r)Ψˆ † (r )V2 (r − r )Ψˆ (r )Ψˆ (r). 2 Section 4.1 further showed that for dilute gases undergoing low-energy collisions V2 (r − r ) may be replaced by a local pseudopotential V2 (r − r ) = gδ(r − r ),

(5.2)

where 4πa h¯ 2 M and a is the s-wave scattering length. g=

(5.3)

3, § 5]

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165

Since ∂Ψˆ /∂t ∼ −igΨˆ † Ψˆ Ψˆ , the complete description of the sample dynamics requires one to simultaneously solve the equations of motion for the full hierarchy of correlation functions of the field operator Ψˆ . The situation is however considerably simpler in the case of a Bose–Einstein condensate well below the critical temperature TC . For T TC , all of the atoms occupy the same singleparticle quantum state and the N -body wave function is well approximated by the Hartree wave function Ψ (r1 , . . . , rN ) =

N 

φ(ri ),

i=1

where φ(r) is the wave function for each atom. Using eq. (2.11) the secondquantized state vector is 1 |ΦN  = √ N!

 d3 r1 · · · d3 rN

N 

φ(ri )Ψˆ † (ri )|0.

(5.4)

i=1

The dynamics of the single-particle wave function are given by the Hartree variational principle (see, e.g., Meystre [2001])   ∂ δ Φ |i h  = 0, (5.5) − H|Φ ¯ N N δφ ∗ (r) ∂t which results in the nonlinear Schrödinger equation  
2
∂φ(r) h¯ 2 2 ih¯ = − ∇ + V (r) φ(r) + g(N − 1)
φ(r)
φ(r). ∂t 2M

(5.6)

This equation correctly predicts a remarkable number of features of Bose– Einstein condensates well below the transition temperature TC . Unfortunately, it cannot account for excitations of the system in which atoms occupy states orthogonal to φ(r). A convenient and expeditious approximation for T → 0 that yields the same result as eq. (5.6) for N  1 is the so-called mean-field approximation, which is the atom-optics analog of the semiclassical approximation in optics. This approach has the added advantage that it can easily be extended to treat excitations above the condensate state. Roughly speaking, one assumes that Ψˆ (r) can be decomposed as the sum of its expectation value, a c-number sometimes called the order parameter or condensate wave function, Φ(r, t), and small fluctuations about this mean. Specifically, we assume that Ψˆ (r, t) = Φ(r, t) + Ψˆ  (r, t),

(5.7)

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[3, § 5

where

  Φ(r, t) = Ψˆ (r, t) .

(5.8)

Ψˆ  (r, t),

which measures the departure of the state of the Clearly, the operator condensate from its mean-field value, must satisfy bosonic commutation relations:    (5.9) Ψˆ (r, t), Ψˆ  † (r , t) = δ(r − r ). Substituting eq. (5.7) into eq. (2.9) and for the moment neglecting the fluctuations, one immediately obtains the nonlinear Schrödinger equation  
2
∂ h¯ 2 2 ih¯ Φ(r, t) = − (5.10) ∇ + V (r) + g
Φ(r, t)
Φ(r, t), ∂t 2M which is known as the Gross–Pitaevskii equation (Pitaevskii [1961], Gross [1961, 1963]). Notice that eq. (5.10) √ is identical to eq. (5.6) for N  1 if one makes the identification Φ(r, t) = N φ(r, t). In addition to the contributions of the kinetic energy and the trap potential, the Gross–Pitaevskii equation contains an additional energy term, g|Φ(r, t)|2 , proportional to the local atomic density |Φ(r, t)|2 of the condensate. The formal analogy between this term and a local Kerr nonlinearity in optics suggests that the atom–atom interactions that are at its origin play a role similar to that of a nonlinear medium for light. It is this term that opens the way to the nonlinear atom-optical effects that are the subject of this review. Because the Hamiltonian (5.1) does not depend on the phase of the field operator, the energy of the ground state is degenerate with respect to changes in the phase of the Schrödinger field. Since there is no preferred phase, the expectation value of the ground state of the system is characterized by a random phase, Ψˆ (r, t) = 0. Giving this quantity a nonzero value instead is equivalent to the assumption that the condensate possesses a well-defined, albeit arbitrary, phase as a result of “spontaneous symmetry breaking”. This is analogous to the situation in lasers, where the phase of the electric field is similarly random, its dynamics being governed in the simplest case by a “Mexican hat” potential. However, the assumption of a well-defined phase implies that the number of atoms is uncertain. This is clear from the definition of Ψˆ (r, t), which can only have nonzero matrix elements between states with N and N −1 atoms. As a result, the quantum state of the condensate must be in a superposition of states with different total numbers of atoms. The question of whether a symmetry-breaking mean-field approach is the correct way to think about a condensate given that the total number of atoms in a closed system must be fixed has a long and contentious history (Leggett [1995], Leggett and Sols [1991], Barnett, Burnett and Vaccaro [1996]). This philosophical problem is not as important in optics since photon number is not, in general, a conserved quantity.

3, § 5]

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167

However, for all practical purposes, a number-conserving treatment such as the Hartree wave function and the mean-field coherent-state wave function yield the same results in the thermodynamic limit N → ∞. In many (but not all) situations a bosonic Fock state can be thought of as an ensemble of coherent states with random phases. Performing a phase-sensitive measurement on the Fock state picks out a single member of the ensemble (Javanainen and Yoo [1996], Castin and Dalibard [1997], Mølmer [1997]). For example, the appearance of interference fringes produced by two overlapping condensates (Andrews, Townsend, Miesner, Durfee, Kurn and Ketterle [1997]) would seem to imply that the condensates have a well-defined phase. However, it has been shown theoretically that the fringes can also be explained in terms of two condensates in Fock states. In this case the interference pattern establishes itself in time as a result of the action of detecting the atoms since the uncertainty as to which condensate the detected atom came from leads to a locking of the relative phase (Javanainen and Yoo [1996], Castin and Dalibard [1997]).

5.2. Quasiparticles Very much like quantum fluctuations are ignored in semiclassical optics, eqs. (5.10) and (5.6) are incapable of describing the fluctuations about the condensate wave function due to finite temperatures and/or external probes that create excitations in the condensate, such as is done with stimulated two-photon laser Bragg scattering of condensate atoms (Stenger, Inouye, Chikkatur, Stamper-Kurn, Pritchard and Ketterle [1999], Stamper-Kurn, Chikkatur, Görlitz, Inouye, Gupta, Pritchard and Ketterle [1999]). The simplest step beyond eq. (5.10) is to assume that the quantum fluctuations remain small and to linearize them about the mean field. This is the essence of the Bogoliubov approach (Bogoliubov [1947]), which, in the context of many-body theory, leads to the concept of quasiparticles. The starting point of the Bogoliubov approach is eq. (5.7) and the assumption that the fluctuations are small enough compared to the mean field that the Heisenberg equations of motion for Ψˆ  (r, t) may be linearized about Φ(r, t). To illustrate how this works, we consider the free-space situation described by the many-body Hamiltonian  g  † H= (5.11) Tk cˆk† cˆk + cˆk+q cˆk† −q cˆk cˆk , 2V  k

k,k ,q

where Tk =

h¯ 2 k 2 , 2M

(5.12)

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Nonlinear and quantum optics of atomic and molecular fields

[3, § 5

and we have expanded Ψˆ (r, t) in terms of plane waves in the quantization volume V as 1  ik·r Ψˆ (r, t) = √ e cˆk (t). V k

(5.13)

For noninteracting bosons in free space at T = 0, all of the particles will be in the zero-momentum state, k = 0. If the interactions are weak enough and the temperature is well below the critical temperature, then it is safe to assume that the state of the system does not differ significantly from that of the T = 0 noninteracting system. More specifically, if N − N0 N, where N is the total number of atoms and N0  1 is the number of atoms in the k = 0 mode, then we can replace cˆ0 and cˆ0† with their c-number matrix ele√ √ ment N0 ≈ N0 ± 1 in eq. (5.11). We then expand the two-body interaction Hamiltonian in powers of N0 and discard those terms that are not at least proportional to N0 . This results in the approximate Hamiltonian H HB =

N02 g  + Tk cˆk† cˆk 2V k=0   N0 g  † 1 1 † † cˆk cˆk + cˆ−k + cˆ−k + cˆk† cˆ−k + cˆk cˆ−k . V 2 2

(5.14)

k=0

1/2

The terms that were discarded in eq. (5.14) are proportional to N0 /V ∼ N 1/2 /V and 1/V and go to zero in the thermodynamic limit N, V → ∞ with N/V = const. The first term in eq. (5.14), N02 g/2V , represents the interaction between condensate atoms while the last term represents scattering of an excited atom off of a condensate atom, which is much more probable than the scattering of two excited atoms. It is convenient to express this Hamiltonian in terms of the total number of atoms N, which is fixed, rather than the number of condensed bosons. The linearized total number operator Nˆ is given by

1  † † Nˆ = N0 + cˆk cˆk + cˆ−k cˆ−k , 2

(5.15)

k=0

where the factor of 12 results from the fact that each mode is counted twice in this expression. If we again neglect terms that contain more than two excited-state

3, § 5]

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169

creation or annihilation operators, then eq. (5.14) becomes

†

 1 1  † HB = gN n + (Tk + ng) cˆk† cˆk + cˆ−k cˆ−k + ng cˆk† cˆ−k + cˆk cˆ−k , 2 2 k=0 (5.16) ˆ where n = N/V is the density. In eq. (5.16), we have replaced N with its eigenvalue, N, for a quantum state with a fixed number of atoms. The Heisenberg equations of motion obtained from eq. (5.16) are linear, † ic˙ˆk = (Tk + ng)cˆk + ng cˆ−k ,

and can therefore be solved exactly. This is accomplished by introducing the Bogoliubov transformation that diagonalizes HB , † cˆk = uk αˆ k − vk αˆ −k ,

(5.17)

† † cˆ−k = uk αˆ −k − vk αˆ k .

The new operators αˆ k represent the quasiparticle excitations above the condensate state and are required to satisfy bosonic commutation relations,   αˆ k , αˆ k† = δk,k . (5.18) With eq. (5.17), these requirements imply that u2k − vk2 = 1,

(5.19)

a condition that is automatically satisfied if the uk and vk are parametrized as uk = cosh ζk ,

(5.20)

vk = sinh ζk .

The transformation defined by eqs. (5.17) and (5.20) is formally the same as the two-photon coherent states that generate squeezed states of the electromagnetic field. This should come as no surprise since eq. (5.16) for a single k is the same as that of a nondegenerate parametric amplifier (Walls and Milburn [1994]). By inserting the Bogoliubov transformation (5.17) into eq. (5.16) and requiring † that the terms proportional to αˆ k† αˆ −k + αˆ k αˆ −k vanish so that the Hamiltonian has the form of uncoupled harmonic oscillators, one finds ng tanh(2ζk ) = (5.21) . Tk + ng Rewriting uk and vk in terms of this expression finally yields the diagonalized form of the Hamiltonian (5.16),  1 1 HB = (5.22) Ek αˆ k† αˆ k + N ng − (Tk + ng − Ek ), 2 2 k=0

k=0

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where Ek =



Tk2 + 2Tk ng,

[3, § 5

(5.23)

with Ek the dispersion relation for the quasiparticle excitations. The last term in eq. (5.22) is the energy of the quasiparticle vacuum and is analogous to the infinite-vacuum energy obtained from the quantization of the electromagnetic field. Equations (5.17) and (5.19) show that the excitations above the condensate ground state consist of quasiparticles that can be thought of as resulting from the coherent addition of particles with momentum k and removal of particles with momentum −k (holes), αˆ k† = uk cˆk† + vk cˆ−k .

(5.24)

It is easily shown that vk ≈ ng/2Tk for large k. In this limit the elementary excitations correspond to the elimination of one particle from the condensate and the creation of one (real) particle of momentum k with energy Ek ≈ Tk + ng,

(5.25)

where ng is known as the mean-field energy and is the result of forward scattering with condensate atoms. At long wavelengths, k → 0, the amplitudes uk and vk diverge as k −1/2 , and the quasiparticle is a nearly equal mixture of a real particle and a hole. For long wavelengths, the mean-field energy is much larger than the kinetic energies of the bare particles and the dispersion relation becomes linear, Ek cs h¯ k, where we have introduced the Bogoliubov sound velocity  cs = ng/M.

(5.26)

(5.27)

Thus at long wavelengths, the excitations correspond to phonons which are nothing more than quantized sound waves. The propagation of these sound waves has been imaged in real time in a trapped condensate and shown to propagate at the Bogoliubov sound velocity that has been averaged over the density profile of the condensate (Andrews, Kurn, Miesner, Durfee, Townsend, Inouye and Ketterle [1997], Andrews, Stamper-Kurn, Miesner, Durfee, Townsend, Inouye and Ketterle [1998]). More recently, the Bogoliubov transformation has been measured directly for the first time using Bragg spectroscopy, where the amplitudes uk and vk for the phonon excitations were shown to be equal mixtures of +k and −k states (Vogels, Xu, Raman, Abo-Shaeer and Ketterle [2002]).

3, § 5]

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171

The transition from long-wavelength sound waves to short-wavelength singleparticle excitations occurs when the kinetic energy Tk equals the mean-field energy. This defines a characteristic length scale corresponding to the wavenumber at which the two contributions to the energy are equal, h¯ 2 /2Mξ 2 = ng, ξ=√

h¯ . 2Mng

(5.28)

ξ is known as the healing length and plays an important role in the spatial properties of Bose–Einstein condensates since, as we will see in our discussion of vortices and solitons, it represents the minimum length over which the order parameter can exhibit significant variations. It has been assumed up to now that the depletion, N − N0 , is small. Using eqs. (5.17) and (5.21) and the fact that the ground state of the system at T = 0 is the quasiparticle vacuum state, |0, we can calculate the depletion of the condensate (Fetter and Walecka [1971]),

1/2 N − N0 1  1  2 8 (5.29) = 0|cˆk† cˆk |0 = vk = √ na 3 . N N N 3 π k=0

k=0

Thus we see that our assumption that the gas was weakly interacting is equivalent to assuming that the gas is dilute, na 3 1. Typical densities in experiments are n = 1013 –1015 cm−3 with bare scattering lengths a = 1–10 nm so that na 3  10−3 . So far, we have considered a free-space situation. In the presence of a trapping potential, the Bogoliubov theory must be modified to account for the spatial variation of Φ(r) and Ψˆ  (r). Instead of using the momentum representation, it is useful in this case to work directly in the coordinate representation, expressing the Schrödinger field as     Ψˆ (r, t) = e−iµt Φ(r) + (5.30) uj (r)αˆ j e−iωj t + vj (r)αˆ j† eiωj t . j =0

Again, the quasiparticle operators αˆ j are required to satisfy the bosonic commutation relations [αˆ j , αˆ † ] = δj  , which leads to the normalization condition    d3 r ui (r)uj (r) − vi (r)vj (r) = δij . (5.31) The free-space transformation (5.17) can be recovered from eq. (5.30) by using 1 uj (r) = √ uk eik·r , V 1 vj (r) = √ vk eik·r . V

(5.32)

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[3, § 6

Starting from eq. (5.1) and again seeking a Hamiltonian for the quasiparticles that is quadratic in creation and annihilation operators and corresponds to inde pendent harmonic oscillators, H ∝ j h¯ ωj αˆ j† αˆ j , one finds that the amplitudes obey the coupled Bogoliubov–DeGennes equations (Pethick and Smith [2002], Meystre [2001]),   h¯ ωj uj (r) = H0 − µ + 2gΦ 2 (r) uj (r) + gΦ 2 (r)vj (r), (5.33)   −h¯ ωj vj (r) = H0 − µ + 2gΦ 2 (r) vj (r) + gΦ 2 (r)uj (r), where H0 = −(h¯ 2 /2M)∇ 2 + Vtrap (r) and the chemical potential µ is determined by the solution of eq. (5.10) for the ansatz Φ(r, t) = Φ(r) exp[−iµt]. Physically, the chemical potential represents the change in energy of the system when a single particle is added, and is defined by the thermodynamic relation µ = ∂E/∂N. In the case of a condensate in free space it is given by µ gn. Equations (5.33) must generally be solved numerically for the trap potential at hand. We note that the Bogoliubov–DeGennes equations can be derived directly from the “classical” Gross–Pitaevskii equation (5.10), by looking at the linearized fluctuations around the condensate wave function with the form   Φ(r, t) = e−iµt/h¯ Φ(r) + u(r)e−iωt + v  (r)eiωt . (5.34) This ansatz is analogous to the weak side-mode probe expansion familiar in nonlinear optics. The classical collective modes obtained in this way can then be quantized as independent harmonic oscillators.

§ 6. Degenerate Fermi gases 6.1. Normal Fermi systems While the mean-field theory and its extensions are of considerable help in the description of condensed bosonic systems, they fail completely in describing normal fermionic systems. Mean-field theory can only be applied to systems that have undergone a phase transition to a highly ordered state that is characterized by a nonzero macroscopic order parameter. However, normal Fermi systems do not undergo any phase transition as the temperature is lowered. Because there is no order parameter to describe normal Fermi gases and there is never more than one particle in each quantum state, the description of fermionic systems is more conveniently achieved in terms of bilinear combinations of atomic annihilation and creation operators, analogous to the particle–hole operators used in microscopic theories of interacting Fermi liquids (Nozières and Pines [1999]).

3, § 6]

Degenerate Fermi gases

173

Because s-wave scattering between identical fermionic atoms is impossible at T = 0, we typically consider systems including atoms in two hyperfine spin states, that we label ↑ and ↓. Consider then for concreteness the free-space Hamiltonian  †



g  † † HF = (6.1) cˆk+q↑ cˆk† −q↓ cˆk↓ cˆk ↑ , Tk cˆk↑ cˆk↑ + cˆk↓ cˆk↓ + 2V k

k

where the annihilation and creation operators now obey fermionic anti-commutation relations. The coefficient g is defined as in eq. (5.3), but a must now be interpreted as the scattering length between two atoms in ↑ and ↓. For noninteracting fermions, the equilibrium state of the Fermi gas at T = 0 is the so-called Fermi sea,  † † |ψ = (6.2) cˆk↑ cˆ−k↓ |0, |k|
where |0 is the vacuum state for the atoms. Here, all states with energy below the Fermi energy, EF = h¯ kF2 /2m, are fully occupied and the average kinetic energy per particle is 3EF /5. For repulsive interactions g > 0, the ground state is well approximated by eq. (6.2) provided that the average interaction energy per particle, gn, is much less than the average kinetic energy of the particles, gn 8 = an1/3 1, EF (36π)1/3 where we have used the fact that n = kF3 /6π 2 (Pethick and Smith [2002], Nozières and Pines [1999]). By using experimental techniques such as Bragg scattering with two laser beams, it is possible to prepare Fermi gases whose momentum distributions are very far from the equilibrium Fermi sea. One can then study the resulting dynamics of the gas as a result of the repulsive interactions between the atoms. As we will discuss in more detail later, the dynamics can be physically interpreted in terms of four-wave mixing of fermionic matter waves. To exploit the fact that all atomic dynamics involve the annihilation of an atom in a specific state and the creation of an atom in another state we introduce the “particle–hole” operators † ρˆk,k ↑ = cˆk+k ˆk↑ , ↑c

(6.3)

and similarly for spin-down atoms. This notation can be trivially extended to situations involving spin-changing collisions. Making explicit use of the fermionic commutation relations and using the identity [A, BC] = [A, B]+ C − B[A, C]+

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Nonlinear and quantum optics of atomic and molecular fields

[3, § 6

then yields the Heisenberg equations of motion for the particle–hole operators: g  d (ρˆk+q,k −q↑ − ρˆk,k −q↑ )ρˆq ,q↓ , (6.4) ih¯ ρˆk,k ↑ = h¯ ωk,k ρˆk,k ↑ + dt 2V  q,q

and similarly for the spin-down atoms. Here, h¯ ωk,k = Tk − Tk+k . Taking the expectation value of the Heisenberg equations of motion (6.4) results in an infinite hierarchy of equations for the higher-order correlations, which normally cannot be solved in closed form. To proceed further one needs to introduce some kind of a factorization scheme, such as, e.g. (Christ, Search and Meystre [2003]), ρˆk,k ↑ ρˆq,q ↓  ≈ ρˆk,k ↑ ρˆq,q ↓ ,

(6.5)

which would result in the truncation of the hierarchy of equations of motion for the higher moments after the first-order moments. Such a truncation scheme is expected to be valid when ρˆk,k σ  have nonzero initial values with no initial coherence between atoms in different spin states, so that the factorization is exact at t = 0. For times t  1/gn, higher-order correlations between the different spin states are expected to have a noticeable effect on the dynamics (Meiser, Search and Meystre [2005]). Equations (6.4) along with a factorization scheme for the higher-order particle–hole correlations are a starting point for numerical treatments of nonlinear wave mixing using fermions.

6.2. Superfluid Fermi systems For attractive interactions, g < 0, the noninteracting ground state is unstable at T = 0 for arbitrarily small |g|. This instability leads to a phase transition to a lower-energy state characterized by finite coherence between time-reversed states, † † cˆ−k↓  = 0, which are known as Cooper pairs. This transition is known as the cˆk↑ Bardeen–Cooper–Schrieffer (BCS) transition and is responsible for the phenomena of superconductivity in metals (Bardeen, Cooper and Schrieffer [1957], Kittel [1987]). Unlike normal Fermi gases, superfluid Fermi systems are characterized by a macroscopic order parameter that corresponds to pair correlations between different spin states, Ψˆ ↓ (r)Ψˆ ↑ (r) = 0. In free space, the ground-state wave function for the BCS state is given by  † †

|ψBCS  = (6.6) uk + vk cˆk↑ cˆ−k↓ |0, k

3, § 6]

Degenerate Fermi gases

where |uk |2 = 1 − |vk |2 =

175

  Tk − E F 1 1+ 2 Ek

(6.7)

and Ek is the dispersion relation for quasiparticle excitations above the BCS ground state,  Ek = (Tk − EF )2 + |∆|2 . (6.8) Here,

    ∆ = |g| Ψˆ ↓ (r)Ψˆ ↑ (r) = |g| cˆ−k↓ cˆk↑  = |g| uk vk k

(6.9)

k

is the zero-temperature energy gap in the excitation spectrum. It represents the minimum energy needed to break up a Cooper pair into uncorrelated atoms. It is this gap in the excitation spectrum that is responsible for the dissipationless flow of currents in a superconductor. The energy gap vanishes as the temperature approaches the critical temperature, TC , for the phase transition. The possibility of achieving the BCS transition in trapped alkali gases was originally discussed by Stoof, Houbiers, Sackett and Hulet [1996] and by Houbiers, Ferwerda, Stoof, McAlexander, Sackett and Hulet [1997]. However, the transition temperature for the BCS transition, TC ≈ 0.28TF e−π/2kF |a| ,

(6.10)

is extremely small compared to the Fermi temperature, TF = EF /kB , since the dilute nature of these gases implies that kF |a|  10−2 for typical scattering lengths of |a| ∼ 100a0 and therefore would seem to preclude all hope of achieving the BCS transition. However, electronically spin-polarized 6 Li interacting via the triplet potential was predicted to have an extremely large negative scattering length that would make kF |a| 0.1 and thereby make the BCS transition an experimental possibility (Stoof, Houbiers, Sackett and Hulet [1996], Houbiers, Ferwerda, Stoof, McAlexander, Sackett and Hulet [1997]). Unfortunately, early experiments with trapped Fermi gases were unable to achieve temperatures below T 0.2TF , where TF 0.6–8 µK (DeMarco and Jin [1999], DeMarco, Bapp and Jin [2001], Truscott, Strecker, McAlexander, Partridge and Hulet [2001], Schreck, Ferrari, Corwin, Cubizolles, Khaykovich, Mewes and Salomon [2001], Schreck, Khaykovich, Corwin, Ferrari, Bourdel, Cubizolles and Salomon [2001], Hadzibabic, Stan, Dieckmann, Gupta, Zwierlein, Görlitz and Ketterle [2002], Granade, Gehm, O’Hara and Thomas [2002]). This was originally attributed to fundamental limits in the evaporative cooling process.

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Nonlinear and quantum optics of atomic and molecular fields

[3, § 6

For direct evaporative cooling of fermions in different spin states, the elastic collision rate necessary for rethermalization goes to zero like (T /TF )2 for T TF due to Pauli blocking (Holland, DeMarco and Jin [2000], O’Hara, Hemmer, Gehm, Granade and Thomas [2002]). For sympathetic cooling by bosons, it was believed that when the heat capacity of the bosons fell below that of the fermions, the bosons were no longer able to effectively cool the fermions (Truscott, Strecker, McAlexander, Partridge and Hulet [2001], Presilla and Onofrio [2003]). This has now been shown not to be the case since this simple argument does not account for the removal of the bosons in the evaporation process (Carr and Castin [2004]). The desire to achieve fermionic superfluidity at experimentally achievable temperatures led to the idea of resonance superfluidity by Timmermans, Furuya, Milonni and Kerman [2001] and independently by Holland, Kokkelmans, Chiofalo and Walser [2001]. Mathematically at least, their proposed theory is very similar to a phenomenological theory of high-TC superconductivity put forward by Friedberg and Lee [1989] more than a decade earlier. The essential idea behind resonance superfluidity is to use a Feshbach resonance to dramatically enhance the strength of the attractive interactions as in eq. (4.29). Unfortunately, BCS theory breaks down when kF |a| 1, and instead one must develop a new theory that properly accounts for the dynamics of the closed-channel molecules close to resonance. The appropriate generalization of eq. (4.30) for s-wave collisions between the fermionic atoms in free space is  

HBF = HF + bˆq† cˆq/2+k↑ cˆq/2−k↓ + H.c. , (Tk /2 + ε)bˆk† bˆk + χ k

k,q

(6.11) where bˆk is a bosonic annihilation operator for molecules with momentum k. When the bare molecular energy, ε, approaches 2EF from above, pairs of atoms near the Fermi surface can resonantly form a molecular condensate with a nonzero mean field, φm = bˆ0 . The small but finite mean field induces strong correlations between the atomic states k↑ and −k↓ to which they are coupled. The induced interactions between the atoms can then lead to a fermionic superfluid state with a transition temperature approaching 0.5TF . Since the system is strongly interacting close to resonance, it is impossible to write down a simple many-body ground-state wave function for the atoms and molecules analogous to eq. (6.6). Instead, φm and the pairing mean field,  † † p = k cˆk↑ cˆ−k↓ , must be calculated numerically in a self-consistent manner at fixed temperature under the constraint that the average number of particles,  † † N = k bˆk† bˆk  + (cˆk↑ cˆk↑ + cˆk↓ cˆk↓ ), is conserved. The superfluid transition occurs at the temperature, TC , when p becomes nonzero.

3, § 6]

Degenerate Fermi gases

177

The nature of the correlated atomic pairs near the resonance is rather complicated since they are in an intermediate regime between purely momentumcorrelated Cooper pairs and a Bose–Einstein condensate of spatially localized bound-state molecules. Despite the fact that the scattering length diverges at the resonance, the transition from weakly correlated momentum pairs to a condensate of bosonic molecules is a continuous function of the strength of the attractive interactions (Leggett [1980], Randeria [1995]). To understand the nature of the BCS–BEC crossover better, we note that for two particles of mass m colliding with zero energy, E → 0+ , via an attractive two-body potential, the scattering length will be negative if the potential does not support a shallow bound state with energy εb that is close to zero. As the strength of the attractive potential increases, a → −∞ until the potential can support a bound state with εb = 0− , at which point a → +∞. For a > 0, the energy of the shallow bound state is εb = −h¯ 2 /ma 2 (Sakurai [1994]). Thus if we start out with a weakly interacting BCS state and increase the strength of the interactions until a bound state emerges, the lowest-energy state of the system will now be a BEC of molecules with binding energy |εb |. The cross-over can be described using a variational wave function that is a generalization of |ψBCS  (Leggett [1980]),
  † †


ψ(η) = (6.12) u(η)k + v(η)k cˆk↑ cˆ−k↓ |0. k

a)−1

For η = (kF → −∞ one recovers the BCS theory, while for η = (kF a)−1 → +∞ one obtains the coherent-state wave function for a molecular BEC,



ψ(η  1) ∝ exp Ξ Bˆ † |0, (6.13) 0 where Bˆ 0 =



fk∗ cˆk↑ cˆ−k↓

k

is the annihilation operator for zero momentum molecules, with the relative wavefunction between the constituent atoms given by 

f (r − r ) = V −1 fk exp ik · (r − r ) , k

Ξ −1 v

fk = k /uk being the normalized wave function in momentum space. Note  that the commutator [Bˆ 0 , Bˆ 0† ] = k |fk |2 (1 − nˆ k↑ − nˆ −k↓ ) is not that of bosons. However, in the limit of very strong interactions, η  1, the momentum distribution becomes very broad, nˆ k↑ + nˆ −k↓  = 2|vk |2 1 and then we have [Bˆ 0 , Bˆ 0† ] ≈ 1.

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Nonlinear and quantum optics of atomic and molecular fields

[3, § 6

Qualitatively, these considerations apply equally well to the case of the Feshbach resonance. Above the resonance, where the energy of the closed-channel molecules lies above the free-atom energies, we have that a(B) → −∞: Here the molecules are unstable with respect to dissociation and there is no long-lived bound state of the system. Below the resonance, B < B0 , where a0  a(B) < +∞, the molecular energy is below that of the free atoms and therefore there exists a stable bound state. Close to the resonance, as a result of the dressing by the atomic-pair states the effective binding energy of this bound state is ε(B) = −h¯ 2 /m[a(B)]2 (Duine and Stoof [2003]). Several recent experiments (Zwierlein, Stan, Schunck, Raupach, Kerman and Ketterle [2004], Regal, Greiner and Jin [2004], Kinast, Hemmer, Gehm, Turlapov and Thomas [2004], Bartenstein, Altmeyer, Riedl, Jochim, Chin, Hecker Denschlag and Grimm [2004], Chin, Bartenstein, Altmeyer, Riedl, Jochim, Hecker Denschlag and Grimm [2004], Kinnunen, Rodriguez and Törmä [2004]) have reported evidence for fermionic superfluidity close to a Feshbach resonance. These experiments were conducted on the BCS side of the resonance, B > B0 , where the effective interaction in eq. (4.29) is attractive. In the first two experiments (Zwierlein, Stan, Schunck, Raupach, Kerman and Ketterle [2004], Regal, Greiner and Jin [2004]), the magnetic field was kept at a fixed value above the resonance for a certain time in order to allow for the formation of strongly correlated pairs of atoms. The magnetic field was then rapidly swept through the resonance, which projects the pairs of atoms onto molecular bound states with the velocity distribution of the molecules reflecting the center-of-mass momenta of the atom pairs in the original atom gas. For a superfluid Fermi gas, the strong correlations between pairs with zero center-of-mass momentum indicate that a significant fraction of the molecules should form a Bose–Einstein condensate (Falco and Stoof [2004], Diener and Ho [2004a, 2004b], Avdeenkov and Bohn [2005]). Indeed, this is what was observed in experiments. The third and fourth experiments instead measured the collective oscillations in a trapped gas of 6 Li on the BCS side of a resonance (Kinast, Hemmer, Gehm, Turlapov and Thomas [2004], Bartenstein, Altmeyer, Riedl, Jochim, Chin, Hecker Denschlag and Grimm [2004]). The measured frequency and damping times of the collective modes were consistent with that of a superfluid Fermi gas. However, the most compelling evidence is probably the direct observation of the energy gap using radio-frequency spectroscopy (Chin, Bartenstein, Altmeyer, Riedl, Jochim, Hecker Denschlag and Grimm [2004], Kinnunen, Rodriguez and Törmä [2004]). This experiment measured the frequency of the radio-frequency field needed to transfer atoms from one of the paired hyperfine states to a third auxiliary hyperfine state in 6 Li. The measured

3, § 7]

Atomic solitons

179

shift in the bare atomic resonant frequency corresponded to the energy needed to break up the Cooper pairs.

§ 7. Atomic solitons We are now in a position to discuss specific examples of nonlinear atom-optical effects that have been studied in the last few years in quantum-degenerate bosonic and fermionic systems. This section briefly reviews some of the work on matterwave solitons. This is followed by sections on four-wave mixing and three-wave mixing. The Gross–Pitaevskii equation is a nonlinear Schrödinger equation. It is known from several areas of physics, in particular from nonlinear optics, that soliton solutions are generic to such equations. Hence, it is natural to study the dynamics of matter-wave solitons in Bose–Einstein condensates. We have seen that the principal parameter governing the nonlinear dynamics of a BEC is the scattering length a. When it is positive, a > 0, the effective interatomic interactions are repulsive and the equilibrium size of the condensate is larger than for a noninteracting (a = 0) case for the same number of atoms. In contrast, when a < 0, the interactions are attractive, and the BEC contracts to minimize its overall energy. The nonlinear term in the Gross–Pitaevskii equation thus indicates a condensate behavior similar to optical “self-defocusing” for condensates with repulsive interactions, and to “self-focusing” when the interactions are attractive. This also implies that large condensates are unstable for the case where a < 0 and will collapse onto themselves. While this prediction is correct in free space, the situation is actually more complicated in traps, where the contraction competes with the “quantum pressure” from the kinetic energy, or the zero-point energy due to the trap. As a result, it is possible to maintain small a < 0 condensates, containing typically a few hundred to a few thousand atoms. But for strong enough attractive interactions or high enough atomic densities, the quantum pressure is insufficient to stabilize the condensate. The Gross–Pitaevskii equation then has no steady-state solution, and the condensate implodes. There is a close analogy between this implosion and beam collapse in nonlinear optics. There, the competition is between diffraction, which tends to expand the beam, and a self-focusing nonlinearity. If the nonlinearity is strong enough, the beam will focus as it propagates until the intensity exceeds the damage threshold of the medium and the beam self-destructs. From these considerations, it would appear that we are faced with an unfortunate situation since the generation of solitons normally requires a self-focusing

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[3, § 7

medium, so that self-focusing can be balanced by diffraction, and this is precisely the situation where only small condensates can be generated. However, there are several ways around this difficulty, including the generation of dark solitons and of gap solitons. To set the stage for the discussion, consider first the generic one-dimensional nonlinear Schrödinger equation ih¯

∂ϕ h¯ 2 ∂ 2 ϕ =− + g|ϕ|2 ϕ. ∂t 2M ∂z2

(7.1)

Soliton solutions are nonspreading solutions of this equation that preserve their shape under propagation (Scott, Chu and McLaughlin [1973]). The kinetic energy term in the nonlinear Schrödinger equation tends to spread wave packets, and the nature of the solitons depends on the sign of the nonlinearity. For the case of repulsive interactions, both the kinetic-energy and nonlinear terms in eq. (7.1) tend to broaden localized wave packets, so we do not expect localized, or bright-soliton, solutions for that case. However, dark solitons can arise, in analogy with what happens in nonlinear fiber optics, see, e.g., Agrawal [1995]. They are localized density dips in an otherwise constant background, given by (Denschlag, Simsarian, Feder, Clark, Collins, Cubizolles, Deng, Hagley, Helmerson, Reinhardt, Rolston, Schneider and Phillips [2000], Burger, Bongs, Dettmer, Ertmer, Sengstock, Sampera, Shlyapnikov and Lewenstein [1999], Reinhardt and Clark [1997])       v2 v 2 1/2 1/2 2 (z − vt) ϕ(z, t) = n 1 − 1 − 2 sech 1− 2 l0 cs cs 

 × exp i φ(z, v, t) − µt/h¯ , 1/2

(7.2)

where n is the background density away from the dark soliton core, µ = n|g|, l0 = (h¯ 2 /Mµ)1/2 is the correlation length that determines the width of the soliton core, cs is the Bogoliubov sound velocity, and v is the dark-soliton velocity whose magnitude is bounded by cs . The soliton phase φ(z, v, t) is given explicitly by     2 1/2  v 2 1/2 cs (z − vt) 1− 2 , (7.3) φ(z, v, t) = − arctan −1 tanh l0 v2 cs which shows that the dark solitons are characterized by the presence of a phase step δ across the localized density dip. This phase step is related to the velocity v and the density nbot at the bottom of the atomic density dip (Denschlag, Simsarian, Feder, Clark, Collins, Cubizolles, Deng, Hagley, Helmerson, Reinhardt, Rolston, Schneider and Phillips [2000], Burger, Bongs, Dettmer, Ertmer, Sengstock, Sampera, Shlyapnikov and Lewenstein [1999]). In particular, one finds

3, § 7]

Atomic solitons

the relation   v nbot δ = , = cos 2 cs n

181

(7.4)

which shows that the phase step δ = π for a stationary soliton v = 0, nbot = 0. Stationary solitons are the only solutions that have a vanishing density at their center. They are referred to as “black” solitons rather than the more generic “gray” solitons. Most of the early soliton experiments in nonlinear atom optics focused on the formation of such dark solitons. In the experimental observations of Denschlag, Simsarian, Feder, Clark, Collins, Cubizolles, Deng, Hagley, Helmerson, Reinhardt, Rolston, Schneider and Phillips [2000] and of Burger, Bongs, Dettmer, Ertmer, Sengstock, Sampera, Shlyapnikov and Lewenstein [1999], dark solitons of variable velocity were launched via “phase imprinting” of a BEC by a light-shift potential, similarly to the way a phase mask can imprint a soliton or vortex on a light beam. The soliton velocity could be selected by applying a laser pulse to only half of the BEC and choosing the laser intensity and duration to select the desired phase step. Subsequent experiments (Anderson, Haljan, Regal, Feder, Collins, Clark and Cornell [2001], Dutton, Budde, Slowe and Hau [2001]) confirmed the predicted onset of dynamical instabilities originating from undulations of the soliton, much like the so-called “snake” instabilities of optical dark solitons. In the first of these experiments, a dark soliton was created in a spherically symmetric BEC in such a way that the soliton nodal region was filled with a different group of condensed atoms, thereby creating a “dark–bright” soliton combination. When the inner soliton-filling component was removed, dynamical instabilities drove the BEC into a more topologically stable configuration in which vortex rings were found to be embedded in the BEC. For the case of attractive interactions, g < 0, the kinetic energy of the BEC can be balanced by the nonlinearity, yielding nonspreading wave packets. These solutions correspond to spatially localized bright solitons (Reinhardt and Clark [1997], Agrawal [1995]). The one-parameter solution to eq. (7.1) then reads       √ v 2 1/2 i(φ(z,v,t)+µt/h¯ ) v 2 1/2 (z − vt) ϕ(z, t) = n 2 − 2 2− 2 e sech , l0 cs cs (7.5) where v is the velocity parameter and µ, l0 , and cs are the same as in the darksoliton case. The phase φ is given in this case by φ(z, v, t) =

(z − vt) v . l0 cs

(7.6)

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[3, § 8

For atom-optics applications such as atom interferometry, it is desirable to achieve the dispersionless transport of a spatially localized ensemble of atoms, rather than propagation of a “hole” within a group of atoms. Bright solitons in a condensate with attractive two-body interactions provide one possible way to achieve this goal, and have recently been demonstrated in experiments by Strecker, Partridge, Truscott and Hulet [2002] at Rice University, and by Khaykovich, Schreck, Ferrari, Bourdel, Cubizolles, Carr, Castin and Salomon [2002] in Paris. In these two experiments, large elongated condensates of 7 Li were created under conditions where the scattering length was positive. Using a Feshbach resonance, the scattering length was then changed to a < 0, at which point condensed atoms were observed to propagate within an effectively onedimensional trap. In the experiment at Rice University, a train of bright solitons was observed, and interactions between the solitons were found to be repulsive, keeping the solitons spatially separated. Our discussion so far indicates that dark solitons arise for positive scattering lengths and bright solitons for negative scattering lengths. However, it is possible to extend the range of options and to realize bright solitons with a positive scattering length, by considering a one-dimensional condensate in a periodic optical lattice. This situation was first discussed by Steel and Zhang [1998] for a scalar condensate. The solitons formed in optical lattices are referred to as gap solitons (Chen and Mills [1987], Christodoulides and Joseph [1989], de Sterke and Sipe [1994], Eggleton, Slusher, de Sterke, Krug and Sipe [1996]). This name derives from the fact that the soliton energies actually lie in the energy gaps of the atomic band structure (Berg-Sørensen and Mølmer [1998], Choi and Niu [1999]). Gap solitons in Bose–Einstein condensates (Zobay, Pötting, Meystre and Wright [1999]) have recently been demonstrated by Eiermann, Anker, Albiez, Taglieber, Treutlein, Marzlin and Oberthaler [2004].

§ 8. Four-wave mixing 8.1. Bosonic four-wave mixing The idea that interactions between ultracold atoms could lead to nonlinear mixing between matter waves was first proposed by Lenz, Meystre and Wright [1993] and independently by Zhang [1993] and by Zhang and Walls [1994]. Both proposals assumed that the two-body interactions originated from the near-resonant dipole– dipole interaction between atoms in which the atomic dipole moments are induced by a driving laser field. This work pre-dated the production of the first BEC and it

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was not known at that time that ground-state collisions between condensate atoms would be strong enough to induce nonlinear wave mixing. Subsequent simulations showed that collisions in a BEC described by the pseudopotential (4.17) were indeed strong enough (Trippenbach, Band and Julienne [2000]), thereby obviating the need for an induced interaction between electronic dipoles and the difficulties associated with this interaction, such as spontaneous emission from the excited state. In particular, the low-energy two-body collisions described by eq. (4.17) are mathematically equivalent to a local Kerr medium with instantaneous response. The first experimental verification of nonlinear mixing of matter-waves was realized by W.D. Phillips and coworkers at NIST in a 23 Na condensate (Deng, Hagley, Wen, Trippenbach, Band, Julienne, Simsarian, Helmerson, Rolston and Phillips [1999]). Their geometry involved atoms in a single hyperfine state, the four modes involved in the mixing process being distinguished by their center-ofmass momentum. The preparation of the condensate in a superposition of three momentum modes was achieved via Bragg diffraction from a moving optical standing wave formed by two overlapping laser beams (Kozuma, Deng, Hagley, Wen, Lutwak, Helmerson, Rolston and Phillips [1999]). The NIST experiments proceeded by first producing a condensate in a very shallow trap so that the condensate has a very narrow momentum distribution. The magnetic trapping potential was then reduced to zero to produce a free BEC. Two optical pulses were used to Bragg-scatter atoms into the momentum states p2 and p3 which, along with the original condensate p1 0, resulted in three overlapping matter-wave packets that were copies of the original condensate wave function. The momentum differences pi −pj were much larger than the momentum spread of the initial condensate wave function so that the three wave packets were well separated in momentum space. As they flew apart, these wave packets interacted nonlinearly to produce a fourth wave at p4 = p1 − p2 + p3 . If we ignore quantum effects such as squeezing, then the dynamics of the fourwave mixing are well described by the semiclassical Gross–Pitaevskii equation given by eq. (5.10). The condensate wave function immediately after the application of the Bragg pulses is (Trippenbach, Band and Julienne [1998, 2000]) Φ(r, t1 ) = Φ(r)

3  i=1

1/2 ipi ·r/h¯

fi

e

,

(8.1)

 where fi = Ni /N is the fraction of atoms in wave packet i, with i fi = 1, and Φ(r) is the ground-state wave function of the condensate just before the Bragg pulse. This initial wave function then evolves according to eq. (5.10), and three wave packets start to separate. During this separation, the mean-field energy

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acts as a cubic nonlinearity and generates a new wave packet of central momentum p4 . It can be shown that for this momentum, both the total energy E(t) and momentum p(t) are conserved during the wave-packet evolution, where   2 2 
2
h¯ ∇ + g
Φ(r, t)
Φ(r, t) d3 r E(t) = Φ ∗ (r, t) (8.2) 2M and

 p(t) = −ih¯

Φ ∗ (r, t)∇Φ(r, t) d3 r.

(8.3)

At a subsequent time t, the condensate wave function may therefore be expressed in the form Φ(r, t) =

4 

ϕi (r, t)ei(ki ·r−ωi t) ,

(8.4)

i=1

where ki = pi /h¯ and 2 ωi = hk ¯ i /2M.

(8.5)

In eq. (8.4) we assumed that the envelopes ϕi (r, t) are slowly varying in space and time on the scales of ωi−1 and ki−1 . Substituting the ansatz (8.4) into the Gross–Pitaevskii eq. (5.10), collecting in the familiar way of nonlinear optics terms multiplying the same phase factors, multiplying these terms by their corresponding complex conjugate phase factors, and finally discarding terms that have oscillating phase factors, yields the set of four coupled partial differential equations for the phase-matched terms,   h¯ ki h¯ ∇ 2 ∂ + ∇−i ϕi (r, t) ∂t M 2M i  =− g δ(ki + kj − km − kn )δ(ωi + ωj − ωm − ωn ) h¯ j,m,n

× ϕj (r, t)ϕm (r, t)ϕn (r, t),

(8.6)

where each of the indices can take any value between 1 and 4. The momentum and energy conservation conditions implicit in the Kronecker δ-functions in these equations, ki + kj − km − kn = 0

(8.7)

ωi + ωj − ωm − ωn = 0,

(8.8)

and

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are a consequence of phase matching and are automatically satisfied if either i = j = m = n or i = m = j = n. In the language of nonlinear optics, the first of these conditions corresponds to self-phase modulation and the second to cross-phase modulation. These processes do not involve any real exchange of particles, hence these contributions cannot lead to the excitation of a new side mode from the three initial wave packets. Indeed, particle exchange between the various wave packets can occur only when all four indices are different, while conservation of momentum and energy are satisfied. The explicit form of the four-wave mixing equations of motion including the self-phase and cross-phase modulation are readily obtained from eq. (8.6) and read   h¯ k1 h∇ ∂ ¯ 2 + ∇−i ϕ1 (r, t) ∂t M 2M

ig ig = − |ϕ1 |2 + 2|ϕ2 |2 + 2|ϕ3 |2 + 2|ϕ4 |2 ϕ1 − ϕ2 ϕ3 ϕ4 (8.9) h¯ h¯ with the other three equations obtained by cyclic permutations. The factors of two in the cross-phase modulation contributions result from nonlinear nonreciprocity, an effect well studied in nonlinear optics. The last term is the source term, which either creates or annihilates particles in the wave packet being propagated. Equations (8.9) were solved numerically by (Trippenbach, Band and Julienne [1998, 2000]) and found to be in excellent agreement with the experiments. The momentum-conservation condition (8.7) implies that ki + kj = km + kn = κ.

(8.10)

It is always possible to construct a reference frame where κ = 0 via the Galilean transformation kl = kl − κ/2.

(8.11)

In this frame we then have ki = −kj and km = −kn , and the four-wave mixing process takes the form of degenerate four-wave mixing. That is, all momenta are equal in magnitude. In the degenerate frame, the four-wave mixing can be interpreted in terms of phase conjugation where two atoms are annihilated from each of the pump beams, p1 and p3 , and a pair of atoms are created in the “probe” beam, p2 , as well as the phase-conjugate “signal” beam, p4 . In this case the signalbeam wave packet, ϕ4 (r, t), is the complex conjugate of the probe wave packet, ϕ1 (r, t), which from the perspective of quantum mechanics corresponds to time reversal of the probe beam. From a quantum-mechanical perspective, four-wave mixing is readily viewed as a stimulated two-body scattering process in which two atoms colliding in the states p1 and p3 scatter off each other into two final

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states. The existence of atoms in the state p2 results in a Bose enhancement by a factor of N2 + 1 of the probability of atoms scattering into this state and the state p4 that conserves the total energy and momentum of the collision. An alternate frame of reference to that of eq. (8.11) is obtained from the condition p2 = −p1 from which it follows that p4 = p3 − 2p2 . In this frame, the wave packets p2 and p1 overlap and interfere with each other to form a density grating. The incident wave packet p3 is then Bragg-diffracted off of the density grating to produce the scattered matter wave p4 . At the same time the grating must absorb a momentum 2p2 , which results in an atom being transferred from the p1 wave packet to the p2 wave packet. We will return to this interpretation of four-wave mixing in the next section where we discuss four-wave mixing with fermions.

8.2. Fermionic four-wave mixing We mentioned that matter-wave four-wave mixing may be interpreted either in terms of diffraction of a matter wave off of a density grating or as a stimulated scattering process. While the first interpretation appears to be essentially classical in nature, the second one is quantum-mechanical, and relies explicitly on the fact that one of the “final” modes is initially populated and the resulting bosonic enhancement. This naturally leads one to ask whether matter-wave four-wave mixing is also possible with fermions. This would appear to be allowed in the “classical” density-grating explanation, but precluded if the boson-enhancement argument holds. In fact, if the four-wave mixing is interpreted in terms of stimulated scattering then the Bose enhancement factor of N2 + 1 in the scattering rate is replaced by N2 − 1 with N2 = 0, 1 for the case of fermions – a direct consequence of the Pauli Exclusion Principle. This issue was addressed in two papers that showed that four-wave mixing and matter-wave amplification should indeed be possible with fermions (Moore and Meystre [2001], Ketterle and Inouye [2001]). To understand how four-wave mixing with fermions is possible it is instructive to look at the rate at which individual test particles scatter off a quantum-degenerate gas. This allows us to examine how the gas collectively responds to a momentum- and energy-changing collision with the particle, based on the initial state and quantum statistics of the gas. Let us assume that the test particle at position r interacts with the gas via the local potential V (r) = g ρ(r), ˆ where ρ(r) ˆ is the density operator for the gas. According to Fermi’s golden rule, the probability per unit time that the particle scatters from the state with momentum k1 and energy ω1 = h¯ k12 /2M to the state with

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momentum k2 = k1 − q and energy ω2 = h¯ k22 /2M is given by

2
2π 

g  −iq ·r †
δ(ωf i − ω12 ) f, k w1→2 = − q| e ρ ˆ |k , i  1 1

q h¯ V  f q  
2 2π g 2 

f |ρˆq† |i
δ(ωf i − ω12 ), = (8.12) h¯ V f where |i and |f  are the initial and final states of the gas, which are eigenstates of the gas Hamiltonian H , with energies h¯ ωi and h¯ ωf , respectively. Here  † ρˆq = (8.13) cˆkσ cˆk+qσ k,σ

is the qth Fourier component of the density, and σ labels the internal states, e.g., spin, of the particles. From eq. (8.12) we define the dynamic structure factor as (Nozières and Pines [1999]) 


f |ρˆ † |i
2 δ(ωf i − ω). S(q, ω) = (8.14) q f

 By using the completeness of the final states, f |f f | = I , the dynamic structure factor can be expressed in terms of the two-time correlations of the density fluctuations in the gas:  1 S(q, ω) = (8.15) dt eiωt i|ρˆq (t)ρˆq† (0)|i, 2π where ρˆq (t) = exp(iH t)ρˆq exp(−iH t). Equation (8.15) is nothing but the power spectrum of the density modulations of the gas. The density modulations can originate either from density variations in the initial state of the gas, i|ρˆq† |i = 0, or from spontaneous fluctuations in the density, i|ρˆq ρˆq† |i = 0. To shed further light on this process, we consider a bosonic system prepared (in one dimension) in two momentum states |±q0 /2 populated by N/2 atoms each, |ψB−F  =

N/2 1 † N/2 † cˆq0 /2 cˆ−q0 /2 |0. (N/2)!

(8.16)

The dominant contribution to the dynamic structure factor involves elastic scattering between the two occupied states, and is given by   S(q, ω) = N 2 δq,0 + N/2(N/2 + 1)δq,q0 + N/2(N/2 + 1)δq,−q0 δ(ω), (8.17)

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with the inelastic terms that correspond to scattering in to unoccupied modes being smaller by a factor of N . This shows that except for forward scattering, scattering is predominantly between the two states that are macroscopically populated initially. In this case the scattering results from the density fluctuations since ψB−F |ρˆq† |ψB−F  = 0 for q = 0. The N/2(N/2 + 1) dependence of the scattering rate may be interpreted as “Bose enhancement” or stimulated scattering due to the macroscopic occupation of the initial state (N/2 term) and the final state (N/2 + 1 term). Besides the Fock state, one can also consider the initial state N 1  † † |ψB−C  = √ (8.18) cˆq0 /2 + cˆ−q |0. 0 /2 N 2 N! This describes the situation where each atom is prepared in a coherent superposition of the two momentum states |±q0 /2. This state can be realized by applying a “π/2” Bragg pulse to the atoms using two lasers with relative momentum q0 that is much larger than the spread in momentum of the initial condensate (Kozuma, Deng, Hagley, Wen, Lutwak, Helmerson, Rolston and Phillips [1999]), as was done in the BEC four-wave mixing experiments (Deng, Hagley, Wen, Trippenbach, Band, Julienne, Simsarian, Helmerson, Rolston and Phillips [1999]). In this case, the condensate density is spatially modulated with the wavenumber q0 ,

1  N 1 + cos(q0 x) , ρ(x) = ψB−C |ρˆq |ψB−C eiqx = V q V and the dominant contribution to S(q, ω) is now a result of diffraction from the density grating,

S(q, ω) = N 2 δq,0 + (N/2)2 [δq,q0 + δq,−q0 ] δ(ω). (8.19) For large N , both |ψB−C  and |ψB−F  give the same scattering rate even though the physical origin of the scattering is quite different for the two cases. The equivalence between Fock states and coherent superposition states for large numbers of bosons has been discussed earlier for the quantum state of the condensate, where it was mentioned that a Fock state can be thought of as a coherent state with a random phase. In this context, eq. (8.17) can be thought of as resulting from diffraction off a density grating that has a random phase. The situation is quite different for fermions, since a single-mode Fock state can have at most an occupation number of one atom. Hence, one should expect that the fermionic “analog” of the first example, i.e. a state of the form |ψF −F  =

N/2  {k}

† cˆk+q 0 /2

N/2  {k}

† cˆk−q |0, 0 /2

(8.20)

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where the products are over N/2 modes about the mean momentum ±q0 /2, will not lead to enhanced scattering. Indeed it is easy to show that the structure factor has the form 

S(q, ω) = N 2 δq,0 δ(ω) + 1 − n(k + q) n(k)δ(ωk+q − ωk − ω), k

(8.21)

where



n(k) = Θ k − |k − q0 /2| + Θ k − |k + q0 /2| and Θ is the unit step function. (Note that for simplicity the widths k of the momentum distributions about ±q0 /2 are taken to be much narrower than q0 , k q0 .) In this case scattering between the occupied states is completely blocked as a result of the Pauli Exclusion Principle, and only incoherent scattering with S(q, ω) ∼ O(N ) occurs. In contrast, one can also generate a fermionic state that is a product of fermionic atoms in coherent superpositions of momentum states separated by q0 , N  1  † † |ψF −C  = √ cˆk+q0 /2 + cˆk−q |0. /2 0 2N {k}

(8.22)

This state produces a density grating ρ(r) =

1  N 1 + cos(q0 x) ψF −C |ρˆq |ψF −C eiqx = V q V

that is identical to the density grating produced by |ψB−C . Note however that since the grating is now formed by fermions with an unavoidable spread of momenta k about ±q0 /2, the grating can be thought of as being “inhomogeneously broadened” and will dephase in a time ∼ m/h¯ q0 k. For short enough times, though, it behaves collectively very much like the bosonic grating and will diffract particles at a rate proportional to the square of the amplitude of the density grating, |ψF −C |ρˆq |ψF −C |2 = (N/2)(N/2). While in the case of a degenerate Bose gas, stimulated four-wave mixing can indeed be attributed to Bose enhancement, in the fermionic case it must therefore be interpreted as originating from constructive quantum interferences between “paths” that lead to indistinguishable final states. Both mechanisms lead to practically the same enhancement, provided the fermionic grating is properly prepared. At a fundamental level, this is due to the fact that Bose enhancement, when viewed in terms of the first-quantized many-particle wave function, is simply a constructive quantum interference where many initial states (the different terms

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under exchange of particle labels) lead to a single final state. Hence, effects that can be interpreted as Bragg scattering from atomic matter-wave gratings, such as atomic four-wave mixing, BEC superradiance, and matter-wave amplification, can in principle work as efficiently in both Bose and Fermi systems. This result is further confirmed by the observation that the states (8.18) and (8.22) are actually momentum-space analogs of the cooperative atomic product states that lead to superfluorescence in atomic systems (Mandel and Wolf [1995]). Consider the simple case of N = 2 atoms for concreteness. In this case, the initial states of the Bose and Fermi density gratings are simply 1  †2 † † †2  |ψB  = √ cˆ−q (8.23) |0 + 2 c ˆ c ˆ + c ˆ −q q q /2 /2 /2 0 0 0 0 /2 8 and 1 † † † |ψF  = cˆ−q cˆ† + cˆ−q cˆ† + cˆq†0 /2 cˆ−q 0 /2 −q0 /2+ε 0 /2 q0 /2+ε 0 /2+ε 2  † † + cˆq0 /2 cˆq0 /2+ε |0  †  1 † + cˆq†0 /2 cˆ−q + cˆq†0 /2+ε |0, = cˆ−q (8.24) /2 /2+ε 0 0 2 where ε q0 /2 is a small momentum shift. If we consider Bragg scattering of the test particle from −q0 /2 to q0 /2, then the ‘gas’ must absorb the momentum −q0 , which must transfer an atom from the q0 /2 group to the −q0 /2 group in order to conserve energy. The resulting final states of the Bose and Fermi gratings are
(f )   1  †2 †
ψ (8.25) = √ cˆ−q |0 + cˆ−q cˆ† B 0 /2 0 /2 q0 /2 4 and
(f )   1  † † †
ψ = √ 2cˆ−q |0, cˆ† + cˆq†0 /2 cˆ−q + cˆ−q cˆ† F 0 /2 −q0 /2+ε 0 /2+ε 0 /2 q0 /2+ε 6 (8.26) (f )

† |ψB,F |2 = respectively. In both cases the resulting scattering rate is |ψB,F |ρˆ−q 0 3 2 , while for momentum transfers other than q0 the corresponding matrix elements for the scattering rate are just 1 for the bosons and fermions. In the case of bosons † the enhanced scattering results from the stimulated transition cˆ−q cˆ† |0 → 0 /2 q0 /2 †2 † |0. In the case of the fermions one finds that both cˆ−q cˆ† |0 and cˆ−q 0 /2 0 /2 q0 /2+ε

† † cˆq†0 /2 cˆ−q |0 lead to the same final state, cˆ−q cˆ† |0 with the two path0 /2+ε 0 /2 −q0 /2+ε ways interfering constructively to produce the enhanced scattering. So far, our discussion has ignored the time dependence of the fermionic grating. Since the atoms forming that grating all have slightly different kinetic energies, their free evolution results in a dephasing that is expected to eventually

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lead to the disappearance of the four-wave mixing signal. This problem was discussed by Christ, Search and Meystre [2003]. Besides dephasing, one must also consider how the back-action of the test particles on the grating effects the longtime dynamics of the four-wave mixing. This corresponds to going beyond the undepleted-pump approximation in nonlinear optics. Miyakawa, Christ, Search and Meystre [2003] give a detailed discussion of the coupled dynamics of the beam of incident particles and the density grating. In particular, it shows that for strong coupling between the beam and grating the effects due to dephasing can be counteracted.

8.3. Fermionic phase conjugation In the previous section we pointed out that four-wave mixing with fermions was possible when the wave-mixing was interpreted in terms of diffraction off a density grating. However, in our discussion of bosonic four-wave mixing we mentioned that in the degenerate frame, the dynamics could also be interpreted in terms of phase conjugation. We recall that in optical phase conjugation, an incident probe beam interacts with a pump field inside a nonlinear medium to generate a signal beam that is the time-reversed or phase-conjugate state of the probe field. In the case of a cubic nonlinearity, two pump photons are destroyed and a photon is created in both the signal and probe fields (Yariv and Pepper [1977]). In classical optics, phase conjugation can be used to correct the phase aberrations incurred by the probe field (Shen [1984]). In this manner, a nonlinear medium that is actively pumped by two lasers can serve as a phase-conjugate mirror, which generates a backward propagating signal field that reverses the wavefront distortions of the probe field when it traverses the distorting medium. In quantum optics, phase conjugation via four-wave mixing can lead to the generation of two-mode squeezed states (Yuen and Shapiro [1979]). Ultimately, the phase conjugation of bosonic matter waves is a result of the phase coherence between the two pump beams, which is a consequence of the mean field of the condensate. However, for fermions in their normal state there is no such phase coherence between overlapping beams, i.e. Ψˆ (r)Ψˆ (r) = 0. However, as we have already discussed, superfluid Fermi gases do possess a nonzero pairing mean field that should make phase conjugation possible. To show why this is so, we consider in one dimension two counter-propagating beams of atomic fermions interacting with a fermionic superfluid in the region 0  z  L. We assume that the superfluid consists of equal numbers of atoms, NF , occupying two hyperfine states that we refer to as spin up (↑) and spin down (↓). In the case of a

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BCS superfluid, the gas is characterized by the nonzero order parameter, ∆, given by eq. (6.9) that characterizes the Cooper-pair correlations. In the resonance superfluidity case, ∆ contains an additional contribution χφm due to the molecular field, φm . Under the circumstances we are interested in, the Hamiltonian (6.1) for the superfluid Fermi gas can be linearized around the pairing mean field as  †

† †

 † H = hω ¯ cˆk↑ cˆ−k↓ + cˆ−k↓ cˆk↑ . (8.27) ¯ k cˆk↑ cˆk↑ + cˆ−k↓ cˆ−k↓ + h∆ k

As before, cˆkσ is the annihilation operator for a fermion of momentum hk ¯ and spin σ , and ωk = h¯ k 2 /2m − hk ¯ F2 /2m and kF = (6π 2 nF )1/3  L−1 is the Fermi wave number of the gas. For convenience we take ∆ to be real. The Hamiltonian H may be diagonalized by the Bogoliubov transformation † , αˆ k↑ = cos(θk /2)cˆk↑ − sin(θk /2)cˆ−k↓

(8.28)

† † αˆ −k↓ = cos(θk /2)cˆ−k↓ + sin(θk /2)cˆk↑ ,

(8.29)

where αkσ  is an annihilation operator for a quasiparticle in the gas with energy h¯ ζk = h¯ ωk2 + ∆2 and tan θk = |∆|/ωk . To show that the superfluid acts like a phase-conjugate mirror (PCM) for the incident beams, we consider a standard geometry where a continuous beam of spin-up fermions with momenta h¯ k = h¯ kˆz, k > 0 impinges on the PCM at z = 0, while a beam of spin-down fermions with momenta h¯ k = −h¯ kˆz, k > 0 impinge on it at z = L. The number of atoms in these beams, NB , is assumed to satisfy NB NF so that the PCM can be treated as an undepleted source of atoms. When the fermions traverse the region holding the superfluid, their de Broglie wavelengths will shorten as a result of the attractive potential that the atoms experience. More specifically, a fermion that is initially described as a wave packet with the group velocity v0 = hk ¯ 0 /m, ψ(z, t) ∼ ψ(z − v0 (t − t0 ), 0) experiences the same confining potential as the trapped atoms, as well as an attractive Hartree potential, −gnF , when it enters the superfluid gas. It therefore propagates with ¯ 0 )/m, where the new group velocity v¯k0 = h¯ k(k 

¯ 0 ) = k 2 + 2m |U0 | + gnF /h¯ 2 , k(k 0 and reaches z = L after a time τk0 = L/v¯k0 . Similarly, a wave packet at z = L with mean momentum −h¯ k0 takes a time τk0 to reach z = 0. The input fields are (t = 0) and cˆ−k↓ [z = then related to the initial conditions, cˆk↑ [z = 0] = cˆk(k)↑ ¯ (t = 0). L] = cˆ−k(k)↓ ¯ By using the Bogoliubov transformation (8.28)–(8.29) and the solution of the equations of motion for the quasiparticles, αˆ kσ = αˆ kσ (0) exp[−iζk t], one readily

3, § 8]

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obtains the output states as † cˆk↑ [L] = Tk(k) ¯ cˆk↑ [0] + Rk(k) ¯ cˆ−k↓ [L],

(8.30)

† ∗ cˆ−k↓ [0] = Tk(k) ¯ cˆ−k↓ [L] + Rk(k) ¯ cˆk↑ [0],

(8.31)

where Tk = cos(ζk mL/h¯ k) − i cos θk sin(ζk mL/h¯ k),

(8.32)

Rk = i sin θk sin(ζk mL/h¯ k).

(8.33)

Phase conjugation, or time reversal, occurs when Rk = 0, the output states then being a superposition of the transmitted input state plus its time-reversed state, † [0] = T cˆk↑ [0]T −1 , where T is the time-reversal operator. Note that Rk = 0 cˆ−k↓ in the absence of a superfluid state, ∆ = 0, so that the existence of this state is essential for the operation of the PCM. An important difference from the optical and bosonic matter-wave cases is that since |Tk |2 + |Rk |2 = 1, there is no amplification of the individual modes of the fermion beams. This is in stark contrast to the case of bosons where one has instead |Tk |2 − |Rk |2 = 1 in order to preserve the commutation relations and as a result the transmitted field is always amplified since |Tk |
1. The lack of amplification for a single mode of the fermion field is a necessary consequence of the Pauli Exclusion Principle. Note however that the total number of fermions in the output beams can be amplified. To see this, we take for definiteness the input state of the fermion beams to be  † cˆk↑ [0]|0, |Ψ  = (8.34) |k−k0 |
k

with k0 > k > 0. This corresponds to a beam of spin-up fermions with momenta centered around hk ¯ 0 incident from z < 0 with no atoms incident from z > L. The † [0]cˆk↑ [0]. Defining the total occupation numbers for this state are nk = cˆk↑ number operators for the input and output fields as  † (in/out) = cˆk↑ [0/L]cˆk↑ [0/L], Nˆ ↑ k>0

(in/out) Nˆ ↓ =



† cˆ−k↓ [L/0]cˆ−k↓ [L/0],

k>0

one finds that their expectation values are  out   in   out  Nˆ ↑ = Nˆ ↑ + Nˆ ↓ ,  out   2 Nˆ ↓ = |Rk(k) ¯ | (1 − nk ). k

(8.35) (8.36)

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Equations (8.35) and (8.36) show that the numbers of atoms in both beams increase after having passed through the gas. However, eq. (8.36) shows that the increase results from the scattering of atoms out of the superfluid gas and into those modes that are not occupied in the incident beam. Consequently, only incoherent amplification of the vacuum fluctuations occurs. Equations (8.30) and (8.31) would seem to indicate that the output beam, † cˆ−k↓ [0], is the phase-conjugate state of the input beam, cˆk↑ [0]. This would undoubtedly be the case for bosonic fields, but this interpretation is questionable for fermionic fields since it is not clear how to define their phase. Since number and phase are canonically conjugate variables, a well-defined phase requires a large uncertainty in particle number. Hence, a straightforward generalization to fermions of the various bosonic phase operators (Dirac [1927], Susskind and Glogower [1964], Pegg and Barnett [1988]) leads to ill-defined results. Therefore, it would appear that if phase is to have any meaning in Fermi systems, it must be associated with a multimode effect such as the order parameter ∆ of a superfluid Fermi gas. Despite these apparent difficulties, it is possible to introduce a fermionic phase in an operational way. As an indication that this might be possible, we note that it is possible in principle to operate atom interferometers with quantumdegenerate fermionic beams, thereby measuring the relative phase of the partial beams (Search and Meystre [2003a], Yurke [1986]). Search and Meystre [2003b] show that by using a PCM for fermions in one arm of a Michelson atom interferometer the interference pattern will be lost completely. This is because the PCM causes the interference pattern to depend on the absolute phase of the fermions rather than just the relative phase between the two arms of the interferometer. Since the phase uncertainty of each mode of the fermionic field is φk ∼ 2π, there will not be any interference pattern on average.

§ 9. Three-wave mixing 9.1. Nonlinear mixing of quasi-particles The Bogoliubov linearization procedure of Section 5.2 leads to the appearance of quasiparticles, a result of a linearization of the Hamiltonian around the macroscopically occupied condensate mode and keeping only terms quadratic in other field modes. A somewhat better approximation consists in also including terms cubic in those modes. The Hamiltonian H of eq. (5.11) then becomes (Ozeri, Katz, Steinhauer, Rowen and Davidson [2003]) H = HB + H3 ,

(9.1)

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where HB is given by eq. (5.17), while 

g  † H3 = N0 Ak,q αˆ k† αˆ q αˆ q−k + αˆ q† αˆ k−q αˆ k 2V q,k

 †

† † + Bk,q αˆ −(k+q) αˆ q αˆ k + αˆ k αˆ −(k+q) αˆ q ,

(9.2)

  Ak,q = 2uk (uq − vq )u|k−q| − uq v|k−q|   − 2vk (vq − uq )v|k−q| − vq u|k−q| ,

(9.3)

  Bk,q = −2uk (uq − vq )v|k+q| + vq u|k+q|   + 2vk (vq − uq )u|k+q| + uq v|k+q| ,

(9.4)

and we have used the transformation to the quasiparticle basis given by eq. (5.17). The first two terms in the Hamiltonian H3 account for three-wave mixing † αˆ k describes processes between Bogoliubov quasiparticles. Specifically, αˆ q† αˆ k−q the annihilation of a quasiparticle from mode k and the creation of a pair of quasiparticles in modes q and k − q and is the analog of frequency down-conversion in optics. Similarly the terms of the form αˆ k† αˆ q αˆ q−k , which describe the annihilation of two quasiparticles and the creation of a new one, are the analogs of sum-frequency generation in optics. The terms proportional to Bk,q describe the spontaneous creation or annihilation of three quasiparticles. Since these terms do not describe energy-conserving processes, they can usually be ignored. In eq. (9.1), higher-order terms of order g/2V that describe four-wave mixing between quasiparticles have been ne√ glected. Since these terms are smaller than the terms in H3 by a factor N0 , they are expected to be negligible compared to the three-wave mixing for N0  1. From the perspective of condensed-matter physics, the terms in H3 have historically been considered to be damping mechanisms for quasiparticle excitations. The spontaneous decay of a quasiparticle into the continuum of unoccupied states † αˆ k is known as Baliev damping (Baliev [1958]) while the coldue to αˆ q† αˆ k−q lision of a quasiparticle with a thermally excited quasiparticle as described by αˆ k† αˆ q αˆ q−k is known as Landau damping (Pitaevskii and Stringari [1997]). However, if one prepares the system in a state with N excitations in state k and M in q, |Nk , Mq , 0k−q , such that 1 N, M N0 , then the stimulated scattering of a quasiparticle into state q, |Nk , Mq , 0k−q  → |(N − 1)k , (M + 1)q + 1, 1k−q , √ k −√ will be a factor of N or M larger than Baliev damping into other modes provided the phase-matching condition Ek = Eq + Ek−q is satisfied. Here, Ek is given by eq. (5.23).

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Ozeri, Katz, Steinhauer, Rowen and Davidson [2003] studied numerically the nonlinear oscillations between the states |(N − i)k , (M + i)q , ik−q  for i = 0, . . . , N including the effects of damping. Because of the phase-matching condition, the states |(N − i)k , (M + i)q , ik−q  are all degenerate eigenstates of HB . H3 can then readily be diagonalized with respect to the |(N −i)k , (M +i)q , ik−q  basis to yield the dressed states of the excitations. Within the dressed-state picture, Baliev damping of quasiparticles into other modes can be interpreted in terms of the transfer of probability between dressed-state manifolds characterized by different N and M, e.g.,

 
(N − i)k , (M + i)q , ik−q →
(N − 1 − i)k , (M + i)q , ik−q . For short times, the system prepared in the state |Nk , Mq , 0k−q  exhibits largeamplitude oscillations in the average occupation numbers of the three modes. However, because of the nonlinearity in the energy spectrum of the dressed states and the damping, the amplitude of the oscillations exhibits collapse and revivals as well as decay over longer time scales. It might appear surprising that the matter-wave analog of optical three-wave mixing should be allowed, since in contrast to photons, the number of atoms is a conserved quantity. The point here is that quasiparticles are collective excitations of the condensate, which are nothing more than phonons in the long-wavelength limit. Just as is the case for photons, their number is not a conserved quantity, hence the possibility of three-wave mixing. Another mechanism that avoids the difficulties associated with atom-number conservation is provided by the coherent generation of molecules starting from an atomic gas, which has recently culminated in the realization of molecular condensates, a topic to which we now turn.

9.2. Coherent molecule formation In Sections 4.2 and 4.3 we discussed the underlying physics of Feshbach resonances and two-photon Raman photoassociation, which can be used to coherently create molecules. These mechanisms are sometimes referred to as examples of superchemistry, the coherent stimulation of chemical reactions via macroscopic occupation of a quantum state (Heinzen, Wynar, Drummond and Kheruntsyan [2000]). Molecule formation can originate from either bosonic or fermionic atomic pairs, but recent work (Petrov, Salomon and Shlyapnikov [2004]) shows that the lifetime of the molecules created from fermions very close to the Feshbach resonance can be orders of magnitude longer than for bosons, a direct consequence of the Pauli Exclusion Principle.

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Since both photoassociation and Feshbach resonances involve the destruction of two atoms and the creation of a molecule, they are described by the same lowenergy effective-field theory, which in the case of bosonic atoms in a single internal state is described by the Hamiltonian (4.30). The generalization of eq. (4.30) for fermions or multi-component bosons that includes s-wave photoassociation or Feshbach resonances is   2 2    h¯ ∇ 3 † ˆ + Vσ (r) Ψˆ σ (r) H= d r Ψσ (r) − 2m σ =↑,↓  2 2  h¯ ∇ † ˆ ˆ + Φ (r) − + V (r) + ε Φ(r) 4m 

1  ˆ χσ,σ  Ψˆ σ† (r)Ψˆ σ† (r)Φ(r) + H.c. , + (9.5) 2  σ,σ

where the Schrödinger field Ψˆ σ (r) describes the σ = ↑, ↓ bosonic or fermionic ˆ atomic species, Φ(r) describes a bosonic molecular field, and ε is the energy difference between a molecule and two atoms. For fermions, χσ,σ  = χδσ,↑ δσ  ,↓ . We have neglected the two-body interactions between atoms and molecules in eq. (9.5) in order to specifically focus on the second-harmonic generation of molecules starting from a quantum-degenerate atomic gas. For the moment, let us consider an initial state consisting of a single-component atomic BEC at zero temperature. The atomic field operator can be expressed in terms of a single mode, Ψˆ (r) = ψ(r)c, ˆ which is coupled to a single centerˆ In this limit we obtain the ˆ of-mass mode for the molecules, Φ(r) = φ(r)b. commonly used two-mode Hamiltonian (Javanainen and Mackie [1999], Vardi, Yurovsky and Anglin [2001])

1 H2 = ε  bˆ † bˆ + χ  bˆ † aˆ aˆ + aˆ † aˆ † bˆ , 2 where   χ = χ d3 r ψ 2 (r)φ ∗ (r),

(9.6)

(9.7)

ε  = ε + εm − 2εa , and εa(m) are the single-particle energies of the atoms (molecules). H2 is familiar from quantum optics as a model for second-harmonic generation and parametric oscillation (Walls and Milburn [1994]). Just as in the case of quantum optics, the dissociation of molecules into atom pairs gives rise to squeezing in the number fluctuations of the atoms (Kheruntsyan and Drummond [2002]).

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The dynamics of aˆ and bˆ given by H2 exhibit nonlinear oscillations in the pop√ ulation between the atomic and molecular modes with a frequency Ω ∼ χ  N0 , where N0 is the initial number of atoms. In terms of mean-field theory, i.e. replacing aˆ and bˆ with their expectation values in the equations of motion (Heinzen, Wynar, Drummond and Kheruntsyan [2000]), full conversion of the atoms into molecules requires exact resonance ε  = 0 and precise knowledge of the oscillation frequency in order that the interaction is switched off at the proper time. Both conditions are difficult to achieve experimentally since Ω depends on N0 , which is not known with sufficient accuracy, and the inclusion of two-body interactions leads to density-dependent energy shifts for the atoms and molecules that vary with time. The two-mode model and the coherent oscillations between atomic and molecular condensates that it predicts will however break down when most of the initial condensate atoms have been converted to molecules. At this point there are no longer any atoms left to bosonically stimulate the dissociation of molecules into the condensate state. The molecules then start to dissociate into atoms in states other than ψ(r) in a process called “rogue photodissociation” (Javanainen and Mackie [1999, 2002], Kostrun, Mackie, Côté and Javanainen [2000], Góral, Gajda and Rz¸az˙ ewski [2001]). This can be remedied by using an extremely tight trapping potential, such as are achievable using optical lattices, so that dissociation into excited states of the potential is energetically unfavorable. Besides rogue photodissociation, quantum effects lead to a collapse in the population oscillations even in the two-mode model (Vardi, Yurovsky and Anglin [2001]). The formation of a molecular condensate is more easily achieved by using adiabatic rapid passage between the two states (Javanainen and Mackie [1999], Mies, Tiesinga and Julienne [2000]). Adiabatic rapid passage (Allen and Eberly [1987]) is much more robust with respect to uncertainties in ε, χ, and N0 and can readily be applied to the full Hamiltonian (9.5). To understand how it works, we note that for ε > 0 and larger than all other energy scales in (9.5), the ground state of the system will be a BEC of atoms. On the other hand, for ε large and negative, the ground state will be a BEC of molecular dimers. If ε is slowly varied between these two limits, then according to the adiabatic theorem in quantum mechanics the system will remain in the ground state corresponding to the instantaneous value of ε and consequently the initial atomic BEC will adiabatically evolve into a molecular BEC. We can obtain an analytic expression for the efficiency of the adiabatic transfer using H√2 if we restrict ourselves to the case of two atoms, namely the states aˆ †2 |0/ 2! and bˆ † |0. If the detuning varies linearly, ε  = −|˙ε  |t from t → −∞ to t → +∞, then we have the Landau–Zener model for an avoided level crossing,

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which can be solved exactly (Bates [1961]). The probability that the state bˆ † |0 is occupied at t → +∞ is  2 /h|˙ ε |

Pmolecule = 1 − e−π|χ |

¯

.

(9.8)

Thus in the limit of |˙ε  | → 0 one forms a molecule with unit probability. In the more general case of N0 atoms, the condition for adiabaticity is h|˙ ¯ ε| |χ|2 (N0 /V ). For Feshbach resonances the detuning can be varied by controlling the external magnetic field in eq. (4.28), while for two-photon Raman photoassociation, ε can be adjusted by controlling the frequency difference of the lasers. The first experimental demonstrations of molecule formation using two-photon Raman photoassociation (Wynar, Freeland, Han, Ryu and Heinzen [2000]) and a Feshbach resonance (Inouye, Andrews, Stenger, Miesner, Stamper-Kurn and Ketterle [1998]) were dominated by the molecular losses due to processes such as inelastic decay to lower-energy molecular vibrational states (Yurovsky, BenReuven, Julienne and Williams [1999]). As a result, the existence of the molecules could only be inferred from the decrease in the number of atoms. The first experimental demonstration of coherent conversion of atoms into molecules was performed by Donley, Claussen, Thompson and Wieman [2002], who exploited a Feshbach resonance in a 85 Rb condensate. They used two magnetic field pulses to briefly tune the molecules into resonance with the atoms, the pulses being separated by a free-evolution period where the molecules were off-resonant and the atom–molecule coupling was negligible as a result. The relative phase between the atomic and molecular states accumulated during this free-evolution period resulted in an interference pattern when the two states were recombined by the second field pulse, in a manner analogous to Ramsey fringes in atomic spectroscopy. The experiment measured the coherent oscillations of the number of atoms remaining in the condensate as a function of the free-evolution period. In subsequent experiments starting from an atomic condensate of 87 Rb, Rempe and coworkers used adiabatic rapid passage to create the molecules. Because the molecules and atoms have different magnetic moments, they could be spatially separated using a magnetic field gradient via the Stern–Gerlach effect (Dürr, Volz and Rempe [2004]). Similar work has been conducted by Xu, Mukaiyama, AboShaeer, Chin, Miller and Ketterle [2003]. Starting from a sodium BEC, they used resonant laser light to blast away the remaining atoms, isolating the molecules. Unfortunately, the conversion efficiency was limited by inelastic losses very close to the resonance so that molecular yields were  10%. Although the resulting phase-space densities for the molecular samples were larger than the criterion for Bose–Einstein condensation given by eq. (3.1), the short lifetimes in comparison

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to the time needed to achieve thermal equilibrium precluded the existence of a molecular BEC. For fermionic atoms close to a Feshbach resonance the inelastic collision rate for relaxation to lower-energy vibrational states of the molecules scales like a(B)−2.55 whereas for bosons it scales like a(B), where a(B) is the scattering length near the resonance (Petrov, Salomon and Shlyapnikov [2004]). This is because close to resonance the effective size of the molecules is of the order a(B), which is comparable to the interparticle spacing. In order for the molecule to decay to a more deeply bound vibrational state with radius Re a(B), the atoms comprising the molecule along with an additional atom must all collide within a distance ∼ Re . Since two of the three atoms are necessarily identical, the collision rate is suppressed for fermions. Consequently, the lifetimes of the molecules formed from quantum-degenerate Fermi gases are sufficiently long to achieve thermal equilibrium. The first molecular Bose–Einstein condensate was achieved by Greiner, Regal and Jin [2003] starting from a quantum-degenerate spin mixture of 40 K using adiabatic rapid passage through a Feshbach resonance with a conversion efficiency of ∼ 80%. The molecular lifetimes were sufficiently long to achieve local thermal equilibrium as well as global phase coherence at least along the radial axis of the trap. At about the same time Zwierlein, Stan, Schunck, Raupach, Gupta, Hadzibabic and Ketterle [2003] and Jochim, Bartenstein, Altmeyer, Hendl, Riedl, Chin, Hecker Denschlag and Grimm [2003] succeeded in producing a BEC of 6 Li dimers. However, instead of using an adiabatic B-field sweep, they produced 2 the molecules by evaporatively cooling the atoms at a constant magnetic field just below a resonance where a(B) is large and positive. The molecules, with binding energy h¯ 2 /ma(B)2 , were formed by three-body recombination whereby three atoms collide and two of the atoms form a molecule (Fedichev, Kagan, Shlyapnikov and Walraven [1996]). The binding energy is converted to kinetic energy that is shared between the molecule and the third atom, which must be present in order to conserve the total momentum of the collision.

9.3. The molecular ‘micromaser’ Mean-field theories provide a description of quantum-degenerate atomic or molecular systems at a level comparable to the semiclassical approximation in optics. But there is considerably more to optics than that, and there is a wealth of new physics that appears as soon as the quantum properties of the light field are brought to the forefront. While it took over half a century after the invention of quantum mechanics for experimentalists to become masters at handling

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the quantum properties of the light field, things have evolved much faster with matter waves: experimentalists and theorists alike started investigating “beyond mean-field” physics, the matter-wave analog of quantum optics, soon after the realization of atomic Bose–Einstein condensates. Just as is the case in optics, one topic of much interest in this context is the realization of matter-wave fields of prescribed quantum-statistical properties. In addition to its intrinsic interest, such a capability could find applications in subshot-noise interferometry and quantum information processing. To achieve a similar goal in optics, it is essential to go past “simple” lasers, as there is little one can do to manipulate the quantum-statistical properties of a laser field. This is because far above threshold, the output of a single-mode laser is well described by a (phase-diffused) coherent state, and that is pretty much the end of the story. Hence, other approaches needed to be developed to generate electromagnetic fields in, say, a number state. One way to achieve this goal in the microwave regime is with micromasers. The micromaser, or single-atom maser, is a quantum-optical device that builds a single-mode cavity field “one photon at a time” by injecting a sequence of twolevel atoms inside a high-Q microwave cavity. It is an important experimental system in quantum optics since it permits one to explore in detail the dynamical interactions between individual atoms and a single mode of the electromagnetic field without the noise associated with conventional lasers and masers that tends to wash out the quantum features of the cavity field. The cavity fields that they generate are characterized by a number of quantum-mechanical features absent in normal lasers and masers. For example, they can exhibit strongly sub-Poissonian photon statistics (Rempe, Schmidt-Kaler and Walther [1990]) and have the ability to dynamically generate Fock states (Scully and Zubairy [1997]). They can also undergo multiple “phase transitions”, or more precisely cross-overs, characterized by sharp changes in the photon statistics from sub-Poissonian to super-Poissonian (Guzman, Meystre and Wright [1989]). Extending these ideas to matter waves, Search, Zhang and Meystre [2003] and Miyakawa, Search and Meystre [2004] have shown that it is possible to design an analog of a micromaser for the molecular field. This result is based on the fact that the photoassociation of two fermions into a bosonic molecule can, under appropriate conditions, be mapped into a generalized version of the Jaynes–Cummings model. In micromasers, the cavity field mode is excited by a sequence of two-level atoms such that only one atom at a time is inside the resonator. One of the considerable challenges of building a micromaser is then the need for an extremely high-Q cavity, so that the field mode is not damped significantly in the inter-

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val between successive atoms. This difficulty is largely removed in the case of atomic and molecular fields: they can easily be trapped for very long times, either in optical or in magnetic traps. Optical lattices, in particular, are extremely attractive and relatively easy to realize. Recent years have witnessed much interest in the properties of ultracold atoms confined in such lattices, the optical potentials created by detuned standing-wave laser beams. This is motivated in part by the desire to build a quantum computer in which the atoms at each lattice site would represent an individual qubit (Deutsch, Brennen and Jessen [2000]), and also by the possibility to realize and investigate in detail analogs of the strongly correlated systems of condensed-matter physics. This is epitomized by the recent experiment (Greiner, Mandel, Esslinger, Hänsch and Bloch [2002], Greiner, Mandel, Hänsch and Bloch [2002]) that produced a Mott-insulator transition (Jaksch, Bruder, Cirac, Gardiner and Zoller [1998], van Oosten, van der Straten and Stoof [2001]) in ultracold 87 Rb confined in an optical lattice. From the point of view of the analogy between de Broglie optics and quantum optics, though, optical lattices can simply be thought of as almost ideal matter-wave resonators, with the additional advantage that optical lattices automatically provide us with arrays of cavities whose coupling, provided by quantum tunneling, can readily be tuned by varying the lattice depth. Photoassociation in these tight optical lattices offers distinct advantages over the quasi-homogeneous systems discussed so far. The low occupation numbers per site minimize the losses due to inelastic collisions between atoms and molecules in vibrational excited states. At the same time, the restriction of the atomic and molecular center-of-mass wave functions to the ground state of each lattice site allows one to avoid the previously mentioned difficulties with molecules dissociating into atoms in the excited states of the potential. Considering specifically the photoassociation of fermionic atoms, as in the preceding section, Search, Zhang and Meystre [2003] have shown that this problem reduces mathematically to a situation almost identical to that of the micromaser (Meschede, Walther and Müller [1985], Filipowicz, Javanainen and Meystre [1986]). The key to understanding the analogy between the problem at hand and micromasers is the close analogy between the theoretical descriptions of fermionic photoassociation and the Jaynes–Cummings model of quantum optics. In the absence of intersite tunneling, we need only consider the Hamiltonian Hˆ = Hˆ 0 + Vˆ at a single lattice site, 1 Hˆ 0 = (h¯ ωb + ε)nˆ b + h¯ ωf (nˆ 1 + nˆ 2 ) + hU ¯ b nˆ b (nˆ b − 1) 2 + h¯ Ux nˆ b (nˆ 1 + nˆ 2 ) + h¯ Uf nˆ 1 nˆ 2 ,

(9.9)

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and Vˆ = h¯ χ  (t)b† c1 c2 + H.c.

(9.10)

Here, b† is the bosonic creation operator for molecules in the Wannier state of the lowest Bloch band of the lattice centered at the lattice site under consideration, φ(r). Similarly, cσ is the fermionic annihilation operator for the state ψ(r)|σ , where |σ = 1, 2 denotes the hyperfine spin. The corresponding number operators nˆ b = b† b and nˆ σ = cσ† cσ have eigenvalues nb and nσ . The terms proportional to Ub , Ux , and Uf in Hˆ 0 describe the on-site two-body interactions between molecules, atoms and molecules, and atoms, respectively. The interaction Hamiltonian Vˆ describes the conversion of atoms into ground-state molecules via two-photon stimulated Raman photoassociation: χ  (t) being given by eq. (9.7) with χ given by eq. (4.33) and ε given by eq. (4.32). The motivation for using photoassociation instead of a Feshbach resonance is that the coupling χ  (t) can be turned on and off at will using the photoassociation lasers. The formal analogy between this problem and the Jaynes–Cummings model of the interaction of a two-level atom with a single-mode quantized electromagnetic field is seen by introducing the mapping (Anderson [1958]), σ − = c1 c2 , σ+ = σ−† = c2† c1† , σz =

c1† c1

+ c2† c2

(9.11) − 1,

where σ+ = |eg| and σ− = |ge| are the familiar raising and lowering operators for a fictitious two-state system, and σz = |ee|−|gg| is the population difference between its upper and lower states. Here, |e = c2† c1† |0 and |g = |0. We note that this mapping only holds if c1 and c2 are fermionic operators, hence, our subsequent discussion does not hold for bosonic atoms. This mapping allows us to re-express Hˆ exactly as

† ∗ Hˆ = h(ω ¯ b + Ux )nˆ b + h¯ (ωf + Ux nˆ b )σz + h¯ χ(t)b σ− + χ (t)bσ+ h¯ + Ub nˆ b (nˆ b − 1), (9.12) 2 where we have dropped constant terms and made the redefinitions ωb +ε/h¯ → ωb and ωf + Uf /2 → ωf . In the absence of two-body collisions, Ub , Ux → 0, this Hamiltonian reduces to the Jaynes–Cummings model of interaction between a quantized, single-mode electromagnetic field and a two-level atom. This model is a cornerstone of quantum optics (Scully and Zubairy [1997]). Because it is exactly solvable, it permits the understanding of detailed aspects of the dynamics of light–matter in-

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teraction. When combined with pump and dissipation mechanisms, the Jaynes– Cummings model provides an accurate description of the micromaser. The mathematical analogy between eq. (9.12) and the Jaynes–Cummings model permits one to conceive of a “molecular micromaser”, a device that builds up the molecular field one molecule at a time. As such, it opens up novel avenues to generate nonclassical states of molecular fields and to study new types of dynamical quantum phase transitions in matter-wave systems. More generally, it permits the extension of the numerous, well-established applications of cavity QED to matter-wave optics. Despite the presence of nonlinear two-body terms in eq. (9.12), the dynamics can be solved for exactly within the two-state manifold {|g, nb + 1, |e, nb } for χ  = const. Within each manifold, the resulting dynamics are quantized Rabi oscillations at the frequency   2 Rnb = (9.13) 2ωf − ωb + (2Ux − Ub )nb + 4|χ  |2 (nb + 1). In particular, if the system starts out in the state |e, nb  then the probabilities for the atom to be in the states |e, nb  and |g, nb + 1 after a time τ are    2




Cg,n +1 (τ )
2 = 1 −
Ce,n (τ )
2 = 4|χ | (nb + 1) sin2 1 Rn τ . (9.14) b b Rnb 2 b As one can see, the only effect of the two-body interactions is to shift the transition frequency by an amount proportional to the number of molecules. The dynamics of the quantum optical micromaser are governed by three mechanisms: the injection of a sequence of individual atoms from a very dilute atomic beam inside the microwave cavity, the Jaynes–Cummings interaction between these atoms and the cavity mode, and cavity dissipation, which can normally be neglected while an atom transits through the cavity, but that dominates the field dynamics during the much longer intervals between atoms. A similar situation can be achieved in the present system: all that is required is to inject a sequence of pairs of fermionic atoms inside the lattice well, turn on the photoassociation lasers for some time interval τ to introduce Jaynes–Cummings-like dynamics – where the electromagnetic field mode is replaced by the molecular field – and finally turn off these fields for a time T  τ and let dissipation take over. This sequence is then repeated to build up the molecular field. Search, Zhang and Meystre [2003] discuss in detail how the pumping of fermionic atoms into the lattice sites during the intervals T can be achieved by using two-photon Raman transition to convert atoms from states that are not trapped by the lattice lasers to hyperfine states that are trapped. At the same time that the atom pairs are being pumped into the lattice sites, the molecular field is damped due to several

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decay mechanisms that include Rayleigh scattering of photons from the lasers forming the lattice potential, inelastic vibrational decay of the molecules due to a collision with an atom, and finally three-body recombination where the atom pair collides with a molecule and the atom pair forms a molecule with enough kinetic energy to escape the lattice.

§ 10. Outlook While it is almost impossible to predict future developments in a field as rapidly developing as the physics of quantum-degenerate atomic and molecular systems and atom optics, some of the exciting developments expected to result from the availability of ultracold and quantum-degenerate molecular systems begin to appear on the horizon: So far, molecular condensation has only been demonstrated when starting from pairs of identical atoms, but two recent experiments have led to the observation of heteronuclear Feshbach resonances in Bose–Fermi mixtures, of 6 Li and 23 Na (Stan, Zwierlein, Schunck, Raupach and Ketterle [2004]) in one case, and of 87 Rb and 40 K in the other (Inouye, Goldwin, Olsen, Ticknor, Bohn and Jin [2004]). There is little doubt that it will soon be possible to exploit these resonances to create ultracold, possibly quantum-degenerate samples of heteronuclear molecules. In this context we should also mention recent experiments by Kerman, Sage, Sainis, Bergeman and DeMille, DeMille [2004, 2004], who produced metastable RbCs molecules in their lowest triplet state starting from a lasercooled mixture of 85 Rb and 133 Cs by photoassociation. Heteronuclear diatomic alkali molecules possess a permanent electric dipole moment of the order of one ea0 . For instance, Kotochigova, Julienne and Tiesinga [2003] found theoretically the permanent electric dipole moment of KRb to be equal to 0.30(2)ea0 . Hence these molecules can be manipulated relatively easily by modest static or quasi-static inhomogeneous electric fields. There have been a number of recent experiments along these lines using various polar molecules (Crompvoets, Jongma, Bethlem, van Roij and Meijer [2002], Tarbutt, Bethlem, Hudson, Ryabov, Ryzhov, Sauer, Meijer and Hinds [2004], Crompvoets, Bethlem, Küpper, van Roij and Meijer [2004], Junglen, Rieger, Rangwala, Pinkse and Rempe [2004]). In experiments carried out with relatively hot molecules, Meijer and collaborators used a combination of electrostatic lenses, such as, e.g., hexapole lenses, and alternating-gradient lenses to produce bunches of slow molecules with a longitudinal temperature of the order of a few milliKelvins and decelerated from their initial velocity of the order of 300 m/s to about 90 m/s. By bending a hexapole into a torus, a storage ring for molecules can be created

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(Katz [1997]). In recent work, Crompvoets, Bethlem, Küpper, van Roij and Meijer [2004] succeeded in injecting molecular bunches into a storage ring with a diameter of the order of 25 cm, and managed to observe about 50 distinct round trips of the bunches in the ring. These experimental demonstrations of focusing, slowing, guiding, storage, and phase-space manipulation of dipolar molecules, combined with the generation of large samples of ultracold heteronuclear diatomic molecules, open up exciting new avenues of fundamental and applied research. On the applied side, major advances in rotation sensors will be achieved if one can build an atom interferometer based on a molecular storage ring with a diameter of tens of centimeters: The Sagnac phase shift for an atom-interferometric gyroscope is   2M ΩA, ΦSagnac = (10.1) h¯ where M is the particle mass, Ω is the rotation rate of the interferometer, and A is its effective area. Current atom interferometers, which involve only a single round trip of the atoms, typically have an area of the order of A 10−3 –10−4 m2 . In contrast, a storage-ring design would have an effective area of the order of 50 × 0.05 m2 = 2.5 m2 , where 50 is an estimate of the number of round trips of the molecules based on the experimental results of Crompvoets, Bethlem, Küpper, van Roij and Meijer [2004]. Accounting also for the increased mass of the molecules, this simple estimate indicates a potential gain in sensitivity of about 4 orders of magnitude. This alone is a major motivation to initiate a study of matter-wave optics with polar molecules. Heteronuclear molecules interact via the electric dipole–dipole potential. This is expected to be both a difficulty and an opportunity. The difficulty is that in interferometric applications, this interaction may lead to the appearance of random density-dependent phase shifts that will act as a source of noise and that will need to be controlled. The opportunity is that the dipole–dipole interaction is expected to lead to fascinating new physics. In particular, the anisotropy of the dipole–dipole interaction should permit us to understand in which way the nature and stability of quantum-degenerate atomic and molecular systems is influenced by the character of interparticle interactions (Santos, Shlyapnikov, Zoller and Lewenstein [2000]). In a less extreme regime, Bochinsky, Hudson, Lewandowski, Meijer and Ye [2003] have demonstrated the phase-space manipulation of dipolar molecules in the presence of dipole–dipole interaction. Under the influence of this interaction, polar molecules modify each other’s trajectory and orientation, leading to complex dynamics as well as intriguing potential applications. For instance, DeMille

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[2002] has proposed to take advantage of the dipole–dipole interaction between heteronuclear diatomic molecules for the realization of a quantum computer. These are just a few examples of the developments likely to be witnessed by atom optics in the near future. If the recent past is any indication of things to come, though, we can expect surprises and developments that will keep this an exciting field for many years to come.

Acknowledgements This work is supported in part by the US Office of Naval Research under Contract No. 14-91-J1205, by the National Science Foundation under Grant No. PHY9801099, by the NASA Microgravity Program Grant NAG8-1775, by the US Army Research Office, and by the Joint Services Optics Program.

Appendix A: Feshbach resonances Here we wish to present the general theory of a Feshbach resonance in order to derive the modification of the collision phase shift due to virtual transitions to the bound state. We follow here the derivation given by Timmermans, Tommasini, Hussein and Kerman [1999]. Let |Ψ  denote the state vector with energy E that includes both open- and closed-channel components so that the time-independent Schrödinger equation is (E − H )|Ψ  = 0.

(A.1)

We define the projection operators P and M that project |Ψ  onto the subspace of open-channel and closed-channel states, respectively, and satisfy P 2 = P , M 2 = M, P M = 0, and P + M = 1. Applying the projection operators to eq. (A.1) yields (E − HP P )|ΨP  = HP M |ΨM ,

(A.2)

(E − HMM )|ΨM  = HMP |ΨP ,

(A.3) † HMP

= P H M. For ulwhere HP P = P H P , HMM = MH M, and HP M = tracold alkali atoms, the coupling between the open and closed channels is given by eq. (4.24), HP M = HMP = H− while the Hamiltonians for the open and closed channels result from projecting H+ given by eq. (4.23) onto the collision channels, |ψj . The open- and closed-channel wave functions are |ΨP  = P |Ψ , respectively.

|ΨM  = M|Ψ ,

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[3, § 10

Equation (A.2) can be formally solved for the outgoing scattered wave function, (+) |ΨP  in terms of |ΨM ,
(+) 
(+) 
Ψ (A.4) =
ψP + G+ (E)HP M |ΨM , P (+)

where |ψP  is the solution of (E − HP P )|ψP  = 0, which represents the scattered wave function in the absence of the coupling to the bound state, 
(+)  r
ψP = −2ikeiδ0 u(r), where u(r) is the regular solution of the radial Schrödinger equation whose s-wave component for r → ∞ is sin(kr + δ0 )/kr and δ0 is the unperturbed collisional phase shift. G+ (E) = 1/(E − HP P + iε) is the propagator for outgoing waves, lim r|G+ (E)|r  =

r→∞

2µ eikr  (+)

  r . ψ h¯ 2 8ikπr P

(A.5)

(+)

By now using eq. (A.4) for |ΨP  in eq. (A.3) to formally solve for |ΨM , |ΨM  =

E − HMM


(+)  1 HMP
ψP , − HMP G+ (E)HP M

(A.6)

and then substituting this back into eq. (A.4), one finally obtains
(+) 
(+) 
Ψ =
ψ + G+ (E)HP M P

P


(+)  × HMP
ψP .

E − HMM

1 − HMP G+ (E)HP M (A.7)

In the observed Feshbach resonances, the energy separation between the resonances is much larger than the width of the resonances, so that in general only a single bound state |φm  has energy close to E, resulting in a small value for the denominator in eq. (A.7). In this case we can approximate the subspace of closedchannel states by |φm  alone, M ≈ |φm φm |. By using the orthogonality of the (+) P and M subspaces we obtain for |ΨP :
(+) 
(+) 
Ψ =
ψ + G+ (E)HP M |φm  P

P


(+)  × φm |HMP
ψP ,

1 E − Em − ∆m + iΓm /2 (A.8)

where Em = φm |HMM |φm  is the unperturbed energy of the bound state, ∆m (E) = Re[φm |HMP G+ (E)HP M |φm ] is the energy shift of the bound state due to coupling to the open-channel continuum, and Γm (E) = 2 Im[φm |HMP × G+ (E)HP M |φm ] is the linewidth due to decay of the bound state into the

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209

open channel. An explicit evaluation of the linewidth gives (Timmermans, Tommasini, Hussein and Kerman [1999])  3/2 √ 2µ 2 1 Γm (E) = |χ| (A.9) E = 2γ k, 2π h¯ 2 where χ is given by  χ = d3 r φm |rHMP (r)u(r)

(A.10)

and for ultracold atoms corresponds to eq. (4.26). We note in passing that the √ E dependence of the decay rate Γm (E) has recently been observed experimentally in the dissociation of molecules whose energies had been ramped from E < 0 to E > 0 (Mukaiyama, Abo-Shaeer, Xu, Chin and Ketterle [2004], Dürr, Volz, Marte and Rempe [2004]). Using eq. (A.5) to evaluate r|G+ (E)HP M |φm ,   (+)
 2µ eikr lim r|G+ (E)HP M |φm  = 2 d3 r  ψP
r HP M (r )r |φm  r→∞ h¯ 8ikπr   2µ eikr iδ0 χe =− (A.11) 4π h¯ 2 r (+)

together with φm |HMP |ψP  = −2ik exp(iδ0 )χ, we finally obtain
(+)  lim r|G+ (E)HP M |φm φm |HMP
ψP   ikr 2µ eikr 2iδ0 2 e 2iδ0 |χ| = ie . = ie k Γ (E) m r r 2π h¯ 2

r→∞

(A.12)

The scattered wave function is then given explicitly by   iΓm (E) eikr e−ikr − 1− e2iδ0 . lim r|ΨP  ∝ r→∞ r E − Em − ∆m (E) + iΓm (E)/2 r (A.13) The term in parentheses can be written as exp(2iδR ), lim r|ΨP  ∝

r→∞

where −1

δR = − tan

eikr e−ikr − e2iδR e2iδ0 , r r



Γm (E)/2 E − Em − ∆m

(A.14)

 (A.15)

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[3

and the total scattering phase shift is δ = δ0 + δR . The scattering length is given by the usual relationship a = lim

k→0

δ γ = a0 − , k EM − E

(A.16)

where a0 is the background scattering length and we have assumed that |E − Em |  γ . We have also absorbed the energy shift into a redefinition of Em , Em + ∆m → Em .

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E. Wolf, Progress in Optics 47 © 2005 Elsevier B.V. All rights reserved

Chapter 4

Space-variant polarization manipulation by

Erez Hasman, Gabriel Biener, Avi Niv, Vladimir Kleiner Optical Engineering Laboratory, Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)47004-3 215

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 2. Formation of space-variant polarization-state manipulations . . . . .

222

§ 3. Geometrical phase in space-variant polarization-state manipulation .

239

§ 4. Applications of space-variant polarization manipulation . . . . . . . .

260

§ 5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

284

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

285

216

§ 1. Introduction One of the important properties of an optical wavefront is the transverse field distribution of the beam, that is, the distribution of its amplitude and phase as well as its polarization state. This field distribution determines the propagation behavior of the beam, and its angular momentum. The ability to generate an arbitrary complex scalar optical wavefront accurately is essential in modern optical applications. Although the ability to generate scalar beams is useful, in an increasing number of cases it is desirable to create arbitrary vectorial beams. In general, vectorial beams are defined as beams having space-variant (transversely nonuniform) polarization state. Polarization is a fundamental property of electromagnetic fields. Accordingly, the state of polarization of light has substantial influence in most optical experiments and in the theoretical models developed to interpret them. In optics the polarization state of a light field can significantly affect the propagation of many fully or partially coherent paraxial light fields. However, its influence can never be ignored in nonparaxial conditions when the field propagates in free space. Significant polarization effects can occur during interaction with material interfaces such as gratings, or arrangements of nanoparticles. Comprehensive background information on polarization optics can be found in textbooks (see, for example, Collett [2003] and Brosseau [1998]). Recent years have witnessed a growing interest, theoretically as well as experimentally, in space-variant polarization-state manipulation that can be exploited in a variety of applications. These include polarization encoding of data (Javidi and Nomura [2000]), neural networks and optical computing (Davidson, Friesem and Hasman [1992a]), optical encryption (Mogensen and Glückstad [2000]), tight focusing (Quabis, Dorn, Eberler, Glöckl and Leuchs [2000]), imaging polarimetry (Nordin, Meier, Deguzman and Jones [1999]), material processing (Niziev and Nesterov [1999]), and atom trapping and optical tweezers (Liu, Cline and He [1999]). The study of polarization manipulation has grown into a new branch of modern physical optics known as polarization singularities (see, for example, Nye [1999] and Soskin and Vasnetsov [2001]). In a scalar field, such singularities ap217

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pear at points or lines where the phase or the amplitude of the wave is undefined or changes abruptly. One class of such dislocations is formed by vortices, which are spiral phase ramps  about a singularity. Vortices are characterized by a topological charge l = 12 π ∇ϕ ds, where ϕ is the phase of the beam and l is an integer. Until recently, research had focused mainly on field dislocations in scalar waves. However, if we allow for the polarization to be space-varying, disclinations can arise (see, for example, Nye [1983], Dennis [2002], and Freund, Mokhun, Soskin, Angelsky and Mokhun [2002]). Disclinations are points or lines of singularity in the pattern or direction of a transverse field. An example is the center of a beam with radial or azimuthal polarization. Different techniques for obtaining space-variant polarization manipulation by use of nonuniform anisotropic polarization elements have been reported in the literature. In general, a polarization optical element is any optical element that can modify the state of polarization of a light beam, such as a polarizer, retarder, rotator, or depolarizer. Space-variant polarization elements can be implemented as space-variant computer-generated sub-wavelength dielectric or metal gratings (Hasman, Bomzon, Niv, Biener and Kleiner [2002]), polarization-sensitive materials such as azobenzene-containing materials (Todorov, Nikolova and Tomova [1984]), and liquid-crystal devices (Davis, McNamara, Cottrell and Sonehara [2000]). For the most general case, the transmission, retardation and optical-axis orientation of such elements depend on the location across the face of the element. In order to analyze the beam emerging from a space-variant polarization element, we must resort to Jones and Mueller polarization-transfer matrix methods. The Jones calculus assumes completely polarized light and coherent addition of waves, whereas the Mueller calculus assumes partially polarized incoherent addition of waves. The space-dependent transfer matrices of Jones and Mueller can be calculated by expressing the local behavior of the element as a polarizer and retarder, while the local orientation of the optical axis can be obtained by applying the rotation matrix (see, for example, Collett [2003]). Moreover, Gori, Santarsiero, Vicalvi, Borghi and Guattari [1998] introduced an important method for analyzing partially coherent sources with space-varying partial polarization utilizing the beam coherence-polarization matrix (BCP). This approach can be viewed as an approximate form of Wolf’s general tensorial theory of coherence (Wolf [1954]). Sometimes the local transmission and retardation of the polarization element cannot be determined in a straightforward manner, for instance when using gratings for which the period is close to or smaller than the incident beam’s wavelength. In this case, a direct solution of Maxwell’s equations is required. This can usually be accomplished by using numerical approaches, such as rigorous coupled wave analysis (RCWA; see Moharam and Gaylord [1986] and Lalanne and

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Morris [1996]), or utilizing finite-difference time-domain methods (FDTD) for analyzing nonuniform polarization optics (see, for example, Mirotznik, Prather, Mait, Beck, Shi and Gao [2000] and Jiang and Nordin [2000]). Complex vectorial fields can be produced either by utilizing polarization elements (see, for example, Bomzon, Biener, Kleiner and Hasman [2002b]) or by using interferometric techniques involving two orthogonally polarized beams (see, for example, Tidwell, Ford and Kimura [1990]). A coherent summation, inside the laser resonator, of two orthogonally polarized TEM01 modes was demonstrated by Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000]. Several designing approaches for obtaining vectorial fields having space-varying polarization distribution have been presented (see, for example, Niv, Biener, Kleiner and Hasman [2004] and Tervo, Kettunen, Honkanen and Turunen [2003]). In a remarkable paper first published in 1956, Pancharatnam [1956] (reprinted in Pancharatnam [1975]) considered the phase of a beam of light whose polarization state is modified. Pancharatnam showed that a cyclic change in the state of polarization of the light is accompanied by a phase shift determined by the geometry of the cycle as represented on the Poincaré sphere (Brosseau [1998]). Therefore, space-variant polarization-state manipulations are accompanied by a phase modification that results from the Pancharatnam–Berry phase (Berry [1987], Bomzon, Kleiner and Hasman [2001d]). In order to investigate the propagation behavior of complex vectorial fields as well as the angular momentum (Allen, Padgett and Babiker [1999]), it is necessary to consider the resulting geometrical phase distribution. The calculation of the space-variant Pancharatnam phase is based on the rule proposed by Pancharatnam [1956] for comparing the phases of two light beams in different states of polarization as the argument of the vectorial projection between the two polarization states. The propagation of paraxial vector fields has been extensively studied theoretically. Several vectorial treatments have been presented in both coherent (see, for example, Gori [2001]) and partially coherent light fields (see, for example, James [1994] and Seshadri [1999]). The simplest approach to studying arbitrarily polarized beams is to decompose the representative field vector at any point of a section into orthogonal linearly or circularly polarized parts. Free-propagation problems can then be performed as the analysis of the propagation of a pair of scalar waves (Goodman [1996]). On the other hand, for specific beams with axial-symmetric polarization distribution, significant results have been obtained by representing the field at a typical point as a superposition of radial and azimuthal components. Jordan and Hall [1994] showed that the propagation process of an azimuthally polarized Bessel–Gauss beam can be analyzed by means of a single one-dimensional propagation integral with a suitable kernel. A general vectorial decomposition

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of electromagnetic fields with application to propagation-invariant and rotating fields was presented by Pääkkönen, Tervo, Vahimaa, Turunen and Gori [2002]. Vectorial Talbot self-imaging and vectorial nondiffracting beams have been investigated for a wave field with periodic variations of the polarization state (e.g., Mishra [1991]), and have been demonstrated experimentally (Arrizón, Tepichin, Ortiz-Gutierrez and Lohmann [1996]). Moreover, Gori, Santarsiero, Borghi and Piquero [2000] introduced an important method for analyzing the propagation of partially coherent beams with space-varying partial polarization by extension of the van Cittert–Zernike theorem using the beam coherence polarization matrix (see, for example, Mandel and Wolf [1995]). In the nonparaxial case, e.g., the propagation of the beam emerging from a lens with high numerical aperture (NA), one must resort to a vectorial formulation that takes into account polarization effects and nonuniformity of the amplitude over the wavefront. A mathematically tractable representation for dealing with polarization was developed by Debye [1909], and a representation for handling apodization was addressed by Hopkins [1943]. These developments were later generalized by Wolf [1959], applied to the analysis of aplanatic refractive lenses (free of spherical aberrations), and then exploited in investigations of the focal distribution in a variety of focusing systems (see, for example, Richards and Wolf [1959], Barakat [1987] and Sheppard and Wilson [1982]). Moreover, in a series of seminal papers Quabis and colleagues (Quabis, Dorn, Eberler, Glöckl and Leuchs [2001] and Dorn, Quabis and Leuchs [2003]) demonstrated both theoretically and experimentally the effectiveness of radially polarized doughnut beams focused by a high-NA lens in achieving significantly tighter focusing in far-field optics than had been possible with linearly polarized beams. The vectorial analysis of such propagation problems can be performed by using the generalized Debye integral, expressed for radially symmetric illumination (see, for example, Davidson and Bokor [2004]). The optical properties of the vectorial beam can be evaluated by measuring the polarization distribution of the waves as well as their amplitude and phase distribution. The polarization state of the beam can conveniently be described geometrically by the polarization ellipse. In this case, a specific polarization state is characterized by the ellipticity and azimuthal angle of the major axis of an ellipse with respect to some reference frame (see, for example, Chapter 3.1 of Brosseau [1998]). The ellipticity and azimuthal angle can be calculated from the Stokes polarization parameters of the beam. There are four Stokes parameters, and these can be used to determine an intensity formulation of a beam’s polarization state (Stokes [1852]). Therefore, they are directly accessible as linear combinations of the intensities measured by transmitting the beam through four different combina-

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tions of optical elements referred to as wave plates and polarizers (see Chapter 5 of Collett [2003]). Henceforth, we will refer to this method as the four-measurements technique. Another useful method is to measure the time-dependent signal after the beam has been transmitted through a rotating optical component (referred to as a quarter-wave plate) and then through a polarizer (see Chapter 13 of Collett [2003]). In this case, the Stokes parameters are derived by Fourier analysis of the detected signal. Many versions of this method have been devised; Jellison [1987], for example, proposed the use of a photoelastic modulator instead of the rotating quarter-wave plate. It is also desirable to be able to measure the transmission matrices of an optical element for either Jones or Mueller formalism. Many optical schemes have been proposed for this purpose. See Jones [1948], Raab [1982] and Brosseau [1985] for examples of Jones matrix measurement, and Lu and Chipman [1998] and Anderson and Barakat [1994] for demonstrations of Mueller matrix measurement. All these methods are based on a set of experiments in which known polarization states are fed into the optical system and the corresponding polarization states of the output beam are then measured. In this chapter, theoretical analysis as well as experimental methods for obtaining space-variant polarization-state manipulation are reviewed along with several related applications. Various vectorial fields having space-variant polarization distributions are discussed in detail, together with the data from experimental studies. The structure of this review is as follows: in Section 2 we review various methods for designing and realizing space-variant polarization-state manipulations. The use of sub-wavelength gratings, polarization interference methods and liquidcrystal devices for this purpose are considered. We also briefly describe the use of polarization-sensitive recording materials and discuss some general design approaches for space-variant polarization optics. In Section 3 we consider optical phase elements based on the space domain Pancharatnam–Berry phase. Unlike with diffractive and refractive elements, the phase is not introduced through optical path differences, but results from the geometrical phase that accompanies space-variant polarization manipulation. Optical elements that use this effect to form a desired phase front are called Pancharatnam–Berry phase optical elements (PBOEs). The ability of PBOEs to generate complex wavefronts is demonstrated by forming helical wavefronts and polarization-sensitive beam splitters as well as polarization-dependent focusing lenses. The effect of the Pancharatnam–Berry phase on the propagation of vectorial beams is investigated in this study through the formation of either one- or two-dimensional vectorial propagation-invariant beams and vectorial Talbot effect.

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Section 4 elaborates on selected applications of space-variant polarization manipulation. Among the topics discussed are: near and far-field polarimetry, light depolarization, polarization encryption, optical computing, and spatial control over polarization-dependent emissivity. Finally, Section 5 presents some concluding remarks. § 2. Formation of space-variant polarization-state manipulations In this section we review the basic formation methods enabling the design and realization of space-variant polarization-state manipulations. We begin by describing the use of sub-wavelength gratings as space-variant polarizers and wave plates. A design method for performing two-dimensional polarization elements utilizing space-variant sub-wavelength gratings is presented. The method is based on determining the local direction and period of the sub-wavelength metal or dielectric gratings to obtain any desired continuous polarization change. As an example, we introduce the formation of linearly polarized axially symmetric beams with various polarization order numbers. We proceed by describing the spacevariant vectorial fields that are obtained by the polarization interference method. In this case vectorial fields are formed by the superposition of orthogonally polarized transverse modes. The superposition is carried out either externally, i.e., by using an interferometer, or within a laser cavity. Next, we proceed by describing space-variant polarization-state manipulations that are obtained using liquidcrystal devices. In this case the liquid-crystal device acts as a space-variant wave plate. Finally, we briefly describe the use of polarization-sensitive recording materials and some general design approaches for space-variant polarization-state manipulations.

2.1. Space-variant polarization-state manipulation by use of sub-wavelength gratings Sub-wavelength gratings have opened up new methods for forming beams with sophisticated phase and polarization distributions. Such gratings are usually used to form homogeneous space-invariant polarizers or wave plates. When the period of the grating is much smaller than the incident wavelength, only the zeroth order is propagating, and all other orders are evanescent. These gratings behave as layers of a uniaxial crystal. Therefore, the use of space-variant (transversely inhomogeneous) sub-wavelength gratings permits the generation of complex vectorial wavefronts with a different polarization state at each location.

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2.1.1. Background The term “sub-wavelength gratings” refers to optical elements comprising typical structures that are smaller than the wavelength for which the elements were designed. Sub-wavelength gratings are usually one- or two-dimensional periodic structures, such as the one-dimensional grating depicted in fig. 1. When light is incident upon a sub-wavelength grating, all diffraction orders become evanescent. The only propagating intensity is due to the zeroth order, and the grating behaves as a uniaxial (or biaxial) crystal with its optical axes parallel and perpendicular to the grating stripes. The threshold period below which only the zeroth order is propagating is given by Λth =

λ n1 sin ζ + (n22 − n21 sin2 θ )1/2

.

(2.1)

Here, θ is the azimuth relative to the grating stripes, ζ is the angle of incidence, n1 and n2 are the refractive indices of the grating stripes, and λ is the wavelength of the incident light. Sub-wavelength gratings can be either spaceinvariant or space-variant, and have been used for fabricating anti-reflection coatings (Grann, Moharam and Pommet [1994]), artificial refractive index distribution (Mait, Prather and Mirotznik [1999]), polarization-selective computer-generated holograms (Xu, Tyan, Sun, Fainman, Cheng and Scherer [1996]), optical filters

Fig. 1. Illustration of a one-dimensional periodic binary sub-wavelength grating. The grating period Λ comprises alternating stripes with refractive indexes n1 and n2 and widths t1 and t2 , respectively. q = t1 /Λ is defined as the duty cycle, ζ is the incidence angle and θ is the azimuthal angle. Note that the period is smaller than the incident wavelength, λ.

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(Puscasu, Spencer and Boreman [2000]), wave plates (Brundrett, Glytsis and Gaylord [1996]) and polarizers (Bird and Parrish [1960]). Their use can be dated back to 1888 when Heinrich Hertz used sub-wavelength metal stripe gratings as radiowave polarizers (Hertz [1893]). Gratings play an important role in optics. Their use has helped lay the foundations of such fields as spectroscopy and diffractive optics. Loosely speaking, when the variations of the surface relief or index modulation are slow compared to the wavelength, λ, the polarization of the incident wave can be neglected and approximate scalar theories can be used (Goodman [1996], Hasman, Davidson and Friesem [1991]). However, as the period of the grating decreases and becomes comparable or smaller than the wavelength, vectorial effects become more dominant, and rigorous electromagnetic theories are needed. Unfortunately, few rigorous analytical solutions are known, and the calculation of diffraction from such gratings generally requires numerical methods (Challener [1996], Guizal and Felbacq [1999]). The most commonly used method for these calculations is rigorous coupled wave analysis (RCWA), which was formulated by Moharam and Gaylord [1986]. An unfortunate drawback of RCWA is that it converges slowly for metallic lamellar gratings. This problem was addressed by Lalanne and Morris [1996] who reformulated the eigenvalues problem to achieve highly improved convergence rates, thereby extending the usefulness of RCWA-related methods. Another common method is the finite-difference time-domain method (FDTD) (see, for example, Mirotznik, Prather, Mait, Beck, Shi and Gao [2000] and Jiang and Nordin [2000]). However, despite the success of RCWA and other numerical approaches, they tend to be calculation-intensive and offer very little intuitive insight into sub-wavelength grating problems. For this reason, approximate methods are often sought. The simplest approximate model for sub-wavelength gratings is the classical form birefringence (see Born and Wolf [1999]). This zero-order approximation gives the effective refractive indices of a binary sub-wavelength grating with the geometry depicted in fig. 1 as n2TE = qn21 + (1 − q)n22 , n2TM =

n21 n22 qn22 + (1 − q)n21

(2.2) ,

(2.3)

where the subscripts TE and TM denote light that has been polarized parallel and perpendicular to the grating stripes, respectively; n1 and n2 denote the refractive indices of the materials that comprise the grating, and q = t1 /Λ is the duty cycle of the grating, i.e. the relative portion of the material with refractive index n1 within the grating. Thus, the grating is replaced by an effective layer consisting

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of a uniaxial crystal. If the grating period is not binary, then it is approximated with a step function, and the effective refractive indices for each step are calculated using eqs. (2.2) and (2.3). The structure is then replaced with a multilayer stack whose properties can be calculated using transfer-matrix methods (Macleod [1989], Yeh [1979]). Some very important results regarding sub-wavelength gratings were presented by Rytov [1956]. He showed that the effective refractive indices for a subwavelength grating could be found from a pair of transcendental equations,  

1/2

1/2 2 λ tan π (1 − q) n21 − n2TE n1 − n2TE Λ   2

1/2 λ 1/2 , = − n2 − n2TE (2.4) tan π q n22 − n2TE Λ   2

1/2 λ 2 (1 − q) n tan π − n TM 1 Λ n21  

1/2 −(n22 − n2TM )1/2 λ 2 2 , q n = tan π − n TM 2 Λ n22

(n21 − n2TM )1/2

(2.5)

where λ is the incident wavelength and Λ is the grating period. Developing these equations into a Taylor series yields second-order approximations,

 2 2

1/2  (0) 2 1 πΛ (2) 2 2 nTE = nTE + (2.6) q(1 − q) n2 − n1 , 3 λ 

 2   (0) 2 1 πΛ 1 1 2  (0) 6  (0) 2 1/2 (2) q(1 − q) n n − , nTM = nTM + TM TE 3 λ n22 n21 (2.7) (0)

(0)

where nTM and nTE are the zero-order solutions of eqs. (2.4) and (2.5) provided by eqs. (2.2) and (2.3). Further research into form birefringence was conducted by Bouchitte and Petit [1985] using homogenization techniques. They rigorously proved that any refractive index distribution can be replaced by a stratified layer as long as the period of the grating tends to zero. The main difficulty in realizing sub-wavelength structures is their small feature size which requires the use of advanced and often creative photolithographic techniques. Roughly speaking, there are three methods for realizing these elements: indirect writing (see Bowden [1994]), direct writing (see Warren, Smith, Vawter and Wendt [1995]) and interference writing (see Enger and Case [1983]). In the indirect writing process, a mask of the element is initially made. This mask is usually a glass substrate onto which the relief pattern of the element has been

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placed. This is usually done by first coating the glass with a metal and then coating the metal with a photoresist. The pattern is developed onto the photoresist using either a laser or an electron beam, and then etched, leaving the desired pattern on the mask. The element can then be realized by imaging the mask onto the photoresist-coated substrate. The photoresist on the substrate is then developed and a copy of the element can be made. If the pattern is not binary, then it is necessary to make separate masks for the different layers. The main advantage of this method is that it enables relatively cheap reproduction of a single element. This technique was used by Deguzman and Nordin [2001] to fabricate a circular polarizer for the mid-infrared regime. The second method used is direct writing. In this technique, instead of making a mask, a laser (usually UV) or electron beam is used to imprint the pattern directly onto the substrate. The substrate is first coated with photoresist, and the pattern is written directly onto this coated substrate. The pattern is then developed, and a single copy of the element is generated. An advantage of this method is that it offers high resolution, especially when electron beams are used. For this reason, it is widely used in academic research of sub-wavelength gratings for the visible region (Lopez and Craighead [1998], Warren, Smith, Vawter and Wendt [1995]). The downside is that the production time is long and the fabrication is very expensive. Therefore, this method is not widely used for commercial production. Interference recording is the most commonly used method for the fabrication of homogeneous one-dimensional gratings (Brundrett, Gaylord and Glytsis [1998], Nordin, Meier, Deguzman and Jones [1999]). The sub-wavelength lines are produced by the interference of ultraviolet or blue laser light, leading to periods of around 200 nm. This technique is very useful in the formation of space-invariant gratings, and simple space-variant structures can be achieved by incorporating simple computer-originated phase masks into the interferometer. However, its use in forming intricate space-variant sub-wavelength gratings is rather limited. There are also differences in the fabrication of metal and dielectric subwavelength gratings. Metal sub-wavelength gratings are usually fabricated using a lift-off process (Doumuki and Tamada [1997]). After the pattern has been transferred to the photoresist-coated substrate by either direct or indirect writing, the photoresist is developed. The substrate is then coated with metal, and the photoresist removed. In this manner, the metal remains only in the areas that were clear of photoresist after development. This technique is very useful in the realization of thin metal stripes with sub-micron features. Since the metal stripes in a subwavelength grating need not be much thicker than the skin depth, this technique is well suited for the realization of space-variant metal-stripe sub-wavelength grat-

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ings for the IR and visible spectra. On the other hand, dielectric gratings are realized using etching techniques. Since the features of the sub-wavelength gratings are very small, and since the depth-feature size aspect ratio is usually large, it is important to choose a technique that has a large degree of anisotropy. Dry etching is usually more suitable than wet etching. In particular, reactive ion etching is especially useful (Lopez and Craighead [1998], Deguzman and Nordin [2001], and Nordin, Meier, Deguzman and Jones [1999]). After the photoresist on the substrate is developed, the element is placed in a vacuum chamber and subjected to a bombardment of a plasma mixture. The characteristics of the plasma are determined by the choice of etching technique. The areas on the substrate that were coated by photoresist remain untouched, whereas the areas that were exposed to the plasma are etched away. In this way a relief pattern is achieved on the substrate and a grating is realized. Bomzon, Kleiner and Hasman [2001a] developed a novel method for designing and realizing nonuniformly polarized beams using computer-generated spacevariant sub-wavelength gratings. Their design is based on determining the local period and direction of the grating at each point, forming space-varying polarizers or wave plates that convert uniformly polarized light into any desired spacevariant polarization. Their gratings are continuous, thereby guaranteeing the continuity of the electromagnetic field. 2.1.2. Space-variant polarization-state manipulation by use of sub-wavelength metal gratings Sub-wavelength metal stripe gratings are usually used as homogeneous spacevariant polarizers (see Glytsis and Gaylord [1992], Honkanen, Kettunen, Kuittinen, Lautanen, Turunen, Schnabel and Wyrowski [1999], Schnabel, Kley and Wyrowski [1999] or Astilean, Lalanne and Palamaru [2000]). Sometimes, however, a different polarization state is required at each location. Bomzon, Kleiner and Hasman [2001b, 2001c, 2001d] demonstrated an innovative method for designing, analyzing and realizing computer-generated space-variant metal-stripe polarization elements. This method is based on determining the local direction and period of a sub-wavelength metal-stripe grating using vectorial optics to obtain any desired continuous polarization change, hence, completely suppressing any diffraction arising from polarization discontinuity. Analysis of the element can then be performed using an original method combining RCWA and Jones calculus, in which the element is represented as a space-varying Jones matrix, which is defined by the local period and orientation of the grating.

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Gratings are typically defined by a grating vector that is perpendicular to the grating stripes. A space-varying grating can therefore be described by the vector



Kg (x, y) = K0 (x, y) cos θ (x, y) xˆ + K0 (x, y) sin θ (x, y) yˆ ,

(2.8)

where K0 (x, y) is the spatial frequency of the grating, θ is the direction of the vector, and xˆ , yˆ are the unit vectors along the x-axis and the y-axis, respectively. In order for such a grating to be physically realizable in a continuous way, Kg should be a conserving vector, i.e., ∇ × Kg = 0, or more explicitly,     ∂K0 ∂θ ∂θ ∂K0 cos(θ ) − K0 sin(θ ) = sin(θ ) + K0 cos(θ ) . ∂y ∂y ∂x ∂x

(2.9)

This necessary restraint is placed on K0 (x, y) to enable a continuous grating with a local groove direction θ (x, y) to exist. Once the grating vector is determined, the grating function φg (x, y) can be found by integrating Kg along any arbitrary path in the x–y plane so that ∇φg = Kg . A Lee-type (Lee [1974]) binary subwavelength structure mask described by the grating function φg (x, y) can be realized using high-resolution laser lithography. The amplitude transmission for such a Lee-type binary mask can be derived as   t (x, y) = Us cos(φg ) − cos(πq) , where Us is the unit step function, defined by 1, η
0, Us (η) = 0, η < 0,

(2.10)

(2.11)

and q is the duty cycle of the grating. Bomzon, Kleiner and Hasman [2001b] have applied this method to the design of a space-variant polarization element, which enables the transformation of circularly polarized light into a wave with a direction of polarization that is a linear function of the x-coordinate. The element was fabricated as metal-stripe gratings on GaAs and ZnSe wafers using lift-off techniques. Figure 2(a) shows the magnified geometry of such a computer-generated mask with the resulting transmission axis varying in the x-direction from 0◦ to 90◦ . The continuity of the grating is clearly apparent. Figure 2(b) shows experimental measurements of the azimuthal angle for a circularly polarized CO2 laser beam transmitted through the spacevariant polarizer. These experimental results, based on complete space-variant Stokes-parameter measurement (see Collett [1993]), revealed 98.6% overall polarization purity, taking into account the azimuthal and ellipticity deviations.

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Fig. 2. (a) Magnified illustration of the computer-generated space-variant polarization element geometry. (b) Experimental measurement of the two-dimensional space-variant polarization orientations. The arrows indicate the direction of the large axis of the local polarization ellipse. (From Bomzon, Kleiner and Hasman [2001a].)

2.1.3. Formation of linearly polarized light with axial symmetry by using space-variant sub-wavelength dielectric gratings Recent years have witnessed a growing interest in beams of a transversally spacevariant polarization state. One of the most interesting types of such beams is the linearly polarized axial symmetric beam (LPASB). LPASBs are characterized by their polarization orientation ψ(ω) = mω + ψ0 , where m is the polarization order number, ω is the azimuthal angle of the polar coordinate system, and ψ0 is the initial polarization orientation for ω = 0. Figure 3(a) illustrates LPASBs of polarization order numbers m = 1 and m = 2. Note that LPASBs have a singularity of their polarization state at the beam axis and, therefore, have a vectorial vortexlike structure. The most renowned members of the LPASB family are the radial (m = 1, ψ0 = 0) and azimuthal (m = 1, ψ0 = 12 π) beams, which are extensively used for the improvement of applications such as particle acceleration, atom trap-

Fig. 3. (a) Illustration of linearly polarized light with axial symmetry with different polarization orders. (b) Geometrical definition of the grating vector. (From Niv, Biener, Kleiner and Hasman [2003].)

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ping, optical tweezers (Allen, Padgett and Babiker [1999]), material processing (Niziev and Nesterov [1999]), and tight focusing (Quabis, Dorn, Eberler, Glöckl and Leuchs [2000]). Two separate conditions have to be met if we wish to convert circularly polarized light into a LPASB using sub-wavelength gratings. The first is converting the circularly polarized light into linearly polarized light by inducing 12 π retardation on the incident wave. The second is creating the proper local polarization direction. The first condition is met by choosing the correct shape for the subwavelength grooves while the second is fulfilled by creating local groove orientation of the form θ (ω) = ψ(ω) − 14 π = mω + ψ0 − 14 π.

(2.12)

An axially symmetric space-variant sub-wavelength grating is typically described by a grating vector of the form 



 Kg (r, ω) = K0 (r, ω) cos θ (r, ω) − ω rˆ + sin θ (r, ω) − ω ωˆ , (2.13) where rˆ , ωˆ are unit vectors in polar coordinates (fig. 3(b)), and K0 (r) = 2π/Λ(r, ω) is the local spatial frequency for a grating of local period Λ(r, ω). Next, to ensure the continuity of the grating, we require that ∇ × Kg = 0, resulting in a differential equation that can be solved to yield the local grating period. The solution to this problem yields K0 (r) = (2π/Λ0 )(r0 /r)m , where Λ0 is the local sub-wavelength period at r = r0 . Integrating Kg over an arbitrary path yields the desired grating function (defined such that ∇φg = Kg ) as    (m − 1)Λ0 φg (r, ω) = 2πr0 (r0 /r)m−1 sin (m − 1)ω + ψ0 − 34 π for m = 1,

(2.14a)

for m = 1.

(2.14b)





 φg (r, ω) = (2πr0 /Λ0 ) ln(r/r0 ) cos ψ0 − 14 π + ω sin ψ0 − 14 π

Niv, Biener, Kleiner and Hasman [2003] realized Lee-type binary grating functions for m = 12 , 1, 32 , 2 and ψ0 = 14 π. The gratings were fabricated on 500 µmthick GaAs wafers for CO2 laser radiation with a wavelength of 10.6 µm, with Λ0 = 2 µm, r0 = 4.7 mm, and a maximum radius of 6 mm. They formed the gratings with a maximum local period of 3.2 µm in order not to exceed the Wood anomaly of GaAs. Figure 4(a) shows the intricate geometry of a sub-wavelength stripe grating designed to convert circularly polarized light into LPASB. Figure 4(b) shows an image obtained by using a linear polarizer as an analyzer. Note that for each beam the polarization state repeats itself 2m times. Experimental values of the local

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Fig. 4. (a) Magnified geometry of the sub-wavelength gratings for different polarization orders, m = 12 , 1, 32 , 2. (b) Experimental intensity distributions, directly after the gratings of different polarization orders, of the beams emerging from a linear polarizer–analyzer. (c) Measured local azimuthal angles of the beams. (From Niv, Biener, Kleiner and Hasman [2003].)

azimuthal angle ψ at each point are shown in fig. 4(c). The manipulation resulted in a high polarization purity of over 98% in the desired direction. Additional insight can be obtained by performing polarization-state and phase analysis of the resulting beam. By representing the element as a space-variant Jones matrix, the resultant wavefront can be found for any incident polarization (see, for example, Hasman, Bomzon, Niv, Biener and Kleiner [2002]). For a space-varying quarter-wave plate and incident right-hand circular polarization, the Jones vector of the resultant beam is     cos(mω + ψ0 ) Eout (r, ω) = (2.15) exp −i(mω + ψ0 ) . sin(mω + ψ0 ) Using the rule proposed by Pancharatnam [1956] for comparing the phases of two light beams in different states of polarization, we can now calculate the spacevariant Pancharatnam phase of the transmitted beam as   ϕp = arg E(r, ω), E(R, 0) , (2.16) where argE(r, ω), E(R, 0) is the argument of the inner product of the two vectors and (R, 0) are the coordinates of the point on the resultant beam with respect to which the phase is measured. This calculation yields   ϕp = arg cos(mω + ψ0 ) − mω. (2.17)

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This phase modification results solely from the polarization manipulation and is purely geometrical in nature (Bomzon, Niv, Biener, Kleiner and Hasman [2002b]). The beam displays a Pancharatnam phase ramp with a helical structure similar to those found in scalar optical vortices; therefore, we define the topological Pancharatnam charge of the beam as  lP = (1/2π) ∇ϕp ds = −m, (2.18) where ds is an infinitesimal distance in the direction of the integration path. Note that in our case the Pancharatnam charge and the polarization order number are equal in magnitude and opposite in sign. This charge can be modified by transmission of the beam through a spiral phase element of the form exp(ild ω), (ld integer), whereby a topological charge of ld is added to the beam. Figures 5(a) and 5(c) show the calculated real parts of the instantaneous fields of LPASBs. Figure 5(a) shows the fields of the beams formed by use of the gratings only, for m = 12 , 1, 32 , 2, and ψ0 = 14 π. Figure 5(c), on the other hand, shows the beams that are created when, in addition to the grating, the waves of fig. 5(a) are also transmitted through spiral phase elements bearing topological charge ld = m. The result is the cancellation of the Pancharatnam phase while

Fig. 5. Calculated real part of the instantaneous vector fields for beams (a) emerging from the gratings only, and (c) with additional spiral phase element of ld = m. (b, d) Experimental far-field images for the beams and their calculated (solid curves) and measured (solid circles) cross-sections. [(b) and (d) correspond to (a) and (c), respectively.] (From Niv, Biener, Kleiner and Hasman [2003].)

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maintaining both the same space-variant polarization directions as well as the same polarization order number m. In cases where the polarization order number is a half-integer (e.g., m = 12 , 32 ), the fields of the beams having Pancharatnam phase (fig. 5(a)) are continuous, whereas the fields of fig. 5(c) without the Pancharatnam phase are discontinuous. This indicates that the formation of continuous LPASBs with a half-integer polarization order is possible only for beams having a topological Pancharatnam charge. Figure 5 also shows the experimental far-field images of these fields, with figs. 5(b) and 5(d) corresponding to figs. 5(a) and 5(c), respectively, as well as the calculated and measured cross-sections. Note that the beams emerging from only the gratings, fig. 5(b), exhibit far-field images with bright centers, while the beams undergoing a cancellation of the Pancharatnam phase exhibit distinct farfield images with dark centers, fig. 5(d). There is a close connection between the instantaneous electric field (figs. 5(a), 5(c)) and the appearance of a bright or dark spot at the far-field image of the beams. When the integral of the real part of the instantaneous electric field around the beam axis is zero,   2π    0 Re (2.19) E(ω) dω = , 0 0 a dark spot at the far-field is obtained; conversely, a nonzero sum reveals a bright spot. The experimental results indicate that LPASBs with identical polarization orders, but of different Pancharatnam phases, propagate in different ways, which emphasizes the relevance of correct phase determination in the propagation of space-variant polarization beams. Another point of interest is the angular momentum of such beams. For a scalar wave, the angular momentum in the direction of propagation per unit energy is given by jz = (l + σ )/(2πν) (see, for example, Allen, Padgett and Babiker [1999]), where l is the topological charge, σ is the helicity (±1 for circular polarization) and ν is the optical frequency of the beam. Using this rule and the decomposition of Eout into circular polarization states yields the angular momentum of LPASBs as 1  jz = (li + σi )/(2πν) = (ld − m)/(2πν) = lp /(2πν), 2 i=L,R

where L and R indicate the components with left and right circular polarization, respectively, and show that the angular momentum of these waves is given by the topological Pancharatnam charge. This result introduces a connection between angular momentum and topological Pancharatnam charge. The formation of both radial and azimuthal beams can also be achieved by the use of methods such as interferometric techniques, intracavity summation of two

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[4, § 2

orthogonally polarized TEM01 modes, or by using liquid-crystal devices, as will be discussed in the following subsections.

2.2. Space-variant vectorial fields obtained by using interference methods In this subsection we present laser resonator configurations and interferometric techniques in which the polarization in different parts of the output beam can be varied, generating, in effect, an output beam with space-variant polarization. Two interferometric techniques for converting a linearly polarized laser beam into a radially polarized beam with uniform azimuthal intensity were presented by Tidwell, Ford and Kimura [1990]. In the first method, linearly polarized beams with intensity profiles tailored using a modified laser or an apodization filter are combined in separate experiments to produce radially polarized light. The linear polarization-combining technique uses a Mach–Zehnder arranged as a 90◦ rotation shear interferometer operating on the null fringe, as shown in fig. 6(a). The second technique, shown in fig. 6(b), uses circularly polarized light instead of linearly polarized light, and a unique spiral phase-delay plate to produce the required phase profile. In a later study, Tidwell, Kim and Kimura [1993] presented a hybrid of these two earlier approaches for the conversion of a linearly polarized CO2 laser beam into a radially polarized beam. The result is a double-interferometer system that is able to convert any linearly polarized laser beam profile into a radi-

Fig. 6. Mach–Zehnder interferometer configurations used to produce radially polarized beams. (a) 90◦ rotational shear interferometer that converts a sinusoidally varying linearly polarized beam. (b) Conventional Mach–Zehnder for converting a general linearly polarized beam using spiral delay of circularly polarized light. (M = mirror, BS = beam splitter, P = polarizer, PS = periscope, SPDP = spiral phase delay plate.) (From Tidwell, Ford and Kimura [1990].)

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ally polarized one with high efficiency. Using this method, the authors were able to generate a beam that was ∼ 92% radially polarized and contained ∼ 85% of the input power. Azimuthal and radial polarizations have also been obtained by inserting polarization-selective elements into a laser resonator. Pohl [1972] inserted a birefringent calcite crystal with the principal axis along the z-axis (z-cut), into a pulsed ruby laser in order to discriminate between azimuthal and radial polarization. Wynne [1974] generalized this method and showed experimentally, with a wavelength-tunable dye laser, that it is possible to select either the azimuthally or the radially polarized modes. Mushiake, Matsumura and Nakajima [1972] used a conical intra-cavity element to select a radially polarized mode. The conical element introduced low reflection losses to the radially polarized mode but high reflection losses to the azimuthally polarized mode. This method is somewhat similar to applying a Brewster window to obtain linear polarization. Similarly, Tovar [1998] suggested using complex Brewster-like windows, of either conical or helical shape, to select radially or azimuthally polarized modes. Nesterov, Niziev and Yakunin [1999] replaced one of the mirrors of a high-power CO2 laser with a sub-wavelength diffractive element. This element consisted of either concentric circles (for selecting azimuthal polarization) or straight lines through a central spot (for selecting radial polarization) to obtain different reflectivities for the azimuthal and radial polarizations. Experimentally, a high output power of 1.8 kW was obtained, but the polarization purity was relatively low, with mixed transverse mode operation. Liu, Gu and Yang [1999] analyzed a resonator configuration into which two sub-wavelength diffractive elements were incorporated, to obtain a different fundamental mode pattern for two different polarizations. Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000] presented a method for efficiently obtaining essentially pure azimuthally and radially polarized beams directly from a laser. The method is based on the selection and coherent summation of two linearly polarized transverse modes that exist inside the laser resonator; specifically, two orthogonally polarized TEM01 modes. Figure 7(a) depicts an azimuthally polarized beam, obtained by coherent summation of a y-polarized TEM01(x) mode and an x-polarized TEM01(y) mode. Figure 7(b) shows a radially polarized beam, obtained by the coherent summation of an x-polarized TEM01(x) mode and a y-polarized TEM01(y) mode. The modes are selected by inserting phase elements that permit significant mode discrimination into the laser resonator when properly combined. The laser resonator configuration in which specific transverse modes are selected and coherently summed is schematically shown in fig. 8.

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Fig. 7. Coherent superposition of two orthogonally polarized TEM01 modes to form azimuthally and radially polarized beams. (a) Azimuthally (θ) polarized doughnut beam. (b) Radially (r) polarized doughnut beam. (From Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000].)

Fig. 8. Laser resonator configuration with a discontinuous phase element (DPE) for forming an azimuthally or radially polarized beam. (From Oron, Blit, Davidson, Friesem, Bomzon and Hasman [2000].)

2.3. Space-variant vectorial fields obtained by using liquid-crystal devices Liquid crystals have been selected as optical materials because of the flexibility they provide in designing new optical components. Stalder and Schadt [1996] suggested the use of a liquid-crystal device for generating linearly polarized light with axial symmetry using two optical effects. In the first case they realigned the incoming linearly polarized light by using the twisted nematic effect. The study made use of a basic cell consisting of one unidirectionally and one circularly rubbed alignment layers and filled with nematic liquid-crystal. The local liquid-

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Fig. 9. Generation of radially and azimuthally polarized light. (From Stalder and Schadt [1996].)

crystal orientation in the cell is that of a twisted cell with a variable twist angle defined by the local alignment layers. The generation of radially and azimuthally polarized light using this type of cell is illustrated in fig. 9. The authors also proposed the generation of light having a polarization order number m = 2 by using the effect of 12 λ wave plates in liquid-crystal devices that exhibit spatially variable alignment layers. In this study, the liquid-crystal cell consists of two circularly rubbed alignment layers with aligned centers of symmetry. The substrate glasses are coated with transparent electrodes. By applying the correct electric field between the electrodes, the cell becomes a 12 λ plate whose fast axis rotates one full rotation around the beam axis, for a given λ. Two-dimensional encoding of the polarization state of a laser beam was demonstrated by Davis, McNamara, Cottrell and Sonehara [2000] using a parallel-aligned liquid-crystal spatial light modulator (LCSLM). Each pixel of the LCSLM acts as a voltage-controlled wave plate that is capable of phase modulation over 2π rad at an argon laser wavelength of 514.5 nm. A liquid-crystal-based polarization grating was reported by Wen, Petschek and Rosenblatt [2002]. In this case, the orientation of one or both of the polarization eigenvectors is altered as light passes through the liquid-crystal cell. In one of its simplest forms, the grating permits only odd-order diffraction peaks. The researchers also developed more complex gratings, including a grating that rotates both polarization components in tandem, while simultaneously applying relative phase retardation. For an appropriate rotation and retardation, the device simulates a blazed grating for circularly polarized light. In addition, because the polar orientation of the liquid-crystal director can be controlled by an electric field applied across semitransparent indium tin oxide electrodes, one can switch the cell from grating mode to straight-throughput mode. A method that permits the accurate generation of arbitrary complex vector wave fields by using a binary optical element was described by Neil, Massoumian,

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Juškaitis and Wilson [2002]. The binary phase modulation was achieved by a reconfigurable ferroelectric liquid-crystal spatial light modulator (FLCSLM). However, the binarization process leads to a maximum efficiency in the first diffracted order of just 40.5%, and the use of only half of the light in the first diffracted order for each polarization reduces this to 20.25%. In addition, the form of particular desired complex fields (the amplitude and phase of the components) could lead to additional reductions in efficiency. Finally, Davis, Adachi, Fernández-Pousa and Moreno [2001] used a parallel-aligned active matrix nematic liquid-crystal spatial light modulator to realize a spatially variable wave plate for which the relative phase retardation varies linearly along the x-direction as φ = 2πx/Λ, where Λ is the period. Interestingly, the diffracted order is linearly polarized in this case, regardless of the incoming polarization state. This is contrary to the case presented in Section 3 and in the work of Hasman, Bomzon, Niv, Biener and Kleiner [2002] in which constant retardation and space-variant sub-wavelength gratings produce diffracted orders that are always circularly polarized (see Cincotti [2003]).

2.4. Alternative methods Another technique to achieve space-variant polarization-state manipulation is by holographic recording using polarization-sensitive materials. In this procedure, a space-variant polarized beam illuminates a polarization-sensitive recording material. The beam is typically generated interferometrically, as described in Section 2.2. When the hologram is illuminated using only one arm of the interferometer, a full reconstruction of the space-variant polarized beam emerges. This method was first described by Kakichashvili [1972], who used the Weigert effect (optical anisotropy induced in a fine-grain silver chloride photographic emulsion by exposure to a beam of linearly polarized light) for holographic recording and reconstruction of the polarization state of a beam. However, further progress in polarization holography was hampered mainly by the low efficiency of the recorded holograms (<1%). This situation remained until Todorov, Nikolova and Tomova [1984] introduced a new polarization-sensitive material: a methyl orange azo dye in a polyvinyl alcohol matrix. They demonstrated that high-efficiency polarization holograms (>35%) can be recorded repeatedly. This seminal work triggered a variety of research aiming to find ever more efficient and stable materials. Among these researchers, Ciuchi, Mazzulla and Cipparrone [2002] demonstrated long-time stability (over a year’s period) in two kinds of azo-dye elastomer films, while Holme, Ramanujam and Hvilsted [1996] reported 10,000 rapid write, read and erase cycles in an azobenzene sidechain liquid-crystalline polyester. Also,

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a veritable wave of applications emerged mainly for the optical storage of data. Among them was that of Kawano, Ishii, Minabe, Niitsu, Nishikata and Baba [1999] which proposed digital holographic storage with polarization multiplexing by use of polyester-containing cyanoazobenzene units in the side chain. In another research Ferrari, Garbusi and Frins [2004] demonstrated phase-shifting interferometry using bacteriorhodopsin film. It is worth mentioning that the ability to perform space-variant polarizationstate manipulations led to the development of several designing techniques. Tervo, Kettunen, Honkanen and Turunen [2003] proposed iterative design algorithms of diffractive elements for paraxial vector fields. Specifically, they showed that utilizing the local polarization state of a beam as an additional design freedom leads to more light-efficient design with minimal trade-off between diffraction efficiency and signal quality. Cincotti [2003] showed that the polarization state of the diffracted waves (higher-order waves) does not depend on the polarization state of the incoming wave but that they are fully determined by the polarization grating. She used this approach to propose a general model for polarization gratings that can be exploited for the design of such elements.

§ 3. Geometrical phase in space-variant polarization-state manipulation The Pancharatnam–Berry phase is a geometrical phase associated with the polarization of light. When the polarization of a beam traverses a closed loop on the Poincaré sphere, the final state differs from the initial state by a phase factor equal to half the Area (Ω) encompassed by the loop on the sphere (see Pancharatnam [1956], Berry [1984], and Shapere and Wilczek [1989]). In a typical experiment, the polarization of a uniformly polarized beam is altered by a series of spaceinvariant (transversely homogeneous) wave plates and polarizers, and the phase, which evolves in the time-domain, is measured by means of interference (Simon, Kimble and Sudarshan [1988], Kwiat and Chiao [1991]). Niv, Biener, Kleiner and Hasman [2003] considered a Pancharatnam–Berry phase in the space domain. Using space-variant dielectric sub-wavelength gratings, they demonstrated conversion of circularly polarized into linearly polarized axially symmetric beams. They showed that the conversion was accompanied by a space-variant phase modification of geometrical origin that affected the propagation of the beams. An earlier study by Bomzon, Kleiner and Hasman [2001d] demonstrated a Pancharatnam–Berry phase in space-variant polarizationstate manipulation using space-variant metal-stripe sub-wavelength gratings. Frins, Ferrari, Dubra and Perciante [2000] described a method for generating arbitrary axial phase discontinuities that is based on Pancharatnam’s theorem.

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Fig. 10. Setup for the generation of phase dislocations. Q1 and Q2 are rectangular QWPs with their respective fast axes (F.A.) orthogonal to each other; P1 is a polarizer at 45◦ with respect to the axes of the QWPs; P2 is a rotatable polarizer (analyzer); ν is the angle between the directions of the transmission axes of P1 and P2 . (From Frins, Ferrari, Dubra and Perciante [2000].)

They utilized this method for converting a bright, nondiffracting beam into a dark one. The basic concept for producing the phase dislocation is shown in fig. 10. The design consists of two quarter-wave plates (QWPs) placed side by side with their fast axes perpendicular to each other, sandwiched by two polarizers. The transmission axis of the first polarizer is at 45◦ with respect to the fast axes of the QWPs. Baba, Murakami and Ishigaki [2001] proposed using a space-variant geometrical phase for applications such as null interferometry. Previously, Bhandari [1997] suggested using a discontinuous spatially varying wave plate as a lens based on similar geometrical phase effects. Zhan and Leger [2002] reported an interferometric measurement of the geometric phase in space-variant polarization manipulation. They experimentally verified it using a dichroic radial polarizer. A dichroic radial polarizer converts a circularly polarized beam into an azimuthally polarized beam with a spiral geometrical phase. A Mach–Zehnder interferometer, which is shown in fig. 11, is used to measure this spiral phase. Conversion of an input polarization state into a space-variant polarization state has been investigated using periodic polarization gratings. Gori [1999] proposed using spatially rotating polarizers as a polarization grating while FernándezPousa, Moreno, Davis and Adachi [2001] proposed using a polarization grating with space-variant retardation realized by a liquid-crystal, spatial light modulator. Tervo and Turunen [2000] proposed using a polarization grating formed by spatially rotating wave plates. These authors showed that the polarization of the

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Fig. 11. Mach–Zehnder interferometer setup for the spiral geometric phase measurement. (From Zhan and Leger [2002].)

diffracted orders could differ from the polarization of the incident beams. The formation of complex vectorial fields by use of polarization gratings has been discussed in greater detail in Section 2. In the present section we consider optical phase elements based on the spacedomain Pancharatnam–Berry phase. Unlike with diffractive and refractive elements, the phase is not introduced through optical path differences, but results from the geometrical phase that accompanies space-variant polarization manipulation. Optical elements that use this effect to form a desired phase front are called Pancharatnam–Berry-phase optical elements (PBOEs). These elements are polarization dependent, thereby permitting the construction of multi-purpose optical elements that are suitable for applications such as optical switching, optical interconnects and beam splitting (see Hasman, Bomzon, Niv, Biener and Kleiner [2002]). Such elements can be realized using computer-generated spacevariant sub-wavelength dielectric gratings. Biener, Niv, Kleiner and Hasman [2002] and Hasman, Kleiner, Biener and Niv [2003] experimentally demonstrated Pancharatnam–Berry-phase diffraction gratings for CO2 laser radiation at a wavelength of 10.6 µm, showing an ability to form complex polarization-dependent phase elements. Figure 12 illustrates the concept of PBOEs on the Poincaré sphere. Circularly polarized light is incident on a wave plate with constant retardation and a continuously space-varying fast axis whose orientation is denoted by θ (x, y). Bomzon, Biener, Kleiner and Hasman [2002c] showed that since the wave plate is space varying, the beam at different points traverses different paths on the Poincaré sphere, resulting in a space-variant phase-front modification originating from the

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Fig. 12. Illustration of the principle of PBOEs by use of the Poincaré sphere. (From Bomzon, Biener, Kleiner and Hasman [2002c].)

Pancharatnam–Berry phase. Their goal was to utilize this space-variant geometrical phase to form novel optical elements. It is convenient to describe PBOEs using Jones calculus. In this representation, a wave plate with a fast axis oriented along the y-axis can be described by the Jones matrix   0 tx , J= (3.1) 0 ty eiφ where tx and ty are the real amplitude transmission coefficients for light which has been polarized perpendicular and parallel to the optical axes, and φ is the retardation of the wave plate. A PBOE which contains wave plates with spacevariant orientation can be described by the space-dependent matrix



TC (x, y) = JR θ (x, y) JJ−1 (3.2) R θ (x, y) , cos θ − sin θ

where θ (x, y) is the local orientation of the optical axis and JR (θ ) = sin θ cos θ is a two-dimensional rotation matrix. For convenience, we adopt the Dirac bra-ket notation, and convert TC (x, y) to the helicity base in which |R = (1 0)T and |L = (0 1)T are the twodimensional unit vectors for right-hand and left-hand circularly polarized light, and T denotes transposition. In this base, the space-variant polarization 1 operator

−1 −1/2 i is described by the matrix T(x, y) = UTC U , where U = 2 1 −i is a unitary conversion matrix. Explicit calculation of T(x, y) yields  

1 0 1 T(x, y) = tx + ty eiφ 0 1 2  

1 0 exp[i2θ (x, y)] iφ . + t x − ty e (3.3) exp[−i2θ (x, y)] 0 2

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Fig. 13. Operation of polarization diffraction gratings. A beam with polarization |Ein  is incident on the polarization grating. The resulting beam comprises three polarization orders: the |Ein  polarization order, which maintains the original polarization and does not undergo phase modification; the |R polarization order that is right-hand circularly polarized, and whose phase is modified by 2θ(x, y); and the |L polarization order that is left-hand circularly polarized, and whose phase is modified by −2θ(x, y). When θ(x, y) is periodic, the |R polarization order and the |L polarization order undergo diffraction, resulting in the appearance of discrete diffraction orders. (From Hasman, Bomzon, Niv, Biener and Kleiner [2002].)

Thus, for an incident plane wave with arbitrary polarization |Ein  the resulting field is |Eout  = ηE |Ein  + ηR ei2θ(x,y) |R + ηL e−i2θ(x,y) |L,

(3.4)

where ηE = 12 (tx + ty eiφ ), ηR = 12 (tx − ty eiφ )Ein |L and ηL = 12 (tx − ty eiφ ) × Ein |R are the complex field efficiencies and α|β denotes the inner product. Figure 13 is a graphic representation of the results of eq. (3.4). It shows that |Eout  comprises three polarization orders: the |Ein  polarization order, the |R polarization order and the |L polarization order. The |Ein  polarization order maintains the polarization and phase of the incident beam, whereas the phase of the |R polarization order is equal to 2θ (x, y), and the phase of the |L polarization order is equal to −2θ (x, y). We note that the phase modification of the |R and |L polarization orders results solely from local changes in polarization and is therefore geometrical in nature. Using eq. (3.4) we can calculate the Pancharatnam phase front of the resulting wave. Pancharatnam’s definition for the phase between two beams of different polarization is ϕp (x, y) = arg[Eout (0, 0)|Eout (x, y)]. For incident |R polarization, ϕp (x, y) = −θ + arctan[cos φ tan θ ] = −θ + arctan[sin(2χ) tan θ ], where χ is the ellipticity of the resulting beam. Geometrical calculations show that ϕp is equal to one half the area of the geodesic triangle, Ω, on the Poincaré sphere defined by the pole |R, |Eout (0), and |Eout (θ ) (as il-

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lustrated in fig. 12), and yields the expected Pancharatnam–Berry phase. Similar results can be obtained for any incident polarization. A case of special interest is φ = π and tx = ty = 1, for which we find that the diffraction efficiency is 100%, and that |R polarization is completely converted into |L polarization. However, despite the fact that the resulting polarization is space-invariant, the Pancharatnam phase, ϕp = −2θ (x, y), is equal to the desired geometrical phase, ϕd . This phase corresponds to one half of the area encompassed by two geodesic paths between the poles that form an angle of 2θ with respect to one another, as illustrated in fig. 12. This proves that the phase added to the incident beam is geometrical in nature. Note that PBOEs operate in different ways on the two helical polarizations. To conclude, unlike conventional elements, PBOEs are not based on optical path differences, but on geometrical phase modification resulting from spacevariant polarization manipulation. In Section 3.1 we will describe the design procedure for a continuous PBOE along with an example of a blazed grating, and demonstrate the ability to form more complex phase fronts such as helical beams. In Section 3.2 we will describe the quantized PBOE (QPBOE) using a different design method. In Section 3.2 we will also demonstrate the ability to form a polarization-dependent lens using a QPBOE, and will analyze propagation invariant beams formed by QPBOEs. In Section 3.3 we will demonstrate an interesting phenomenon – vectorial Talbot beams and vectorial nondiffracting beams generated by using a PBOE.

3.1. Continuous Pancharatnam–Berry-phase optical elements PBOEs can be realized by using space-variant sub-wavelength gratings. When the period of the grating is much smaller than the incident wavelength, the grating acts as a uniaxial crystal (see Section 2). Therefore, by correctly controlling the depth, structure and orientation of the grating, the desired PBOE can be made. To design a PBOE, we need to ensure that the direction of the grating stripes, θ (x, y), is equal to half of the desired geometrical phase, which we denote as ϕd (x, y). Next we define a grating vector Kg = K0 (x, y)[cos(ϕd (x, y)/2)ˆx + sin(ϕd (x, y)/2)ˆy], where xˆ and yˆ are unit vectors in the x and y directions, K0 = 2π/Λ(x, y) is the spatial frequency of the grating (Λ is the local sub-wavelength period) and 1 2 ϕd (x, y) is the space-variant direction of the vector defined so that it is perpendicular to the grating stripes at each point. Next, to ensure the continuity of the grating, thereby ensuring the continuity of the resulting optical field, we require ∇×Kg = 0, resulting in a differential equation that can be solved to yield the local

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Fig. 14. Geometry of the space-variant sub-wavelength grating as well as the geometrical phases for incident |R and |L polarizations. (From Bomzon, Biener, Kleiner and Hasman [2002c].)

grating period. The grating function φg (defined so that ∇φg = Kg ) is then found by integrating Kg over an arbitrary path (see, for example, Bomzon, Kleiner and Hasman [2001a]). The realization of the grating function can be done by a Leetype binary mask. The formation of a Lee-type binary mask was explained in Section 2.1.2. An interesting example is introduced in the work of Bomzon, Biener, Kleiner and Hasman [2002c]. They designed a PBOE that acts as a diffraction grating by requiring that ϕd = (2π/d)x|mod 2π , where d is the period of the structure. They realized a Lee-type binary grating describing the grating function, φg . The grating was fabricated for a CO2 laser radiation with a wavelength of 10.6 µm. Two types of gratings were formed on a GaAs wafer to yield retardations of φ = 12 π and φ = π. Figure 14 illustrates the geometry of the grating, as well as the geometrical phase for incident |L and |R polarization states as calculated by eq. (3.4). The geometrical phases resemble blazed gratings with opposite blazed directions for incident |L and |R polarization states, as expected from our previous discussions. Following their fabrication, the PBOEs were illuminated with circular and linear polarizations. Figure 15 shows the experimental images of the diffracted fields for the resulting beams, as well as their cross-sections for retardations of φ = 12 π and φ = π, respectively. When the incident polarization is circular, and φ = 12 π, close to 50% of the light is diffracted according to the geometrical phase added to the |L or |R polarization orders (the direction of diffraction depends on the incident polarization), whilst the other 50% remains undiffracted in the |Ein  polarization order as expected from eq. (3.4). Furthermore, the polarization of the diffracted order has switched helicity as expected. For φ = π, no energy appears in the |Ein  polarization order, and the diffraction efficiency is close to 100%.

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Fig. 15. Measurements of the transmitted far-field for the sub-wavelength PBOE grating for retardations of φ = 12 π and φ = π respectively, for incident (a) circular left, (b) circular right and (c) linear polarizations. (From Bomzon, Biener, Kleiner and Hasman [2002c].)

When the incident polarization is linear, |Ein  = 2−1/2 (|R + |L), the two helical components of the beam are subject to different geometrical phases of opposite sign, and are diffracted to the |R and |L polarization orders in different directions. When φ = 12 π, the |Ein  polarization order maintains the original polarization, in agreement with eq. (3.4), whereas for retardation π the diffraction is 100% efficient for both circular polarizations, and no energy is observable in the |Ein  polarization order. The elements proposed here can be utilized for polarization-sensitive beam splitting and optical switches. In Section 3.1.1 we will demonstrate the ability to form helical wavefronts using continuous PBOEs. The example provided will demonstrate the ability to form a complex wavefront by using the design procedure for continuous PBOEs described in here. 3.1.1. Formation of helical beams by Pancharatnam–Berry-phase optical elements Recent years have witnessed a growing interest in helical beams and their use in a variety of applications. These include trapping of atoms and macroscopic parti-

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cles (see, for example, Paterson, MacDonald, Arlt, Sibbett, Bryant and Dholakia [2001] and Allen, Padgett and Babiker [1999]), transfer of orbital angular momentum to macroscopic objects (Mair, Vaziri, Weihs and Zeilinger [2001]), rotational frequency shifting, the study of optical vortices (Sacks, Rozas and Swartzlander [1998]), and specialized alignment schemes. Beams with helical (or spiral) wave fronts are described by complex amplitudes u(r, ω) ∝ exp(−ilω), where r and ω are the cylindrical coordinates – the radial coordinate and azimuthal angle, respectively – and l is the topological charge of the beams. At the center, the phase has a screw dislocation, also called a phase singularity, or optical vortex. Typically, helical beams are formed by manipulating the light after it emerges from a laser by the superposition of two orthogonal (nonhelical) beams, or by transforming Gaussian beams into helical beams by means of computer-generated holograms (see Sacks, Rozas and Swartzlander [1998]), cylindrical lenses or spiral phase elements (SPEs) (Beijersbergen, Coerwinkel, Kristensen and Woerdman [1994]). Alternatively, a helical beam can be generated inside a laser cavity by inserting SPEs into the laser cavity (Oron, Davidson, Friesem and Hasman [2001]). The common approaches to forming SPEs are as refractive or diffractive optical elements using a milling tool, a single-stage etching process with a gray-scale mask, or a multistage etching process (Oron, Davidson, Friesem and Hasman [2001]). In general, such helical beam formations either are cumbersome or suffer from complicated realization, high aberrations, low efficiency, or large and unstable setups. Biener, Niv, Kleiner and Hasman [2002] considered spiral phase elements based on the space-domain Pancharatnam–Berry phase. They showed that such elements could be realized using continuous computer-generated space-variant sub-wavelength dielectric gratings. Moreover, they experimentally demonstrated SPEs with different topological charges, based on Pancharatnam–Berry phase manipulation, with an axially symmetric local sub-wavelength groove orientation, for CO2 laser radiation at a wavelength of 10.6 µm. To design a PBOE with a spiral geometrical phase, we need to ensure that the direction of the grating grooves is given by θ (r, ω) = lω/2. By following the design procedure given in Section 3.1 the grating function, φg , would result in  (2πr0 /Λ0 )(r0 /r)l/2−1 cos[(l/2 − 1)ω]/[l/2 − 1] for l = 2, φg (r, ω) = φg (r, ω) = (2πr0 /Λ0 ) ln(r/r0 ) for l = 2, (3.5) where Λ0 is the local sub-wavelength period at r = r0 . For convenience, we use polar coordinates in the design procedure instead of the Cartesian coordinates used in Section 3.1. Biener, Niv, Kleiner and Hasman [2002] realized a Lee-type

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Fig. 16. (a) Interferogram measurements of the spiral PBOEs. (b) The corresponding spiral phases for different topological charges. (From Biener, Niv, Kleiner and Hasman [2002].)

binary grating describing the grating function, given by eq. (3.5), for l = 1, 2, 3, 4. The grating was fabricated for CO2 laser radiation with a wavelength of 10.6 µm. The geometry of the gratings for different topological charges are identical to those presented in fig. 4 in Section 2.1.3. The elements were fabricated on 500 µm thick GaAs wafers using contact photolithography. Following the fabrication, the spiral PBOEs were illuminated with a right-hand circularly polarized beam, |R, at 10.6 µm wavelength. In order to provide experimental evidence of the resulting spiral phase modification of their PBOEs, they used “self-interferogram” measurement using PBOEs with retardation of φ = 12 π. For such elements, the transmitted beam comprises two different polarization orders: |R polarization state, and |L with phase modification of −lω, according to eq. (3.4). The near-field intensity distributions of the transmitted beams followed by a linear polarizer were then measured. Figure 16(a) shows the interferogram patterns for various spiral PBOEs. The dependence of the intensity on the azimuthal angle is of the form I ∝ 1 + cos(lω), whereas the number of fringes is equal to l, the topological charge of the beam. Figure 16(b) illustrates the phase fronts resulting from the interferometer analysis, indicating spiral phases with different topological charges. Figure 17 shows the far-field images of the transmitted beams through the spiral PBOEs with retardation φ = π, having various topological charges, as well as the measured and theoretically calculated cross-sections. The experimental results were achieved by focusing the beams through a lens. Dark spots can be observed at the center of the far-field images, providing evidence of the phase singularity in the center of the helical beams. Excellent agreement between theory and experi-

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Fig. 17. Experimental far-field images as well as their calculated and measured cross-sections for helical beams with l = 1–4. (From Biener, Niv, Kleiner and Hasman [2002].)

mental results was found, clearly indicating spiral phases for beams with different topological charges. In summary, we have demonstrated the formation of a helical wavefront using a PBOE, thereby proving the ability to form a complex geometrical phase using space-variant polarization manipulation.

3.2. Quantized Pancharatnam–Berry-phase diffractive optics One of the most successful and viable outgrowths of holography involves diffractive optical elements (DOEs). These diffract light from a generalized grating structure having nonuniform groove spacing. They can be formed as thin optical elements that provide unique functions and configurations. High diffraction efficiencies for DOEs can be obtained with kinoforms that are constructed as surface-relief gratings on some substrate (d’Auria, Huignard, Roy and Spitz [1972]). However, in order to achieve a high efficiency, it is necessary resort to complex fabrication processes that provide the required accuracies for controlling the graded shape and depth of the surface grooves. Specifically, in a single process one photomask with variable optical density is exploited for controlling the etching rate of the substrate to form the desired graded relief gratings, or multiple binary photomasks are used so the graded shape is approximated by multilevel binary steps (see, for example, d’Auria, Huignard, Roy and Spitz [1972] and Dammann [1970]). Both fabrication processes rely mainly on etching techniques that are difficult to control accurately. As a result, the shape and depth of the grooves may differ from those desired, leading to reduced diffraction efficiency and poor repeatability of performance (Hasman, Davidson and Friesem [1991]). Researchers have begun to investigate polarization diffraction gratings consisting of spatially rotating polarizers (Gori [1999]) or wave plates (Bhandari [1995]). Bomzon, Biener, Kleiner and Hasman [2002c] demonstrated simple polarization diffraction gratings based on continuous space-variant computer-generated subwavelength gratings. However, applying constraints on the continuity of the sub-

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wavelength grating leads to a space variation of the local period. As a result, the elements are restricted in their ability to form a desired complex phase function in addition to being limited in their physical dimensions. Moreover, the result of space-varying periodicity complicates the optimization of the photolithographic process. In this subsection we present an approach for generating polarization-dependent DOEs based on quantized Pancharatnam–Berry-phase diffractive optics. Hasman, Kleiner, Biener and Niv [2003] have shown that such elements can be realized with a discrete geometrical phase, using a computer-generated space-variant subwavelength dielectric grating. By discretely controlling the local orientation of such grating, which has uniform periodicity, they were able to form more complex and sophisticated phase elements. They experimentally demonstrated quantized Pancharatnam–Berry-phase optical elements (QPBOEs) as a blazed diffraction grating and a polarization-dependent focusing lens, for the 10.6 µm wavelength from a CO2 laser. In addition, they showed that high diffraction efficiencies can be attained by utilizing a single binary computer-generated mask. This enabled the formation of multipurpose polarization-dependent optical elements that are suitable for applications such as optical interconnects, polarization beam splitting, optical switching and polarization-state measurements. In the QPBOE approach, the continuous phase function ϕd (x, y) is approximated in discrete steps, leading to the formation of a PBOE with discrete local grating orientation. In the scalar approximation, an incident wavefront is multiplied by the phase function of the quantized phase element described by exp[iFN (ϕd )], where ϕd is the desired phase and FN (ϕd ) is the actual quantized phase. The division of the desired phase ϕd into N equal steps is shown in fig. 18, where the actual quantized phase FN (ϕd ) is given as a function of the desired phase. The Fourier expansion of the actual phase front is given by  exp[iFN (ϕd )] = p Cp exp(ipϕd ), where Cp is the pth-order coefficient of the Fourier expansion. The diffraction efficiency, ηp , of the pth-diffracted order is given by ηp = |Cp |2 . Consequently, the diffraction efficiency ηp for the first diffracted order for such an element is related to the number of discrete levels N by η1 = [(N/π) sin(π/N)]2 . This equation indicates that for 2, 4, 8, and 16 phase quantization levels, the diffraction efficiency will be 40.5, 81.1, 95.0, and 98.7%, respectively. The creation of a QPBOE is done by discrete orientation of the local sub-wavelength grating as illustrated in fig. 18. One of the objectives of the study by Hasman, Kleiner, Biener and Niv [2003] was to design a blazed polarization diffraction grating, i.e., a grating for which all the diffracted energy is in the first order, when the incident beam is |R polarized. They designed a QPBOE with retardation φ = π that acted as a diffraction grating

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Fig. 18. Actual quantized phase F (ϕd ) as a function of the desired phase ϕd , as well as the discrete local grating orientation. Inset: scanning-electron microscope image of a region of the sub-wavelength structure of the focusing lens. (From Hasman, Kleiner, Biener and Niv [2003].)

by requiring that ϕd = (2π/d)x|mod 2π . This formed the quantized phase function, FN (ϕd ), depicted in fig. 18, where d is the period of the diffraction grating. In order to illustrate the effectiveness of their approach, they realized quantized diffraction gratings with various number of discrete levels, N = 2, 4, 8, 16, 128. The grating was fabricated for CO2 laser radiation with a wavelength of λ = 10.6 µm, with the diffraction grating period d = 2.5 mm and the sub-wavelength grating period Λ = 2 µm. The dimensions of the elements were 30 mm × 3 mm and consisted of 12 grating periods. The magnified geometry of the grating for the case N = 4, and the predicted geometrical quantized phase distribution, are presented in fig. 19. The elements were fabricated on 500 µm thick GaAs wafers using a single binary mask by means of contact photolithography. The insets in fig. 19 show scanning-electron microscopy images of some regions of the fabricated grating with a number of discrete levels, N = 4. Following the fabrication, the QPBOEs were illuminated with a right-handed circularly polarized beam, |R, at 10.6 µm wavelength. Figure 19 shows the measured and predicted diffraction efficiency for the first diffracted order for the different QPBOEs. The efficiencies are normalized relative to the total transmitted intensity for each element. The measured diffraction efficiency for N = 16 was 99 ± 1%, the theoretical value being 98.7%. The excellent agreement between the experimental results and the predicted efficiency confirms the expected quantized phase. In addition, Hasman, Kleiner, Biener and Niv [2003] formed a quantized Pancharatnam–Berry-phase focusing element for a 10.6 µm wavelength, having a

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Fig. 19. Magnified geometry of the grating for N = 4, as well as the predicted geometrical quantized phase distribution, and scanning-electron microscopy images of some regions of the grating. Also shown are the measured (triangles) and predicted (dashed curve) diffraction efficiency as a function of the number of discrete levels, N . (From Hasman, Kleiner, Biener and Niv [2003].)

quantized spherical phase function of FN (ϕd ) = FN [(2π/λ)(x 2 + y 2 + f 2 )1/2 ] with a diameter of 10 mm, a focal length of f = 200 mm, the number of discrete levels N = 8 and retardation φ = π. Figure 20 illustrates the magnified geometry of a focusing lens based on a QPBOE with N = 4, as well as the predicted quantized geometrical phase. A scanning-electron microscope image of a region on the sub-wavelength structure that they had fabricated is shown in the inset of fig. 18. A diffraction-limited focused spot size for |L transmitted beam was measured, while illuminating the element with |R polarization state. The inset in fig. 20 shows the image of the focused spot size as well as the measured and theoretically calculated cross-section. The measured diffraction efficiency was 94.5±1%, in agreement with the predicted value. The geometrical phase of a PBOE is polarization dependent, and this allowed them to experimentally confirm that their element is a converging lens for incident |R state, and a diverging lens for incident |L state, as indicated by eq. (3.4). For incident |L state, the measured focal length was f = −200 mm as expected, and the measured diffraction efficiency was identical to the measured incident |R state. Moreover, it is possible to form a bifocal lens as a PBOE with a retardation phase of φ = π by illuminating with a linear polarization beam and inserting a refractive lens following the PBOE. A trifocal lens can also be created as a PBOE with a retardation phase of φ = 12 π, resulting in three distinct focuses for |R, linear, and |L polarization states. As can be seen, the introduction of space-varying geometrical phases through QPBOEs enables new approaches in the fabrication of polarization-sensitive optical elements.

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Fig. 20. Illustration of the magnified geometry of a quantized-PBOE focusing lens with N = 4, as well as the predicted quantized geometrical phase. Inset: the image of the focused spot size as well as the measured (dots) and theoretically calculated (solid curve) cross-section. (From Hasman, Kleiner, Biener and Niv [2003].)

3.2.1. Propagation-invariant vectorial beams obtained by use of quantized Pancharatnam–Berry-phase optical elements Propagation-invariant scalar fields have been extensively studied, both theoretically and experimentally, since they were first proposed by Durnin, Miceli and Eberly [1987]. These fields were employed in applications such as optical tweezers and the transport and guiding of microspheres (Garcés-Chávez, McGloin, Melville, Sibbett and Dholakia [2002]). While recently there has been considerable theoretical interest in propagation-invariant vectorial beams (Tervo and Turunen [2001]), experimental studies of such beams have remained somewhat limited (see Pääkkönen, Tervo, Vahimaa, Turunen and Gori [2002] or Bomzon,

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Niv, Biener, Kleiner and Hasman [2002a]). One of the most interesting types of propagation-invariant vectorial beams is the linearly polarized axially symmetric beam (LPASB) (see, for example, Niv, Biener, Kleiner and Hasman [2003]). These vectorial beams are characterized by their polarization orientation, ψ(ω) = mω + ψ0 , where m is the polarization order number, ω is the azimuthal angle of the polar coordinates, and ψ0 is the initial polarization orientation for ω = 0. Methods for forming LPASBs have been discussed extensively in Section 2. In this subsection we propose the formation of propagation-invariant vectorial Bessel beams by the use of QPBOEs followed by an axicon. Niv, Biener, Kleiner and Hasman [2004] demonstrated the formation of LPASBs with different polarization order numbers by using QPBOEs. They realized the QPBOEs by using computer-generated space-variant sub-wavelength gratings upon GaAs wafers for 10.6 µm laser radiation. The optical performance of the elements was experimentally evaluated by measuring the polarization distribution of the emerging beam through the QPBOE, verifying high quality LPASBs. Subsequently, propagationinvariant vectorial Bessel beams were achieved by inserting an axicon after the QPBOEs. As a final step, the resulting beams were transmitted through a polarizer which produced a unique propagation-invariant scalar beam. This beam had a propeller-shaped intensity pattern that could be rotated by simply rotating the polarizer, which makes it suitable for optical tweezers (MacDonald, Paterson, Volke-Sepulveda, Arlt, Sibbett and Dholakia [2002]). The Jones vector of a LPASB is given by   √ |Pm  = exp(imω)|R + exp(−imω)|L / 2, (3.6) where the |Pm  state represents the linearly polarized beam whose polarization azimuthal angle is given by ψ = mω (let us choose the reference axis so that ψ = 0 at ω = 0). Propagation of the |Pm  state when transmitted through an axicon can be approximated by the stationary phase method (Pääkkönen, Tervo, Vahimaa, Turunen and Gori [2002]) to yield  |Bm  = Kz fa (r)|Pm 





 ∼ πγ z/λ exp ik 1 − γ 2 /2 z + r 2 /2z − λ/8 = × (−i)m Jm (kγ r)|Pm ,

(3.7)

where Kz is the Fresnel free space propagation operator for propagation distance z, r is the radial polar coordinate, k is the wave number, and Jm is the mth-order Bessel function of the first kind. In this case, the axicon phase function is paraxially approximated by fa (r) = exp(−ikγ r), where γ = β(n − 1) and β and n are the inclination angle and refractive index of the axicon, respectively.

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This paraxial calculation confirms propagation invariance of the polarization state as well as the Bessel intensity distribution, except for a linear growth function of z that can be removed by appodizing the incoming intensity (Davidson, Friesem and Hasman [1992b]). For this vectorial Bessel beam, the intensity profile is determined by m, the polarization order number of the original LPASB, while the local polarization state is unchanged by the axicon. When illuminating a linearly polarized beam upon a QPBOE with retardation of π radians, the Jones vector of the emerging beam will be     1 1 |Eout  = √ exp i2θ (r, ω) |R + √ exp −i2θ (r, ω) |L. 2 2

(3.8)

According to eq. (3.8), the emerging beam, |Eout , comprises two scalar waves of orthogonal circular polarizations, as expected from eq. (3.4). By selecting a local sub-wavelength groove orientation such as θ = 12 mω, eq. (3.8) will be identical to eq. (3.6), and thus the desired |Pm  state will be obtained. LPASBs with polarization order numbers m = 1, 2, 3 and 4 were formed by use of QPBOEs, as computer-generated space-variant sub-wavelength gratings. These elements were illuminated with a linearly polarized plane wave at a wavelength of 10.6 µm from a CO2 laser. Scanning-electron microscope images of the elements’ central sections are provided in fig. 21(a) for elements with polarization order numbers m = 2, 3. The local azimuthal angle was observed by inserting a polarizer directly behind the QPBOEs. The resulting intensities are depicted in fig. 21(b) for polarization order numbers m = 2, 3. Note that a specific azimuthal angle returns 2m times within each trip around the beam axis. Propagation-invariant vectorial beams were obtained by inserting a ZnSe axicon (β = 3◦ , n = 2.4) following the QPBOEs. Figure 21(c) shows the intensities at 8 cm beyond the axicon for beams of polarization order number m = 2, 3. The double arrows, arranged along the circumference of the beams, illustrate the local azimuthal angles. The space-variant polarization state of the propagating beams was measured at different distances, verifying the vectorial propagation invariance of the beams. Finally, the ability to achieve a controlled rotation of the intensity pattern by inserting a polarizer behind the axicon was demonstrated. It can be shown, again using stationary phase approximation, that transmittance of propagation-invariant LPASBs through a polarizer results in an amplitude of ∝ Jm (kγ r) cos(mω). This beam is propagation-invariant with a propeller-shaped intensity pattern given by I ∝ Jm2 (kγ r)(1 + cos(2mω)). These propeller-shaped intensities are depicted in fig. 22(a). If the polarizer is rotated by an angle of ω0 , the fringes rotate by an angle of ω0 /m. This behavior is demonstrated in fig. 22(b), where the polarizer

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Fig. 21. (a) Scanning-electron microscope images of the central parts of the QPBOEs for m = 2, 3 polarization orders. (b) Experimental intensity distributions, directly after the element, for beams emerging from the linear polarizer for m = 2, 3. (c) Intensity distributions at 8 cm beyond the QPBOE followed by an axicon for m = 2, 3. The strips arranged along the circumference of the beam illustrate the local azimuthal angles.

was rotated by 90◦ . The dashed and dotted lines indicate the resulting rotation of the propellers. It can be seen that rotations of 90◦ , 45◦ , 30◦ and 22.5◦ were obtained for m = 1, 2, 3 and 4, respectively.

3.3. Polarization Talbot self-imaging The Talbot effect is a well-known interference phenomenon in which coherent illumination of a periodic structure gives rise to a series of self-images at welldefined planes (Talbot [1836]). This effect has many applications to fields such as wavefront sensing (Siegel, Loewenthal and Balmer [2001]), spectrometry (Kung, Bhatnagar and Miller [2001]) and Talbot laser resonators (Wrage, Glas, Fischer, Leitner, Vysotsky and Napartovich [2000]). Although most studies of the Talbot effect relate to waves for which the polarization is uniform, several contemporary papers have dealt with the Talbot effect in fields with space-variant polarization (see Arrizón, Tepichin, Ortiz-Gutierrez and Lohmann [1996] or Rabal, Furlan

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Fig. 22. (a) Propeller-shaped intensity patterns of the beams emerging from the QPBOEs followed by an axicon and a polarizer for four polarization orders, m = 1, 2, 3, 4, from left to right. (b) Controlled rotation of the propeller-shaped intensities by rotating the polarizer by 90◦ ; the dashed lines and the dotted curves indicate the rotation angles of the patterns. (From Niv, Biener, Kleiner and Hasman [2004].)

and Sicre [1986]). However, the experimental discussions were usually limited to simple binary anisotropic gratings or other discontinuous polarization distributions. Arrizón, Tepichin, Ortiz-Gutierrez and Lohmann [1996] have shown that an anisotropic grating with two alternate linear perpendicular states of polarization is transformed by free propagation at one fourth of the Talbot distance into another grating with a circular polarization state. They formed this anisotropic grating by superposing two Ronchi-type diffraction gratings with a relative shift of half of the period, and with different polarization states. Figure 23 schematizes the theoretical space-variant polarization state at the near-field of the grating and at the quarter of the Talbot distance. Further, in this section, we demonstrate a Talbot effect involving a PBOE for which the orientation of the fast-axis varies linearly in the x-direction. We show that for any incident polarization the resulting field undergoes self-imaging and fractional Talbot effects involving polarization, intensity and phase. Bomzon, Niv, Biener, Kleiner and Hasman [2002b] presented a theoretical analysis of the phenomenon and experimentally demonstrated the effect using a continuous spacevariant sub-wavelength dielectric structure designed for CO2 laser radiation at a

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Fig. 23. (a) Ronchi grating with period d, (b) anisotropic grating with linear polarization, and (c) anisotropic grating with circular polarization state, observed at the distance 14 ZT behind the grating in (b). (From Arrizón, Tepichin, Ortiz-Gutierrez and Lohmann [1996].)

wavelength of 10.6 µm. Moreover, when a circularly polarized beam is incident upon the proposed PBOE a one-dimensional nondiffracting effect occurs, thus the beam emerging from the PBOE conserves its space-varying polarization and intensity as it propagates. Let us assume a PBOE with tx = ty = 1 and local grating orientation function θ = πx/d|mod π , where d is the rotation period of the space-variant wave plates’ fast axis. In this case the emerging field can be calculated by using eq. (3.4). To prove that the emerging field, |Eout , undergoes self-imaging, we calculate the propagation of each of the diffracted orders using the Fresnel approximation (see, for example, Goodman [1996]) to yield,   

Eout (x, z) = cos φ |Ein  − i sin φ ηL |L exp − i2πx − iπλz 2 2 d d2     i2πz iπλz i2πx exp − 2 , + ηR |R exp (3.9) d λ d from which we find that |Eout (x, z = 0) = |Eout (x, z = mZT ), where z = 0 corresponds to the plane just after the grating, ZT = 2d 2 /λ is the Talbot distance, and m is an integer. This proves that |Eout (x, z = 0) is reconstructed at the Talbot planes. Further analysis shows that |Eout (x, z = 12 ZT ) = |Eout (x + 12 d, z = 0). Thus, at half the Talbot distance the field is shifted in the x-direction by half a period compared to the field at z = 0, demonstrating a fractional Talbot effect. We can expect additional interesting effects at other fractional Talbot planes. Figure 24 presents the concept of the vectorial Talbot effect along with the cal-

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Fig. 24. Diffraction from the PBOE. The Talbot effect occurs in the region where the diffracted polarization orders overlap (the striped region). The polarization state at planes z = 0, 14 ZT , 12 ZT and ZT are also depicted for linearly polarized incident beam and retardation phase of the grating, φ = 12 π . The planes are symbolized by the dashed lines in the striped region.

culated space-variant polarization state at the planes z = 0, 14 ZT , 12 ZT and ZT , for a linearly polarized incident beam, and using grating with retardation phase φ = 12 π. Bomzon, Niv, Biener, Kleiner and Hasman [2002b] have used a grating similar to that one presented in Section 3.1. They illuminated the elements with linearly polarized light and measured the Stokes parameters at various planes along the z-axis using the four-measurement technique (see Section 4 or Collett [1993] for a detailed discussion of the Stokes parameters). The experimental results agree with the predictions. At z = 0 just after the grating, the polarization varies periodically and continuously in the x-direction from linear polarization to nearly circular polarization, and the intensity is constant. This field is reconstructed at z = ZT , thereby demonstrating the Talbot effect. At the plane z = 12 ZT a shifted field can be observed as predicted by eq. (3.9). A fractional Talbot effect is also demonstrated at z = 14 ZT . At this plane a clear periodic variation in intensity is observable. Although the polarization at this plane is space varying, the ellipticity is zero and the beam is linearly polarized at all points. A case of special interest occurs when PBOE is illuminated with off-axis circularly polarized light at a small incident angle of ζ ≈ λ/2d. Using eqs. (3.4) and (3.9) we find that the resulting field when φ = 12 π is

Eout (x, z)         π 2πx π 2πz π 2πx + xˆ + sin + yˆ exp i + , = −cos d 4 d 4 λ 4

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Fig. 25. Illustration of a PBOE for forming propagation-invariant fields. The insets show (top) the geometry of the PBOE, as well as (bottom) the vector-field formed by it. (From Bomzon, Niv, Biener, Kleiner and Hasman [2002a].)

where xˆ and yˆ are Cartesian unit vectors transverse to the direction of propagation. The resultant beam has uniform intensity and a constant space-variant polarization that is retained throughout its propagation. The beam is essentially a one-dimensional vectorial nondiffracting beam, analogous to a scalar nondiffracting cosine beam. Bomzon, Niv, Biener, Kleiner and Hasman [2002a] formed a vectorial nondiffracting beam by using a PBOE based on computer-generated space-variant sub-wavelength metal-stripe grating. Figure 25 illustrates a nondiffracting periodically space-variant polarization beam by using sub-wavelength gratings with φ = 12 π. The transmitted beam comprises two polarization orders which travel in different directions. The interference of the two polarization orders in the region where they overlap results in a propagation-invariant beam. The uniqueness of the vectorial solution lies in its space-varying polarization and uniform intensity, which makes it better suited for applications such as metrology and three-dimensional scanning. § 4. Applications of space-variant polarization manipulation Space-variant polarization-state manipulation has been found useful in a wide range of research fields. Quabis, Dorn, Eberler, Glöckl and Leuchs [2000] showed theoretically that the focal area is reduced when a radially polarized instead of a linearly polarized light annulus is used. This reduction of the focal area can be utilized for improving the spatial resolution in imaging systems. Liu, Cline and

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He [1999] used a radially polarized, high-power, Gaussian CO2 laser beam interacting with a high-quality electron beam located at the Brookhaven accelerator test facility, in order to perform laser acceleration in vacuum. They noted that a radially polarized, Gaussian laser beam can produce a stronger longitudinal electric field than a linearly polarized one. Niziev and Nesterov [1999] investigated the influence of the beam’s polarization on laser cutting efficiency. They concluded that in the case of cutting metals with a large ratio of sheet thickness to width of cut, the laser cutting efficiency for radially polarized beams is 1.5–2 times higher than for plane p-polarization (TM-polarization) and circularly polarized beams. In this section we elaborate on several applications using space-variant polarization manipulation performed by polarization-dependent optical elements such as polarization gratings. A polarization grating is defined, according to Gori [1999], as a transparency in which the polarization of the incident wave is changed periodically. In Section 4.1 we focus on polarization measurements, including near-field polarimetry and far-field polarimetry by use of spatially varying polarization manipulation and imaging polarimetry. In Section 4.2 we review depolarization methods with special attention to depolarizers based on space-variant polarization manipulation. Section 4.3 discusses the application of polarization encryption, emphasizing geometrical phase encryption, and polarization encoding, as well as optical computing as an example of an interesting encoding application. Section 4.4 describes the possibility of spatial control of polarization-dependent emissivity using sub-wavelength-structured elements.

4.1. Polarization measurements In this subsection we discuss polarization measurements performed by spacevariant polarization-state manipulation. Optical polarization measurement has been widely used for a wide range of applications such as ellipsometry (Lee, Koh and Collins [2000]), bio-imaging (Sankaran, Everett, Maitland and Walsh [1999]), imaging polarimetry (Nordin, Meier, Deguzman and Jones [1999]), and optical communications (Chou, Fini and Haus [2001]). A commonly used method is to measure the time-dependent signal once the beam has been transmitted through a photoelastic modulator (see Jellison [1987]) or a rotating quarter-wave plate (QWP) followed by a polarizer–analyzer (see, for example, Collett [1993]). The polarization state of the beam can be derived by Fourier analysis of the detected signal. An increasing demand for faster and simpler methods has led to the development of the simultaneous four-channel ellipsometer (Azzam [1987]). Oka and Kato [1999] reported on a method for spectroscopic measurement of the spectrally resolved polarization state. In their scheme, the light is passed successively

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through a pair of thick birefringent retarders and a polarizer–analyzer. The light emerging from the polarizer–analyzer is then fed into a spectrometer followed by a photodetector. Finally, the signal obtained by the photodetector is analyzed by a computer. The light that is being measured is assumed to have a broad-band spectrum. 4.1.1. Near-field polarimetry Bomzon, Biener, Kleiner and Hasman [2002a] presented a Fourier-transform polarimeter that is a space-domain analogue to the rotating QWP polarimeter method using a continuous space-variant dielectric sub-wavelength grating. Biener, Niv, Kleiner and Hasman [2003b] and Hasman, Biener, Kleiner and Niv [2003] later proposed a Fourier-transform polarimeter using discrete spacevariant sub-wavelength dielectric gratings. The grating of this type of element is divided into equal-sized zones. The sub-wavelength grooves are of uniform orientation and period within each zone and are rotated at discrete angles from zone to zone. The measurements for this type of polarimeter are performed in the near field, therefore this polarimeter is referred to as a near-field polarimeter. A Fourier-transform polarimeter using discrete space-variant sub-wavelength dielectric gratings is less sensitive to statistical errors because of the increased number of measurements, it is suitable for real-time applications, and it can be used in compact configurations. In addition, it is possible to integrate this polarimeter on a two-dimensional detector array for lab-on-chip applications. The high throughput achieved and the low cost make it useful as a commercial polarimeter for biosensing. The concept of near-field polarimetry based on sub-wavelength gratings is presented in fig. 26. Uniformly polarized light is incident upon a polarization- sensitive medium (e.g., biological tissue, an optical fiber, a wave plate, etc.) and then transmitted through a space-variant sub-wavelength grating that acts as a space-variant wave plate, followed by a polarizer. The space-variant wave-plate element is a particular case of polarization grating. The resulting intensity distribution is detected by a camera and captured for further analysis. The emerging intensity distribution is uniquely related to the polarization state of the incoming beam. This dependence is given by a spatial Fourier-series analysis, wherein the resulting Fourier coefficients completely determine the polarization state of the incoming beam. The polarization state within the Stokes representation is described by a Stokes vector S = (S0 , S1 , S2 , S3 )T , where S0 is the intensity of the beam, and S1 , S2 , S3 represent the polarization state. In general, S02
S12 + S22 + S32 , where the equality holds for fully polarized beams. The degree of polarization (DOP) of a

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Fig. 26. Schematic presentation of near-field spatial Fourier-transform polarimetry based on a discrete space-variant sub-wavelength dielectric grating followed by a sub-wavelength metal polarizer. Also shown is the measured intensity distribution captured in a single camera frame when the fast axis of the rotating QWP was at an angle of 20◦ . (We used a CO2 laser that emitted linearly polarized light and replaced the polarization-sensitive medium with a rotating QWP.) (From Hasman, Biener, Kleiner and Niv [2003].)

beam is defined by DOP = (S12 +S22 +S32 )1/2 /S0 . The polarization state emerging from an optical system (e.g., wave plates, polarizers, etc.) is linearly related to the incoming polarization state through S = MS, where M is a 4-by-4 real Mueller matrix of the system and S and S are the Stokes vectors of the incoming and outgoing polarization states, respectively (see Collett [1993] for further reading). The optical system under consideration consists of a polarization grating followed by a polarizer. This composite element can be described in Cartesian coordinates by a Mueller matrix M(θ ) = MP MR (−θ )MWP (φ)MR (θ ),

(4.1)

where θ represents the discrete rotation angle of the retarder (e.g., sub-wavelength dielectric grating) as a function of its location along the x-axis,   1 0 0 0 0 cos 2θ sin 2θ 0  MR (θ ) =  (4.2) 0 − sin 2θ cos 2θ 0 0

0

0

1

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is the Mueller matrix that represents rotation of the axis frame by angle θ (x),  2 tx + ty2 tx2 − ty2 0 0  1 tx2 − ty2 tx2 + ty2 0 0  MWP (φ) =  (4.3)  0 0 2tx ty cos φ −2tx ty sin φ  2 0

0

2tx ty sin φ

2tx ty cos φ

is the Mueller matrix of a transversally uniform retarder, with retardation φ and real transmission coefficients for two eigen-polarizations tx and ty , and   1 1 0 0 1 1 1 0 0  MP =  (4.4) 2 0 0 0 0 0

0

0

0

is the Mueller matrix of an ideal horizontal polarizer. The outgoing intensity can be related to the incoming polarization state of the beam by calculating the Mueller matrix given above and using the linear relation between the incoming and the outgoing Stokes vectors, yielding 1 1  AS0 + (A + C)S1 + B(S1 + S0 ) cos 2θ (x) S0 (x) = 4 2 + (BS2 − DS3 ) sin 2θ (x)

  1 + (A − C) S1 cos 4θ (x) + S2 sin 4θ (x) , (4.5) 2 where A = tx2 + ty2 , B = tx2 − ty2 , C = 2tx ty cos φ, and D = 2tx ty sin φ. Equation (4.5) describes the intensity of the outgoing beam as a truncated Fourier series with coefficients that depend on the Stokes parameters of the incident beam. S0 –S3 would be extracted using Fourier analysis (for a detailed discussion see Biener, Niv, Kleiner and Hasman [2003b]). We note that the grating coefficients A, B, C and D should be determined by direct measurement of the polarization grating parameters, tx , ty , and φ, or by performing a suitable calibration process. Biener, Niv, Kleiner and Hasman [2003b] fabricated a discrete space-variant sub-wavelength grating element for CO2 laser radiation of 10.6 µm wavelength. The dimensions of the element were 30 mm × 3 mm and consisted of 12 periods of d. They used the setup shown in fig. 26 to demonstrate experimentally the ability of their method to measure the polarization state for fully and partially polarized light (see figs. 27 and 28). The ability of their device to conduct polarization measurements of fully polarized light was tested by using a CO2 laser that emitted linearly polarized light and replacing the polarization-sensitive medium with a rotating QWP. Figure 27 shows the experimental and theoretical azimuthal

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Fig. 27. Measured (circles) and predicted (solid curves) values for azimuthal and elipticity angles as a function of orientation of the QWP. (From Hasman, Biener, Kleiner and Niv [2003].)

Fig. 28. Calculated (solid curve) and measured (circles) DOP as a function of the intensity ratio of two independent lasers having orthogonal linear polarization states, as used in the setup depicted in the top inset. The bottom inset shows calculated (solid curves) and measured (circles) intensity cross-sections for two extremes, I1 = I2 (DOP = 0.059) and I2 = 0 (DOP = 0.975). (From Hasman, Biener, Kleiner and Niv [2003].)

angle, ψ, and the ellipticity χ, calculated from the measured data, by use of the relations tan(2ψ) = S2 /S1 and sin(2χ) = S3 /S0 . The partially polarized beam was constructed by combining two CO2 lasers with orthogonal polarization (the setup is shown in the inset of fig. 28). Figure 28 shows the measured and predicted DOP as a function of the intensity ratio, I1 /I2 , of the combined lasers. The inset shows the experimental intensity distributions for two extreme cases. The first is for equal intensities (I1 = I2 ), in which the measured DOP is 0.059, indicating unpolarized light. The second is for illumination by a single laser only (i.e., I2 = 0), in which the measured DOP is 0.975, indicating fully polarized light.

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Fig. 29. Experimental interferograms generated by the OBPI filter of fig. 1. Alongside each interference pattern is the associated incident Stokes vector. A 50 µm scale bar is shown in the lower right-hand corner of each image. (From Van Delden [2003].)

Van Delden [2003] proposed an interferometric approach to polarization measurement by the use of an ortho-Babinet, polarization-interrogating filter (OBPI). His polarimeter is composed of four birefringent wedges and a linear polarizer comprising an OBPI filter assembly. Operationally, the OBPI filter is characterized by a three-stage optical system consisting of two modified Babinet compensators (i.e., spatially varying linear retarders) and a linear polarizer. The most important feature of the resulting interferogram is that a unique pattern is generated for any polarization state of the incident beam. Figure 29 shows experimental interferograms produced by the assembled OBPI filter with varying conditions of the polarized Koehler illumination. 4.1.2. Far-field polarimetry Gori [1999] proposed measuring the Stokes parameters by means of a polarization grating comprised of a linear polarizer whose orientation varied periodically along a line. His analysis was done using Jones calculus. A more general case of polarization grating of a periodically rotating wave plate analyzed using Jones calculus was presented in Section 3. The analysis of a beam emerging from such a polarization grating is given by eq. (3.4). As noted in Section 3, the beam emerging from a polarization grating comprises three polarization orders. The first maintains the original polarization state and phase of the incident beam, the second

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is right-handed circularly polarized and has a phase modification of 2θ (x), and the third has a polarization direction and phase modification opposite to those of the former polarization order. Gori assumed θ (x) to be a linear function of x, in which case the second and third polarization orders become 1 and −1 diffraction orders, respectively, and the first polarization order becomes the zeroth diffraction order. Gori showed that the decomposition performed by the polarization grating can be used for evaluating the Stokes parameters of a light beam. Suppose that the beam to be analyzed is sent through the polarization grating followed by a polarizer. At a suitable distance from the grating the three diffraction orders are spatially separated. By setting the polarizer orientation angle at the two values of 0◦ and 45◦ and measuring the corresponding intensities of the undiffracted-order beam together with the intensities of the 1 and −1 diffracted beams (at any angle of the polarizer) the Stokes parameters of the beam can be obtained. Figure 30 depicts the far-field intensities measured for a beam being passed through the polarization grating, a lens and a polarizer oriented at 0◦ and 45◦ . The polarization grating, which was realized by dielectric space-variant sub-wavelength gratings, was illuminated by a CO2 laser at the wavelength of λ = 10.6 µm. The inset in fig. 30(a) shows a scanning-electron microscopy image of a region on the subwavelength structure on the GaAs wafer. The measurements were taken in the focal plane of the lens as illustrated in the inset of fig. 30(b). The polarization state of the measured incident beam was generated by transmitting a linearly polarized beam oriented at 0◦ through a QWP with its fast axis oriented at 20◦ with

Fig. 30. Cross-section of the measured intensity in the Fourier plane of a beam emerging from a spatially rotating QWP polarimeter followed by a polarizer oriented at (a) 0◦ and (b) 45◦ . The polarization of the incident beam was generated by illuminating a QWP oriented at 20◦ with a horizontal linear polarized CO2 laser beam at the wavelength of λ = 10.6 µm. The inset depicted in (b) describes the concept of far-field polarimetry. The far-field polarimeter contains a discrete space-variant sub-wavelength dielectric polarization grating (D), a lens (L) and a polarizer (P). The inset in (a) is a scanning-electron microscope image of a region of the sub-wavelength structure.

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respect to the x-axis. The measured azimuthal angle, ψ, and the ellipticity, χ, of the incident beam were 16◦ and 20.7◦ , respectively, which is in good agreement with the predicted results. 4.1.3. Imaging polarimetry A more general application of polarization measurement is imaging polarimetry, which is being investigated as a means to extend the capabilities of infrared (IR) systems beyond conventional amplitude imaging. For example, some polarization matrices used in imaging polarimetry offer the capability to highlight or suppress different materials in a scene, or objects in different orientations. As a result, imaging polarimetry offers a means to extend the capabilities of conventional IR imaging and to provide new imaging modalities. Nordin, Meier, Deguzman and Jones [1999] and Guo and Brady [2000] proposed a micropolarizer array for IR imaging polarimetry. The polarization-imaging camera proposed by Nordin, Meier, Deguzman and Jones [1999] consisted of a 128 × 128 array of unit cells, each of which was composed of a 2 × 2 array of sub-wavelength metal strips in different orientations that acts as a 2 × 2 array of micropolarizers. Figure 31 depicts a unit cell within the micropolarizer array that contains two upper micropolarizers oriented at 90◦ , and two lower micropolarizers oriented at 0◦ and 45◦ . The unit cell illustrated in fig. 31 is considered to comprise a single image pixel. Oka and Kaneko [2003] proposed an alternative design for imaging polarimetry

Fig. 31. Schematic diagram of unit cell containing a 2 × 2 array of micropolarizers. (From Nordin, Meier, Deguzman and Jones [1999].)

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Fig. 32. Configuration of the block of polarimetric devices. (From Oka and Kaneko [2003].)

as illustrated in fig. 32. Their instrument consisted of two pairs of birefringent wedge prisms cemented together and a polarizer–analyzer.

4.2. Spatial polarization scrambling In this subsection we present methods for depolarizing light, based on spacedomain polarization-state scrambling. Depolarizers are optical elements that reduce the degree of polarization (DOP) of a beam, independent of its incident polarization state. These components are essential for removing undesired polarization sensitivity in optical systems, such as for long-haul transmission systems that use erbium-doped fiber amplifiers (Mazurczyk and Zyskind [1994]), and for optical measurement equipment (Kersey, Marrone and Dandridge [1990]). Totally unpolarized light is described by a Stokes vector of the form S = (S0 , 0, 0, 0)T , where the angle brackets denote the average value over the space domain. Therefore, for a uniform incident beam, the components of the Mueller matrix of a dep dep perfect depolarizer, Mdep , are given by mij  = 0, except for m11  = 1. Stokes–Mueller calculus has been described above, in Section 4.1. Several approaches for depolarizing light based on the scrambling of the polarization state in the time or wavelength domains have been suggested and experimentally demonstrated. Lyot [1928] was the first to propose an approach for reducing the DOP of a beam. His method relies on polarization scrambling over the wavelength. Billings [1951] and Heismann and Tokuda [1995] proposed the possibility of depolarizing monochromatic beams using a temporally varying retarder. McGuire and Chipman [1990] suggested a crystal-based depolarizer for scrambling the polarization state in the space domain. They suggested using two identical Babinet compensators oriented at 45◦ to each other. Each Babinet compensator, in turn, consisted of two prisms cemented together, one with the fast axis horizontal and the other with the fast axis vertical. Figure 33 shows the concept of the depolarizer using double Babinet compensators. Spatial polarization-state

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Fig. 33. Construction of the dual Babinet compensator pseudodepolarizer. H and V denote birefringent material with the fast axis horizontal and vertical, respectively. (From McGuire and Chipman [1990].)

scramblers are compact, passive components, and suitable for use in real-time applications and with monochromatic laser radiation. Biener, Niv, Kleiner and Hasman [2003a] proposed a complete depolarizer based on space-domain polarization-state scrambling performed by cascaded, computer-generated, space-variant sub-wavelength dielectric gratings, as shown in fig. 34(a). The first is a space-variant quarter-wave plate (QWP) with a rotation

Fig. 34. (a) Schematic presentation of our concept for depolarizing over the space domain. The insets illustrate the geometry of the sub-wavelength gratings. (b) SEM image of a typical cross-section of the grating profile of the QWP. (From Biener, Niv, Kleiner and Hasman [2003b].)

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Fig. 35. Illustration of the outgoing beam’s polarization state when the polarization of the incoming beam is (a) vertically linear, (b) horizontally linear, (c) linear at 45◦ or (d) circular. The spheres show the trajectories of the outgoing polarization states onto the Poincaré sphere. (From Biener, Niv, Kleiner and Hasman [2003b].)

period of d1 = 14 d; the second is a space-variant half-wave plate (HWP) with a rotation period of d2 = d. The Mueller matrix of a wave plate of which the fast axis rotates periodically with respect to the position along the x-axis is described in Section 4.1 with θ = πx/d, where d is the rotation period. Calculating the av d/2 erage value along the x-axis as mij  = (2/d) 0 mij (x) dx yields the Mueller matrix of an ideal depolarizer. The depolarization effect is achieved by spatially scrambling the beam’s polarization state. Figure 35 illustrates the local outgoing polarization states for different incoming beams, along with an illustration of the local polarization state as a trajectory on a Poincaré sphere (see Brosseau [1998] for a detailed explanation on Poincaré spheres). As shown in fig. 35, the resulting space-variant polarization state includes polarization ellipses of different orientation and ellipticity. These polarization ellipses demonstrate the principle of the scrambling procedure. Biener, Niv, Kleiner and Hasman [2003a] realized Lee-type gratings, which describe their grating functions. The first grating was a spatially rotating QWP with d1 = 2.5 mm; the second was a spatially rotating HWP with d2 = 10 mm. The elements were fabricated on a GaAs wafer. Figure 34(b) shows a scanningelectron microscope image of a typical cross-section of the grating profile of the QWP at a period of about 2 µm. The measured phase retardations of the elements were 0.46π and 0.96π for the appropriate QWP and HWP, respectively. The retardation of the elements was measured for 10.6 µm wavelength radiation. These

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Fig. 36. Measured and predicted DOP as a function of the orientation of the QWP, through which the incident beam has been transmitted. (From Biener, Niv, Kleiner and Hasman [2003b].)

results are in good agreement with the theoretical predictions achieved by rigorous coupled-wave analysis, utilizing the measured profiles of the gratings. Subsequently, their depolarizer was experimentally tested using linearly polarized CO2 laser radiation at a wavelength of 10.6 µm. They illuminated a rotating QWP in order to manipulate the polarization state of the beam incident on the depolarizer. The polarization state of the beam emerging from the depolarizer was measured using the four-measurement technique (see Collett [1993]). Each measurement was obtained by summing the intensity over the x-axis over the interval 0 < x < 12 d2 . Figure 36 shows the measured and predicted DOP as a function of the orientation of the QWP. The experimental DOP attained was less than 0.16. When the polarization state of the incident beam is known, the use of a simple pseudo-depolarizer is sufficient. Biener, Niv, Kleiner and Hasman [2003a] have demonstrated that a single, spatially rotating QWP or HWP based on spacevariant sub-wavelength dielectric polarization gratings can completely depolarize incident light with circular and linear polarization states, respectively. They used the same sub-wavelength polarization grating previously described for the cascaded gratings. The experimentally measured DOPs for the QWP and HWP scramblers were 0.021 and 0.075, respectively.

4.3. Polarization encryption and polarization encoding In this subsection we present several concepts for polarization encryption as well as numerous methods for the polarization encoding of data by using space-variant polarization-state manipulation. In the past few years there has been increasing interest in data security and a growing need for improved methods for encrypting data. One of the processes that has been extensively investigated is the optical encryption technique. Different optical encryption schemes have been suggested, for example schemes involving pure amplitude image encryption suggested by

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Fig. 37. Schematic diagram showing (a) the generic system architecture; and (b) the polarization encoding and decoding geometry in greater detail. The system operates as a direct phase-only mapping of the encrypted mask and decrypting key. A pair of crossed polarizers (P1 and P2) are used to generate an intensity pattern at the output. The polarization directions of the various components are indicated, the decrypting key (D) and encrypted mask (E) are aligned so that they act to phase-shift only one orthogonal component of the polarized wavefront (a). (From Mogensen and Glückstad [2000].)

Unnikrishnan, Joseph and Singh [1998]. Other encryption schemes involving phase-only images were explored by Towghi, Javidi and Luo [1999] in order to improve the visibility of the decrypted image. Both methods use double-random phase encryption, a technique first presented by Refregier and Javidi [1995]. Polarization encryption has been investigated by several groups, each employing slightly different concepts. Polarization encryption provides additional flexibility in the key encryption design by adding polarization-state manipulation to the conventional phase and amplitude manipulation used in earlier methods. This feature is advantageous as it makes the polarization encryption method more secure. Mogensen and Glückstad [2000] proposed polarization encryption using spatially modulated retardation. Their optical decryption system is shown schematically in fig. 37(a). The polarization procedure for encoding and decoding the phase information is shown in greater detail in fig. 37(b). A laser beam linearly polarized along the 45◦ axis is aligned with the polarizer P1. The encrypted phase mask (E) and the decrypting phase key (D) are aligned such that their fast axes are parallel to the y-axis. A second polarizer (P2) at an angle of 135◦ , crossed with respect to the first, is used to produce an intensity read-out of the phase-shifting information. The phase masks were implemented using a pair of parallel-aligned liquid-crystal spatial light modulators supplied by Hamamatsu Photonics.

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Fig. 38. Schematic of a system for polarization encryption; P1 , input polarizer; P2 , output polarizer; L1 , L2 , imaging lenses. (From Unnikrishnan, Pohit and Singh [2000].)

A second scheme for polarization encryption was proposed by Unnikrishnan, Pohit and Singh [2000] using a ferroelectric liquid-crystal spatial light modulator. Their encryption is done by an exclusive-OR (XOR) operation between the image and a random phase code (a key used to encrypt the data). The XOR operation is carried out in the polarization domain of coherent light by using two ferroelectric liquid-crystal spatial light modulators. The decryption of the encrypted data is done by a second XOR operation between the encrypted image and the key. Figure 38 shows a schematic representation of the concept of polarization encryption using ferroelectric liquid-crystal spatial light modulator. A different recording method for polarization encryption was suggested by Tan, Matoba, Okada-Shudo, Ide, Shimura and Kuroda [2001] using bacteriorhodopsin (a polarization recording medium). This method uses interference to record the spatially scrambled polarization field onto a medium that is sensitive to the electromagnetic field but not the intensity. Figure 39 shows the experimental setup using a polarization mask formed by polarizer films, a spatial light modulator for encrypting the polarization data, and a polarization recording medium (bacteriorhodopsin). The experimental setup shown in fig. 39 includes the decryption process. A concept of double-random polarization encryption was suggested by Matoba and Javidi [2004]. Their method includes one polarization scrambling mask located at the image plane and a second polarization scrambling mask located at the Fourier plane. This is considered to be a more secure method for encrypting data due to the use of two scrambling keys. Figure 40 shows the schematic of the proposed double-random polarization encryption technique. A different concept of polarization encryption involving geometrical phase was proposed by Biener,

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Fig. 39. Experimental setup for encryption using bacteriorhodopsin: SP, spatial filter; M, mirror; BS, beam splitter; P, polarizer; L, lens; CCD, CCD camera; RMM, random modulation mask; BR, bacteriorhodopsin; PC, personal computer. (From Tan, Matoba, Okada-Shudo, Ide, Shimura and Kuroda [2001].)

Fig. 40. Schematic of the double-random polarization encryption technique: f , focal length. (From Matoba and Javidi [2004].)

Niv, Kleiner and Hasman [2005]. The connection between geometrical phase and spatially varying polarization state manipulation was discussed in Section 3. The concept of geometrical phase encryption involves a PBOE that encodes the image intensity added by a random key function. (The term PBOE is explained in detail in Section 3.) The proposed PBOE, which is a space-variant rotating wave plate, imprints the image intensity plus the random key function in the local orientation of the wave plate’s fast axes. Let us assume that a PBOE with a spacevarying wave-plate orientation function of θi (x, y) encodes the primary image

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Fig. 41. (a) Schematic representation of the concept of geometrical phase encryption. (b) Primary image intensity to be encrypted. (c) Encrypting PBOE wave plate’s orientation function, θi + θk , in grayscale. (d) Sub-wavelength grating mask of the PBOE. (e) Polarization state of the beam emerging from the PBOE. (d) and (e) are taken from a small region near the eyebrow of Einstein, which is depicted in (b).

of young Einstein, depicted in fig. 41(b). In order to further encrypt the encoded primary image information embedded in the PBOE, we add a random rotation function, θk (x, y), to the space-varying wave plates’ orientation. This random rotation factor serves as an encryption/decryption key. The total orientation function of the wave plates, comprising the encrypted PBOE, is shown in grayscale in fig. 41(c). Decryption is performed by illuminating the encrypted element with circularly polarized light and then analyzing the emerging Stokes parameters with the appropriate key to retrieve the primary image. The scheme for this process is shown in fig. 41(a). The beam emerging from a PBOE, which is a rotating QWP, illuminated by |R-polarized light comprises two polarization orders, as can be seen in eq. (3.4). The first maintains the original polarization state and phase of the incident beam, and the second is left-hand circularly polarized, |L, and has a phase modification of −2θ (x, y). The phase added to the |L-polarized beam, which is geometrical in nature, equals −(ϕi + ϕk ), where ϕi = 2θi and ϕk = 2θk denote the geometrical phase added by the encoded primary image’s intensity and the encoded key re-

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Fig. 42. (a–c) Three intensity pictures generated by the decryption process for the polarizer in the different orientations: (a) 0◦ , (b) 45◦ and (c) 90◦ . The arrows indicate the orientation angle of the polarizer. (d) Decrypted image achieved by the decryption process using the intensities shown in (a–c).

spectively. Figure 41(e) depicts the space-variant polarization direction emerging from a PBOE with optical parameters of tx = ty = 1 and φ = 12 π. The emerging field, which is a result of the vectorial self-interference, is a space-varying polarized field. As can be seen, the orientation of the arrows is random. In order to retrieve the primary image’s geometrical phase we need to measure the Stokes parameters of the emerging beam. The Stokes parameters are measured by using a polarizer oriented in three different orientations. These measurements have been discussed extensively by Biener, Niv, Kleiner and Hasman [2005]. By using the measured Stokes parameters and by applying the geometrical phase key, we can retrieve the phase function of the primary image. For the realization of the optical concept, we can implement a method first discussed in Section 2 for space-variant polarization-state manipulations using computer-generated sub-wavelength structures. Figure 41(d) is a magnified illustration of the sub-wavelength grating mask of the encrypted element. A computer simulation was performed to test the geometrical phase encryption. Figures 42(a)–42(c) show the three intensity pictures obtained by measuring the encrypted image after being transmitted through a simulated polarizer oriented at three different orientations. The decrypted image shown in fig. 42(d) was generated by calculating the Stokes parameters when applying the simulated intensities, and by applying the correct geometrical phase key, ϕk . Polarization encryption can be considered to be a specialized form of polarization encoding, which is a general application of space-variant polarization-state manipulation. There are several methods for encoding the space-variant polarization state of a vectorial field. Eriksen, Mogensen and Glückstad [2001] and Davis, McNamara, Cottrell and Sonehara [2000] proposed the use of dynamic modulation of the space-variant polarization state by use of spatial light modulators. Figure 43 depicts the concept of polarization encoding by using spatial light modulators. Figure 44 shows the image produced by polarization encoding using

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Fig. 43. An optical system for converting incident polarized light into an arbitrary state of elliptically polarized light with the major axis of the elliptically polarized light rotated by an arbitrary angle. The lines denote the extraordinary axis of the SLMs, the quarter-wave plates ( 14 λ) and the polarization direction of the linear polarizer. (From Eriksen, Mogensen and Glückstad [2001].)

Fig. 44. (a) Intensity pattern with the analyzer–polarizer perpendicular to the input polarizer. (b) Intensity pattern with the analyzer–polarizer parallel to the input polarizer. (c) Intensity pattern with the analyzer–polarizer parallel to the ordinary axis of the LCSLM. (d) Intensity pattern with the analyzer–polarizer parallel to the extraordinary axis of the LCSLM. (From Davis, McNamara, Cottrell and Sonehara [2000].)

a polarizer–analyzer in different orientations. Another approach, discussed extensively throughout this chapter, is utilizing space-variant sub-wavelength gratings. Zeitner, Schnabel, Kley and Wyrowski [1999] demonstrated diffractive elements with polarization multiplexing for visible light constructed with metal-stripe sub-

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Fig. 45. Scanning-electron microscope picture of the element structure etched into the chromium layer. (From Zeitner, Schnabel, Kley and Wyrowski [1999].)

wavelength period gratings. They introduced different functions of the element for two orthogonal polarization directions using polarization-dependent pixel transmission, which was realized by sub-wavelength gratings within a pixel. Figure 45 shows a scanning-electron microscope picture of a region in the structured element. Another interesting application that utilizes polarization encoding is optical computing. Lohmann and Weigelt [1987] proposed a spatial filtering logic approach based on polarization. Using polarization logic instead of diffraction logic or scattering logic operations has several advantages. There is no loss of energy for both logic values 0 or 1, which is desirable for cascading elements, and polarization logic has a high space-bandwidth product. Figure 46 shows three different logical operations using a polarization-based spatial filtering logic method. Hashimoto, Kitayama and Mukohzaka [1989] proposed space-variant operations using optical parallel processor based on polarization encoding. A liquid-crystal spatial light modulator residing in the processor is used as an operational kernel. It enables a programmable space-variant operation to be performed on a real-time basis by spatially filtering the encoded light, pixel by pixel. Free-space optical

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Fig. 46. Truth table for all sixteen binary logic operations and the corresponding filters in the second Fourier plane. The right column shows results of laboratory experiments, which were made visible by an analyzer (white = logical level 1, black = logical level 0). See also pages 134 and 135. Partly taken from fig. 4 in the work of Lohmann and Weigelt [1987].

interconnects offer low cross talk, high bandwidth, and parallel operation, and are therefore attractive for use in digital optical computers. Two particularly useful interconnect schemes are based on the perfect-shuffle transform and its inverse. Davidson, Friesem and Hasman [1992a] proposed realizing inverse perfect shuffle by use of space-variant polarization-state manipulation. Their arrangement for optical implementation of one-dimensional inverse perfect shuffle is shown in fig. 47. The input is coded with an interlaced polarizing mask. The odd pixels are covered with vertical polarizers whereas the even pixels are covered with horizontal polarizers. The input is illuminated with diffuse laser light that is derived from an argon laser (λ = 514.5 nm). The holographic optical element is composed of two sub-holograms. The first sub-hologram is covered with a vertical polarizer so as to transmit light coming from only the odd pixels, and the second sub-hologram is covered with a horizontal polarizer so as to transmit light coming only from the even pixels. As this method for perfect shuffle uses only a single holographic optical element, it is simple, lightweight and compact.

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Fig. 47. Optical arrangement for implementing a one-dimensional PS−1 transform. (From Davidson, Friesem and Hasman [1992a].)

4.4. Space-variant polarization-dependent emissivity Thermal emission from the bulk of a smooth, absorbing material is considered to be incoherent and unpolarized, thus, it is correlated to spontaneous emission. The surface properties of the absorbing material have a profound impact on its optical properties, and can lead to partially coherent and partially polarized radiation emission. Raether [1988] argued that in the case of materials with a dielectric constant that has a negative real part, surface waves provide the connection between the emission or absorption properties of the material and the surface properties. There are two kinds of materials, that support surface waves: conductive materials that support surface plasmon polaritons, and dielectric materials, that support surface phonon polaritons. Surface plasmon polaritons are due to an acoustic type of oscillation of the electron gas. Therefore, the surface electromagnetic waves are actually charge-density waves. The underlying microscopic origin of the surface phonon polariton is the mechanical vibration of the atoms or phonons (Marquier, Joulain, Mulet, Carminati, Greffet and Chen [2004]). A surface polariton has a longer wavevector than the light-waves propagating along the surface with the same frequency. Therefore, they are called “nonradiative” surface polaritons. Their electromagnetic fields decay exponentially into space perpendicular to the surface and have their maximum value in the surface, as is characteristic of surface waves. In order to couple a propagating wave with the surface polariton, an additional prism or grating is needed. In this case, the

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coupling is obtained for a certain frequency at a well-defined, propagating wave direction. When using a grating, the relationship between the emission angle ζ and the wavelength λ is simply given by the usual grating law, 2π 2π sin ζ = ksp + p , (4.6) λ Λ where p is an integer, Λ is the grating period, and ksp is the wavevector of the surface wave. The connection between ksp , λ and the real part of the substrate’s dielectric constant, ε  , is given by  ε 2π ksp = (4.7) . λ 1 + ε The coupling between the surface polaritons and the propagating wave can lead either to an increased resonant absorption or to directional emission. Surface polaritons can be coupled only with TM-polarized propagating waves, and as a result the absorption or emission is polarization dependent (see Setälä, Kaivola and Friberg [2002]). Greffet, Carminati, Joulain, Mulet, Mainguy and Chen [2002] used surface wave theory to design and optimize a grating, ruled on a SiC substrate, that produced a strong peak of the emissivity around a wavelength λ = 11.36 µm. They measured the spectral reflection in various directions in order to obtain the emissivity using Kirchhoff’s law ε = α = 1 − R, where ε, α and R denote the emissivity, the absorption and the reflectivity, respectively. Their results are shown in fig. 48. The emission spectra of the thermal source are directionally dependent, as

Fig. 48. Emissivity of a SiC grating in TM-polarization. Line (a): λ = 11.04 µm; line (b): λ = 11.36 µm; line (c): λ = 11.86 µm. The emissivity was deduced from measurements of the specular reflectivity R using Kirchhoff’s law. The data have been taken at ambient temperature using a Fourier-transform infrared (FTIR) spectrometer as a source and a detector mounted on a rotating arm. The angular acceptance of the spectrometer was reduced to a value lower than the angular width of the dip. The experimental data are indicated by circles; the lines show the theoretical results. (From Greffet, Carminati, Joulain, Mulet, Mainguy and Chen [2002].)

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predicted by Wolf [1987]. They approximated the coherence length as λ/ζ to be 60λ. Furthermore, Marquier, Joulain, Mulet, Carminati, Greffet and Chen [2004] showed that in certain frequencies the emission could be frequency-resonant and nondirectional. Spatial variation of the emissivity can be obtained by using space-variant gratings embedded in a polar material (Dahan, Niv, Biener, Kleiner, Hasman [2005]). Accordingly, by spatially controlling the emissivity, we can generate spatially varying polarized fields. These can be used in various applications such as thermal polarization imaging, optical encryption, spatially modulated heat transfer, and the formation of high-efficiency thermal sources. We realized four space-variant elements with local groove orientations of θ = 12 mω, where m is the polarization order number and ω is the azimuthal angle of the polar coordinates. Such an element forms a spiral-like intensity and is appropriately called a spiral element. The elements formed were designed for polarization order numbers m = 1, 2, 3 and 4. We optimized such a grating using RCWA to receive maximum emission at a wavelength of 9 µm on a fused silica substrate, which is a polar material. The grating period was 2 µm with a fill factor of 0.3 and a depth of 0.8 µm. It was fabricated using advanced photolithographic techniques. In order to reduce signal-to-noise ratio the realized element was heated to 80 ◦ C. An image of the four elements, captured using a thermal camera with and without a polarizer– analyzer, is shown in fig. 49. Evidently, space-variant intensity modulation was obtained using the polarizer–analyzer due to space-variant polarization-dependent emissivity.

Fig. 49. Thermal image of four SiO2 spiral elements. The elements are at uniform temperature of 80 ◦ C hence the intensity is proportional to the emissivity. (a) With analyzer one can observe space-variant intensity. The polarization direction of the radiation from the dark sectors is perpendicular to the bright sectors. (b) Without analyzer, one observes uniform intensity.

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§ 5. Concluding remarks Computer-generated holograms (Brown and Lohmann [1966], Lee [1974, 1978]) and diffractive optics had revolutionized the field of optics by allowing the formation of scalar fields with arbitrary phase structures. More recently, space-variant polarization manipulation using computer-generated polarization elements such as sub-wavelength gratings and liquid-crystal modulators has led to new approaches for obtaining complex fields. In this review we have explored the nature of beams with space-variant polarization-state distributions. We began by discussing several possible methods for forming beams. These included computer-generated sub-wavelength metal or dielectric gratings, polarization interferometric methods, liquid-crystal devices and polarization-sensitive recording materials. The extensive discussion on subwavelength gratings included a theoretical background of sub-wavelength gratings along with theoretical analysis and experimental demonstration of spacevariant sub-wavelength metal and dielectric gratings. Several general design approaches for space-variant polarization optics were reviewed. Space-variant polarization-state manipulations are necessarily accompanied by a geometrical phase – the Pancharatnam–Berry phase. We have demonstrated the generation of optical phase elements based on a space-domain Pancharatnam– Berry phase (Pancharatnam–Berry-phase optical elements, PBOEs). We then discussed the ability to utilize this phase to form sophisticated scalar as well as vectorial wavefronts. Following this, the effect of this geometrical phase on the propagation of a vectorial beam was explored. We believe that PBOEs will advance a variety of applications in modern optics and will lead to novel approaches in nano-optics as well. We then moved on to review several applications involving space-variant polarization-state manipulation. These applications included near-field and farfield polarimetry, imaging polarimetry, spatial polarization scrambling, namely depolarizers, polarization encryption and polarization encoding. A preliminary study of space-variant polarization-dependent emissivity was presented. Theoretical research involving space-variant polarization distribution is still in the primary stages and experimental demonstrations are somewhat limited. Nevertheless, this field has great potential to influence several other fields such as bioimaging and biosensing, optical tweezing and optical computing among others. To indicate the great interest in space-variant polarization manipulation, we can cite several preliminary works in these fields. Hielscher, Eick, Mourant, Shen, Freyer and Bigio [1997] reported on measuring the Mueller matrix of a cancerous and a noncancerous cell suspension. More recently, Galajda and Ormos [2003] de-

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scribed the effect of using polarized light on the trapping of nonspherical beads. There are numerous other studies dealing with the influence of polarized light on light–matter interactions. It would seem that there are countless possible areas to explore dealing with space-variant polarized beams. One such area is that of optical computing, and we cited two studies that exploited the properties of polarization – the logical operation suggested by Lohmann and Weigelt [1987] and the interconnect proposed by Davidson, Friesem and Hasman [1992a]. This review has focused mainly on polarized, coherent and monochromatic light beams. Further experimental and theoretical investigations should be conducted on partially polarized, partially coherent, polychromatic light beams with space-variant polarization-state distribution (Gori, Santarsiero, Borghi and Piquero [2000]). These investigations should also include nonparaxial beams. Finally, more comprehensive research should be conducted in the field of polarization thermal emissivity, with the emphasis on the interaction between spacevariant polarized coherent emission and the excitation of surface phonon or plasmon polariton resonance.

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E. Wolf, Progress in Optics 47 © 2005 Elsevier B.V. All rights reserved

Chapter 5

Optical vortices and vortex solitons by

Anton S. Desyatnikov, Yuri S. Kivshar Nonlinear Physics Center and Center for Ultra-high bandwidth Devices for Optical Systems, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia http:// www.rsphysse.anu.edu.au/ nonlinear

and

Lluis Torner ICFO-Institut de Ciencies Fotoniques, and Department of Signal Theory and Communications, Universitat Politecnica de Catalunya, Barcelona ES 08034, Spain http:// www.icfo.es

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)47006-7 291

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293

§ 2. Self-trapped vortices in Kerr-type media . . . . . . . . . . . . . . . .

298

§ 3. Composite spatial solitons with phase dislocations . . . . . . . . . . .

310

§ 4. Multi-color vortex solitons . . . . . . . . . . . . . . . . . . . . . . . .

324

§ 5. Stabilization of vortex solitons . . . . . . . . . . . . . . . . . . . . . .

336

§ 6. Other optical beams carrying angular momentum . . . . . . . . . . .

344

§ 7. Discrete vortices in two-dimensional lattices . . . . . . . . . . . . . .

354

§ 8. Links to vortices in other fields . . . . . . . . . . . . . . . . . . . . .

359

§ 9. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . .

371

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

372

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

372

292

§ 1. Introduction In physics, wave propagation is traditionally analyzed by means of regular solutions of wave equations. However, solutions of wave equations in two and three dimensions often possess singularities, that is, points or lines in space at which mathematical quantities that describe physical properties of waves become infinite or change abruptly (Berry [2000]). For example, at the point of phase singularity, the phase of the wave is undefined and wave intensity vanishes. Phase singularities are recognized as important features common to all waves. They were first discussed in depth in a seminal paper by Nye and Berry [1974]. However, the earliest known scientific description of phase singularity was made in the 1830s by Whewell, as discussed by Berry [2000]. While Whewell studied the ocean tides, he came to the extraordinary conclusion that rotary systems of tidal waves possess a singular point at which all cotidal lines meet and at which tide height vanishes. Waves that possess a phase singularity and a rotational flow around the singular point are called vortices. They can be found in physical systems of different nature and scale, ranging from water whirlpools and atmospheric tornadoes to quantized vortices in superfluids and quantized lines of magnetic flux in superconductors (Pismen [1999]). In a light wave, the phase singularity is known to form an optical vortex: The energy flow rotates around the vortex core in a given direction; at the center, the velocity of this rotation would be infinite and thus the light intensity must vanish. The study of optical vortices and associated localized objects is important from the viewpoint of both fundamental and applied physics. The unique nature of vortex fields is expected to lead to applications in many areas that include optical data storage, distribution, and processing. Optical vortices propagating, e.g., in air, have been suggested also for the establishment of optical interconnects between electronic chips and boards (Scheuer and Orenstein [1999]), as well as free-space communication links (Gibson, Courtial, Padgett, Vasnetsov, Pas’ko, Barnett and Franke-Arnold [2004], Bouchal and Celechovsky [2004]), based on the multidimensional alphabets afforded by the corresponding angular momentum states (Molina-Terriza, Torres and Torner [2002]). The ability to use light vortices to create reconfigurable patterns of complex inten293

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sity in an optical medium could aid optical trapping of particles in a vortex field (Gahagan and Swartzlander [1999]), and could enable light to be guided by the light itself, or in other words by the waveguides created by optical vortices (Truscott, Friese, Heckenberg and Rubinsztein-Dunlop [1999], Law, Zhang and Swartzlander [2000], Carlsson, Malmberg, Anderson, Lisak, Ostrovskaya, Alexander and Kivshar [2000], Salgueiro, Carlsson, Ostrovskaya and Kivshar [2004]). Thus, singular optics, the study of wave singularities in optics (Nye and Berry [1974], Vasnetsov and Staliunas [1999], Soskin and Vasnetsov [2001]), is now emerging as a new discipline (for an extended list of references, see http://www.u.arizona.edu/~grovers/SO/so.html). In a broad perspective, the study of optical vortices brings inspiring similarities between different and seemingly disparate fields of physics; the comparison of singularities of optical and other origins leads to theories that transcend the confines of specific fields. Vortices play an important role in many branches of physics, even those not directly related to wave propagation. An example is the Kosterlitz–Thouless phase transition (Kosterlitz and Thouless [1973]) in solid-state physics models, characterized by creation of tightly bound pairs of point-like vortices that restore the quasi-long-range order of a two-dimensional model at low temperatures. Such vortex-induced phase transitions can be observed in superfluid helium films, thin superconducting films, and surfaces of solids, as well as in models of interest to particle physicists and cosmologists. The Bose–Einstein condensate (BEC), a state of matter in which a macroscopic number of particles share the same quantum state, constitutes a well-researched example of a superfluid in which topological defects with a circulating persistent current are observed. Nearly 75 years ago, Bose and Einstein introduced the idea of condensate of a dilute gas at temperatures close to absolute zero. The BEC was experimentally created in 1995 by the JILA group (Anderson, Ensher, Matthews, Wieman and Cornell [1995]), who trapped thousands (later, millions) of alkali 87 Rb atoms in a 10-µm cloud and then cooled them to a millionth of a degree above absolute zero. The extensive study of vortices in BEC (Williams and Holland [1999], Matthews, Anderson, Haljan, Hall, Wieman and Cornell [1999], Madison, Chevy, Wohlleben and Dalibard [2000], Raman, Abo-Shaeer, Vogels, Xu and Ketterle [2001]) promises a better understanding of deep links between the physics of superfluidity, condensation, and nonlinear singular optics. To introduce the notion of optical vortices, we recall that a light wave can be represented by a complex scalar function ψ (e.g., an envelope of an electric field), which varies smoothly in space and/or time. Phase singularities of the wave func-

5, § 1]

Introduction

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tion ψ appear at the points (or lines in space) at which its modulus vanishes, i.e., when Re ψ = Im ψ = 0. Such points are referred to as wave-front screw dislocations or optical vortices, because the surface of constant phase structurally resembles a screw dislocation in a crystal lattice, and because the phase gradient direction swirls around the singular line much like fluid in a whirlpool. Optical vortices are associated with zeros in light intensity (black spots) and can be recognized by a specific helical wave front. If the complex wave function is presented as ψ(r, t) = ρ(r, t) exp{iθ (r, t)}, in terms of its real modulus ρ(r, t) and phase θ (r, t), the dislocation strength (sometimes referred to as the vortex topological charge) is defined by the circulation of the phase gradient around the singularity,  1 S= (1.1) ∇θ dl 2π here dl is the element of an arbitrary counter-clockwise path closed around the dislocation. The result is an integer because the phase changes by a multiple of 2π. Under appropriate conditions, it also measures an orbital angular momentum of the vortex associated with the helical wave-front structure. If a light wave is characterized by an extra parameter, e.g., the wave polarization, its mathematical representation is no longer a scalar but a vector field. In vector fields, several types of line singularity exist; for example, those analogous to disclinations in liquid crystals, which could be edge type, screw type, or mixed edge-screw type, that could move relative to background wave fronts and could interact in several different ways (Nye and Berry [1974], Soskin and Vasnetsov [2001]). In the linear theory of waves, each wave dislocation could be understood as a simple consequence of destructive wave interference. In this review we mostly address screw phase singularities existing in scalar wave fields and thus we concentrate our analysis on the corresponding vortices. However, other types of singularities whose analysis falls beyond the scope of this review, such us polarization singularities (Freund [2004a, 2004b]), do exist and exhibit fascinating properties. A laser beam with a phase singularity generally has a doughnut-like shape and diffracts when it propagates in free space. However, when the vortex-bearing beam propagates in a nonlinear medium, a variety of interesting effects can be observed. Nonlinear optical media are characterized by an electromagnetic response that depends on the strength of the propagating light. The polarization of such a medium can be described as P = χ (1) E + χ (2) E 2 + χ (3) E 3 , where E is the amplitude of the light wave’s electric field, and the coefficients characterize both the linear and the nonlinear response of the medium (Shen [1984], Butcher and Cotter [1992], Boyd [1992]). The χ (1) coefficient describes the linear refractive index of the medium. When χ (2) vanishes (as happens in the case

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Fig. 1. Propagation of the Gaussian beam with a phase dislocation generated by the input beam E(r, ϕ, z = 0) = 2r exp(−r 2 /4 + iϕ), in (a) linear medium, χ (3) = 0, and (b) self-focusing Kerr medium, χ (3) > 0. Shown is the field intensity. Note the difference in the intensity scales.

of centro-symmetric media), the main nonlinear effect is produced by the third term that can be presented as an intensity-induced change of the refractive index proportional to χ (3) E 3 . An important consequence of such intensity-dependent nonlinearity is the spontaneous focusing of a beam that is due to the lensing property of a self-focusing medium (i.e. when χ (3) > 0). This focusing action of a nonlinear medium can precisely balance the diffraction of a laser beam, resulting in the creation of optical solitons, which are self-trapped light beams that do not change shape during propagation (Kivshar and Luther-Davies [1998]). A stable bright spatial soliton is radially symmetric, and it has no nodes in its intensity profile. If, however, a beam with elaborate geometry carries a topological charge and propagates in a self-focusing nonlinear medium, it has a doughnut like structure. However, such a doughnut beam is unstable, and it decays into a number of more fundamental bright spatial solitons, such an example is shown in fig. 1. The resulting field distribution does not preserve the radial symmetry, and the vortex beam decays into several solitons that repel and twist around one another as they propagate. This rotation is due to the angular momentum of the vortex beam transferred to the splinters. Remarkably, the behavior of a laser beam in a self-defocusing nonlinear medium (i.e. when χ (3) < 0) is distinctly different, see an example in fig. 2. Such a medium cannot produce a lensing effect and therefore cannot support bright solitons. Nevertheless, a negative change of the refractive index can compensate for spreading in light intensity of the dip, thus creating a dark soliton (Kivshar and Luther-Davies [1998]), a self-trapped, localized low-intensity state

5, § 1]

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297

Fig. 2. Propagation of the paraxial super-Gaussian beam with a phase dislocation generated by the input beam E(r, ϕ, z = 0) = tanh(r/3) exp(−(r/18)10 /2 + iϕ), in (a) linear medium, χ (3) = 0, and (b) self-defocusing Kerr medium, χ (3) < 0. Shown is the field intensity.

(a dark hole) in a uniformly illuminated background. Vortices of single and multiple topological charges can be created in both linear and nonlinear media by use of, e.g., computer-generated holograms or spatial light modulators. Propagating through a nonlinear self-defocusing medium, such as a vortex-carrying beam, creates a self-trapped state, a dark vortex soliton. Dark vortex solitons have been observed experimentally in different materials with self-defocusing nonlinearity, such as slightly absorbent liquids, vapors of alkali metals, and photorefractive crystals (Swartzlander and Law [1992], Mamaev, Saffman and Zozulya [1996], Chen, Segev, Wilson, Muller and Maker [1997]). In this review paper, we describe the basic concepts of the nonlinear physics of optical vortices in the context of the propagation of singular beams in nonlinear media. In particular, we overview the recent advances in the study of optical vortices propagating in different types of self-defocusing and self-focusing nonlinear media, but concentrate mostly on the case of self-focusing nonlinearity which leads to the azimuthal instability of a vortex-carrying beams (Kivshar and Agrawal [2003]). We summarize different physical settings where such a nonlinearity can support novel types of stable or quasi-stable self-trapped beams carrying nonzero angular momentum, such as vortex solitons, necklace beams, soliton clusters, etc. We also describe the properties of vortex beams created by partially incoherent light and of discrete vortices in periodic photonic lattices. In addition, we present some of the experimental results demonstrating the propagation of singular optical beams in nonlinear media.

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§ 2. Self-trapped vortices in Kerr-type media In this section, we describe the conventional scalar optical vortices in Kerr-like nonlinear media. In a defocusing media, a diffracting core of an optical vortex may get self-trapped and the resulting beam with a singular core should be classified as a vortex soliton. In contrast, as discussed above, a vortex-carrying beam itself may become self-trapped in a focusing nonlinear medium, and it is know to suffer azimuthal modulational instability.

2.1. Vortices in defocusing nonlinear media In a self-defocusing nonlinear medium, a screw dislocation in the wave phase can create a stationary beam structure with a phase singularity resulting in a selftrapped vortex beam or a vortex soliton. To describe the major properties of vortex solitons, we consider the propagation of a continuous wave (CW) beam in a bulk self-defocusing medium governed by a (2+1)-dimensional nonlinear Schrödinger (NLS) equation. In the specific case of the Kerr nonlinearity, this equation can be written in the normalized form ∂u 1 2 + ∇ u − |u|2 u = 0. (2.1) ∂z 2 We can eliminate the background of constant amplitude u0 through the transformation, z = u20 z, x  = u0 x, y  = u0 y, u = u0 ψ exp(−iu20 z), and obtain the following equation for the normalized field ψ: i

1 ∂ψ + ∇ 2 ψ + 1 − |ψ|2 ψ = 0, (2.2) ∂z 2 where we have dropped the primes for simplicity of notation. The dimensionless field should satisfy the boundary conditions |ψ| → 1 as x and y → ±∞. The existence of vortex solutions for the (2 + 1)-dimensional cubic NLS equation (2.2) can be established using an analogy between optics and fluid dynamics. Using the so-called Madelung transformation (Spiegel [1980], Donnelly [1991], Nore, Brachet and Fauve [1993]),   ψ(r, z) = χ(r, z) exp ϕ(r, z) , (2.3) i

where r is a two-dimensional vector with coordinates x and y, we can transform the NLS equation (2.2) into the following set of two coupled equations:

∂χ 2 + ∇ · χ 2 ∇ϕ = 0, ∂z

(2.4)

5, § 2]

Self-trapped vortices in Kerr-type media

1 ∂ϕ ∇ 2χ + (∇ϕ)2 = 1 − χ 2 + . ∂z 2 2χ

299

(2.5)

These equations can be viewed as the equations governing the conservation of mass and momentum for a compressible inviscid fluid of density ρ = χ 2 and velocity v = ∇ϕ, with the effective pressure defined as p = ρ 2 /2. More importantly, this kind of analogy between optics and fluid mechanics remains valid even for the generalized NLS equation with an arbitrary form of g(|u|2 ), provided the effective pressure is defined as  dg(ρ) dρ. p(ρ) = ρ (2.6) dρ The analogy, however, is not exact because, in addition to the standard pressure, eq. (2.5) includes a second term that has no analog in fluid mechanics. This term results from the so-called quantum-mechanical pressure in the context of superfluids. The Madelung transformation is singular at the points where χ = 0. Around such points on the plane (x, y), the circulation of v is not zero but equals 2π. These points present topological defects of the scalar field, and they are called vortices. To find the stationary solution corresponding to a vortex soliton, also called a dark soliton with circular symmetry, see fig. 3(a), we look for solutions of the cubic NLS equation (2.2) in the polar coordinates r and θ , ψ(r, θ ; z) = U (r)eimθ ,

(2.7)

Fig. 3. (a) Schematic of the intensity distribution in an optical beam carrying a vortex (mesh) and its helical wave front (gray surface). After Kivshar and Ostrovskaya [2001]. (b) Vortex profiles in a self-defocusing Kerr medium for four first values of the integer vortex charge n (Neu [1990]).

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where the integer m is the so-called vortex winding number, sometimes also called the vortex charge, and the real function U (r) satisfies the amplitude equation,

d2 U 1 dU m2 + − 2 U + 1 − U 2 U = 0, 2 r dr dr r

(2.8)

with the boundary conditions U (0) = 0,

U (∞) = 1.

(2.9)

The continuity of U at r = 0 forces the first condition, while U (∞) = 1 is consistent with a uniform background of intensity U02 as r → ∞. Equation (2.8) can be solved numerically to find the shape U (r) of the vortex soliton for different values of m, shown in fig. 3(b) (Neu [1990]). Alternatively, the approximate envelopes converge to the stationary states in the numerical simulation of the full eq. (2.2) (Velchev, Dreischuh, Neshev and Dinev [1997]). The region in the vicinity of r = 0, where U (r) is significantly less than 1, is called the vortex core. The functional form of U (r) near r = 0 and r = ∞ can be established directly from eq. (2.8) by taking the appropriate limit and is found to be 

 ar |m| + O r |m|+2 , r → 0, U (r) ∼ (2.10)  1 − m2 + O 1/r 4 , r → ∞. 2 2r The structure of the vortex soliton for an arbitrary form of the nonlinearity can be found using the same method and solving numerically for the amplitude function U (r). No qualitatively new features are found when the nonlinearity is allowed to saturate (Chen and Atai [1992]). However, the effective diameter of the vortex core increases almost linearly with the saturation parameter s = I0 /Is , where Is and I0 are the saturation and background intensities, respectively (Tikhonenko, Kivshar, Steblina and Zozulya [1998]). Stability of vortex solitons associated with the generalized NLS equation has not yet been fully addressed. However, it is believed that vortices with the winding numbers m = ±1 are topologically stable but that those with larger values of |m| are unstable and should decay into |m| single-charge vortices. In the context of superfluids, the multi-charged vortex solitons are found to survive for a relatively long time (Aranson and Steinberg [1996]). A similar behavior is found to occur in optics (Dreischuh, Paulus, Zacher, Grasbon, Neshev and Walther [1999]). For this reason, multi-charged vortices are usually classified as metastable. As an example, an intentional perturbation of a triply charged vortex leads to its incomplete decay to a long-lived doubly charged vortex and a singly charged vortex

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(Dreischuh, Paulus, Zacher, Grasbon and Walther [1999]), confirming that saturation of nonlinearity can effectively suppress the instability. We should, however, stress that multi-charged vortices are strongly unstable in anisotropic nonlinear media (Mamaev, Saffman and Zozulya [1997]). Experimentally, a vortex soliton appears as a dark region that maintains its shape on a diffracting background beam displaying a nontrivial dynamical behavior (Swartzlander and Law [1992], Tikhonenko and Akhmediev [1996]). To generate the vortex input beam, one images the waist of the Gaussian beam onto the surface of a singly charged phase mask with a telescope. The first diffracted order of this mask is then imaged onto the plane of the input window of the nonlinear cell, providing an initial condition consisting of a singly charged vortex nested centrally at the waist of a Gaussian beam. The position of the vortex in the initial field is controlled by translation of the phase mask across the beam. Figure 4(a) shows a typical intensity distribution at the output with the vortex nested at the approximate center of the beam. Figure 4(b) shows the output under the same conditions, apart from a translation of the phase mask. There is little

Fig. 4. Intensity distributions at the output of a nonlinear medium (a rubidium vapor). The position of the vortex in (a) and (b) was controlled by translation of the vortex mask. (c) The intensity profile at the cross-section made through the vortex core in the cases (a) (solid) and (b) (dashed), respectively (Christou, Tikhonenko, Kivshar and Luther-Davies [1996]).

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change in the beam away from the core of the vortex, as is seen in fig. 4(c), upon which cross sections on a line through the vortex cores of fig. 4(a, b) are overlaid. To measure the size of the output background, we should remove the vortex from the profile and calculated the average 1/e2 radius of undisturbed background. Phenomena such as the rotation and radial drift of the vortex relative to the background CW beam are often observed experimentally, even though they cannot always be predicted from a casual analysis of the stationary solution of the NLS equation. A proper theoretical description of these effects requires analytical techniques capable of analyzing the vortex motion (Christou, Tikhonenko, Kivshar and Luther-Davies [1996], Kivshar, Christou, Tikhonenko, Luther-Davies and Pismen [1998]). Physically, specific dynamical features such as vortex rotation and drift result from a nonuniform intensity profile of the background field which is typically a Gaussian beam. The rotation rate of optical vortices can be controlled by introducing a modulated phase gradient of the background beam, when the slope of the helical wave front is not uniform in the azimuthal direction (Kim, Lee, Kim and Suk [2003]). Phase profile determines not only the dynamics of a single vortex but also interaction between two vortices, such as attraction and repulsion of counterand co-rotating vortices, respectively (Luther-Davies, Powles and Tikhonenko [1994], Velchev, Dreischuh, Neshev and Dinev [1996]). A pure phase modulation, obtained by using computer-synthesized holograms, was used to create dark ring solitary waves (Kivshar and Yang [1994], Baluschev, Dreischuh, Velchev, Dinev and Marazov [1995], Dreischuh, Fliesser, Velchev, Dinev and Windholz [1996], Dreischuh, Kamenov and Dinev [1996], Kamenov, Dreischuh and Dinev [1997], Neshev, Dreischuh, Kamenov, Stefanov, Dinev, Fliesser and Windholz [1997], Dreischuh, Neshev, Paulus, Grasbon and Walther [2002]). Similar examples of phase patterns with singularities include vortex arrays and lattices in self-defocusing Kerr type media studied by Neshev, Dreischuh, Assa and Dinev [1998], Kim, Jeon, Noh, Ko, Moon, Lee and Chang [1998], Kim, Jeon, Noh, Ko and Moon [1998], Dreischuh, Chervenkov, Neshev, Paulus and Walther [2002].

2.2. Ring-like beams in focusing nonlinear media In an isotropic optical medium with a local nonlinear response the propagation of a paraxial light beam is described by the well-known generalized NLS equation (Kivshar and Agrawal [2003]). In the dimensionless form, the generalized NLS equation has the form (cf. eq. (2.1)), i

∂E + ⊥ E + F (I )E = 0, ∂z

(2.11)

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where E is the complex envelope of the electric field, ⊥ = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 is the transverse Laplacian, and z is the propagation distance measured in units of the diffraction length LD . Function F (I ) describes the nonlinear properties of an optical medium, and it is assumed to depend on the total beam intensity, I ≡ |E|2 . Examples of these Kerr-like materials include two-level model of resonant gases, F = I , or pure Kerr nonlinearity; the so-called saturable nonlinearity, F = I (1+αI )−1 , or its low-intensity expansion, the cubic-quintic model, F = I −αI 2 , etc. Here the parameter α defines the nonlinearity saturation. In a self-focusing medium, i.e. F (I )
0, the diffraction of a light beam can be compensated by the nonlinearity and the balance between these two counteracting “forces” corresponds to the stationary state. Spatial optical solitons are stationary spatially localized solutions of the NLS equation (2.11) which do not change their intensity profile during propagation (Segev [1998], Stegeman and Segev [1999], Kivshar and Stegeman [2002]). Such a definition covers many different types of stationary beams with a finite power and, in general, the spatial solitons can be found in a generic form,   E(x, y, z) = U (x, y) exp ikz + iφ(x, y) , (2.12) where the real functions U and φ are the soliton amplitude and phase, respectively, and k is the soliton propagation constant. Substituting eq. (2.12) into eq. (2.11), we arrive at the system of coupled equations for the soliton amplitude and phase,

⊥ U − kU − (∇φ)2 U + F U 2 U = 0, (2.13) ⊥ φ + 2∇φ∇ ln U = 0.

(2.14)

We start with the solutions of the system (2.13), (2.14) with a constant phase, taking φ = 0 without restriction of generality. In this case, it can be shown that the only type of a structure localized in both transverse dimensions should pos sess a radial symmetry, i.e. U (x, y) = U (r), where r = x 2 + y 2 . Solutions of this type include the fundamental (bell-shaped) soliton (see fig. 5(b) for m = 0) and higher-order modes with several rings surrounding the central peak (Haus [1966], Yanauskas [1966], Soto-Crespo, Heatley, Wright and Akhmediev [1991], Edmundson [1997]). The number of radial nodes, defined by the index n, distinguishes the higher-order radially-symmetric spatial solitons. The main parameter characterizing the spatial soliton is its power   P = |E|2 dr = U 2 dr, (2.15) being the integral of motion associated with phase invariance of a solution to eq. (2.11). Radial modes with n-rings in the intensity profile, Un , belong to the

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Fig. 5. Examples of the stationary radially symmetric soliton solutions of eq. (2.11) with saturable nonlinearity F = I /(1 + αI ), characterized by the topological charge m indicated next to the curves, the case m = 0 corresponds to the fundamental soliton; (a) intensity distribution of a single-charged vortex, m = 1, (b) radial profiles for α = 0.5 and k = 1, (c) soliton power vs. soliton parameter k for α = 0.2, after Skryabin and Firth [1998a]. The absolute values for soliton power in (c) correspond to the diffraction coefficient 1/2 and should be doubled for eq. (2.11) with diffraction coefficient 1.

different branches of the dependence P (k), i.e. they form a discrete set of soliton families, Pn (k). The generalization of each soliton family includes transversely moving solitons, obtained by applying the Galilean transformation, r → r − 2qz and φ → φ + q(r − qz), where v = 2q is the soliton transverse velocity. Such moving solitons can be characterized by the soliton linear momentum   L = Im E ∗ ∇E dr = ∇φU 2 dr, (2.16) which is defined for the fundamental soliton as L = qP . A novel class of spatially localized beams in self-focusing nonlinear media, associated with the rotation of the field phase, was introduced by Kruglov and Vlasov [1985]. The beam phase has a spiral structure with a singularity at the origin, as the one shown in fig. 3(a), representing a phase dislocation of the wave front in the form of an optical vortex (Kivshar and Ostrovskaya [2001]). The intensity of such a beam vanishes at the beam center, and, at the same time, the beam remains localized (i.e. its intensity decays at infinity) propagating in the form of a ring-like beam, see fig. 5(a, b). The existence of the ring-profile solitary waves can be explained intuitively. Indeed, quasi-one-dimensional solitary waves, i.e. the (1 + 1)-dimensional solitary waves embedded into a (2 + 1)-dimensional bulk medium, undergo the transverse modulational instability (Kivshar and Pelinovsky [2000]). One of the possible ways to suppress this instability is to consider a ringprofile structure created from a (1 + 1)-dimensional soliton stripe wrapped around its tail (Lomdahl, Olsen and Christiansen [1980]). It can be shown that such a soliton ring with the phase depending on the radial coordinate can display a stabilization effect, and even an initially expanding beam can shrink and eventually col-

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lapse (Lomdahl, Olsen and Christiansen [1980], Afanasjev [1995], Anastassiou, Pigier, Segev, Kip, Eugenieva and Christodoulides [2001]). The beams studied by Kruglov, Logvin and Volkov [1992] provide another example of nonstationary ring-profile solitary waves. Introduced by Kruglov and Vlasov [1985] ring-profile vortex solitons represent the first example of a spatial soliton with the field depending on the azimuthal coordinate ϕ = tan−1 (y/x). They can be found as the solutions to eqs. (2.13), (2.14) with a rotating spiral phase φ in the form of a linear function of the polar angle ϕ, i.e. φ = mϕ. Substituting this expression into eqs. (2.13), (2.14) we obtain

d2 U 1 dU m2 + − 2 U − kU + F U 2 U = 0, 2 r dr dr r

(2.17)

where the radially symmetric amplitude U (r) vanishes at the center, U ∼ r |m| at r → 0. The rotation velocity should be quantized by the condition of the field univocacy eq. (1.1), and therefore m is integer (see fig. 5(b, c)). The index m stands for a phase twist around the intensity ring, and it is usually called winding number, topological index or topological charge of the solitary wave. Such winding number distinguishes azimuthal higher-order stationary states, in addition to the radial modes, so that the full set of radially-symmetric spatial soliton families can be denoted by Pn,m (k), with radial and azimuthal quantum numbers n and m. Figure 5(b) illustrates the envelopes U (r) for single ring (n = 0) vortex solitons with different topological charges m. Corresponding families are divided by the minimal threshold power, necessary for the formation of corresponding higher-order states, the dependencies P0,m (k) for a fixed nonzero saturation α = 0.2 are shown in fig. 5(c). The threshold power is defined in the limit of zero saturation α = 0, i.e. in pure Kerr medium F (I ) = I , where soliton constant k plays a role of a scaling parameter, see horizontal lines in fig. 5(c). Subsequently, vortex solitons were re-discovered in other studies by Kruglov, Logvin and Volkov [1992], Atai, Chen and Soto-Crespo [1994], Afanasjev [1995] and for other types of nonlinear optical media, including quadratic nonlinear media (Torner and Petrov [1997a, 1997b], Firth and Skryabin [1997], Skryabin and Firth [1998a]), the latter nonlinear model is discussed in Section 4. The important integral of motion associated with this type of the solitary waves is the beam angular momentum,  Mez = Im E ∗ (r × ∇E) dr, which can be expressed through the soliton amplitude and phase,  ∂φ 2 M= U dr. ∂ϕ

(2.18)

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It is important to point out that the nonvanishing angular momentum is an overall property of the light beam, not necessarily directly connected to the quantum properties at the single photon level. Similarly, the angular momentum is not necessarily associated with optical singularities although in practice the two phenomena may occur together (Allen [2002], Berry, Dennis and Soskin [2004], Padgett, Courtial and Allen [2004]). The angular momentum of a paraxial light beam can be separated into spin and orbital parts (see, e.g., Cohen-Tannoudji, Dupont-Roc and Grynberg [1989], Barnett [2002]), where the spin momentum is associated with the polarization structure of the light, and the orbital momentum is associated with the spatial structure of the beam, in particular, the beam carrying optical vortices. Therefore, the angular momentum of a scalar vortex soliton defined by eq. (2.18) should be identified as an orbital angular momentum. However, as we describe below, the ring-profile vortex optical beams experience the azimuthal instability in nonlinear media, and they decay into a number of moving fundamental solitons. Because the input beams carry the overall angular momentum given by eq. (2.18), the splinters fly off the ring along the tangential trajectories. Therefore, in the soliton community it has become customary to refer to the soliton spin angular momentum, and thus to spinning solitons, and to use orbital angular momentum in the case of several interacting solitons. This notion leads to the description of the break-up of a vortex due to modulational instability in terms of the transformation of the initial soliton spin angular momentum to the net orbital angular momentum of moving splinters (Firth and Skryabin [1997], Skryabin and Firth [1998a]). The ratio of the soliton angular momentum to its power can be identified with the soliton spin, S = M/P , and for a vortex soliton the spin is equal to its nonzero topological charge S = m (cf. eq. (1.1)). We notice, however, that such a notation might be confusing when it is used in other areas where the concepts of spin angular momentum and orbital angular momentum are employed in the rigorous, proper sense.

2.3. Azimuthal modulational instability As was shown in many numerical and analytical studies, the ring-like vortex beams in self-focusing nonlinear media are subject to the azimuthal symmetrybreaking modulational instability, a specific type of transverse modulational instability similar to one which is responsible for filamentation of the beams and generation of trains of optical solitons (Kivshar and Pelinovsky [2000]). This effect should not be confused with the well-known collapse instability (Berge [1998]) which is eliminated in the nonlinear media with saturable and quadratic

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nonlinearity. As a result, stable fundamental solitons have been found in two and three spatial dimensions and the instability criterion was established by Vakhitov and Kolokolov [1973]. It states that the principal mode of the nonlinear wave equation (2.11) is stable if its integrated intensity eq. (2.15) has a positive slope, ∂P /∂k > 0, and the latter is possible if the nonlinearity grows monotonically with intensity, i.e. for dF /dI
0. Both these conditions are satisfied for the vortex solitons in, e.g., self-focusing media with saturation, see fig. 5(c). However, this property does not guarantee the vortex stability against azimuthal perturbations. So far, no universal criterion, similar to the Vakhitov–Kolokolov stability criterion for the fundamental optical solitons, has been suggested for the stability of vortex solitons, and the vortex stability should be addressed separately for different models. Below, we present the stability analysis developed by Soto-Crespo, Wright and Akhmediev [1992], Firth and Skryabin [1997], Torres, Soto-Crespo, Torner and Petrov [1998a]. We assume that there exists a stationary solution E = E0 solving eq. (2.11). The linear stability of this solution can be established by the behavior of an additional small perturbation |p| |E0 |. Substituting the perturbed solution E = E0 + p into eq. (2.11) and linearizing it with respect to a small perturbation p, we obtain i

∂p + ⊥ p + F0 + |E0 |2 F0 p + E02 F0 p ∗ = 0, ∂z

(2.19)

where F0 = F (|E0 |2 ) and F0 = (dF /dI )|I =|E0 |2 . Equation (2.19) describes the evolution of initially small perturbation p corresponding to the solution E0 : if p does not grow with the beam propagation, the stationary solution E0 is linearly stable. The linear equation (2.19) can be solved by the separation of variables, depending on the geometry of underlying stationary point E0 . For the case of radially symmetric vortices determined by (2.17), i.e. E0 = U (r) exp(imϕ + ikz), the perturbation p should posses an azimuthal periodicity and, therefore, it can be represented as a Fourier series p(r, ϕ, z) =

∞ 

pn (r, z) exp(inϕ).

(2.20)

n=−∞

Substituting this decomposition into eq. (2.19) and matching terms with equal angular dependence, we obtain the infinite set of equations for complex modal functions pn . However, for any integer s, only two modes pm+s and pm−s are

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actually coupled and build a closed system:

∂ ∗ i + Lˆ ± pm±s + exp(i2kz)Apm∓s = 0, ∂z

[5, § 2

(2.21)

where A ≡ U 2 F0 and Lˆ ± ≡ d2 /dr 2 + r −1 d/dr − (m ± s)2 r −2 + F0 + A. Solution to these equations is given by pm+s (r, z) = us (r) exp(ikz + γs z) and pm−s (r, z) = vs∗ (r) exp(−ikz + γs∗ z), with complex perturbation wave number γs and the modes us and vs that solve eigenvalue problem      −A us us k − Lˆ + iγs (2.22) = . − ˆ vs vs A −k + L In these notations, if the eigenvalue γs has a positive real part for some s, the perturbation modes pm±s grow exponentially with the growth rate Re γs > 0; and such modes are called instability modes. The equivalent representation of perturbation p(r, ϕ, z) = exp(imϕ+ikz){us (r) exp(isϕ+γs z)+vs∗ (r) exp(−isϕ+ γs∗ z)} guarantees, due to the completeness of the basis of azimuthal harmonics eq. (2.20), that all possible perturbations are taken into account. The instability growth rate depends on the vortex power, as is shown in fig. 6(a, b) for a particular case of spatial vortex solitons with the topological charges m = 1 and m = 2 in a saturable medium with α = 1. For any s, the growth rate vanishes in the linear limit k → 0, and also in the opposite limit of

Fig. 6. Growth rate, Re γs (k), of the instability modes for (a) single- and (b) double-charged vortex solitons in saturable medium, F (I ) = I /(1 + I ). Corresponding values of perturbation index s are shown next to the curves. After Skryabin and Firth [1998a]. Break-up of the vortex solitons with the maximum growth rate and the charges (c) m = 1 and (d) m = 2. Dashed curves at z = 50 show the peak intensity of the initial rings and trajectories of the solitons flying away after the decay.

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infinite power, k → 1/α, but at least one mode has a nonzero positive value in the whole domain of the vortex soliton existence, k ∈ [0, 1/α]. Thus, all vortex solitons are linearly unstable in saturable media. Similar results hold for the parametric interaction in a quadratic medium, as we discuss in detail in Section 4. The index of the modes with the highest growth rate depends on the model and the mode topological charge. For example, a single-charged vortex always has the s = 2 mode growing faster than other modes in a saturable medium, while this can be the s = 3 mode in a quadratic medium, and, as shown in fig. 1(b), also in a pure Kerr medium with α = 0. In general, the higher-charge vortices allow for competition between different modes with close values of the growth rate, such as the modes with s = 3 and s = 4 for m = 2, see fig. 6(b). The symmetry-breaking instability of the ring-like vortex beams has been observed experimentally in saturable vapors (Tikhonenko, Christou and LutherDavies [1995], Tikhonenko, Christou and Luther-Davies [1996], Bigelow, Zerom and Boyd [2004]), photorefractive (Chen, Shih, Segev, Wilson, Muller and Maker [1997]) and quadratic (Petrov, Torner, Martorell, Vilaseca, Torres and Cojocaru [1998]) nonlinear media. In all such cases, the generation of different numbers of fundamental solitons due to the ring instability was observed. Figure 6 shows a numerical example of the breakup scenario of the ring-like vortex solitons, when the initial stationary ring-profile structure decays, under the action of a numerical noise, into two (the case m = 1, see fig. 6(a)) or four (the case m = 2, see fig. 6(b)) fundamental solitons. The number of splinters coincides exactly with the topological index of the instability mode with highest growth rate, see fig. 6(a, b), thus the predictions of a linear stability analysis are in an excellent agreement with the numerical solution of the full system. The detailed stability analysis of solitary waves with central phase dislocation were reported by Skryabin and Firth [1998a] for both saturable self-focusing and quadratic nonlinear media, and for the latter case also by Torres, Soto-Crespo, Torner and Petrov [1998a]. In the experiments, the excitation of the ring-profile vortex beams is conducted by pumping the nonlinear media by suitable approximations of Laguerre–Gaussian modes, and it was shown earlier that not only the spontaneous symmetry-breaking instability of the vortex solitons could be observed in that geometry but also that such excitation conditions offer additional possibilities by inducing suitable instabilities (Torner and Petrov [1997a, 1997b]), as we discuss in detail in Section 4. An analytical approach to the study of the filament dynamics after the breakup, based on the conservation of the beam angular momentum and Hamiltonian, was developed by Skryabin and Firth [1998a]. Given initial values of the conserved quantities, it is possible to predict the features of the filament trajectories and

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estimate their number. Two different analytical expressions for the velocity of the filaments in the transverse plane were derived, both formulas giving a reasonably good quantitative predictions for the velocities. The formula based on angular momentum being particularly simple and effective: the escape velocity can be estimated as v ≈ |m|/R, where R is the initial radius of the vortex ring. The important conclusion, giving an insight into the underlying physics of the beam breakup, is that when filaments move out along tangents to the initial ring, they carry away its angular momentum. In our notation we can describe this effect as the transformation of initial spin angular momentum of the vortex soliton to the angular momentum of the splinters spiraling out. Stabilization of coherent vortex solitons against the azimuthal instabilities remains a major challenge in the physics of spatial vortex solitons. Several theoretical models were suggested which support stable vortex solitons, including the formation of vortex solitons in the presence of competing nonlinearities (see Section 5), nonlocal nonlinear media (Yakimenko, Zaliznyak and Kivshar [2004], Breidis, Petersen, Edmundson, Krolikowski and Bang [2005]), or Bessel photonic lattices (Kartashov, Vysloukh and Torner [2005b]), without experimental observations so far.

§ 3. Composite spatial solitons with phase dislocations In this section we describe optical vortices (and other closely related higherorder spatial solitons) composed of several beams (components) interacting via cross-phase modulation (XPM). Because all of those components contribute to the total beam intensity, the refractive index change is usually referred to as commonly induced waveguide, and such composite solitons can be thought of as corresponding guided modes. In the simplest case of two interacting beams, e.g., two orthogonally polarized components of a vector soliton, this system posses radially-symmetric vector vortices as well as azimuthally modulated multipole and necklace-ring vector solitons. For large number of mutually incoherent components the composite solitons are closely related to partially coherent selftrapped beams and vortices.

3.1. Soliton-induced waveguides Spatial solitons may be understood as the modes of the effective waveguides they induce in a nonlinear medium (Kivshar and Agrawal [2003]). A natural exten-

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sion of this concept is to assume that the waveguide induced by relatively powerful soliton beam may guide and control another, weak beam. This concept may be applicable to bright (Delafuente, Barthelemy and Froehly [1991]) as well as dark (Luther-Davies and Xiaoping [1992], Luther-Davies and Yang [1992]) spatial solitons. The possibility of effective waveguiding of a weak probe (or signal) beam via the XPM type interaction can be analyzed within the coupled system of NLS equations for two wave envelopes E1,2 ,

∂E1 + ⊥ E1 + σ c11 |E1 |2 + c12 |E2 |2 E1 = 0, ∂z

∂E2 i + ⊥ E2 + σ c21 |E1 |2 + c22 |E2 |2 E2 = 0, ∂z

i

(3.1) (3.2)

where the coefficient σ = ±1 defines a focusing and defocusing nonlinear medium, respectively. In a general case with different carrier frequencies and polarization states of two interacting beams, the contributions cnn from self-phase modulation (SPM) and the XPM coefficients cnj are all different, cnj = cj n = cnn (n, j = 1, 2 and j = n). If the two waves have the same carrier frequency but different polarization states, the interaction is symmetric, cnj = cj n and cjj = cnn , and one of them can be scaled away, e.g., SPM coefficients cnn = 1. In the latter case, the XPM coefficients cnj = 23 , for linearly polarized components, cnj = 2, for circular polarized components, and in the case of elliptically polarized components this parameter satisfied the relation: 23 < cnj < 2. Transition to the Manakov system cnj = 1, completely integrable in 1D geometry (Manakov [1974]), occurs at ellipticity angle 35◦ (Menyuk [1989]). When the signal beam, e.g., E2 , is much weaker than the soliton beam |E2 | |E1 |, the system (3.1), (3.2) can be linearized with respect to E2 . Then eq. (3.1) transforms to the NLS equation (2.11) with the nonlinear potential F (I ) = σ c11 |E1 |2 which supports stationary soliton solutions, while the signal wave E2 propagates in effectively linear regime in the waveguide with the refractive index σ c21 |E1 |2 . Depending of the actual shape of the soliton beam E1 and its peak intensity, the induced waveguide can be single-moded, i.e. supporting only the fundamental bell-shaped mode E2 , as well as multi-moded, when the higher-order guided modes, including vortices, can be guided. In focusing nonlinear media, such as anisotropic photorefractive crystals, the fundamental soliton has a bell-shaped form (Segev, Crosignani and Yariv [1992]). Similar to the 1D case (Morin, Duree, Salamo and Segev [1995], Shih, Chen, Mitchell, Segev, Lee, Feigelson and Wilde [1997]), the (green) light beam from a frequency-doubled Nd:YAG laser creates a two-dimensional spatial soliton and

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Fig. 7. (a) Higher-order modes guided by a fundamental soliton in the self-focusing photorefractive medium: plot shows the calculated distribution of the refractive index induced by the fundamental soliton and experimental photos demonstrate the intensity of three first modes of a guided red probe beam (Petter, Denz, Stepken and Kaiser [2002]). (b) Radially symmetric fundamental (top) and vortex (bottom) modes localized in the waveguide induced by the vortex soliton (thick lines) in self-defocusing medium (Carlsson, Malmberg, Anderson, Lisak, Ostrovskaya, Alexander and Kivshar [2000]).

induces an effective waveguide for the probe beam at less photosensitive wavelength; in SBN crystal it can be taken from the HeNe laser (red beam). The effective contribution to the refractive index from the red beam is negligible, i.e. both the SPM c22 ∼ 0 and XPM coefficient c12 ∼ 0 in eqs. (3.1), (3.2). At the same time the guidance of a probe beam by a single two-dimensional soliton as well as by the pair of interacting solitons is indeed possible, it was demonstrated by Petter and Denz [2001]. Furthermore, because of the anisotropy of photorefractive screening nonlinearity (Zozulya, Anderson, Mamaev and Saffman [1998]), the effective waveguide is highly anisotropic and may support higher-order modes. Figure 7(a) shows the induced refractive index profile and corresponding higherorder TEM guided modes, generated experimentally by Petter, Denz, Stepken and Kaiser [2002]. In defocusing nonlinear media, spatial solitons are associated with a nontrivial phase structure, and a two-dimensional soliton is a dark vortex, as discussed in Section 2.1. In this case, the spatial profile of the induced refractive index (“induced fiber”) supports radially symmetric spatially localized modes (Snyder, Poladian and Mitchell [1992]). The waveguiding properties of dark optical vortex solitons in self-defocusing Kerr media have been analyzed by Sheppard and Haelterman [1994], Law, Zhang and Swartzlander [2000], Carlsson, Malmberg, Anderson, Lisak, Ostrovskaya, Alexander and Kivshar [2000]. It was shown that

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these properties depend crucially on the relative strength of the cross- and selfphase modulation effects. Families of composite solitons formed by a vortex and its guided mode with or without a topological charge have been identified. Examples of these modes are presented in fig. 7(b). A soliton-waveguiding experiment has been conducted by Truscott, Friese, Heckenberg and Rubinsztein-Dunlop [1999] in an atomic vapor. A weak probe beam tuned near one atomic resonance is guided through a waveguide written by an intense pump vortex beam at a different atomic resonance. As the pump beam is tuned close to resonance, it creates a nonlinear refractive index profile in the atomic vapor with which the weak probe beam interacts. The efficiency of the guiding is found to depend strongly on the power and frequency of the guiding beam. Moreover, since the guiding takes place in an atomic vapor, it is possible to tune to both sides of the atomic resonance. This has the distinct advantage that it allows the guiding of light into either bright or dark regions of the guiding beam. The theory of waveguides electromagnetically induced in Rb vapors was developed by Kapoor and Agarwal [2004]. Their density matrix approach was based on the three-level V-system, and it was generalized later to the fivelevel model by Andersen, Friese, Truscott, Ficek, Drummond, Heckenberg and Rubinsztein-Dunlop [2001], who took into account the hyperfine structure of the D-line of rubidium as well as the presence of the two major isotopes. The results allow one to deduce which frequency combinations are likely to give successful guiding. Systematic analysis of the waveguiding properties of the vortex solitons and vortex-mode vector solitons in saturable nonlinear media, for both selfdefocusing and self-focusing nonlinearities, was performed recently by Salgueiro and Kivshar [2004a]. Following the earlier analysis of Carlsson, Malmberg, Anderson, Lisak, Ostrovskaya, Alexander and Kivshar [2000], the authors examine two major regimes of the vortex waveguiding. The most interesting nonlinear regime corresponds to large intensities of the guided beam, and it gives rise to composite (or vector) solitons with a vortex component, that have been identified and analyzed numerically. In quadratic media, the simultaneous guidance of both fundamental and second-harmonic waves by an optical vortex soliton has been analyzed by Salgueiro, Carlsson, Ostrovskaya and Kivshar [2004]. These authors describe novel types of three-component vector soliton created by a vortex beam together with both fundamental and second-harmonic parametrically coupled localized modes and determine conditions for a potential enhancement of the conversion efficiency.

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3.2. Higher-order vector solitons When the guided beam is weak, i.e. in the linear waveguiding regime, the analysis of the soliton waveguiding properties can be carried out by using approximate analytical methods, and reducing the problem to the well-known analysis of the linear guided-wave theory (for example, see Law, Zhang and Swartzlander [2000] and also the theory of optical vortices in optical fibers by Volyar [2002]). For a finite-amplitude probe beam, the linearization in eqs. (3.1), (3.2) is no longer valid, and the nonlinear theory of soliton-induced waveguides should be developed (Ostrovskaya and Kivshar [1998]); this theory takes into account mutual effect of the interacting waves on each other. Several light beams generated by a coherent source can be combined to produce a vector soliton with a complex internal structure. Properties of two-component vector solitons have been extensively studied in both self-focusing (Manakov [1974], Christodoulides and Joseph [1988], Snyder, Hewlett and Mitchell [1994], Christodoulides, Singh, Carvalho and Segev [1996], Krolikowski, Akhmediev and Luther-Davies [1996]) and self-defocusing (Kivshar and Turitsyn [1993], Haelterman and Sheppard [1994]) nonlinear media. The structure of multicomponent vector soliton may become rather complicated and, for example, the soliton intensity profile may display several peaks. These so-called “multihump” solitons propagate as the corresponding higher-order modes of the soliton-induced waveguides. The first experimental demonstration of such multicomponent solitons has been reported by the Princeton group (Mitchell, Segev and Christodoulides [1998]), who observed both single and multi-peak spatial solitons. Linear stability of the two-component vector solitons has been studied by Ostrovskaya, Kivshar, Skryabin and Firth [1999], and it has been shown that the two-peak structures can be stable in propagation while three-peak soliton structures are unstable. Recently, the concept of multi-hump spatial solitons has been extended to two transverse dimensions. First, Musslimani, Segev, Christodoulides and Soljacic [2000], Musslimani, Segev and Christodoulides [2000] studied stationary propagation of the vortex-mode vector soliton which has a nodeless shape in one component and a vortex in the other component, see fig. 8(b). However, it appears that such a radially symmetric, ring-like vector soliton (which is analogous to the Laguerre–Gaussian modes of cylindrical waveguide) may undergo a symmetry-breaking instability (Garcia-Ripoll, Perez-Garcia, Ostrovskaya and Kivshar [2000], Malmberg, Carlsson, Anderson, Lisak, Ostrovskaya and Kivshar [2000]) which transforms it into a radially asymmetric dipole-mode vector soliton, even in a perfectly isotropic nonlinear medium, see fig. 8(a). The dipole-mode

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Fig. 8. Examples of constituents of (a) dipole-mode (s = 0.3) and (b) vortex-mode (s = 0.65) two-component vector solitons (top row: |E1 |2 , middle row: |E2 |2 ) shown with the phase distributions needed to generate the E2 -modes experimentally (bottom row), after Krolikowski, Ostrovskaya, Weilnau, Geisser, McCarthy, Kivshar, Denz and Luther-Davies [2000]. Delayed-action interaction between the vortex-mode vector solitons and formation of spiraling dipole (Musslimani, Soljacic, Segev and Christodoulides [2001a]).

vector soliton is a novel type of an optical vector soliton that originates from trapping of a dipole HG01 -type mode by a fundamental-soliton-induced waveguide created by another, incoherently coupled, co-propagating beam. Moreover, it has been shown that, while many other topologically complex structures may be created, it is only the dipole mode that is expected to generate a family of dynamically robust vector solitons. Very recently, the rigorous stability analysis performed by Yang and Pelinovsky [2003], has shown that very close to the bifurcation line, where the vortex and dipole components are small, both solutions are linearly stable. Far from the bifurcation line, the family of vortex solitons becomes azimuthally unstable, while the dipole-mode solitons remain stable in the whole domain of their existence. Dipole-mode vector solitons have been observed experimentally in photorefractive media by Krolikowski, Ostrovskaya, Weilnau, Geisser, McCarthy, Kivshar, Denz and Luther-Davies [2000]. Two different methods have been used, one is based on a phase imprinting technique, and the other uses the symmetrybreaking instability of vortex-mode soliton. In the latter case, the initial angular momentum of vortex component is transformed to the dipole and leads to its rotation during propagation after the break-up. Skryabin, McSloy and Firth [2002]

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showed that rotational velocity provides an additional parametrization for the dipole-soliton family, and, with the help of a generalized Vakhitov–Kolokolov stability criterion, they predicted stability thresholds for spiraling solutions. Rotating dipole-mode soliton can be viewed as an optical “propeller” because of the mutually tilted phase fronts of the dipole lobes (Carmon, Uzdin, Pigier, Musslimani, Segev and Nepomnyashchy [2001]). In anisotropic photorefractive media, however, the rotation of the dipole is limited because of a preferable direction for its orientation (Neshev, McCarthy, Krolikowski, Ostrovskaya, Kivshar, Calvo and Agullo-Lopez [2001], Motzek, Stepken, Kaiser, Beli´c, Ahles, Weilnau and Denz [2001]). It appears that the interaction of composite solitons, determined by the effective interaction potential (Malomed [1998]), depends significantly on their angular momenta. Musslimani, Soljacic, Segev and Christodoulides [2001a] demonstrated numerically that collisions of vortex-mode solitons with opposite topological charges (“spins”) can lead to mutual trapping of two composite beams and the formation of bound state with a prolonged lifetime of about 35 diffraction lengths. The metastable bound state eventually disintegrates giving rise to new vector solitons, the process is characterized as the “delayed action interaction”. If both colliding vortex-mode solitons have the same topological charge, the resulting object is a stable rotating dipole-mode soliton, as is shown in fig. 8(c). These authors draw an analogy with spin–orbit coupling in interaction of soliton as effective “particles”. The comprehensive study of collisions between vortex-mode solitons was reported by Musslimani, Soljacic, Segev and Christodoulides [2001b], and out-of-plane scattering of the dipole-mode solitons was studied by Pigier, Uzdin, Carmon, Segev, Nepomnyashchy and Musslimani [2001], Krolikowski, McCarthy, Kivshar, Weilnau, Denz, Garcia-Ripoll and Perez-Garcia [2003]. The dipole-mode vector soliton is the first example of azimuthally-modulated spatial solitons. As we noted above, close to the bifurcation line this composite structure can be described as a linear mode guided by a scalar fundamental soliton. Natural extension of this approach is to search for higher-order modes such as multipole-mode vector solitons. Indeed, the vortex-mode soliton can be described as a coherent superposition of two dipole components twisted in space and shifted in phase by π2 with respect to each other. Similarly, the higher-charge optical vortex might be constructed from multi-poles, e.g., double charged vortex consists of two quadrupoles. This analogy with linear waveguide theory suggests the general ansatz for the azimuthally modulated component of a composite spatial soliton, E2 (x, y, z) = U (r){cos mϕ + ip sin mϕ} exp(ikz),

(3.3)

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Fig. 9. Experimental demonstration of multipole-mode solitons: (a) quadrupole vector soliton, and (b) hexapole vector soliton. Intensities of two components shown at the output after independent (middle columns) and simultaneous (right columns) nonlinear propagation. In all cases, the propagation distance is z = 10 mm. After Desyatnikov, Neshev, Ostrovskaya, Kivshar, McCarthy, Krolikowski and Luther-Davies [2002].

where the parameter p is real, m is integer, and k is the propagation constant. For p = 1, eq. (3.3) describes a vortex-mode soliton with the charge m and the radially symmetric intensity U 2 (r). This ansatz has been used by Desyatnikov, Neshev, Ostrovskaya, Kivshar, Krolikowski, Luther-Davies, Garcia-Ripoll and Perez-Garcia [2001] as a variational approximation to the modes (E2 -component in equations similar to eqs. (3.1), (3.2)) of the waveguide, induced by fundamental soliton in the E1 -component. Experimental results of the generation of quadrupole and hexapole vector solitons are shown in fig. 9. The corresponding parameters in the ansatz eq. (3.3) are m = 2 and m = 3 correspondingly, with p = 0 in both cases. For p = 0, the angular momentum of the azimuthally modulated solitons is nonzero and this leads to the soliton rotation. Incoherent interaction between the components of a composite (or vector) ringlike beam allows to compensate for repulsion of beamlets, creating a new type of quasi-stationary self-trapped structure exhibiting the properties of the necklacering beams and ring vortex solitons. The physical mechanism for creating such composite vector ringlike solitons is somewhat similar to the mechanism responsible for the formation of the so-called soliton gluons (Ostrovskaya, Kivshar, Chen and Segev [1999]) and multi-hump vector solitary waves (Ostrovskaya, Kivshar, Skryabin and Firth [1999]), and it is explained by a balance of the interaction forces acting between the coherent and incoherent components of a composite soliton. In that case, the mutual repulsion of out-of-phase beamlets in the E2 component is balanced by the incoherent attraction of the mutually coupled E1 component.

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3.3. Multi-component vortex solitons The effective optical waveguide induced by the fundamental soliton or dark vortex soliton in a self-focusing or self-defocusing nonlinear media, has a relatively simple bell-like shape whose modes are well known from the linear guided-wave theory. In contrast, the doughnut shape of a bright vortex soliton does not allow simple predictions about its guided modes, and, consequently, the possible structure of multi-component vortex solitons. Moreover, similar to the one-dimensional multi-hump vector solitons, their two-dimensional counterparts have a complex nonmonotonous radial envelope (Musslimani, Segev, Christodoulides and Soljacic [2000]). For example, in addition to the well-understood “first” bifurcation from the scalar fundamental soliton, giving rise to the vortex-mode solitons, the corresponding stationary solutions have the second bifurcation point where the “guided” component transforms to the scalar vortex (Desyatnikov, Neshev, Ostrovskaya, Kivshar, McCarthy, Krolikowski and Luther-Davies [2002]). This situation can also be regarded as guiding of a simple bell-shaped beam by a ringlike waveguide. Very recently, the comparison between the so-called single- and double-vortex solitons has been carried out by Salgueiro and Kivshar [2004b]. Here the notation “single-vortex” is used to distinguish a vortex-mode soliton, which consists of a strong fundamental component and a weak guided vortex (see fig. 8(b)), from the other case (“second” bifurcation) with a strong vortex component and a small fundamental beam. At the same time, the ring vortex waveguide also supports double-vortex (or “vortex-vortex”) vector solitons. Several types of the double-vortex solitons are shown in the fig. 10(a). Extension of the concept of two-dimensional multi-hump vector solitons to the case of larger number of mutually incoherent components was proposed by Musslimani, Segev and Christodoulides [2000], who studied a radially symmetric potential commonly created by a strong fundamental soliton and several vortex beams. Similar idea applied to the ring vortex beams result in the so-called “necklace-ring” vector solitons (Desyatnikov and Kivshar [2001]). In general, the interaction of N paraxial beams via XPM can be described by the system of coupled NLS equations for the envelopes En (x, y, z) (n = 1, 2, . . . , N ), ∂En 2 + ∇⊥ En + F (I )En = 0, (3.4) ∂z where, similar to eq. (2.11), the function F (I ) describes the nonlinear refractive  2 index, and the total beam intensity is defined as I = N n=1 |En | . In contrast to the system (3.1), (3.2), all the SPM and XPM contributions are taken to be equal here for simplicity. In addition to a variety of radially symmetric solutions, such as those displayed in fig. 10(a), there exists a special class of azimuthally modulated i

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stationary states. The simplest route to find this class of spatial solitons is to search for the modes of a radially symmetric potential with I = U 2 (r). This is possible if all components have the same radial envelope U (r), i.e. we are looking for the stationary solution in the form En (x, y, z) = U (r)Φn (ϕ) exp(ikz),

(3.5)

with the complex azimuthal envelopes Φn (ϕ) = an cos mϕ+bn sin mϕ selected in   2 such a way that N Re(an bn∗ ) = 0 n=1 |Φn | = 1. The latter condition requires   2 2 and |an | = |bn | = 1. These equations define exact solutions to the system (3.5) for any N and, in the particular case N = 1, they describe a scalar vortex of the charge m with a = 1 and b = i. Radially symmetric multi-component vortices correspond to the symmetric case, bn = ±ian ; in this case the total soliton spin is integer m or zero, for the counter-rotating vortices. In a general case, the components resemble azimuthally modulated optical necklaces, introduced by Soljacic, Sears and Segev [1998] (see Section 6.2), and the total spin may take a fractional value. The simplest two-component solution of the necklace-ring type with m = 1 is given by a1 = b2 = 1, a2 = b1 = 0, and it represents two crossed dipoles, shown in fig. 10(b). Although such a structure has no vorticity and carries zero angular momentum, the total intensity has a profile of a single-charge scalar vortex soliton. This perfect symmetry helps to find such solutions, but it is not crucial for their existence. Indeed, similar solutions have been found numerically and generated

Fig. 10. (a) Envelopes of different types of double-vortex vector solitons (Salgueiro and Kivshar [2004b]). (b) Intensities of two components of a dipole–dipole vector soliton. The total intensity of this soliton is a perfect ring as in fig. 5(a) (Desyatnikov and Kivshar [2001]).

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experimentally by Ahles, Motzek, Stepken, Kaiser, Weilnau and Denz [2002] in an anisotropic photorefractive medium, which posses no radially symmetry. The main outcome of these studies, presented by Desyatnikov and Kivshar [2001], is that the vectorial interaction allows for additional stabilization of otherwise nonstationary beams, such as expanding necklace beams. Nevertheless, no linearly stable necklace-ring solitons have been found so far. Similar stabilization of counter-rotating vector vortices was reported for the case of a self-focusing saturable medium by Bigelow, Park and Boyd [2002], and was demonstrated experimentally in a self-defocusing photorefractive medium by Mamaev, Saffman and Zozulya [2004]. The total angular momentum for interacting vortices with opposite topological charges is less than that of co-rotating vortices, and this was found to be the main reason for long lifetimes. Theoretically, the maximal growth rate of the azimuthal instability is significantly smaller for zero-spin vector solitons (Ye, Wang, Dong and Li [2004]). The azimuthal instability can be eliminated completely in the so-called cubic-quintic (CQ) model, as discussed in Section 5.1, and there exists also the stability domain for the corotating vortices even in the presence of four-wave mixing (Mihalache, Mazilu, Towers, Malomed and Lederer [2002]). The counter-rotating vortices with zero total angular momentum have smaller stability domain and exhibit an interesting “internal” instability dynamics in CQ medium (Desyatnikov, Mihalache, Mazilu, Malomed, Denz and Lederer [2005]). Due to the exchange of the angular momentum between interacting components, the soliton slowly reverse the topological charges, keeping the zero total angular momentum and perfectly stable total intensity. This phenomenon is related to “charge-flipping” effect predicted by Alexander, Sukhorukov and Kivshar [2004] to occur for the discrete vortices in two-dimensional optical lattices (see Section 7). Very recently, Park and Eberly [2004] showed that the necklace-type solutions exist for the model of two-component Bose–Einstein condensates where the symmetry between the SPM and XPM contributions to the nonlinear interaction is broken. These nontopological vortices exhibit “spin” dynamical behavior and may accumulate the Berry phase under an adiabatic change of external fields that control the trapping potential (see also Section 8.2). The generalization of the concept of necklace-ring vector solitons includes the temporal effects in dispersive media, where the three-dimensional spatiotemporal vortex solitons, or spinning light bullets, have been predicted to exist (Desyatnikov, Maimistov and Malomed [2000]). Corresponding solutions were found by Andersen and Kovachev [2002] for nonlinear Maxwell equations and by Kovachev [2004] for Maxwell–Dirac equations. The internal structure of the

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three-dimensional vector vortices with radially symmetric total intensity is described in this case by spherical harmonics. Finally, combining the counter-rotating vortices with strong guiding from the fundamental soliton leads to a possibility to stabilize completely the necklace-ring type solutions, as was shown by Desyatnikov, Kivshar, Motzek, Kaiser, Weilnau and Denz [2002], Motzek, Kaiser, Weilnau, Denz, McCarthy, Krolikowski, Desyatnikov and Kivshar [2002] in the case of three-component composite solitons. With further increasing the number of interacting components, the composite beams posses a complex internal structure, and they can serve as the modal approximation for spatially incoherent and partially coherent beams. The latter however posses the distinctive features which we describe below.

3.4. Partially coherent vortices As was demonstrated in all examples presented above, optical vortices occur in coherent systems having a vanishing intensity at the vortex position and welldefined phase front topology being associated with the circulation of momentum around the helix axis. If a vortex-carrying beam is partially incoherent, the phase front topology is not well defined, and statistics are required to quantify the phase. In the incoherent limit neither the helical phase nor the characteristic zero intensity at the vortex center is observable. However, several recent studies have shed light on the question of how phase singularities can develop in incoherent light fields and how these phase singularities can be unveiled (Gbur and Visser [2003], Schouten, Gbur, Visser and Wolf [2003], Palacios, Maleev, Marathay and Swartzlander [2004]). In particular, Palacios, Maleev, Marathay and Swartzlander [2004] used experimental and numerical techniques to explore how a beam transmitted through a vortex phase mask changes as the transverse coherence length at the input of the mask is changed. Assuming a quasi-monochromatic, statistically stationary light source and ignoring temporal coherence effects, they demonstrated that robust attributes of the vortex remain in the beam, most prominently in the form of a ring dislocation in the cross-correlation function. Propagating in nonlinear coherent systems, optical vortices become highly unstable when the nonlinear medium is self-focusing, see Section 2.3. However, when spatial incoherence of light exceeds a certain threshold, the stable propagation of optical vortices in self-focusing nonlinear media is possible and has been recently demonstrated in experiments conducted with a biased photorefractive SBN crystal. Jeng, Shih, Motzek and Kivshar [2004] generated partially incoherent vortices and vortex solitons, and then inspected their stability. First, a cw laser

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light beam (at 488 nm) of the extraordinary polarization was made partially incoherent by passing it through a lens and then through a rotating diffuser. The rotating diffuser introduced random-varying phase and amplitude on the light beam every 1 µs, which is much shorter than the response time (about 1 s) of photorefractive crystal. By adjusting the position of the diffuser to near (away from) the focal point of the lens in front the diffuser, it is possible to increase (decrease) the degree of the light coherence, and collect the light after the rotating diffuser by a second lens and then pass is through a computer-generated hologram to imprint a vortex phase. Since the partially incoherent light beam can be considered as a superposition of many mutually-incoherent light beams, the first-order diffracted light beam after the hologram becomes a superposition of many mutuallyincoherent vortex beams. Then, the partially coherent vortex beam was launched into the SBN crystal along its a-axis. The total power of the vortex beam is of 0.17 µW, which results in the nonlinearity of the photorefractive crystal falling into the Kerr region when the ratio of the peak intensity of the vortex beam to the background intensity is much less than unity. A lens was used to project the images at the input and output faces onto a CCD camera. When the diffuser is removed from the experimental setup, the vortex beam at the input face of the crystal is shown as fig. 11(a). While a 2.5 kV biasing

Fig. 11. Numerical and experimental results showing the stabilization of the vortex with growing incoherence: (a) input intensity, (b) vortex after 9 mm of propagation for the coherent case, (c) vortex after 9 mm for the partially incoherent case, θ0 = 0.14 (less coherent), and (d) vortex after 9 mm for the partially incoherent case, θ0 = 0.29 (least coherent). The incoherence stabilize the vortex soliton when voltage of 2.5 kV is applied.

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voltage is applied on the photorefractive crystal creating a Kerr-type self-focusing nonlinear medium, the vortex beam breaks up into two pieces (fig. 11(b)). This vortex break-up is due to the azimuthal instability. As the rotating diffuser is used, fig. 11(c) clearly shows that the vortex light beam is stabilized by the reduction of the degree of coherence though two very unclear bright spots still can be seen on the opposite sides (top and bottom) of the ring-like intensity distribution. The rotating diffuser has been then moved further away from the focal point of the lens to make the light more incoherent, fig. 11(d) shows the generated stable partially incoherent vortex soliton at the output face of the crystal. The propagation of partially incoherent optical vortices in a photorefractive medium has been studied numerically by Jeng, Shih, Motzek and Kivshar [2004] using the coherent density approach developed by Christodoulides, Coskun, Mitchell and Segev [1997], Anastassiou, Soljacic, Segev, Eugenieva, Christodoulides, Kip, Musslimani and Torres [2000]. The coherent density approach is based on the fact that partially incoherent light can be described by a superposition of mutually incoherent light beams that are tilted with respect to the z-axis at different angles. One thus makes the ansatz that the partially incoherent light consists of many coherent, but mutually incoherent light beams Ej : I =  √ 2 2 2 2 j |Ej | . By setting |Ej | = G(j ϑ)I , where G(θ ) = (1/ πθ0 ) exp(−θ /θ0 ) is the angular power spectrum, one obtains a partially incoherent light beam whose coherence is determined by the parameter θ0 , i.e. less coherence means larger θ0 . Here, j ϑ is the angle at which the j th beam is tilted with respect to the z-axis. A set of 1681 mutually incoherent vortices was used in simulations, all initially tilted at different angles. The top row in fig. 11(a–d) shows numerical results for the propagation of an input Gaussian beam carrying a phase dislocation (a) after the total propagation (9 mm) in a nonlinear medium for the coherent light (b) and two different partially incoherent beams (c, d), corresponding to the values θ0 = 0.14 and θ0 = 0.29, respectively. The most obvious difference to the scenario of the propagation the coherent vortex is that the vortex decay undergoes a visible delay when the degree of incoherence grows. Furthermore, in the incoherent case the vortex changes its profile only very slowly as it propagates and thus can be considered as being in a transition stage between the decay and stabilization. Spatial coherence properties of optical vortices created in partially coherent light were studied by Motzek, Kivshar and Swartzlander [2004], Motzek, Kivshar, Shih and Swartzlander [2004], who revealed the existence of phase singularities in the spatial coherence function of a vortex field that can characterize the stable propagation of vortices through nonlinear media. Thus, the phase singularities of the spatial coherence function predicted to exist in incoherent vortices propagat-

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ing in linear media (Palacios, Maleev, Marathay and Swartzlander [2004]) also survive the propagation through nonlinear media. The intensity distribution in the far field shows a local minimum in the center of the beam, contrary to what one would obtain if the vortex was propagating through a linear medium, and also in contrast to the result we would obtain if we were propagating a light beam without topological charge. This emphasizes the importance of the interaction between coherence and nonlinearity. Not only the phase structure, but also the intensity distribution strongly depends on the initial form of the coherence function of the light beam. The interaction of a coherent vortex beam with partially coherent fundamental soliton, similar to the vortex-mode soliton discussed in Section 3.2, was considered recently by Motzek, Kaiser, Salgueiro, Kivshar and Denz [2004]. Strong destabilization and enhancement of azimuthal instability of vortex component is observed for a low-amplitude incoherent beam. In the opposite limit, vortex can be stabilized by a large-amplitude fundamental beam with the value of its incoherence above a certain threshold. These results are consistent with the stabilization dynamics of a coherent vortex- and dipole-mode solitons (Yang and Pelinovsky [2003]).

§ 4. Multi-color vortex solitons Similar to Kerr-type (or χ (3) ) nonlinear media, self-induced trapping of light occurs in quadratic (χ (2) ) nonlinear media (Karamzin and Sukhorukov [1974, 1975], Kanashov and Rubenchik [1981]). In this case, both spatial and temporal multicolor solitons form through the mutual focusing and trapping of the waves parametrically interacting in the nonlinear medium. Occurrence of self-focusing effects in quadratic nonlinear processes were sporadically suggested under specific conditions, namely when the parametric interaction is weak resulting in an effective third-order effect for the pump wave (Ostrovskii [1967], Flytzanis and Bloembergen [1976]). However, it took two decades before such effective nonlinearityinduced phase shift was identified experimentally (Belashenkov, Gagarsky and Inochkin [1989], DeSalvo, Hagan, Sheik-Bahae, Stegeman, Vanstryland and Vanherzeele [1992]), and until the importance of the associated, so-called cascaded nonlinearities was properly appreciated by Stegeman, Sheik-Bahae, Vanstryland and Assanto [1993] (for a review, see Stegeman, Hagan and Torner [1996]). Since then, formation of spatial and temporal multi-color solitons has been observed experimentally in a variety of physical settings (Torner and Stegeman [2001], Buryak, Di Trapani, Skryabin and Trillo [2002]) after the pioneering observations

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in the case of second-harmonic generation (SHG) by Torruellas, Wang, Hagan, Vanstryland, Stegeman, Torner and Menyuk [1995] in a crystal of potassium titanium phosphate (KTP), and by Schiek, Baek and Stegeman [1996] in a planar waveguide made of lithium niobate (LiNbO3 ). Cascaded nonlinearities can be modeled by the standard theory of the χ (2) mediated three-wave mixing described in detail in several books on nonlinear optics (Shen [1984], Butcher and Cotter [1992], Boyd [1992]), including more complex multistep parametric processes in quadratic media (Saltiel, Sukhorukov and Kivshar [2004]). A comprehensive review on optical quadratic solitons was published by Buryak, Di Trapani, Skryabin and Trillo [2002], and can be also found in a recent book by Kivshar and Agrawal [2003]. A number of overview papers on the theory and experimental generation of spatial parametric optical solitons in quadratic nonlinear media were published during the last years (Stegeman, Schiek and Fuerst [1997], Kivshar [1997], Torner [1998], Etrich, Lederer, Malomed, Peschel and Peschel [2000], Stegeman [2001], Sukhorukov [2001], Torruellas, Kivshar and Stegeman [2001]), and summaries of the advances in the field were reported by Kivshar [1998], Stegeman [1999], Torner and Stegeman [2001], Torner and Sukhorukov [2002], Torner and Barthelemy [2003]. In this section, we describe briefly the physics and the salient properties of the so-called quadratic vortex solitons, i.e., self-trapped multi-color optical beams, composed of several waves carrying nested optical vortices and parametrically interacting in a phase-matchable quadratic crystal under conditions close to phasematching. Most studies and experiments have been conducted for the simple case of SHG, or frequency doubling, and parametric down-conversion, thus we concentrate on the corresponding families of ring-like vortex solitons. We discuss the spontaneous and induced modulational-instabilities that affect the ring-shaped beams, the potential stabilization of quadratic vortices by competing nonlinearities, and we summarize the result of the available experimental observations. Vortex solitons are two-dimensional light beams, therefore we concentrate in quadratic vortex solitons in bulk media.

4.1. Model The SHG process for generating a double-frequency wave is a special case of a more general three-wave mixing parametric processes which occur in a dielectric medium with a quadratic nonlinear response. The typical three-wave mixing and SHG processes require only one phase-matching condition to be satisfied and, therefore, they can be classified as single phase-matched parametric processes.

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Following the detailed derivations presented by Menyuk, Schiek and Torner [1994], Bang [1997], and in Buryak, Di Trapani, Skryabin and Trillo [2002], we consider parametric interaction between three stationary quasi-monochromatic  waves with the electric fields E = ej Ej (x − ρj z, y, z) exp(ikj z − iωj t) + c.c. (where j = 1, 2, 3), with the three frequencies satisfying the energy-conservation condition, ω1 + ω2 = ω3 . We assume that these three beams propagate along the same direction z. However, when appropriate, the energy walk-off due to birefringence should be taken into account by introducing the angles ρj 1 between the Poynting vector and the wave vectors. For concreteness, we take the special case of type II SHG when two fundamental waves ω1 = ω2 = ω with the ordinary (E1 , ρ1 = 0) and extraordinary (E2 , ρ2 = 0) polarizations interact with the secondharmonic (SH) wave, E3 (ω3 = 2ω), ρ3 = 0. We assume that waves propagate under conditions close to perfect phase-matching, with a small mismatch between the three wave vectors given by the parameter k = k1 (ω1 ) + k2 (ω2 ) − k3 (ω3 ). In the slowly varying envelope approximation, one can derive the following set of three parametrically coupled equations: ∂a1 1 (4.1) + a1 + a3 a2∗ e−iβz = 0, ∂z 2 α2 ∂a2 ∂a2 − iδ2 + a2 + a3 a1∗ e−iβz = 0, i (4.2) ∂z ∂x 2 ∂a2 ∂a3 α3 i (4.3) − iδ3 + a3 + a1 a2 e+iβz = 0. ∂z ∂x 2 Here the Laplace operator  acts on a transverse coordinates (x, y) normalized to a characteristic beam width r, and the propagation coordinate is measured in units twice the diffraction length, so that the normalized phase mismatch is β = k1 r 2 k. A typical value of the dimensionless mismatch parameter, β = ±3, is obtained for a focused beam with r 15 µm, and the mismatch π/|k| 2.5 mm. Other parameters are: δj = k1 rρj , αj = k1 /kj (we use α1 ≡ 1); in practice, α2 1 and α3 0.5. Finally, a normalized field amplitude of some |a|2 ∼ 10 corresponds to an actual power flow in the range of 1–10 GW/cm2 in a typical quadratic nonlinear crystal, such as KTP. Of course, the above estimates greatly depend on the particular properties of the material employed and on the pump light conditions, thus for a detailed derivation of the governing equations and estimates for different materials, we refer to the several reviews mentioned above. In the case of type I SHG, only a single beam at the pump frequency ω interacts with a field at the frequency 2ω. Equations (4.1)–(4.3) can then be reduced by setting a1 = a2 = u, and δ2 = 0, δ3 = δ, and v = a3 exp(iβz). One obtains: i

i

∂u 1 + u + u∗ v = 0, ∂z 2

(4.4)

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∂v ∂v 1 (4.5) − iδ − βv + v + u2 = 0. ∂z ∂x 4 Indeed, in what is sometimes refereed as the cascading limit, corresponding to large values of the phase mismatch parameter, β  1, and to weak SH signals, eq. (4.5) approximately reduces to a simple relation, v u2 /β, and the fundamental harmonic satisfies the familiar cubic NLS equation, i

i

1 ∂u 1 + u + |u|2 u 0, ∂z 2 β

(4.6)

which poses well-known stable soliton solutions in one-dimensional geometries. However, it is worth stressing that most quadratic solitons occur under conditions where the above reduction does not hold. One obvious example is the case analyzed here of ring-profile vortex solitons, which exist in two-dimensional geometries. Similarly, the above derivation does not hold near phase-matching, whenever the second-harmonic waves are intense, and in general with high enough light intensities. However, these are the conditions where most quadratic solitons are generated in practice. Therefore, the analogy indicated by eq. (4.6) must be used with the proper understanding of its important limitations in the interpretation of most experiments.

4.2. Frequency doubling with vortex beams Second-harmonic generation is a particular case of frequency conversion processes associated with the energy transfer between several waves propagating in a nonlinear medium. The nonlinear wave-mixing obeys conservation of energy, linear momentum, and, under proper conditions, angular momentum. In the general case of frequency mixing in which two fields of optical frequencies ω1 and ω2 combine to produce a third field of frequency ω3 , conservation of energy requires the condition ω1 + ω2 = ω3 . Conservation of the linear momentum leads to the phase-matching requirement k1 + k2 = k3 . Under proper conditions and definitions, the angular momentum carried by the light beam must also be conserved. Conservation of spin angular momentum imposes constraints in the polarization of the input and output light beams. In the case of co-linear interaction of paraxial light beams in a quadratic medium, conservation of the paraxial beam orbital angular momentum holds too, and affects the spatial shape of the generated beams. The simplest situation occurs in the case of type I frequency doubling of single Laguerre–Gaussian (LG) pump modes (Basistiy, Bazhenov, Soskin and Vasnetsov [1993], Dholakia, Simpson, Padgett and Allen [1996], Soskin and Vasnetsov [1998], Allen, Padgett and Babiker

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[1999]). In this case, the winding number of the LG pump modes must double, because of the azimuthal symmetry of the governing equations. In the general case of type II phase-matching or frequency-mixing of single LG modes with winding numbers mj , j = 1, 2, 3, the symmetry of the equations leads to m1 + m2 = m3 , as it is observed experimentally by Berzanskis, Matijosius, Piskarskas, Smilgevicius, and Stabinis [1997]. However, conservation of orbital angular momentum in parametric processes with arbitrary input beams and general phase-matching geometries does not necessarily translate into simple algebraic rules between the winding numbers of the vortices present in the beam. Illustrative cases are multi-mode and complex pump beams (Petrov and Torner [1998], Berzanskis, Matijosius, Piskarskas, Smilgevicius and Stabinis [1998], Beržanskis, Piskarskas, Smilgeviˇcius, Stabinis and Di Trapani [1999], Petrov, Molina-Terriza and Torner [1999], Molina-Terriza and Torner [2000], Jarutis, Matijosius, Smilgevicius and Stabinis [2000], Stabinis, Orlov and Jarutis [2001]), or the generation of vortex-streets by the presence of Poynting-vector walk-off (Molina-Terriza, Torner and Petrov [1999], MolinaTerriza, Petrov, Recolons and Torner [2002]). Actually, the orbital angular momentum of a light beam is not necessarily directly related to the properties of the vortices that it contains. Notice also that, in general, the angular momentum at the classical level is an overall property of the light beam, and must be clearly distinguished from the quantum angular momentum at the single photon level. That in the latter case the conservation of orbital angular momentum in parametric processes is not necessarily given by simple algebraic rules is most clearly illustrated in noncollinear geometries (Molina-Terriza, Torres and Torner [2003]), or in the socalled transverse-emitting processes (Torres, Osorio and Torner [2004]). In what follows we concentrate in vortex solitons, which are intense light beams carrying energies which might exceed tens of µJ at visible or near-infrared wavelengths, thus far from any single-photon effects. In the next subsections, we discuss the families and stability properties of quadratic vortex solitons – nonlinear optical beams with strong energy exchange between its constituents.

4.3. Families of the vortex solitons Because the type II phase-matching process involves three beams, there is a wider variety of different solutions than in the case of the type I SHG geometries, which we will regard as a limit of degeneracy of a former model when both fundamental frequency (FF) beams are of the same polarization, see Section 4.1. The type II

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model described by eqs. (4.1)–(4.3) defines an infinite-dimensional Hamiltonian system with a conserved Hamiltonian which in the absence of walk-off (δj = 0) is given by     1 1 2 2 ∗ ∗ ∗ αj |∇⊥ Aj | + β|A3 | − A1 A2 A3 − A1 A2 A3 dr⊥ , H = 2 2 (4.7) where A1,2 = a1,2 , and A3 = a3 exp(−iβz). We use two additional conserved quantities of the beam evolution, namely the total beam power or energy flow I ,     1 1 |A1 |2 + |A2 |2 + |A3 |2 dr⊥ , I= Ij = (4.8) 2 2 and the energy imbalancing Ii , 

1 Ii = |A1 |2 − |A2 |2 dr⊥ . 2

(4.9)

The conservation of Ii means that the energy transfer between the fundamental waves and the second-harmonic wave cannot favor any of the two orthogonal polarizations that compose the fundamental beam. Notice that strictly speaking the above expressions hold only for continuous-wave light propagation, and that the temporal effects on pulsed light might introduce important new features in the imbalancing of the quadratic solitons (Minardi, Yu, Blasi, Varanavicius, Valiulis, Berzanskis, Piskarskas and Di Trapani [2003]). In the absence of the Poynting vector walk-off, the total beam orbital angular momentum, defined as  

, 1 *+ A∗j ∇⊥ Aj − Aj ∇⊥ A∗j ez dr⊥ , r⊥ × M= (4.10) 4i is also conserved during the beam evolution. For our purposes, it is convenient to investigate configurations without walk-off, and we hereafter set δj = 0. Spatial optical solitons are optical beams with constant transverse profile along the propagation direction, defined as stationary solutions of the corresponding propagation equations (4.1)–(4.3). From this definition, it follows that we may look for the soliton solutions in a form of the generic ansatz, aj = Vj (x, y) exp(iκj z), where κj are the nonlinear corrections to the corresponding propagation constants. Stationary propagation of the multi-color solitons requires vanishing power exchange between the fundamental and second-harmonic waves; to avoid this exchange the condition κ3 = κ1 + κ2 + β applies. Radially symmetric solutions are found by separation of variables in the form Vj (x, y) = Uj (ρ) exp(imj ϕ),

(4.11)

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 with real amplitudes Uj depending on the polar radius ρ = x 2 + y 2 and phases being linear functions of the azimuthal coordinate ϕ = arctan(y/x). Using this ansatz and the algebraic constraint to topological charges m1 + m2 = m3 , we obtain the z-independent (stationary) version of eqs. (4.1)–(4.3):   m2j αj d2 1 d (4.12) − 2 Uj + Up Uq = κj Uj , + 2 dρ 2 ρ dρ ρ with j, p, q = 1, 2, 3, and j = p = q. For a fixed set of the topological charges, solutions depend on two parameters (e.g., κ1 and κ2 ), which correspond to different total and relative (imbalancing) energy flows between the three interacting waves (Buryak, Kivshar and Trillo [1996], Buryak and Kivshar [1997], Peschel, Etrich, Lederer and Malomed [1997]). Properties of the fundamental (bell-shaped) quadratic spatial solitons have been described in a number of studies (see, e.g., the review paper by Buryak, Di Trapani, Skryabin and Trillo [2002]), and instabilities of higher-order vorticityless modes with central peak and one or more surrounding rings are also known (Skryabin and Firth [1998b]). We do not consider these issues here and focus on the “doughnut”-shaped vortex solitons. Families of vortex solitons as solutions of eq. (4.12), for different combinations of the topological charges, wave vector mismatches, and both zero and nonzero imbalancing Ii , have been found by Firth and Skryabin [1997], Torres, SotoCrespo, Torner and Petrov [1998b], Skryabin and Firth [1998a], Molina-Terriza, Torres, Torner and Soto-Crespo [1998]. Figure 12 shows typical shapes of vortex solitary waves with different combinations of topological charges. We choose a particular case of zero imbalancing Ii = 0 because it also covers the solutions with equal amplitudes for both FF beams, such as that shown in fig. 12(c). This particular branch corresponds to stationary solutions for the type I geometry eqs. (4.4)–(4.5), consisting of only two components. The latter ones have been studied in detail by Firth and Skryabin [1997], Torres, Soto-Crespo, Torner and Petrov [1998a], Skryabin and Firth [1998a]. Useful information about the soliton families is given by integrals of motion defined above; for the stationary solutions under consideration the angular momentum and Hamiltonian can be expressed in terms of the beam powers, 1 (4.13) (m1 + m2 )I + (m1 − m2 )Ii , 2 1 H = − (κ1 + κ2 )I + (κ1 − κ2 )Ii − βI3 . (4.14) 4 Note that the vectorial nature of three-wave multi-color vortex solitons allows combinations including vorticity-less beam, e.g., m1 = 1 and m2 = 0, or having M=

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Fig. 12. (a–d) Typical solutions for the vortex solitary waves with different combinations of topological charges, m3 = m1 + m2 . Solid line: ordinary-polarized fundamental beam; dashed line: extraordinary-polarized fundamental beam; dotted line: second-harmonic field. Parameters are: β = 3, κ1 = 2, and Iu = 0. Notice that in (c) the curves corresponding to two fundamental beams are identical. (e) Families of the lowest-order vortex solitons presented through the energy flow-Hamiltonian diagram. Plot (f) is a zoom of the corresponding region of plot (e). Numbers in parentheses stand for the topological charges of the two fundamental beams, i.e., (m1 , m2 ). The curve labeled (0, 0) corresponds to the family of lowest-order, vorticity-less bright solitons (Torres, Soto-Crespo, Torner and Petrov [1998b]).

zero total angular momentum M = 0 (e.g., m1 = −m2 and Ii = 0), similar to their χ 3 counterparts described in Section 3. However, the two-component solutions in the type I model are always limited by the constraints κ1,2 ≡ κ, m3 = 2m with m1,2 ≡ m, and Iu ≡ 0, therefore M = mI , similar to the scalar χ 3 spatial soliton. Figure 12(e–f) shows some examples of the Hamiltonian dependencies eq. (4.14) in the case Ii = 0, the similar plots for the nonzero imbalancing are available in Molina-Terriza, Torres, Torner and Soto-Crespo [1998].

4.4. Spontaneous break-up: azimuthal instability Generation of different vortex patterns due to the frequency conversion, described in Section 4.2, occurs for the input powers of the FF pump beam below some threshold. For higher input powers, which are sufficient for the soliton formation, the generation of sets of simple fundamental solitons was predicted numer-

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ically for SHG by Petrov and Torner [1997], Torner and Petrov [1997a, 1997b]. Such phenomenon is related to the azimuthal modulational instability of the corresponding stationary states – ring-shaped vortex solitons. The linear stability analysis of the type I vortex solitons was performed by Firth and Skryabin [1997], Skryabin and Firth [1998a] and Torres, Soto-Crespo, Torner and Petrov [1998a]. The experimental confirmation of the spontaneous break-up of optical vortex solitons in quadratic crystal was reported by Petrov, Torner, Martorell, Vilaseca, Torres and Cojocaru [1998]. To outline briefly the main steps of these calculations, we examine the stability of the vortex solitary waves against azimuthal perturbations and seek the perturbed solutions of the form    aj = Uj (r) + % fj,s (r, z) exp(isϕ) + gj,s (r, z) exp(−isϕ) × exp(iκj z + imj ϕ),

(4.15)

where s, fj,s , and gj,s stand for the azimuthal index and the envelopes of the perturbation eigenfunctions, respectively. Inserting eq. (4.15) into eqs. (4.1)–(4.3) and linearizing the equations in respect to small perturbations, we obtain a set of six coupled linear partial differential equations for f and g at a given value of s. Such equations have many different solutions; some of them, the so-called instability modes, display exponential growth along the propagation direction. To obtain such solutions, one can use the method of averaging the growth rate of perturbation over the propagation direction described by Soto-Crespo, Heatley, Wright and Akhmediev [1991], Soto-Crespo, Wright and Akhmediev [1992], or further reduce the problem by setting {f ; g} = exp(Γ z){f˜(r); g(r)} ˜ and solving the corresponding boundary value problem for f˜ and g˜ (see also the discussion in Section 2.3). Both methods give identical results obtained for the type I vortices by Firth and Skryabin [1997], Skryabin and Firth [1998a] and Torres, Soto-Crespo, Torner and Petrov [1998a]. In fig. 13, we show typical values of the instability growth rate Re(Γ ) for the solutions with different sets of topological indices. Instability induced break-up of the ring-like solitons that follows qualitatively the similar scenario in all the cases, namely the splitting of the initial ring to a number of the fundamental solitons flying off the ring with the further propagation. Varying an initial perturbation results in different light patterns, where positions of the soliton splitters differ, as is seen in fig. 13(e–f), however, their number corresponds exactly to the azimuthal index of the instability mode with the largest growth rate. Number of solitons created through the decay is generally robust to small imbalancing of energy Ii = 0, while for large imbalancing, several perturbations with similar growth rate can come into play (Molina-Terriza, Torres, Torner and Soto-Crespo [1998]).

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Fig. 13. (a–d) Instability growth rate for perturbations with different azimuthal indices as a function of the nonlinear wave number shift κ1 , for the various families of vortex solitary waves shown in fig. 12 (Torres, Soto-Crespo, Torner and Petrov [1998b]). (e–f) Stroboscopic view of the decay of an exact vortex solitary wave solution in the presence of the corresponding exact azimuthal perturbation of a given index. The plots show the light patterns of the ordinary polarized fundamental beams calculated at z = 4, 8, 12, 16 propagation units, when the input is the field of fig. 12 plus the corresponding symmetry-breaking perturbation with azimuthal index s = 3. Amplitude of the added perturbation: in (e) % = 10−2 and in (f) % = 10−3 . The extraordinary polarized fundamental and the second-harmonic beams exhibit similar features and thus are not shown (Torner, Torres, Petrov and Soto-Crespo [1998]).

4.5. Induced break-up: soliton algebra The process of vortex beam break-up by azimuthal modulational instabilities is spontaneous, and thus governed by the perturbations that happen to have the highest growth rate. This process produces beautiful patterns of solitons flying off the input vortex ring, but leaves little control over the number of spots present in such soliton pattern. Vortex ring-shaped solitons might be broken in a controllable manner by inducing their splitting in a way that favors predetermined azimuthal symmetries. One way to favor a given azimuthal symmetry is to impose a specific phasepattern to the input beam. In the case of parametric interactions in quadratic media, such goal can also be accomplished by seeding the SHG process with a vortex in the SH frequency: The azimuthal phase-varying relation between the interact-

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ing beams generates a prescribed azimuthal symmetry of energy exchange between the beams that produces the desired soliton pattern (Torner, Torres, Petrov and Soto-Crespo [1998]). Such a process was termed soliton algebra, and is implemented by changing the topological charge of a weak, co-linear input SH seed beam. If this topological charge m3 satisfies the condition m3 = m1 + m2 , where m1,2 are the charges of two orthogonally polarized FF pump beams, the generation and subsequent spontaneous azimuthal instability of three-wave vortex solitons occurs. However, if the condition above is violated, m3 = m1 + m2 , the beam break-up into solitons is induced by the local, azimuthally-varying phase difference that exists between the pump and the seed signals and hence by the initial local direction of the energy flow between the FF and SH waves. The number of solitons formed in each portion depends on the input light conditions, such as the total energy flow and SH seed intensity. As a result, the information coded in the value of the input array (m1 , m2 , m3 ) is transformed into a certain number of output soliton spots (Torner, Torres, Petrov and Soto-Crespo [1998]). Different typical output patterns are presented in fig. 14.

Fig. 14. Results of the induced break-up of the input beams containing different topological charges: (a) [0, 0, 1], (b) [1, 1, 0], (c) [1, 1, −1], (d) [2, 2, 0], (e) [2, 2, −1], and (f) [2, 2, −2]. Input energy flows: at the fundamental frequency I1 = I2 = 36π for (a–c) and I1 = I2 = 128π for (d–f); at the second harmonic I3 = 2π . The plots show the ordinary-polarized beams at z = 10 (Torner, Torres, Petrov and Soto-Crespo [1998]).

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The experimental demonstration of the concept of soliton algebra were performed in a type I SHG geometry by Minardi, Molina-Terriza, Di Trapani, Torres and Torner [2001]. Under properly chosen operating conditions, the number of generated solitons n was shown to follow the rule n = |2mFF − mSH |, with mFF and mSH being the topological charges of the input FF and SH beams, respectively. When 2mFF = mSH , the beam break-up is known to occur through spontaneous azimuthal modulation instability, therefore the input was designed in such a way that 2mFF = mSH . Both processes, spontaneous or induced, require features similar to those needed for the generation of solitons with Gaussian-like beams and thereby is found to be very robust. In the context of soliton control by the presence of phase dislocations in the input beams, we notice the observation of the deflection of multi-color solitons generated by edge-like topological amplitude and phase dislocations reported by Petrov, Carrasco, Molina-Terriza and Torner [2003]. The experiments were conducted near phase-matching in a bulk potassium titanium phosphate crystal pumped with picosecond light pulses at 1064 nm, and the angular deflection of the solitons was found to be controllable through the position of the edge dislocation.

4.6. Dark multi-color vortex solitons Existence of multi-color dark vortex solitons has been discussed by Alexander, Buryak and Kivshar [1998] who analyzed also some basic properties of such beams which were found to be highly unstable against modulational instabilities of their nonvanishing background (Buryak, Di Trapani, Skryabin and Trillo [2002]). Such instability is known to be suppressed by a strong effect of competing nonlinearities (Alexander, Buryak and Kivshar [1998], Alexander, Kivshar, Buryak and Sammut [2000]), but such prediction did not find yet experimental verification. Nevertheless, significant efforts have been put to overcome modulational instability due to parametric wave interaction and to generate dark vortexcarrying beams. The most important advance was reported by Di Trapani, Chinaglia, Minardi, Piskarskas and Valiulis [2000], who used a large walk-off between the components to quench the modulational instabilities. In the reported experimental observations, the SH beam broke-up and formed many spikes, having an energy content much larger than the rest of the SH beam carrying a phase dislocation. Because of the large walk-off, the spikes propagated out of the beam rapidly. The generation of such spikes was claimed to be essential for self-quenching of the instability process. The experiment was performed with negative large phase-mismatch, where at moderate powers the cascading non-

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linearities leads to an effective defocusing Kerr nonlinearity (eq. (4.6)), known to support stable dark vortices. In the experiments reported by Di Trapani, Chinaglia, Minardi, Piskarskas and Valiulis [2000], the generated SH vortex dislocation exhibited a size much smaller than the host diffracting beam, indicating at least a transient trapping effect. The corresponding numerical simulations performed by Di Trapani, Chinaglia, Minardi and Valiulis [1999] confirmed such transient self-trapping and the propagation of the dislocation in the fundamental frequency beam for a distance of the order of several diffraction lengths without core spreading. In contrast, the linear diffraction of the beam results in strong spreading of the vortex core. The vortex beams observed in the experiments exhibited some features expected from a dark vortex soliton, but a comprehensive investigation of these important observations is not yet fully developed. Thus, the existence of true stable dark vortices in a quadratic nonlinear medium remains an open problem.

§ 5. Stabilization of vortex solitons In this section we discuss several theoretical predictions of the stabilization of bright vortex solitons in nonlinear media. Several models supporting stable vortex solitons were suggested, e.g., the Kerr media made of alternating self-focusing and self-defocusing layers (Towers and Malomed [2002], Montesinos, PerezGarcia and Michinel [2004], Montesinos, Perez-Garcia, Michinel and Salgueiro [2005], Adhikari [2004]) and nonlocal self-focusing medium (Yakimenko, Zaliznyak and Kivshar [2004], Breidis, Petersen, Edmundson, Krolikowski and Bang [2005]). In particular, we discuss here two distinct models with so-called competing nonlinearities. The first model includes self-focusing cubic and selfdefocusing quintic terms in the power-law Kerr-type nonlinearity, whereas the second model includes phase-dependent quadratic and self-defocusing cubic nonlinear interaction. We also summarize numerical results demonstrating stable spatiotemporal vortex solitons in the (3 + 1)-dimensional geometry, the so-called spinning light bullets.

5.1. Cubic-quintic nonlinearity Nonlinear models discussed in previous sections correspond to the lowest order nonlinearities available, namely to the first two nonlinear terms in expansion of optical medium polarization P = χ (1) E + χ (2) E 2 + χ (3) E 3 + · · · . For the centrosymmetric media all even terms vanish and taking into account higher-order terms

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one can represent the refractive index as a power-law Kerr-type nonlinearity, n = n0 + n2 I + n4 I 2 + · · · , here the intensity I ≡ |E|2 . Obviously, if nonlinear coefficients n2 and n4 have the same signs, corresponding models exhibit simply increasing strength of nonlinear self-action, self-focusing for n2,4 > 0 or self-defocusing for n2,4 < 0. More interesting situation occurs when two nonlinear contributions have opposite signs, n2 n4 < 0, this case usually refereed as “competing” cubic-quintic (CQ) nonlinearity: corresponding nonlinear terms in propagation equation being of the third and fifth orders: i

∂E + E + n2 |E|2 E + n4 |E|4 E = 0. ∂z

(5.1)

Let n4 be of self-focusing type, n4 > 0. Then, for any sign of Kerr contribution n2 , there will be a threshold power of (transversely two-dimensional) light beam E, when higher-order self-focusing will predominate both the linear diffraction and n2 contribution. In this case the beam will collapse, similar to the pure Kerr case with n2 > 0 and n4 = 0 (see recent review by Berge [1998]). However, if n4 < 0, than the collapse can be stopped, because the parts of light beam with high enough intensity I > Ith experience effectively self-defocusing environment, dn/dI < 0. Here the threshold intensity is given by Ith = −n2 /(2n4 ). Furthermore, the CQ nonlinearity can be regarded as a power-law expansion for any collapse-free nonlinearity with saturation, for example the phenomenological one discussed above, n = n0 + n2 I /(1 + sI ). In this case n4 = −sn2 , and the CQ medium is refereed also as a saturable one. Saturable nonlinearity offered a great advantage over the conventional cubic one because it supports stable spatial solitons, free of collapse instability. That is why its simplest version, the CQ nonlinearity, attracts significant attention of theoreticians from the early days of nonlinear optics (Zakharov, Sobolev and Synakh [1971]). Numerical studies have confirmed the stability of fundamental spatial solitons in this system (Wright, Lawrence, Torruellas and Stegeman [1995], Dimitrevski, Reimhult, Svensson, Ohgren, Anderson, Berntson, Lisak and Quiroga-Teixeiro [1998], Quiroga-Teixeiro, Berntson and Michinel [1999]). Experimentally a CQ nonlinear dielectric response with positive cubic and negative quintic contributions has been observed in chalcogenide glasses (Smektala, Quemard, Couderc and Barthelemy [2000], Boudebs, Cherukulappurath, Leblond, Troles, Smektala and Sanchez [2003]), and in organic materials (Zhan, Zhang, Zhu, Wang, Li, Li, Lu, Zhao and Nie [2002]). However, in all these cases the quintic nonlinearity is accompanied by significant higher-order multiphoton processes such as two-photon absorption, therefore the validity of the CQ models to light propagation in these materials requires additional explorations. Very recently, the

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Fig. 15. Stationary solutions to the eq. (5.1) for different topological charges m and soliton parameter κ (shown next to the corresponding curves). For m = 1, we also show the example of two types of optical vortices, localized (bright) and nonlocalized (dark), to coexist in media with competing nonlinearities. Similar solutions exist for any number m. Note that the slopes of envelopes for both types, i.e. the size of the vortex core, is sufficiently different for the case of κ = 0.1, and almost exactly the same for κ = 0.18, indicating the transformation of bright to dark vortex with increase of power, see text for the details.

criteria for the experimental observation of multidimensional solitons in CQ type saturable media were developed by Chen, Beckwitt, Wise and Malomed [2004]. The “auto-waveguide” propagation of the “spiral beams” with nonzero topological charge has been predicted by Kruglov and Vlasov [1985] for pure Kerr model. As early as 1988, it was found that saturation in the form of CQ nonlinearity stabilize optical vortices against collapse, and “the data from the computer experiment show that these beams are stable” (after Kruglov, Volkov, Vlasov and Drits [1988]). Further study, however, reveal that the azimuthal instability of CQ vortices may take place (Kruglov, Logvin and Volkov [1992]). A key insight was put forward by Quiroga-Teixeiro and Michinel [1997], who found by numerical simulations that the vortex solitons with charge m = 1 could be stable provided that the power of the beam is over some critical value. Figure 15 shows several examples of the bright and dark vortex soliton solutions to eq. (5.1) coexisting in the CQ model (Berezhiani, Skarka and Aleksic [2001]). Note that with increase of power the amplitude of the solutions saturates and starting from some value of the soliton parameter (or, equivalently, some value of soliton power), the slope of the bright vortex soliton coincide with the one for dark vortex. That may indicate the transition from unstable vortices in self-focusing regime to the stable ones in effectively self-defocusing regime. The critical power for this transition has been found analytically by Michinel, Campo-Taboas, Quiroga-Teixeiro, Salgueiro and Garcia-Fernandez [2001], it was shown to exceed four times the threshold power of the generation for vortex soliton with charge m = 1. The issue of the stability of vortex solitons in CQ model was put on more solid mathematical grounds by calculating the linear stability spectrum by Towers, Buryak, Sammut, Malomed, Crasovan and Mihalache [2001] and Skarka, Alek-

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sic and Berezhiani [2001]. The growth rate of small azimuthal perturbations was found to be nonzero for a limited domain 0 < κ < κstab , with linearly stable solutions above the critical value κ > κstab . In addition, very small instability with respect to shift of the dislocation core (central dark spot) was found by Towers, Buryak, Sammut, Malomed, Crasovan and Mihalache [2001], Malomed, Crasovan and Mihalache [2002]. Nevertheless, the supercritical vortex solitons appear to be strong attractors, the Gaussian beam with nested phase dislocation may initially break to several splitters but than restores radially symmetric vortex shape (Skarka, Aleksic and Berezhiani [2001]). A mathematically rigorous stability analysis was performed by Pego and Warchall [2002], who predicted the stability of higher-order vortex solitons with m > 1 as well, this issue was also addressed recently by Davydova and Yakimenko [2004]. A detailed study of the stability of (2 + 1)-dimensional vortex solitons in both conservative and dissipative CQ models can be found in Crasovan, Malomed and Mihalache [2001b], the related issue of the stability of spatiotemporal (3 + 1)-dimensional spinning light bullets we discuss in Section 5.3. Formal analogies between the CQ vortex solitons and quantum fluids have been discussed by Michinel, Campo-Taboas, Garcia-Fernandez, Salgueiro, and Quiroga-Teixeiro [2002] (see also comments by Coffey [2002], Weiss [2003]). In particular, analogies have been drawn between the collisional dynamics of vortex solitons and surface tension properties by Paz-Alonso, Olivieri, Michinel and Salgueiro [2004]. Such features can be accurately explained by the internal oscillations of spatial solitons in the domain of their stability (Dong, Ye, Wang, Cai and Li [2004]). To conclude this section we note that competing nonlinearities of different kinds have been suggested. Examples include thermal mechanism, studied by Kruglov, Logvin and Volkov [1992], or nonlinearity of the form n = 1 + n2 I − nK I K (Skarka, Aleksic and Berezhiani [2003]). In the latter case, it was suggested that for intense laser pulses in air the parameter K might be as high as K = 10, albeit propagation of intense pulses in air usually involves a variety of strong multiphoton processes not captured by the above reduced model.

5.2. Quadratic-cubic nonlinearity As discussed previously, bright doughnut-shaped vortex solitons in pure χ (2) media are unstable against azimuthal symmetry-breaking perturbations, similar to their χ (3) counterparts in self-focusing media. This property might be a generic feature of bright vortices in the nonlinear models where the balance between

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counteracting self-focusing type nonlinearity and repulsive diffraction “forces” allows stationary radially-symmetric states, but it is not sufficient to damp the azimuthal modulational instability along the ring. The dark vortices in selfdefocusing χ (3) medium, however, are the stable entities provided their nonzero constant background, the stationary plane wave solutions, is stable against small modulation. The competing χ (3) –χ (5) nonlinearities, discussed in Section 5.1, were shown to posses both, the ability to localize the bright solitons, and to guarantee additional stabilization in certain parameters region. Expanding the analogy to the case of parametric solitons, one may expect stable vortex solitons to exist in χ (2) mediated wave mixing, if there will be possible additional nonlinearity to compete, for example of the χ (3) self-defocusing type. To start with, the dynamical equations that govern the interaction between a weakly modulated plane wave and its second harmonic for materials with asymmetric crystal structure, in which the effects of both the quadratic and the cubic nonlinear susceptibility tensors must be considered, were derived by Bang [1997]. Following these derivations and taking into account the diffraction in two transverse dimensions and paraxial approximation, one can describe soliton-like propagation of narrow beams:

∂u i (5.2) + u − κ1 u + u∗ v − |u|2 /4 + 2|v|2 u = 0, ∂z

∂v + v − κv + u2 /2 − 4|v|2 + 2|u|2 u = 0, 2i (5.3) ∂z where κ1 is the nonlinear contribution to the propagation constant for the fundamental wave u, and parameter κ combines it with the phase-mismatch k, κ = 2(k + 2κ1 ). Stable fully localized (2+1)-dimensional ring solitons with intrinsic vorticity in optical media with competing quadratic and self-defocusing cubic nonlinearities have been found by Towers, Buryak, Sammut and Malomed [2001]. It is noteworthy that properties of the stationary solutions to eqs. (5.2)–(5.3) are very similar to those shown in fig. 15: with increasing of power, the amplitude saturates and soliton width diverges to the dark state. Stability windows for sufficiently broad ring solitons with the spin m = 1 and 2 have been found, both in direct dynamical simulations and analyzing eigenvalues of the linearized equations. Similar to their χ (3) –χ (5) counterparts, stable two-color vortex solitons survive strong perturbations such as collisions, as it was shown by Malomed, Peng, Chu, Towers, Buryak and Sammut [2001]. It is necessary to say that conventional nonlinear materials with strong χ (2) nonlinearity do not satisfy the requirement of the model to have a negative χ (3) co-

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efficient at both the fundamental and second-harmonic frequencies. Different possibilities to create a necessary effective χ (3) nonlinearity have been proposed. For example, Malomed, Peng, Chu, Towers, Buryak and Sammut [2001] suggested by creating a layered medium in which layers providing for the χ (2) nonlinearity periodically alternate with others that account for the self-defocusing Kerr nonlinearity. Engineering χ (2) quasi-phase-matched gratings (Bang, Clausen, Christiansen and Torner [1999]) also produces effective χ (3) nonlinearities. However, notice that in this case the higher-order nonlinearities are induced on average over all the Fourier components associated to the quasi-phase-matching modulation, thus the models only hold under proper conditions when averaging is justified and thus these models might not be able to stabilize otherwise unstable solitons. A modulationally stable branch of a plane two-wave solutions to the system eqs. (5.2)–(5.3) and the corresponding stable dark vortex solitons were found by Alexander, Buryak and Kivshar [1998]. Later on, Alexander, Kivshar, Buryak and Sammut [2000] reported existence of novel vortex states on infinite background, the so-called “halo-vortex” and “ring-vortex”. It is interesting to note that in a similar model there exist a kind of bright vortex states, localized in transverse plane by additional harmonic trapping potential; such system might be perhaps used to describe some features of hybrid atomic-molecular Bose–Einstein condensates (Alexander, Ostrovskaya, Kivshar and Julienne [2002]).

5.3. Spatiotemporal spinning solitons Three-dimensional optical spatiotemporal solitons, the so-called “light bullets” (LB, this term was introduced by Silberberg [1990]), attract a growing interest, as they represent a new fundamental physical object. They have been suggested to implement ultra-fast all-optical switching in bulk media (McLeod, Wagner and Blair [1995], Liu, Beckwitt and Wise [2000], Wise and Di Trapani [2002]). Physical content of this new object lies in the spatiotemporal analogy, which allows one to consider in the same way both, the temporal dispersion of the short light pulse, and the diffraction (or “spatial dispersion”) of the narrow beam (see, e.g., paper by Kanashov and Rubenchik [1981]). In the presence of group-velocity dispersion, the evolution (along propagation direction z) of slowly-varying envelope of the electromagnetic field E(x, y, z; t) is described by the paraxial equations similar to the eq. (5.1) and eqs. (5.2)–(5.3), but with time-dependent Lapla2 + κDE cian  = ∇⊥ T T . Here κ is the propagation constant (wave number), 2 D = −d κ/dω2 > 0 is the coefficient of the temporal dispersion assumed anomalous, T ≡ t − z/vg (vg being the group velocity of the carrier wave) is

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2 = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 represents spatial diffraction. the “reduced time”, and ∇⊥ √ Normalizing reduced time τ = T / κD, one obtains spatiotemporal Laplacian  = ∂ 2 /∂x 2 + ∂ 2 /∂y 2 + ∂ 2 /∂τ 2 , completely symmetrical with respect to the spatial coordinates (x, y) and reduced time τ . In nonlinear media, the self-focusing may balance out both, the temporal and the spatial broadenings of sufficiently short pulses of light. The combination of these two effects, responsible for the formation of (1+1) temporal and (2+1) spatial solitons, supports stationary states known as (3 + 1)-dimensional solitons, or light bullets. For the details of physics of LB, the review on theoretical and experimental progress in this field, as well as for the description of fundamental (bellshaped) three-dimensional solitons, we refer to the recent paper by Malomed, Mihalache, Wise and Torner [2005]. Here we consider only the higher-order LB with phase dislocations, or optical vortices in spatiotemporal domain. Similar to the (2 + 1)-dimensional case, stationary solutions corresponding to the fundamental (3 + 1)D soliton can be obtained using radially-symmetrical ansatz of the form E = U (r) exp(ikz), where k is a soliton parameter and the radius is given by r 2 = x 2 + y 2 + τ 2 . The same ansatz describes higher-order spherically-symmetrical modes consisting of several concentric shells surrounding inner core (Edmundson [1997]). Angle-dependent higher-order states, however, do not allow simple separation of variables, such as E = U (r) exp(imφ + ikz), used for two-dimensional vortex solitons. Therefore, the search for higherorder LB with phase dislocations requires solving the full multidimensional stationary equation, quite a nontrivial task. The so-called “three-dimensional spinning solitons”, introduced by Desyatnikov, Maimistov and Malomed [2000], is only known example of LB with phase dislocation, and it may represent much broader class of possible (and yet not known) spatiotemporal optical vortices. In other systems, such example include the “smoke rings” of vortex lines in nondegenerate optical parametric oscillator (Weiss, Vaupel, Staliunas, Slekys and Taranenko [1999]), and the “parallel vortex rings” in matter waves trapped by the three-dimensional external potential (Crasovan, Perez-Garcia, Danaila, Mihalache and Torner [2004]). The structure of the spinning LB can be well understood by using the approximate (e.g., variational) solutions. We suppose that stationary optical pulse E = A(x, y, τ ) exp(ikz) (localized in time τ , so that A → 0 for τ → ±∞) has a phase-dislocation located in the transverse plane, A(x, y, τ ) = V (x, y, τ ) exp(imϕ) with azimuthal angle ϕ = tan−1 (y/x), similar to the twodimensional CW vortex beams. Because of this dislocation, field should vanish in the origin and produce a “doughnut” shape in transverse plane, V → 0 for ρ → 0, here ρ is a polar radius, ρ 2 = x 2 + y 2 . Thus spinning LB can be thought

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as a CW beam modulated in time by an additional multiplier, e.g., of the sech(τ ) shape, V = Ucyl (ρ) sech(τ ),

(5.4)

and we note that this ansatz offers the separation of variables in cylindrical coordinates, which does not satisfy the nonlinear equation of course. Physically, the ansatz eq. (5.4) can follow as a result of temporal (longitudinal) modulational instability of initially continuous (CW) optical vortex beam, responsible for generation of soliton trains (this so-called “neck”-instability in the spatial domain was experimentally observed by Fuerst, Baboiu, Lawrence, Torruellas, Stegeman, Trillo and Wabnitz [1997]). Similar shape can be also modeled by a spherical harmonic, V = Usph (r) cos(θ ),

(5.5)

with spherical coordinates, (x, y, τ ) → (r, ϕ, θ). Two model envelopes eqs. (5.4) and (5.5) can be used as the trial functions for variational method. The advantage of this method is that it actually allows one to separate the variables in corresponding nonlinear equation and greatly simplifies the problem (variational methods in optics were reviewed recently by Malomed [2002]). Partial-differential equation for stationary envelope V (x, y, τ ) is than reduced to ordinary differential equations for the envelope Ucyl (ρ) or Usph (r). Solutions to these equations and their analysis for the model with cubic-quintic nonlinearity show that radial envelopes Ucyl (ρ) and Usph (r) are qualitatively similar to the ones shown in fig. 15 for twodimensional case, and they define very close values for the parameters of stationary spinning LB solution, for example the minimal threshold energy of soliton formation (Desyatnikov, Maimistov and Malomed [2000]). However, variational solutions do not provide a full information about stationary states family and their stability, and direct numerical modeling is necessary. Exact numerical solutions for spinning LD were obtained by Mihalache, Mazilu, Crasovan, Malomed and Lederer [2000a] in the type I SHG model, and by Mihalache, Mazilu, Crasovan, Malomed and Lederer [2000b] for the CQ medium. Authors tested the stability of solutions in numerical propagation and observed azimuthal modulational instability which leads to the breakup of doughnut solitons into several fragments, each being a stable moving zero-spin soliton (see fig. 16). The general conclusion based on direct numerical simulations was that the spinning LBs are always unstable against azimuthal perturbations (Crasovan, Malomed and Mihalache [2001b]). Later on, however, more accurate study of the associated linear stability problem in the CQ model revealed first ever found completely stable spatiotemporal vortex soliton (Mihalache, Mazilu, Crasovan,

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Fig. 16. Examples of stable and unstable spinning light bullets in CQ model for topological charges m = 1 in (a), (b), and m = 2 in (c), (d). Top row – soliton solutions (perturbed in (b)) at z = 0. Bottom row – after propagation of z = 60 in (a); z = 100 in (b); z = 50 in (c), and z = 90 in (d). After Mihalache, Mazilu, Towers, Malomed and Lederer [2003].

Towers, Buryak, Malomed, Torner, Torres and Lederer [2002]). Example of stable propagation of initially strongly perturbed spinning LB is shown in fig. 16(b). The reason for stabilization of spinning LBs was found in the competition between nonlinearities. Following these predictions, stable spinning LBs were found in the model with competing quadratic and cubic nonlinearities (Mihalache, Mazilu, Crasovan, Towers, Malomed, Buryak, Torner and Lederer [2002]) and in twocomponent vectorial CQ system (Mihalache, Mazilu, Towers, Malomed and Lederer [2003]).

§ 6. Other optical beams carrying angular momentum In this section we summarize some theoretical and experimental results on the study of self-trapped optical beams carrying angular momentum, which differ from the optical vortex beams discussed above. In general, such beams do not necessarily correspond to the stationary states, and their angular momentum manifests itself in the complex interaction of simple spatial solitons and leads to their spiraling. Multi-soliton complexes with an imposed angular momentum, such as

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necklaces and soliton clusters, can also be regarded as multi-soliton spiraling beams.

6.1. Soliton spiraling Spiraling of two spatial solitons was suggested theoretically by Poladian, Snyder and Mitchell [1991]. This should occur when two fundamental solitons collide with trajectories that are not lying in a single plane, so that they form a two-body system with nonzero orbital angular momentum. Then, if the mutual interaction is attractive, the centrifugal repulsive force can be balanced out, and two solitons orbit about each other in a double-helix structure, as illustrated in fig. 17. It is interesting to note, that similar predictions concerning three-dimensional solitons, or light bullets, were made by Edmundson and Enns [1993], based on the particle-like nature of soliton mutual interaction (Edmundson and Enns [1995]). In parallel, the experimental and numerical study of azimuthal instability of vortex solitons, described in Section 2.3, revealed the spiraling behavior of splitters, first demonstrated experimentally in rubidium vapors by Tikhonenko, Christou and Luther-Davies [1995, 1996]. In quadratic media, soliton spiraling was predicted and studied in detail theoretically by Steblina, Kivshar and Buryak [1998], Buryak and Steblina [1999]. The mechanical model, based on potential of soliton interaction (Malomed [1998]) has been derived, with extremal points of effective potential corresponding to spi-

Fig. 17. (Left) An illustration of the soliton spiraling process. The arrows indicate the initial direction of the two soliton beams. After Shih, Segev and Salamo [1997]. (Right) Spiraling of solitons with initially skewed trajectories in photorefractive crystal. (a, c) Initial position of the beams; cross and circle denote output positions of solitons A1 and A2 , respectively, during individual propagation. (b, d) Output positions of the solitons during simultaneous propagation. After Stepken, Belic, Kaiser, Krolikowski and Luther-Davies [1999].

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raling bound states. Similar to any phase-sensitive soliton interaction the effective mass corresponding to the phase degree of freedom was shown to be always negative. As a result, corresponding stationary points are of saddle type, i.e. spiraling bound states are unstable. Depending on initial soliton states, such as soliton velocities, relative phase, and the impact parameter, soliton can reflect, spiral, and fuse. More recently, experimental observation of related phenomena has been reported by Simos, Couderc and Barthelemy [2002]. The spiraling of mutually incoherent spatial solitons was observed experimentally (Shih, Segev and Salamo [1997]) and studied theoretically (Desyatnikov and Maimistov [1998], Schjodt-Eriksen, Schmidt, Rasmussen, Christiansen, Gaididei and Berge [1998]) as a possible scenario for a dynamically stable two-soliton bound state formed when two solitons are launched with initially twisted trajectories. Mutually incoherent solitons always attract each-other in isotropic Kerr-type medium, independently of their relative phase, and effective potential minimum corresponds to stable bound state. Note the very similar results obtained by Ren, Hemker, Fonseca, Duda and Mori [2000] for “braided light” in plasmas. As a matter of fact, the soliton spiraling due to an effectively vectorial beam interaction is associated with large-amplitude oscillations of a dipole-mode vector state generated by the interaction of two initially mutually incoherent optical beams (Skryabin, McSloy and Firth [2002]). In anisotropic photorefractive medium, however, the mutually incoherent solitons demonstrate much complicated anomalous interaction (Krolikowski, Saffman, Luther-Davies and Denz [1998], Stepken, Kaiser, Belic and Krolikowski [1998], Krolikowski, Denz, Stepken, Saffman and Luther-Davies [1998]). Anomalous interaction results in complex trajectories which typically show partial mutual spiraling, followed by damped oscillations and the fusion of solitons. The rotation can be propelled to prolonged spiraling by the skewed launching of beams. This nontrivial behavior is caused by the anisotropy of the nonlinear refractive index change in the crystal, as was shown by Stepken, Belic, Kaiser, Krolikowski and Luther-Davies [1999], Belic, Stepken and Kaiser [1999] and summarized by Krolikowski, Luther-Davies, Denz, Petter, Weilnau, Stepken and Belic [1999], Denz, Krolikowski, Petter, Weilnau, Tschudi, Belic, Kaiser and Stepken [1999]. Nevertheless, the fascinating analogy between spiraling solitons and mechanical two-body system is applicable if one takes into account the anisotropic nature of spatial screening solitons interacting in photorefractive medium (Belic, Stepken and Kaiser [2000]), the comparison of the above model to the isotropic one was published recently by Belic, Vujic, Stepken, Kaiser, Calvo, Agullo-Lopez and Carrascosa [2002].

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6.2. Optical necklace beams Since the decay of the ring-profile vortex solitons is associated with the growing azimuthal modulation of their intensity and the symmetry-breaking instability, one may try to stabilize the ring structure by imposing the initial intensity and phase modulation. The azimuthally modulated rings resemble “optical necklaces”; they are closely related to the higher-order guided modes, as we discussed in Section 3.1, and also to suitable superpositions of Laguerre–Gaussian beams. In Kerr media, the first experimental results on the self-trapping of necklacetype beams were reported by Barthelemy, Froehly, Shalaby, Donnat, Paye and Migus [1993]. Figure 18(a) shows the experimental data for the case in which an input beam in the form of a higher-order Laguerre–Gaussian mode was launched at the input of a CS2 cell. The beam diameter was 260 µm, with a petal thickness of about 80 µm. The output pattern after 5 cm of propagation is shown when the intensity was low enough that the nonlinear effects were negligible (middle). As expected, the beam diameter increased to 265 µm, because of diffraction, while the petal thickness remained close to 80 µm. As the beam power was gradually increased, the petal thickness decreased, because of self-focusing. The petal size reduced to 30 µm at an intensity level of 5 × 107 W/cm2 (Barthelemy, Froehly and Shalaby [1994]). Such self-trapped structures are remarkably stable and allow

Fig. 18. (a) Experimental demonstration of the necklace beams. Propagation of a Laguerre–Gaussian beam inside a Kerr medium: (from left to right) input beam, diffracted beam at low intensity, and the self-trapped necklace beam at high intensities (Barthelemy, Froehly, Shalaby, Donnat, Paye and Migus [1993]). (b) Rotating and expanding necklace with integer spin. After Soljacic and Segev [2001].

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one to transport optical beams with powers several times the critical power at which a Gaussian beam would otherwise collapse because of self-focusing; they disintegrate for input intensities lower than the self-trapping intensity. The concept of necklace beams was developed by Soljacic, Sears and Segev [1998], who studied numerically the propagation of azimuthally modulated bright rings in Kerr medium. These authors demonstrated that, in contrast to all previous studies in this model, the self-trapped beams localized in transverse plain can preserve their shape during the propagation and escape the collapse instability. It is possible if such beams are constructed as a “necklace” of the out-of-phase “pearls”, each carrying the power less than critical power of catastrophic selffocusing. Each petal thus slowly diffracts, if it propagates alone, and this diffraction is greatly suppressed within the ring, because of collective self-trapping of the ring as a whole. Due to the repulsion between neighboring petals the ring expands self-similarly. The shape of the necklace can be approximated by the ansatz similar to eq. (3.3) with p = 0. To slow down the expansion of the ring, its radius should be taken as large as possible, so that the radial envelope U (r) in eq. (3.3) can be approximated by the corresponding 1D soliton of sech shape, which is a bright stripe in twodimensional spatial domain. Extensive numerical simulations and semi-analytical analysis performed by Soljacic and Segev [2000] showed that the dynamics of the necklace can be controlled and reduced to the quasi-stationary if the radius of the ring, its width, the amplitude, and the order of azimuthal modulation (the winding number m in eq. (3.3)), minimize corresponding action integral. These “quasi-solitons” have a shape of a thin modulated stripe wrapped to a large ring, and they propagate stably over several tens of diffraction lengths. Necklaces with additional phase modulation, introducing a nonzero angular momentum, exhibit a series of phenomena typically associated with rotation of rigid bodies and centrifugal force effects (Soljacic and Segev [2001]). The simplest way to explore these novel features is to consider the ansatz eq. (3.3) with p = 0, which, from one hand, describes a vortex with the topological charge m at p = 1, and, from the other hand, it includes the varying modulation parameter p (0 < p < 1). The spin of this nonstationary structure is defined as S=

2mp , 1 + p2

(6.1)

and it vanishes for p → 0 (see the definition of spin after eq. (2.18)). When the ring vortex is only slightly modulated (p ≈ 1), it decays into a complex structure of filaments because of a competition between different instability modes of the corresponding vortex soliton (Desyatnikov and Kivshar [2002b]).

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When the modulation becomes deeper, e.g., for p ≈ 0.5, the initial vortex transforms into a necklace-like structure (see fig. 18(b)), and its dynamics is modified dramatically. The modulated ring-profile structure does not decay but, instead, it expands with small rotation, the rotation is much weaker because the initial angular momentum (spin) is much smaller than that in the case of p ≈ 1. These ringlike structures were introduced by Soljacic and Segev [2001] as the first example of optical beams with fractional spin (see eq. (6.1)) and rotating intensity. Note, that the anticlockwise direction of the rotation is determined by the gradient of phase, which grows anticlockwise for m > 0, see fig. 18(b). The angular velocity vanishes as the ringlike structure expands, in analogy with Newtonian mechanics and “scatter on ice” effect.

6.3. Soliton clusters In saturable media, the fundamental solitons are stable and demonstrate the particle-like robust interaction, e.g., spiraling out of the initial vortex ring after it breaks, see Section 2.3. The analogy with particles and forces between them applied to spatial solitons allows one to search for the bound states of several solitons corresponding to the balance between all acting forces, as in classical mechanics. In order to create nonexpanding configurations of N solitons in a bulk medium, first we recall the basic physics of coherent interaction of two spatial solitons. It is well known (Kivshar and Agrawal [2003]) that such an interaction depends crucially on the relative soliton phase, say θ , so that two solitons attract each other for θ = 0, and repel each other for θ = π. For the intermediate values of the soliton phase, 0 < θ < π, the solitons undergo an energy exchange and inelastic interaction. Here we follow the original paper by Desyatnikov and Kivshar [2002a] and analyze possible stationary configurations of N coherently interacting solitons for a ringlike geometry. It is easy to understand that such a ringlike configuration will be radially unstable due to an effective tension induced by bending of the soliton array. Thus, a ring of N solitons will collapse, if the mutual interaction between the neighboring solitons is attractive, or expand otherwise, resembling the expansion of the necklace beams. Nevertheless, a simple physical mechanism will provide stabilization of the ringlike configuration of N solitons, if we introduce an additional phase on the scalar field that twists by 2πm along the soliton ring. This phase introduces an effective centrifugal force that can balance out the tension effect and stabilize the ringlike soliton cluster. Due to a net angular momentum induced by such a phase distribution, the soliton clusters will rotate with an angular velocity which depends on the number of solitons and phase charge m.

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To describe the soliton clusters analytically, we consider a coherent superposition of N solitons with the envelopes Gn (x, y, z), n = 1, 2, . . . , N , propagating in a self-focusing bulk nonlinear medium. The equation for the slowly varying field  envelope E = Gn can be written in the form of the NLS equation (2.11). For a ring of identical weakly overlapping solitons launched in parallel, we can calculate the integrals of motion employing a Gaussian ansatz for a single beam Gn ,   |r − rn |2 Gn = A exp − (6.2) + iαn , 2a 2 where rn = (xn ; yn ) describes the soliton location, and αn is the phase of the nth beam. We assume that the beams Gn are arranged in a ring-shaped array of radius R, i.e. rn = {R cos ϕn ; R sin ϕn } with ϕn = 2πn/N . Analyzing many-soliton clusters, we remove the motion of the center of the mass and put L = 0, here the linear momentum is given by eq. (2.16). Applying this constraint, we find the conditions for the soliton phases, αi+n − αi = αk+n − αk , which are satisfied provided the phase αn has a linear dependence on n, i.e. αn = θ n, where θ is the relative phase between two neighboring solitons in the ring. Then, we employ the periodicity condition in the form αn+N = αn + 2πm, and find: 2πm θ= (6.3) . N In terms of the field theory, eq. (6.3) gives the condition of the vanishing energy  flow L = 0, because the linear momentum L = j dr can be presented through the local current j = Im(E ∗ ∇E). Therefore, eq. (6.3) determines a nontrivial phase distribution for the effectively elastic soliton interaction in the ring. In particular, for the well-known case of two solitons (N = 2), this condition gives only two states with the zero energy exchange, when m is even (θ = 0, mutual attraction) and when m is odd (θ = π, mutual repulsion). For a given N > 2, the condition (6.3) predicts the existence of a discrete set of allowed states corresponding to a set of the values θ = θ (m) with m = 0, ±1, . . . , ±(N − 1). Here, two states θ (±|m|) differ by the sign of the angular momentum, similar to the case of vortex solitons. Moreover, for any positive (negative) m+ within the domain π < |θ | < 2π, one can find the corresponding negative (positive) value m− within the domain 0 < |θ | < π, so that m+ and m− describe the same stationary state. For example, in the case N = 3, three states with zero energy exchange are possible: θ (0) = 0, θ (1) = 2π/3, and θ (2) = 4π/3, and the correspondence is θ (±1) ↔ θ (∓2) . The number m determines the full phase twist around the ring, and it plays a role of the topological charge of the corresponding phase dislocation, see fig. 19.

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Fig. 19. (Left) Intensity and phase distribution for a four-soliton cluster in saturable medium. Note that in terms of the azimuthal coordinate ϕ = tan−1 (y/x), the vortex phase is given as a linear function mϕ with integer m, while the staircase-like phase of the cluster is a nonlinear phase dislocation (Desyatnikov, Denz and Kivshar [2004]). (Right) Cluster composed of six spatiotemporal two-color solitons. The topological charge m of the soliton cluster is equal to one. (a) The fundamental frequency field and (b) the second harmonic field. (c) The phase distribution at fundamental frequency and (d) the phase distribution at the second harmonic (Crasovan, Kartashov, Mihalache, Torner, Kivshar and Perez-Garcia [2003]).

Applying the effective-particle approach, Desyatnikov and Kivshar [2002a] derived an effective interaction energy for the soliton cluster and have shown that it can be classified in a simple way by its extremal points. The existence of a minimum point suggests that such a configuration describes a stable or long-lived ring-like cluster of a particular number of solitons. This prediction was verified by a series of numerical simulations for different N -soliton rings and their propagation in a saturable medium. For example, the effective potential is always attractive for m = 0, and thus the ring of in-phase solitons exhibit oscillations and fusion. Another scenario of the mutual soliton interaction corresponds to the repulsive potential, e.g., for the case θ > π/2. In the numerical simulations corresponding to this case, the ringlike soliton array expands with the slowing down rotation, similar to the rotating necklace beams, see fig. 18(b). Finally, the stationary soliton bound state that corresponds to a minimum of the effective potential is shown in fig. 19 for particular case N = 4 and m = 1. Here the angular momentum is nonzero, and it produces a repulsive centrifugal force that balances out an effective attraction of π/2-out-of-phase solitons. The general rule, generalizing a two-soliton phase sensitive interaction, predicts the existence of a bound state of N solitons if the

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nonzero phase step θ is equal or less π/2, i.e. the interaction is attractive and can be balanced out by the net centrifugal force for 0 < θ  π/2. Therefore, soliton cluster with topological charge m can be quasi-stationary only for N
4m, see eq. (6.3). In addition, “excited” states are possible with the radius of the cluster oscillating near the minimum of interaction potential during the propagation. The concept of soliton clusters was extended to the case of light propagation in quadratically nonlinear media by Kartashov, Molina-Terriza and Torner [2002], where it is intimately related to the concept of induced beam break-up or soliton algebra discussed above. Due to the phase relation between the fundamental wave and the second harmonic beam (see Section 4), the topological charge in the second-harmonic field is double the fundamental wave charge. Therefore, the corresponding phase jump between neighboring solitons eq. (6.3) is doubled, see fig. 20. Different regimes of cluster propagation were found to be possible.

Fig. 20. (a) Rotating vector cluster consisting of four fundamental solitons. The angle of rotation, corresponding to the distance z = 225LD , is ∼ 11.25π . Note that the direction of rotation is opposite to the one of necklace in fig. 18(b), despite the fact that in both cases the angular momentum is positive. After Desyatnikov and Kivshar [2002b]. (b) Iso-surface plot of two counter-propagating vortices. Breakup into three rotating beamlets is visible. Because of the topological charge, the beamlets start to spiral, which is only weakly visible due to the short propagation distance. The beamlets rotate in the direction indicated by the arrow on the right (Motzek, Jander, Desyatnikov, Belic, Denz and Kaiser [2003]).

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Metastable, robust propagation of clusters in media with competing quadratic and self-defocusing nonlinearities was reported by Kartashov, Crasovan, Mihalache and Torner [2002]. These results were followed by similar findings in the cubic-quintic model as well by Mihalache, Mazilu, Crasovan, Malomed, Lederer, and Torner [2003]. Similar to the extension of the concept of two-dimensional vortex solitons into spatiotemporal domain (see paper by Desyatnikov, Maimistov and Malomed [2000] and Section 5.3), the bound states of three-dimensional solitons, or light bullets, can be constructed using the approach described above for two-dimensional cw beams. Three-dimensional, robust “soliton molecules” were introduced by Crasovan, Kartashov, Mihalache, Torner, Kivshar and Perez-Garcia [2003], Mihalache, Mazilu, Crasovan, Malomed, Lederer and Torner [2004] using this concept. In fig. 20 such a molecule is shown, constructed from six two-color “atoms”. It turns out that the special initial condition for N solitons in the ring, namely the “inverse” picture of the vortex splitting fig. 6, allows one to reconstruct the vortex soliton. Desyatnikov, Denz and Kivshar [2004] demonstrated how several initially well separated solitons, being launched toward the target ring in the absence of perturbations, can excite a metastable vortex ring. Mutual trapping of several solitons on the collision can be regarded as a synthesis of soliton molecules, and it corresponds to a transfer of an initial angular momentum of a system of solitons to angular momentum stored by the optical vortex. Similar results were obtained by Mihalache, Mazilu, Crasovan, Malomed, Lederer and Torner [2004] for 3D clusters of light bullets, launched with additional azimuthal tilts which mimics the vortex phase. The analogy with particles and their bound states, employed to develop the concept of scalar soliton clusters, can be applied to the composite solitons, constructed from incoherently coupled beams, see Section 3. In this case, two mutually incoherent beams can be regarded as atoms of different sorts, always attracting each other (see, e.g., papers by Segev [1998], Stegeman and Segev [1999]). Now, the combination of three types of forces acting between solitons allows to construct a great variety of bound states, or quasi-solitons. These forces are of different origin, namely the coherent phase-dependent attraction/repulsion, incoherent attraction, and the repulsive centrifugal force in clusters with angular momentum. The important issue of the effective interaction potential was investigated by Malomed [1998], Maimistov, Malomed and Desyatnikov [1999], and conservation of angular momentum for interacting two-dimensional and threedimensional solitons discussed by Desyatnikov and Maimistov [2000]. Combining the ringlike clusters of N fundamental solitons with the staircase phase distribution and the stabilizing effect of the vectorial beam interaction,

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Desyatnikov and Kivshar [2002b] introduced a concept of the vector soliton clusters. Indeed, if we take a scalar cluster of solitons and combine it with the other beam that interacts incoherently with the primary beam supporting the cluster structure, the resulting vector soliton will demonstrate the surprising long-lived rotational dynamics, as clearly seen in fig. 20(a). Moreover, the vectorial interaction can allow to trap not only a large number of solitons with N
4, but also three and even two solitons. In the latter case, this structure is, as a matter of fact, the rotating dipole-mode vector soliton or optical propeller (Krolikowski, Ostrovskaya, Weilnau, Geisser, McCarthy, Kivshar, Denz and Luther-Davies [2000], Carmon, Uzdin, Pigier, Musslimani, Segev and Nepomnyashchy [2001]), where two out-of-phase solitons are trapped by the other beam interacting incoherently. Therefore, the rotating structure presented in fig. 20(a) is somewhat similar to the four-soliton propeller. Similar ideas are applicable in other fields, such as light in plasmas (Ren, Hemker, Fonseca, Duda and Mori [2000], Berezhiani, Mahajan, Yoshida and Pekker [2002]) or atomic mixtures of Bose–Einstein condensates (Perez-Garcia and Vekslerchik [2003]). The latter case of trapped matter-waves is especially interesting because of recent breakthroughs in experimental realization of vortices, we discuss it in Section 8.2. Recent results of interaction of counterpropagating vortices presented by Motzek, Jander, Desyatnikov, Belic, Denz and Kaiser [2003], also point to the existence of robust stationary points resembling soliton clusters, see fig. 20(b). This particular system is known to be a rich source of the dynamic instabilities and chaos, thus the clustering of vortices resulted from this instability may indicate the presence of “islands” of order attracting unstable system. Finally, we mention here clusters of dissipative solitons in lasers and externally driven cavities (Vladimirov, McSloy, Skryabin and Firth [2002], Skryabin and Vladimirov [2002]), however, these bound states have different physical origin and we briefly explain this difference in Section 8.1.

§ 7. Discrete vortices in two-dimensional lattices The optical vortices discussed above propagate in homogeneous nonlinear media. When refractive index is periodically modulated, it modifies the wave diffraction properties, and it can affect strongly both nonlinear propagation and localization of light (Kivshar and Agrawal [2003], Christodoulides, Lederer and Silberberg [2003]). As a result, periodic photonic structures and photonic crystals recently attracted a lot of interest due to the unique ways they offer for controlling light propagation. In particular, many nonlinear effects, including formation of lattice solitons, have been demonstrated experimentally for one- and

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two-dimensional optically-induced photonic lattices (Fleischer, Carmon, Segev, Efremidis and Christodoulides [2003], Neshev, Ostrovskaya, Kivshar and Krolikowski [2003], Fleischer, Segev, Efremidis and Christodoulides [2003]). The concept of optically-induced lattices arises from the possibility to modify the refractive index of a nonlinear medium with periodic optical patterns, and use a weaker probe beam to study scattering of light from the resulting periodic photonic structure. Current experiments employ photorefractive crystals with strong electro-optic anisotropy to create a linear optically-induced lattice with a polarization orthogonal to that of a probe beam, which also eliminates the nonlinear interaction between the beam and the lattice. A vortex beam propagating in an optical lattice can be stabilized by the effective lattice discreteness in a self-focusing nonlinear media creating a twodimensional discrete vortex soliton. This has been shown in several theoretical studies of the discrete (Johansson, Aubry, Gaididei, Christiansen and Rasmussen [1998], Malomed and Kevrekidis [2001], Kevrekidis, Malomed, Bishop and Frantzeskakis [2002], Kevrekidis, Malomed and Gaididei [2002], Kevrekidis, Malomed, Chen and Frantzeskakis [2004], Kevrekidis, Malomed, Frantzeskakis and Carretero-Gonzalez [2004]) and continuous models with an external periodic potential (Yang and Musslimani [2003], Baizakov, Malomed and Salerno [2003], Yang [2004]), and such vortices have been also generated experimentally by Fleischer, Neshev, Bartal, Alexander, Cohen, Ostrovskaya, Manela, Martin, Hudock, Makasyuk, Chen, Christodoulides, Kivshar and Segev [2004]. Below, we discuss some of the basic properties of the discrete vortices and summarize the major experimental observations. We consider two-dimensional optically-induced lattices created in photorefractive crystals. In this case, the evolution of a laser beam can be described by the generalized nonlinear Schrödinger-type equation,   2

∂Ψ ∂ Ψ ∂ 2Ψ i (7.1) − G x, y, |Ψ |2 Ψ = 0, +D + ∂z ∂x 2 ∂y 2 where Ψ (x, y, z) is the normalized envelope of the electric field, the transverse coordinates x, y and the propagation coordinate z are normalized to the characteristic values x0 and z0 , respectively, D = z0 λ/(4πn0 x02 ) is the beam diffraction coefficient, where n0 is the average medium refractive index and λ is the vacuum wavelength. The function G(x, y, |Ψ |2 ) accounts for both lattice potential and nonlinear beam self-action effects,  −1 G = γ Ib + I0 sin2 (πx/d) sin2 (πy/d) + |Ψ |2 (7.2) , where γ is proportional to the external biasing field, Ib is the dark irradiance, and I0 is the intensity of interfering beams that induce a square lattice of period

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d through the photorefractive effect (see details by Fleischer, Segev, Efremidis and Christodoulides [2003], Neshev, Alexander, Ostrovskaya, Kivshar, Martin, Makasyuk and Chen [2004], Fleischer, Bartal, Cohen, Manela, Segev, Hudock and Christodoulides [2004]). Similar mathematical models exist for describing the self-action effects in nonlinear photonic crystals (Mingaleev and Kivshar [2001]), and the nonlinear dynamics of atomic BEC in optical lattices (Ostrovskaya and Kivshar [2003]). Self-focusing nonlinearity in eq. (7.1) can compensate for the diffractioninduced beam spreading in the transverse directions, leading to the formation of stationary structures in the form of spatial solitons, Ψ (x, y, z) = ψ(x, y)eiβz , where ψ(x, y) is the soliton envelope, and β is a soliton parameter, a nonlinear shift of the beam propagation constant. In order to analyze the vortex-like structures in a periodic potential, we present the field envelope in the form, ψ(x, y) = |ψ(x, y)| exp[iϕ(x, y)], and assume that the accumulation of the phase ϕ around a singular point (at ψ = 0) is 2πM, where the integer M is a topological charge of the phase singularity. We consider spatially localized structures in the form of vortex-like bright solitons with the envelopes decaying at infinity. Such structures may exist when the soliton eigenvalue β is inside a gap of the linear Floquet-Bloch spectrum of the periodic structure (Mingaleev and Kivshar [2001], Ostrovskaya and Kivshar [2003]). More importantly, a self-induced waveguide created by the vortex soliton is double-degenerated, and it supports simultaneously two modes, |ψ(x, y)| cos ϕ and |ψ(x, y)| sin ϕ, for the same value of β. For symmetric vortex-like configurations, i.e. those possessing a 90◦ rotational symmetry, this is always the case. The profiles of stable symmetric vortex solitons, mentioned above, resemble closely a ring-like structure of the soliton clusters in homogeneous media, see Section 6.3. Using the reduced Hamiltonian approach developed by Desyatnikov and Kivshar [2002a] in homogeneous media, Alexander, Sukhorukov and Kivshar [2004] have generalized it to construct the discrete vortex solitons as a superposition of a finite number of the fundamental (no nodes) solitons, similar to eq. (6.2). Here, in contrast to the case of a homogeneous medium, the positions of individual solitons are fixed by the lattice potential, provided the lattice is sufficiently strong. This approximation is valid when the overlapping integrals between solitons with numbers n and m are the small parameter, cn=m 1. It was rigorously demonstrated by MacKay and Aubry [1994], Aubry [1997] that under such conditions the soliton amplitudes are slightly perturbed due to their interaction, and one can seek stationary solutions using the perturbation approach. In the first order a

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Fig. 21. Examples of the vortex-type soliton structures with various symmetries in a square lattice potential (a), where (b, c) symmetric square vortex solitons; (d) rectangular structure that can only have a trivial phase profile; (e) rhomboid configuration with a topological charge +1. Shown are the intensity profiles (top) and phase structure (bottom). After Alexander, Sukhorukov and Kivshar [2004].

simple constraint for the soliton phases αn have been obtained, cf. eq. (6.3), N −1 

cnm sin(αm − αn ) = 0.

(7.3)

m=0

In the sum (7.3), each term defines the energy flow between the solitons with numbers n and m, so that eq. (7.3) introduce a condition for a balance of energy flows which is a necessary condition for stable propagation of a soliton cluster and the vortex-soliton formation. These conditions are satisfied trivially when all the solitons are in- or out-of-phase. The nontrivial solutions of eq. (7.3) correspond to the vortex-like soliton clusters have been analyzed only for symmetric configurations (Eilbeck, Lomdahl and Scott [1985], Eilbeck and Johansson [2003]), and even then some important solutions have been missed. Using this approach, Alexander, Sukhorukov and Kivshar [2004] introduced different novel types of asymmetric vortex solitons, some of them are shown in fig. 21. Moreover, the existence properties of asymmetric vortex-like solutions are highly nontrivial, due to specific properties of the coupling coefficients calculated for realistic periodic structures. Note also that the symmetric modes can be represented as the angular Bloch modes (Ferrando [2004]). In a number of recent publications, authors develop the concept of higher-order lattice solitons, such as dipole solitons (Yang, Makasyuk, Bezryadina and Chen [2004]), “quasivortices” (Kevrekidis, Malomed, Chen and Frantzeskakis [2004]), or even periodic soliton trains (Chen, Martin, Eugenieva, Xu and Bezryadina [2004], Kartashov, Vysloukh and Torner [2004b]). These multi-humped states

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can be constructed as the bound states of individual lowest order lattice solitons (Kevrekidis, Malomed and Bishop [2001]). Though the simplest stationary solutions with in-phase solitons also exist, usually the stable configuration requires the edge-type π-phase dislocations between neighboring sites (Kartashov, Egorov, Torner and Christodoulides [2004]). The approach developed by Alexander, Sukhorukov and Kivshar [2004] includes into this rich family the higher-order solitons with nested screw phase dislocations and nonzero angular momentum. To provide evidence for the existence of the discrete vortex solitons two separate groups performed independent experimental investigations (Neshev, Alexander, Ostrovskaya, Kivshar, Martin, Makasyuk and Chen [2004], Fleischer, Bartal, Cohen, Manela, Segev, Hudock and Christodoulides [2004]). Both relied on optical induction to create a nonlinear lattice in a photosensitive (photorefractive) material (Efremidis, Sears, Christodoulides, Fleischer and Segev [2002]). In this technique ordinarily polarized light is periodically modulated (by interference or by imaging a mask) to induce a 2D array of waveguides in an anisotropic photorefractive crystal. A separate probe beam of extraordinary polarization acquires a vortex structure by passing through a phase mask and is then launched into the array. The degree of nonlinearity is controlled by applying a voltage across the c-axis of the crystal (photorefractive screening nonlinearity) and controlling the intensity of the probe beam. Typical experimental results are summarized in fig. 22. A two-dimensional square lattice was first created, with its principal axes oriented in the diagonal directions shown in the top panel of fig. 22(a). The resulting periodic structure acts as a square array of optically induced waveguides for the probe beam. The vortex beam, shown in the bottom panel of fig. 22(a), was then launched straight into the middle of the lattice “cell” of four waveguides, as indicated by a bright ring in the lattice pattern. Due to the coupling between closely spaced waveguides of the lattice, the vortex beam exhibits discrete diffraction when the nonlinearity is low (fig. 22(b)), whereas it forms a discrete vortex soliton at an appropriate level of higher nonlinearity (see fig. 22(c, d), the top-middle panel). As predicted, the observed discrete diffraction and discrete self-trapping of the vortex beam in the photonic lattice is remarkably different from that in a homogeneous medium (see fig. 22 (bottom row)). In addition, we mention that the concept of lattice solitons has been recently explored theoretically in lattices induced by Bessel beams Kartashov, Vysloukh and Torner [2004a], Kartashov, Egorov, Vysloukh and Torner [2004a]. In this case the optically induced potential possesses a cylindrical symmetry and support stable soliton complexes in the form of ring-shaped multipoles and necklaces (Kartashov, Egorov, Vysloukh and Torner [2004b]). Remarkably, such lattices al-

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Fig. 22. Experimental observation of an optical vortex propagating with (top panel) and without (bottom panel) an optically-induced lattice. Middle panel – three dimensional representation. (a) Input intensity pattern of the lattice (top) and input (middle) and linear output vortex beam (bottom). (b–d) Intensity patterns of the vortex beam at crystal output with a bias field of 600, 1200, and 3000 V/cm, respectively. The bright ring in the lattice pattern (a) indicates the location of the input vortex. After Neshev, Alexander, Ostrovskaya, Kivshar, Martin, Makasyuk and Chen [2004].

low stable ring vortex soliton to exist even in self-defocusing medium (Kartashov, Vysloukh and Torner [2005b]). Further generalization of the concept of lattice vortex solitons include higherband vortex solitons (Manela, Cohen, Bartal, Fleischer and Segev [2004]), made of two components from different bands, composite vortices in Bessel lattices (Kartashov, Vysloukh and Torner [2005a]), and in conventional honeycomb lattices made in quadratic nonlinear media (Xu, Kartashov, Crasovan, Mihalache and Torner [2005]). We also notice recent investigations of higher-order antiguiding modes (Yan and Shum [2004]) and optical vortices (Ferrando, Zacares, de Cordoba, Binosi and Monsoriu [2004]) in photonic crystal fibers which feature interesting similarities with higher-order solitons and vortices in optically-induced lattices.

§ 8. Links to vortices in other fields Many diverse concepts in physics, ranging from the vortex clusters in quantum dots (Saarikoski, Harju, Puska and Nieminen [2004]) to the data vortex switch

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architecture in the networks of optical waveguides (Yang and Bergman [2002]), may find their analogies with the physics of optical vortices. The concepts of linear singular optics can be expanded to the femtosecond regime (Bezuhanov, Dreischuh, Paulus, Schatzel and Walther [2004]) as well as to apply in the entirely new wavelength domain, e.g., the X-ray regime (Peele, McMahon, Paterson, Tran, Mancuso, Nugent, Hayes, Harvey, Lai and McNulty [2002], Peele and Nugent [2003], Turner, Dhal, Hayes, Mancuso, Nugent, Paterson, Scholten, Tran and Peele [2004]). The fundamental physical concept of phase singularities finds many promising applications, such as extensively studied manipulation of micro-objects by optical tweezers and spanners (Simpson, Allen and Padgett [1996], Friese, Nieminen, Heckenberg and Rubinsztein-Dunlop [1998], Gahagan and Swartzlander [1999], Koumura, Zijlstra, van Delden, Harada and Feringa [1999], Paterson, MacDonald, Arlt, Sibbett, Bryant and Dholakia [2001], MacDonald, Paterson, Volke-Sepulveda, Arlt, Sibbett and Dholakia [2002], Grier [2003]). Moreover, in many cases, the physics of optical vortices is useful for getting a deeper insight into the novel phenomena described by similar nonlinear models. Here, we mention only three of such rapidly growing fields that include vortex states in dissipative systems, vortices in Bose–Einstein condensates, and the application of singular optical beams to quantum information. Each of these topics is rather extended and diverse, and deserves a separate overview.

8.1. Vortices in dissipative optical systems Above, we described the vortex solitons and related phenomena in optical systems with the help of the conservative NLS-type equations. The NLS equation can be linked to a more general dissipative Ginzburg–Landau (GL) model as its conservative limit. The theoretical approach of Ginzburg and Landau [1950] was introduced to describe the phenomena of superconductivity (Cyrot [1973]). Abrikosov [1957] developed further this approach for the type II superconductors, more common in nature, and showed that the flux penetrates the superconductor in the form of a regular array of flux tubes or vortices (Abrikosov vortices). In two dimensions, the complex GL equation (CGL) admits extensively studied quantized vortices (Cross and Hohenberg [1993], Pismen [1999]), the stationary limit of the latter equation is also known as the stationary Gross–Pitaevskii equation. A large number of publications is devoted to the study of vortices in boson condensates (such as superconductors and superfluids), described by the CGL equation (see, e.g., Ovchinnikov and Sigal [1997, 1998a, 1998b, 2002], and references therein).

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In the context of nonlinear optics, the latter model corresponds to the stationary NLS with self-defocusing type nonlinearity, discussed above in Section 2.1. It is interesting to note that related integrable complex sine-Gordon model admits the continuous families of the nonradially symmetric dark vortex solitons (Barashenkov, Shchesnovich and Adams [2002]), while their existence remains an open question for the nonintegrable CGL equation (Ovchinnikov and Sigal [2000]). The analogy between superfluids and laser optics was recognized as early as 1970 (Graham and Haken [1970]). In particular, the vortex solutions to laser equations were found by Coullet, Gil and Rocca [1989], Tamm and Weiss [1990] and intensively studied latter, both theoretically and experimentally (Weiss, Vaupel, Staliunas, Slekys and Taranenko [1999]). The concept of vortices in lasers is connected to the dissipative optical solitons (DOS), or auto-solitons. This kind of soliton was initially predicted for wide-aperture nonlinear interferometers excited by external radiation (Rozanov and Khodova [1988]) and for laser systems with saturable absorbers (Rozanov and Fedorov [1992]). If the material relaxation time is much smaller than that for the field in optical resonator (the so-called class-A laser), the master Maxwell–Bloch equations can be reduced to a CGL equation (Mandel [1997])

∂E (8.1) = (δ + i)d E + f |E|2 E + Ei . ∂ζ Here the evolution variable ζ stands for time, in the resonator schemes, or the propagation coordinate z, in a bulk medium, δ is the effective diffusion coefficient, and the diffraction operator d is acting in d = 1, 2, 3 “transverse” coordinates. The parameter Ei represents an external driving plane-wave field, and the nonlinearity f (|E|2 ) is a complex function, so that the conservative limit eq. (2.11) is given by δ = Ei = 0 and f (|E|2 ) = iF (|E|2 ). The wide-aperture interferometers, filled with passive or active nonlinear media (optical cavities) and excited by external radiation Ei , exhibit optical bistability (Gibbs [1985]). Due to the spatial hysteresis (Rozanov [2002]) there possible domain walls connecting two stable and otherwise spatially homogeneous transmitted waves. Domain walls are usually identified with switching waves, and they represent building blocks for DOSs – the localized solutions emerging as bound states of switching waves. This implies the discreetness of the DOS spectrum in contrast to the continuous “families” of conservative solitons. The external radiation Ei determines the frequency and the phase of DOSs, or “cavity solitons”; they exist on the nonzero background and have oscillating tails. Several DOSs may interact (Afanasjev, Malomed and Chu [1997], Ramazza, Benkler, Bortolozzo, Boccaletti, Ducci and Arecchi [2002], Schapers, Feldmann, Ackemann and Lange

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[2000], Schapers, Ackemann and Lange [2003], Tlidi, Vladimirov and Mandel [2003]) and form bound states or clusters of cavity solitons (Vladimirov, McSloy, Skryabin and Firth [2002]). In contrast to the conservative model discussed in Section 6.3, the clusters may appear due to the oscillating effective interaction potential (Malomed [1991, 1998]) with a discrete set of the equilibrium distances between cavity solitons. A large number of publications is devoted to the study of cavity solitons and their link to spontaneous pattern formation, as also discussed in the recent review papers (Arecchi, Boccaletti and Ramazza [1999], Rozanov [2000], Firth and Weiss [2002], Peschel, Michaelis and Weiss [2003], Lugiato [2003]), and reflected in the comprehensive list of references prepared by Mandel and Tlidi [2004]. Interesting application of vortices as the pump beams in externally driven cavities was suggested by Rozanov [1992, 2002]. For degenerate optical parametric oscillator it was demonstrated by Oppo, Scroggie and Firth [2001] that the stable domain walls appear as being trapped in the beam. In the vertical-cavity surfaceemitting lasers the cavity solitons perform a uniform rotary motion along the crater of a doughnut-shaped holding beam (Barland, Brambilla, Columbo, Furfaro, Giudici, Hachair, Kheradmand, Lugiato, Maggipinto, Tissoni and Tredicce [2003]). Micro-cavities offer novel possibilities for the cavity soliton generation and control (Barland, Tredicce, Brambilla, Lugiato, Balles, Giudici, Maggipinto, Spinelli, Tissoni, Knödl, Miller and Jäger [2002], Debernardi, Bava, di Sopra and Willemsen [2003], Maggipinto, Brambilla and Firth [2003], Vahala [2003]), and the examples include the spontaneous generation of the “optical vortex crystals” (Scheuer and Orenstein [1999]). In lasers, the phase of the field is free (Ei = 0 in eq. (8.1)) and the topological solitons are possible (Firth and Weiss [2002]). The comprehensive overview of the theoretical studies and the experimental generation of laser vortices can be found in Weiss, Vaupel, Staliunas, Slekys and Taranenko [1999], Weiss, Staliunas, Vaupel, Taranenko, Slekys and Larionova [2003]. In lasers with saturable absorbers, i.e. when nonlinearity in eq. (8.1) is given by f (|E|2 ) = −1 + g0 /(1 + |E|2 ) − a0 /(1 + b|E|2 ), novel types of solitons may appear, such as transversely asymmetric and rotating structures without phase dislocations and radially symmetric vortices with higher-order topological charges (Rozanov, Fedorov, Fedorov and Khodova [1995], Fedorov, Rosanov, Shatsev, Veretenov and Vladimirov [2003]). Furthermore, bright dissipative vortex solitons can form strongly coupled “vortex clusters”, as shown in fig. 23(a, b), as well as weakly coupled bound states (Rozanov, Fedorov and Shatsev [2004]). The latter exhibit spontaneous rotation during the evolution, see fig. 23(c).

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Fig. 23. (a, b) The instantaneous transverse distributions of intensity of laser radiation for the regime of rotating chains with (a) different number of single-charged dislocations with m = +1, and (b) the opposite topological charges of the links m = −1, +1 (on the left) and m = −1, +1, +1 (right). After Rozanov, Fedorov and Shatsev [2003]. (c) Interaction between laser vortices. The instantaneous transverse intensity distributions of laser radiation for the regime of three solitons with topological charges m = −1 during the evolution at dimensionless time t = 0 and t = 1250 (Rozanov, Fedorov, Shatsev and Loiko [2004]).

Stabilization of dissipative vortex solitons in the cubic-quintic CGL and new types of radially symmetric solitons, such as erupting, flat-top, and composite vortices, were reported recently by Crasovan, Malomed and Mihalache [2001a, 2001c]. In the same model, the existence of stable clusters of dissipative solitons rotating around a central vortex core was predicted by Skryabin and Vladimirov [2002]. This extends the results presented in Section 6.3 to the case of dissipative CGL systems. Another example of the dissipative optical systems supporting topological spatial solitons is given by the nonlinear interferometer formed by a liquid crystal light valve with a feedback. The so-called “triangular solitons” with the rich structure of phase singularities were found by Ramazza, Bortolozzo and Pastur [2004], Bortolozzo, Pastur, Ramazza, Tlidi and Kozyreff [2004] in this system. The vectorial generalization of the CGL model and corresponding patterns of phase dislocations studied by Hernandez-Garcia, Hoyuelos, Colet and Miguel [2000], Hoyuelos, Hernandez-Garcia, Colet and San Miguel [2003]. Similar polarization patterns and vectorial defects were observed in numerical simulations of the three-wave optical parametric oscillators by Santagiustina, Hernandez-Garcia, San Miguel, Scroggie and Oppo [2002]. Finally, we mention the three-dimensional generalization of cavity solitons, namely the “bubbles with a dark skin” and 3D Turing structures in synchronously

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pumped degenerate, and 3D vortex rings in nondegenerate optical parametric oscillators (Weiss, Vaupel, Staliunas, Slekys and Taranenko [1999]).

8.2. Vortices in matter waves The concept of optical vortices can take us away from optics itself and emphasize the relevance of optical solitons and optical vortices to other fields of nonlinear physics. In particular, the study of vortices is an important research topic in the rapidly developing field of coherent matter waves and nonlinear atom optics. In particular, vortices appear in the nonlinear dynamics of the Bose–Einstein condensates and they provide a close link between self-focusing of light in nonlinear optics and the nonlinear dynamics of matter waves. The phenomenon known as Bose–Einstein condensation (BEC) was actually predicted in 1924 for systems whose particles obey the Bose statistics and whose total particle number is conserved. It was shown that there exists a critical temperature below which a finite fraction of all particles condenses into the same quantum state. Since 1995, the BEC phenomenon has been observed using several different types of atoms, confined by a magnetic trap and cooled down to extremely low temperatures (Anderson, Ensher, Matthews, Wieman and Cornell [1995], Bradley, Sackett, Tollett and Hulet [1995], Davis, Mewes, Andrews, Vandruten, Durfee, Kurn and Ketterle [1995], Fried, Killian, Willmann, Landhuis, Moss, Kleppner and Greytak [1998]). From a mathematical point of view, the dynamics of BEC wave function can be described by an effective mean-field equation known as the Gross–Pitaevskii (GP) equation (Dalfovo, Giorgini, Pitaevskii and Stringari [1999]). This is a classical nonlinear equation that takes into account the effects of particle interaction through an effective mean field. As a matter of fact, the complete theoretical description of a BEC requires a quantum many-body approach (Dalfovo, Giorgini, Pitaevskii and Stringari [1999]). The many-body Hamiltonian describing N inˆ teracting bosons is expressed through the boson field operators Φ(r) and Φˆ † (r) that, respectively, annihilate and create a particle at the position r. A mean-field approach is commonly used for the interacting systems to overcome the problem of solving exactly the full many-body Schrödinger equation. Apart from the convenience of avoiding heavy numerical work, mean-field theories allow one to understand the behavior of a system in terms of a set of parameters that have a clear physical meaning. Actually, most of the experimental results show that the mean-field approach is very effective in providing both qualitative and quantitative predictions for the static and dynamic properties of the trapped ultracold gases.

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Because of the similarities between the GP equation in the BEC theory and the NLS equation in nonlinear optics, many of the phenomena predicted and observed in nonlinear optics are expected to occur for the BEC macroscopic quantum states, even though the underlying physics can be quite dissimilar. In particular, this includes the dynamics of BEC vortices (Williams and Holland [1999], Garcia-Ripoll and Perez-Garcia [2000], Alexander and Berge [2002]) recently reviewed by Fetter and Svidzinsky [2001] (see also the introduction to the theory of vortices in BEC by Ghosh [2004]). Historically, quantum vortices in trapped atomic gases were first observed in 1999, using two-component condensates (Matthews, Anderson, Haljan, Hall, Wieman and Cornell [1999]). The possibility of trapping more than one BEC component arises from the hyperfine atomic structure. Atoms in internal states with different total angular momentum may coexist in the BEC fraction, and it is possible to induce transitions between their different states. To form a vortex soliton, a phase gradient was imprinted in one of the BEC components, which caused it to rotate. The system was stabilized at a configuration in which the nonrotating component was localized at the center of the trap acting as an effective potential on the rotating component, which resided in the outer region. The main properties of a two-component BEC can be described using a system of two incoherently coupled GP equations, similar to the two coupled NLS equations that describe optical vector solitons, except for the presence of a trapping potential that prevents the condensate with repulsive interaction from spreading. The two-component GP equation has solutions with remarkable properties (Garcia-Ripoll and Perez-Garcia [2000]). In a situation where the two components overlap considerably, the creation of a vortex in just one of the components is not dynamically stable. The reason is that the angular momentum can be transferred after some time from one component to the other one, initiating a cyclic process. If one only monitors the density profile of the two atomic species, it may seem that the vortex disappears and eventually reappears in a periodic manner. If a nearly two-dimensional trap is made to rotate, the situation changes qualitatively. In the rotating frame the Coriolis force manifests itself through an additional term in the Hamiltonian, −ΩLz , where Lz is a component of the angular momentum operator L and Ω is the angular frequency (Fetter and Svidzinsky [2001]). This additional force produces the centrifugal barrier proportional to L2z z, where Lz = mh¯ is the angular momentum per particle. Thus, it is always energetically costly to have a high angular momentum. However, for nonzero values of Ω, it may be energetically favorable to develop small positive values of Lz . If Ω is sufficiently large, solutions with Lz greater than h¯ may have the lowest energy; a vortex is created in this situation.

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For Ω greater than a critical frequency Ωc = 0.22ω0 , where ω0 is the trap frequency, the lowest energy corresponds to the state with Lz = h¯ . As the frequency increases, states with increasingly high angular momentum become the effective ground state in the rotating frame, thereby creating vortices with a topological charge m > 1. Such configurations are unstable and decompose into several single-charge vortices (Butts and Rokhsar [1999]). The interaction between vortices is generally believed to be repulsive. Thus, at rotating frequencies high enough to generate many stable vortices, vortices tend to move apart and drift toward the borders of the condensate, where they would disappear if their existence were not favored by the rotation. The result is that, at very high angular frequencies, vortices tend to form a regular array, as also observed experimentally. Figure 24 shows examples of vortices for a rotating Rb-vapor BEC. The array formation is akin to what has been long known for type II superconductors, where it is the presence of a magnetic field that forms a triangular vortex crystal – the so-called Abrikosov lattice (Abrikosov [1957]). In another experiment by Madison, Chevy, Wohlleben and Dalibard [2000], the formation of a regular vortex array was observed in a 87 Rb BEC as the number of stable vortices was raised from 0 to 4 by increasing the angular frequency. The formation of a triangular vortex lattice with as many as 130 vortices was observed

Fig. 24. Generation of vortices and vortex lattices in a rotating 87 Rb condensate. (Courtesy P. Engels.)

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in an experiment where BEC was obtained using sodium atoms (Abo-Shaeer, Raman, Vogels and Ketterle [2001]). The optimum configuration in terms of size and regularity is achieved after 500 ms. For times shorter than that, regular order is not completely established, and a blurry structure is formed because of the misalignment of some vortices with respect to the rotation axis. For times much longer than 500 ms, inelastic collisions induce atom losses and a decrease in the number of vortices. The spin texture of various vortex-lattice states at higher rotation rates and in the presence of an external magnetic field has been presented recently by Mizushima, Kobayashi and Machida [2004], and intriguing reshaping from the triangular to a square lattice has been observed by Schweikhard, Coddington, Engels, Tung and Cornell [2004]. Several experiments have studied the nucleation of vortices in a BEC stirred by a laser beam. In the experiment by Raman, Abo-Shaeer, Vogels, Xu and Ketterle [2001], vortices were generated in a BEC cloud stirred by a laser beam and observed with time-of-flight absorption imaging. Depending on the stirrer size, either discrete resonances or a broad response was visible as the stir frequency was varied. Stirring beams that were small compared to the condensate size generated vortices below the critical rotation frequency for the nucleation of surface modes, suggesting a local mechanism of vortex generation. In addition, it was observed that the centrifugal distortion of the condensate induced by a rotating vortex lattice led to bending of the vortex lines. Recent developments in the topic of vortices in BEC were summarized by Kevrekidis, Carretero-Gonzalez, Frantzeskakis and Kevrekidis [2004], including discrete vortices in periodic lattices (Baizakov, Salerno and Malomed [2004]). Corresponding continuous model is very similar to the optically induced photonic lattices, described in Section 7, and supports the “gap vortex soliton” identified by Ostrovskaya and Kivshar [2004], Sakaguchi and Malomed [2004]. However, there are examples of the phenomena which have no analogy in optics, for example the structural transitions of the lattice of vortices in the rapidly rotating periodic potential, reported by Pu, Baksmaty, Yi and Bigelow [2003]. Similarly, even though the properties of a single vortex soliton in BEC and self-defocusing Kerr media are very similar, the stable vortex dipoles in nonrotating BEC, introduced by Crasovan, Vekslerchik, Perez-Garcia, Torres, Mihalache and Torner [2003], cannot exist in optical system without the external trapping potential, see Section 2.1. Different from singular vortices, the nonsingular topologically nontrivial states, or skyrmions, have attracted attention since the creation of BEC spinors (Matthews, Anderson, Haljan, Hall, Wieman and Cornell [1999]). Two-dimensional (2D) particlelike solitons of this kind are sometimes referred to as coreless vortices. Skyrmions can be created out of the ground state, in which all the spins

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are aligned, by reversing the average spin in a finite region of space (Al Khawaja and Stoof [2001]). Then the skyrmion is characterized by the winding number, the analog of a topological charge. The half-charge skyrmion has been successfully generated in experiment with three-component BEC by Leanhardt, Shin, Kielpinski, Pritchard and Ketterle [2003], and possible spin textures for anti- and ferromagnetic interactions in this system were compared by Mueller [2002]. The stability of 3D (Al Khawaja and Stoof [2001]) skyrmions, composed of two coaxial tori, and 2D (Zhai, Chen, Xu and Chang [2003]) single-charged skyrmions has been studied in two-component BEC. The general conclusion been that skyrmions can exist as a metastable state only, though the stabilizing mechanisms were suggested for single- and multiply-quantized skyrmions by Savage and Ruostekoski [2003], Ruostekoski [2004]. Skyrmions have important applications in nuclear physics and quantum-Hall effect (Makhankov, Rubakov and Sanyuk [1993]), it is expected that observation of these structures in BEC would enable a direct comparison between theory and experiment.

8.3. Optical vortices and quantum information During recent years a new fascinating avenue for applications of optical vortices in the area of quantum optics has been identified. As discussed extensively throughout this chapter, light beams with nested optical vortices carry orbital angular momentum, a property that holds as well for the mode functions that describe the photon quantum states, including states corresponding to single photons or to entangled pairs. The quantum angular momentum of light contains a spin and an orbital contribution, and in general only the total angular momentum is an observable quantity (Cohen-Tannoudji, Dupont-Roc and Grynberg [1989]). However, within the paraxial regime, both contributions can be measured and manipulated separately (Van Enk and Nienhuis [1994a, 1994b], Simpson, Dholakia, Allen and Padgett [1997], Barnett [2002], Neil, MacVicar, Allen and Padgett [2002], Leach, Courtial, Skeldon, Barnett, Franke-Arnold and Padgett [2004]). The spin contribution is described by a two-dimensional state, thus can be employed to generate qubits, whereas the orbital contribution can generate multidimensional quantum entangled states, or qudits, with an arbitrarily large number of entanglement dimensions. While the spin angular momentum is a workhorse of quantum optics and quantum information (Bouwmeester, Ekert and Zeilinger [2000]), only recently the orbital angular momentum has been added to the toolkit (Arnaut and Barbosa [2000], Mair, Vaziri, Weihs and Zeilinger [2001], MolinaTerriza, Torres and Torner [2002]).

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Allen, Beijersbergen, Spreeuw and Woerdman [1992] showed a decade ago that paraxial Laguerre–Gaussian laser beams, with a nested vortex, carry a welldefined orbital angular momentum associated to their spiral wave fronts (Allen, Padgett and Babiker [1999]). The formal analogy between paraxial optics and quantum mechanics implies that such modes are the eigenmodes of the quantum mechanical angular momentum operator. The Laguerre–Gaussian modes form a complete Hilbert set and can thus be used to represent the quantum photon states within the paraxial regime of light propagation. The quantum angular momentum number carried by the photon is then represented by the topological charge, or winding number m, of the corresponding mode, and each mode carries an orbital angular momentum of mh¯ per photon. Multi-dimensional vector photon states can be constructed with controllable projections into modes with well-defined winding numbers, thus providing higher-dimensional alphabets Molina-Terriza, Torres and Torner [2002]. In particular, mode functions in the form of vortex-pancakes allow the manipulation, including the addition and removal, of specific projections of the vector states. It is worth stressing that beams without nested vortices, or alternatively with a complex topological structure, can also carry orbital angular momentum. A beautiful illustrative example was shown by Santamato, Sasso, Piccirillo and Vella [2002] in the classical regime, by studying the optical angular momentum transfer to transparent particles using light beams carrying zero average angular momentum. The concept applies as well to the quantum regime in a variety of beams shapes and geometries. However, because Laguerre–Gaussian modes carrying nested optical vortices are eigenstates of the quantum orbital angular momentum operator, vortices play a central role in the area. The generation of quantum states entangled in orbital angular momentum relies in the process of spontaneous parametric down conversion (Klyshko [1967], Arnaut and Barbosa [2000], Mair, Vaziri, Weihs and Zeilinger [2001], FrankeArnold, Barnett, Padgett and Allen [2002], Padgett, Courtial, Allen, FrankeArnold and Barnett [2002]). The generated entangled two-photon states can be prepared in desired states by making use of transverse engineering of quasi-phasematched geometries, Torres, Alexandrescu, Carrasco and Torner [2004] or by appropriate tailoring of the spatial characteristics of the pump beam. The latter can be accomplished by a variety ways, including by pumping the nonlinear crystal with several nested optical vortices (Torres, Deyanova, Torner and Molina-Terriza [2003]). An illustrative example of the potential engineering of quantum entangled states with pump beams with nested vortex pancake is shown in fig. 25. The characterization of the entangled photon pairs in terms of eigenstates of the orbital angular momentum operator yields the concept of quantum spiral bandwidth, which was found to depend on the shape of the beam that pumps the down-

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Fig. 25. Intensity profile of the pump beam that generates the maximally entangled qu-quart. (a) Intensity profile, (b) the generated quantum state. In (a), the transverse coordinates are normalized to the beam width. In (b), the bars show the weight distribution and the clocks show the phase distribution of the quantum state. After Torres, Deyanova, Torner and Molina-Terriza [2003].

converting crystal, and on the material properties and length on the crystal (Torres, Alexandrescu and Torner [2003]). All these schemes are awaiting experimental demonstration, although the related concept of entanglement concentration put forward by Torres, Deyanova, Torner and Molina-Terriza [2003] has been observed experimentally using an alternative approach, as mentioned below. A fundamental question that arises is the conservation of OAM in the process of photon down-conversion (Arnaut and Barbosa [2000], Eliel, Dutra, Nienhuis and Woerdman [2001], Visser, Eliel and Nienhuis [2002], Barbosa and Arnaut [2002], Caetano, Almeida, Souto Ribeiro, Huguenin, Coutinho dos Santos and Khoury [2002]). In collinear down-conversion, the two-photon entangled state constituted by the signal and idler photons is described by a transverse mode function that is globally paraxial. Therefore, the orbital angular momentum of all the involved photons is a well-defined quantity that in the absence of momentum transfer between light and matter must be conserved, a feature that within the experimental accuracy is consistent with the observations by Mair, Vaziri, Weihs and Zeilinger [2001], in the quasi-collinear geometry used. In noncollinear geometries the relation between the orbital angular momentum and the vorticity of the mode function is not necessarily given by simple algebraic rules, as most clearly illustrated

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in highly noncollinear settings (Molina-Terriza, Torres and Torner [2003], Torres, Osorio and Torner [2004]). The quantum applications of optical vortices holds promise for exciting developments in the near future, in particular to the proof-of-principle demonstration of quantum algorithms and to explore fundamental quantum features in higherdimensional Hilbert spaces. Significant advances along this direction have been already achieved during the last years after the observation of OAM entanglement by Mair, Vaziri, Weihs and Zeilinger [2001]. Important steps include the development of interferometric schemes that might be used to sort out single photons according to their OAM (Vasnetsov, Slyusar and Soskin [2001], Leach, Padgett, Barnett, Franke-Arnold and Courtial [2002]), the generation of qutrits encoded in OAM and their use to observe violation of Bell inequalities in three-dimensional Hilbert spaces (Vaziri, Weihs and Zeilinger [2002]), the demonstration of concentration of higher-dimensional entanglement (Vaziri, Pan, Jennewein, Weihs and Zeilinger [2003]), the triggered production, transmission and reconstruction of qutrits for different quantum communication protocols (Molina-Terriza, Vaziri, Rehacek, Hradil and Zeilinger [2004]), the use of qutrits for quantum bit commitment (Langford, Dalton, Harvey, O’Brien, Pryde, Gilchrist, Bartlett and White [2004]), the proposal of innovative set-ups to efficiently measure highdimensional entanglement (Oemrawsingh, Aiello, Eliel, Nienhuis and Woerdman [2004]), and the demonstration of coin-tossing algorithms based in qutrits (Molina-Terriza, Vaziri, Ursin and Zeilinger [2005]). Vortices are also being used to explore fundamental quantum features, like the uncertainty principle for angular position and angular momentum (Franke-Arnold, Barnett, Yao, Leach, Courtial and Padgett [2004]), or the effects induced by the quantum vacuum on the perfect zero of a classical field (the vortex core) (Berry and Dennis [2004]). Much more is expected to come soon, as this area of research is in its infancy.

§ 9. Concluding remarks We have presented a comprehensive overview of exciting research in the field of nonlinear singular optics that studies the propagation of optical vortices and optical beams carrying an angular momentum in nonlinear media. Understanding and controlling the properties of optical vortices could lead to applications in the near future, ranging from optical communications and data storage to the trapping, control, and manipulation of particles and cold atoms. Indeed, optical vortices provide an efficient way to control light by creating reconfigurable waveguides in bulk media. The study of phase singularities in optical parametric processes not

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Fig. 26. The number of papers cited in this review vs. the publication year.

only suggests novel directions of fundamental research in optics but also provides links to other branches of physics. For example, the recent discovery of a rich variety of exotic topological defects in unconventional superfluids (such as 3 He-A) and superconductors points to the likelihood that deep analogies exist between vortices in complex superfluids and singularities in light waves.

Acknowledgements We thank many of our colleagues for collaboration and useful discussions, most especially T. Alexander, L. Berge, S. Carrasco, Z. Chen, L. Crasovan, C. Denz, Y. Kartashov, W. Krolikowski, B. Luther-Davies, B.A. Malomed, D. Mihalache, S. Minardi, G. Molina-Terriza, D. Neshev, E. Ostrovskaya, D. Petrov, N.N. Rozanov, M.F. Shih, M. Soskin, A. Sukhorukov, G. Swartzlander, and J.P. Torres for sharing their knowledge with us and for their constant support and collaboration. ASD acknowledges the hospitality of ICFO during his visit to Barcelona. This work was produced with the assistance of the Australian Research Council (ARC); the Center for Ultra-high bandwidth Devices for Optical Systems (CUDOS) is an ARC Center of Excellence. LT acknowledges support by the Generalitat de Catalunya and by the Spanish Government.

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E. Wolf, Progress in Optics 47 © 2005 Elsevier B.V. All rights reserved

Chapter 6

Phase imaging and refractive index tomography for X-rays and visible rays by

Koichi Iwata Technology and Research Institute of Osaka Prefecture, 2-7-1 Ayumino, Izumi, Osaka 594-1157, Japan

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)47008-0 393

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

395

§ 2. Properties of X-ray and visible ray . . . . . . . . . . . . . . . . . . .

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§ 3. Formation of intensity and phase images . . . . . . . . . . . . . . . .

400

§ 4. Phase imaging methods . . . . . . . . . . . . . . . . . . . . . . . . .

402

§ 5. Reference type interferometers . . . . . . . . . . . . . . . . . . . . .

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§ 6. Shearing type interferometers . . . . . . . . . . . . . . . . . . . . . .

411

§ 7. Refractive index tomography . . . . . . . . . . . . . . . . . . . . . .

419

§ 8. Discussion on interferometers and refractive index tomography . . .

424

§ 9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

430

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

430

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

430

394

§ 1. Introduction A hard X-ray is transmitted through many kinds of material with low absorption and it travels almost straight even at the boundary of the objects. These features make it an important tool to examine the internal structures of objects. In many X-ray imaging methods, such as contact radiography, images are displayed as intensity distribution of the transmitted X-ray (Hendee and Ritenour [1992]). Spatial variation of X-ray absorption in the interior of the objects produces intensity distribution in the image. In the computed tomography the intensity data is used to calculated the distribution of absorption coefficient in the interior of the object. Internal structure is displayed as the distribution of absorption coefficient (Hounsfield [1973], Cormack [1963]). However, as far as the imaging method is based on this principle, weakly absorbing materials are difficult to form high contrast images (Momose, Takeda, Yoneyama, Koyama and Itai [2001]). As a lot of carbon-based compounds have low absorption coefficient, we cannot obtain high contrast images for polymers and soft biological tissues with hard X-rays. In order to overcome this difficulty, we can use information of phase variation of X-rays that travels through the object. In recent years many researches on the phase imaging techniques appeared in the X-ray region (Fitzgerald [2000]). The phase variation is caused by the spatial variation of refractive index for the X-ray in the object. Therefore the internal structure of the object is displayed as a distribution of refractive index for the X-ray (Momose, Takeda, Itai and Hirano [1996]). There are some review articles on phase imaging in X-ray region (Fitzgerald [2000], Momose [2003]). But there is no article which describes the subject in relation to the corresponding techniques in the visible spectrum region. The comparison between the techniques in the two quite different wavelength regions may be beneficial to the development of both techniques. This is an attempt toward this aim. In the visible spectrum region, phase imaging has a long history (Jenkins [1980a], Hecht [1987]). It has been used for investigating the interior of transparent fluid for a long time (Merzkirch [1974]). They include shadowgraphy, 395

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[6, § 2

Schlieren method and interferometry in the field of gas dynamics. In the visible spectrum phase imaging is also used for investigating the profiles of reflecting surfaces (Malacara [1978]). For improving these phase imaging, many techniques has been investigated. In this situation in mind this review will describes the X-ray phase imaging techniques in relation to the corresponding optical techniques.

§ 2. Properties of X-ray and visible ray 2.1. Absorption coefficient and refractive index Both X-rays and visible rays are electromagnetic waves. The main difference is their wavelength. The wavelength of X-rays described here is in the range around 0.1 nm, which corresponds to the photon energy larger than about 10 keV. The wavelength of visible rays is in the range from 350 to 700 nm. The wavelength is more than 1000 times longer than the wavelength of the X-rays. In visible ray region many applications use rays reflected from a surface of a solid object for surface profiling. In these case, phase imaging caused by the height change of the surface is used. But this review will not refer to these techniques. The discussion will be limited to transparent material and transmitted rays. This is because in X-ray region phase imaging is used in similar situation. Material transparent for visible ray is not so many, but many materials are transparent for X-ray. For transparent material, refractive index and absorption coefficient are two important material constants. In the visible region, refractive index of an ordinary solid is larger than that of vacuum by about 10%. For this value of refractive index, rays deviate appreciably from a straight line due to refraction. The nonstraight path makes it difficult to analyze the resultant phase image. Therefore, phase imaging for transparent materials are limited to fluid, especially to gas. Many gases are mixtures of several components, like air. Refractive index n of the mixture is given by  n=1+ Kk ρk , (2.1) where Kk and ρk are the Gladstone–Dale constants and partial densities of the individual atomic or molecular components indicated by the suffix k (Merzkirch [1974]). When the mass fractions of a gas are known, refractive index is written as n = 1 + Kρ,

(2.2)

where K is the Gladstone–Dale constant and ρ is the density of the gas. The Gladstone–Dale constant for air at temperature of 288 K is about 0.225 (cm3 /g)

6, § 2]

Properties of X-ray and visible ray

397

for visible spectrum. Therefore refractive index of air at the density of 1.29 × 10−3 g/cm3 (1 atm) is 1.00029. Thus the refractive index change due to pressure and temperature is smaller than 10−4 . In this case visible rays travels almost straight. Therefore many examples of phase imaging in this review is limited to flow visualization. For hard X-rays with photon energy of 10 keV, refractive index of the ordinary material is smaller than that of vacuum by about 10−6 to 10−7 (Center for X-ray Optics, website). This difference is smaller than that for the visible spectrum by the ratio of 10−2 to 10−3 . Therefore the X-rays transmit through the material almost straight. Straight path makes it easy to explain the effect of absorption in the interior of the object. Therefore many nondestructive measurement and diagnostics have been made using hard X-rays. However, these measurement techniques were limited to absorption measurement. Intensity of the transmitted X-ray is measured. In computed tomography, distribution of absorption coefficient in the interior of the human body is used as an image. However, fluid visualization techniques shows that phase imaging or refractive index imaging is more effective to visualize a transparent object. As many materials are transparent for hard X-rays, phase and refractive index imaging is expected to be an effective tool. Absorption coefficient and refractive index for the hard X-ray region is shown in papers by Momose and Fukuda [1995] and Momose, Takeda, Yoneyama, Koyama and Itai [2001]. The complex refractive index n is represented by n = 1 − n − iµ.

(2.3)

Note that the refractive index is a little smaller than 1. The real part n and imaginary part µ are expressed by the phase-shift cross-section pk and the atomic absorption cross-section µk for the element k n =

λ  N k pk , 2π k

λ  Nk µk , µ= 4π

(2.4)

k

where Nk is the atomic density of element k. Variation of pk and µk by the elements of different atomic number are shown in fig. 1. As seen from this figure materials of low atomic number show very little absorption. Therefore the contrast of image is low for the object consisting of these elements. However, refractive index does not decrease remarkably with atomic number compared with absorption. For instance, the values of n and µ of water are

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Phase imaging and refractive index tomography for X-rays and visible rays

[6, § 2

Fig. 1. Absorption and phase-shift cross-sections in relation to atomic number (Momose Lab, website).

5.8 × 10−7 and 6.0 × 10−10 for 20-keV X-rays. The ratio n/µ is a thousand. Soft biological tissues are made of elements of low atomic numbers. Therefore phase imaging is expected to be a new diagnostic tool for them. In the next section, with these background in mind, the effects of diffraction and refraction will be investigated for visible rays and X-rays and their difference will be discussed.

2.2. Deviation of ray direction 2.2.1. Diffraction effect In order to estimate the effect of diffraction, we shall investigate the diffraction by an a straight edge of an opaque obstacle. Due to the diffraction, waves enter into the part of the shadow. According to Fresnel diffraction theory, the diffraction intensity is as shown in the right part of fig. 2. For a parallel incident wave the distance ∆ from the edge to the effectively diffracted area is expressed by a expression √ ∆ = 2λR, (2.5) where λ is the wavelength and R is the distance from the edge to the observation point (Jenkins [1980b]).

6, § 2]

Properties of X-ray and visible ray

399

Fig. 2. Effect of diffraction.

If we observe the intensity at R = 100 mm, ∆ = 1.4 × 10−3 mm for X-ray with λ = 0.1 nm and ∆ = 3 mm for visible ray with wavelength 0.5 µm. Thus for X-ray the diffraction does not cause appreciable deviation as far as the convenient scale for a daily life is concerned. Thus the X-ray can be considered to travel straight. On the contrary, for visible light it becomes appreciable value and diffraction has to be taken into account. On the other hand, we have lenses forming an image in the visible spectrum. Thus we can form an image that corresponds to the plane just after the object to reduce R. With the image formation diffraction effect is reduced appreciably. But in the X-ray region we do not have such a convenient image forming element. 2.2.2. Refraction effect In order to estimate the effect of refraction, we shall investigate the refraction by a straight boundary between two transparent media shown in fig. 3. The refractive indices of the two media are n and n + n. The incident angle is expressed by θ and the refraction angle is expressed by θ + θ . Then the maximum deviation occurs when θ = 90◦ , where Snell’s law tells us that the displacement ∆ of the

Fig. 3. Effect of refraction.

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Phase imaging and refractive index tomography for X-rays and visible rays

refracted ray from the incident ray is  n ∆ = Rθ = R 2 n

[6, § 3

(2.6)

at the observe position R from the incident point. If we observe the ray at R = 100 mm, ∆ = 0.04 mm for X-ray with n/n = 10−7 and ∆ = 1.4 mm for visible ray with n/n = 10−4 which is a plausible value corresponding to pressure change in air. The above discussions show that the deviation of the ray direction may be considered small for X-rays. Thus usually the X-rays are considered to travel straight as far as the required spatial resolution of the image is smaller than the deflection of the beams. Under this condition, even when the observation plane is somewhat far from the object, the observed image is considered as a projection along the straight line. However, if we want higher resolution, we have to use image forming element. But in the X-ray region, we have no good image forming element such as refractive lenses, reflective curved mirrors and diffractive zone plate. As the refractive effect is larger than the diffractive effect, we have to use matching liquid whose refractive index matches that of the object. In the visible spectrum image forming lens is necessary to form images on the observation plane because the small distance from the object caused the deviation of rays. However, as the ray refraction is larger, straight path assumption is limited to images of lower resolution.

§ 3. Formation of intensity and phase images 3.1. Intensity image As shown above, if the refractive index variation is small, rays can be considered straight. The allowable deviation may depends on the required spatial resolution of the image. This is a straight approximation of rays (Mewes, Herman and Renz [1994]). As shown in the preceding chapter, X-rays conform to this approximation as far as the required spatial resolution is around 0.1 mm. Under this assumption, we shall consider X-rays in the plane X–Y in fig. 4. The intensity of the rays transmitted through the object is mainly affected by the absorption coefficient µ(x, y) of the object. As shown in fig. 4, parallel rays are incident on the object in the direction of Y . In the figure (x, y) is the coordinate fixed to the object and ξ is the angle between X and x. Then the intensity of the

6, § 3]

Formation of intensity and phase images

401

Fig. 4. Straight approximation and coordinates.

penetrated rays is expressed by the equation    I (X, ξ ) = I0 exp − µ(x, y) dY ,

(3.1)

where I0 is the intensity of the incident ray. By defining

R(X, ξ ) = log I0 /I (X, ξ )

(3.2)

we obtain the expression  R(X, ξ ) = µ(x, y) dY.

(3.3)

3.2. Phase image If the refractive index difference between the object and the surrounding medium is expressed as n(x, y), the difference of optical path length g(X, ξ ) between the ray transmitted through the object and the ray outside the object is expressed as  g(X, ξ ) = n(x, y) dY (3.4) and this can be converted to phase difference φ by the equation g(X, ξ ) . (3.5) λ As far as the optical path length difference is continuous with respect to X, the deviation angle θ of the ray from straight line is expressed as φ(X, ξ ) = 2π

θ =

dg . dX

(3.6)

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[6, § 4

Variation of refractive index of gas is 10−4 for visible rays and variation of refractive index of solids is 10−6 for X-rays. Thus deviation angle of X-rays are small. On the other hand, phase change for the X-ray region is larger than that for the visible spectrum region, because ratio of wavelength for visible rays and X-rays is 10−4 . In the visible region, intensity image have not been used for transparent solid because refraction effect is large for general solid. Fluid can be the only object useful for measurement with reasonable resolution. In the X-ray region intensity images resulted from eq. (3.3) have been used for observing the interior of many objects since the discovery of X-rays. Rather recently phase imaging draws attention. The reason is that coherent X-rays from a large scale synchrotron radiation source becomes available. § 4. Phase imaging methods 4.1. Shadowgraphy Figure 5 shows wavefront and rays after penetrating through an object when parallel rays is incident on the object. In this figure the refractive index inside the sphere is assumed to be a little larger than that of the surrounding medium. The wavefront is a surface on which path length expressed by eq. (3.4) is constant. The rays are perpendicular to the wavefront. The angle deviation θ of the penetrated rays from the incident rays are expressed by eq. (3.6).

Fig. 5. Object and penetrating rays.

6, § 4]

Phase imaging methods

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When the objects are transparent without absorption, no intensity variation is observed immediately after the object. But as shown in fig. 5 ray directions depends on the position X. When these rays are propagated some distance through a uniform medium after the object, the intensity variation appears. The intensity variation can be expressed by calculating the ray density at the observation point. When the intensity of the wave immediately after the object is denoted by I0 and the path difference there is denoted by g(X), the intensity in the plane at the distance of R is approximated as (Ishisaka, Ohara and Honda [2000])   d2 g(X) I (X) = I0 1 − R (4.1) . dX 2 In the derivation of this expression, parallel incident rays and one-dimensional variation of the path difference is assumed. Similar equation stands for a rays diverging from a point source. As shown in eq. (4.1), the intensity changes corresponding to the second derivative of path length when an observation plane is set at a suitable distance R behind the object. This technique can be used when the incident rays are parallel or divergent from a point source. This technique is said to be shadowgraphy in the field of flow visualization (Saunder [1994]). This phase imaging technique for visible region is used, for example, to visualize supersonic shock wavefront. In the X-ray region, X-rays from synchrotron radiation or from a very small X-ray source have to be used. With this technique contrast of the transparent object can be enhanced (Wilkins, Gureyev, Gao, Pogany and Stevenson [1996]). An example for X-ray region is shown in fig. 6 (Fitzgerald [2000]). To obtain this figure an X-rays source with the diameter of 10 µm was used. Figure 6(a) is taken at the observation plane 10 mm from the object. Figure 6(b) is taken at the distance R = 1 m.

(a)

(b)

Fig. 6. X-ray images of a grass hopper. (a) Intensity image. (b) Shadowgraph. (Reprinted with permission from Fitzgerald [2000]. Copyright 2000, American Institute of Physics.)

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[6, § 4

If the source is not a point, intensities from different positions of the source have to be superposed and the resultant intensity is averaged out. The effect of source size was examined (Ishisaka, Ohara and Honda [2000]) and a phase contrast mammography equipment is already available. This method is very useful for obtaining phase imaging without a complicated optical system. But as eq. (4.1) is an approximation based on geometrical optics, it is difficult to obtain quantitative data.

4.2. Schlieren method 4.2.1. Visible ray region Schlieren method is a term for visible spectrum region. It is also used for flow visualization (Saunder [1994]). Principle of the method is to convert the ray deviation angle into intensity. The optical system for the technique is shown in fig. 7. The state of rays at the object is reproduced at the image with a lens. This image forming relation is shown by the rays in broken lines. The lens also converts parallel beam to converge at a point in the focal plane. Different position in the focal plane corresponds to different direction of the beam incident on the lens. When inserting a mask in the plane, some beams which converge to the mask position are prevented to travel to the image plane as shown a bold line. Thus the place in the image corresponding to the masked ray direction becomes dark. Thus ray direction can be displayed as intensity variation in the image plane. The ray direction corresponds to the derivative of path length dg(X) dX . 4.2.2. X-ray region In the X-ray region, no convenient image forming element is available. Different approach is adopted using Bragg diffraction by a crystal plate. It is shown in fig. 8.

Fig. 7. Optical system for Schlieren method.

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Fig. 8. X-ray Schlieren system.

The X-rays are diffracted by the two crystal plates. The first plate produces collimated monochromatic rays. The second plate diffracts those rays which satisfy the Bragg condition. The condition is denoted by the expression 2d sin θ = λ,

(4.2)

where d is the lattice spacing of the crystal plates, θ is the Bragg angle and λ is the wavelength. The Bragg diffraction does not occur when the angel of incident beam is deviated from this angle by several second of arc. Collimated monochromatic X-rays incident on the object are deviated due to phase change after penetrating the object. The deviation depends on the position in the object. The deviated rays are not diffracted by the analyzer crystal plate producing dark parts in the observation plane. But nondiffracted rays are diffracted and the corresponding position in the observation plane becomes bright. Thus image intensity corresponds to the derivative of path length. As shown in Section 2, refraction and diffraction are very small for X-rays. Therefore if the distance between the object and observation plane is small, position in the object corresponds to the position in the observation plate. This is assured as far as the deviation is smaller than the required spatial resolution. An example of an image taken by this method is shown in fig. 9 with a mosquito as an object. Here X-rays of wavelength 0.154 nm are used. In this experiment X-rays from a source is reflected by a Si crystal plate used as a monochromator. The reflected parallel X-ray is incident on the object. Figure 9(a) is taken in front of the analyzer crystal plate. This is an intensity image. Figure 9(b, c) is “Schlieren image” taken after the analyzer crystal plate with different rotation angles (Davis, Gao, Gureyev, Stevenson and Willkins [1995]). As this method is depend on the Bragg condition, collimated monochromatic X-ray is necessary. Moreover, it does not seem suitable to obtain quantitative data because width of the Bragg diffraction angle is difficult to control.

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Fig. 9. X-ray images of a mosquito. (a) Radiograph taken before the crystal plate. (b) Image taken by “Schlieren” method with the crystal plate is rotated by 0.4 arcsec and (c) by 1.6 s (Davis, Gao, Gureyev, Stevenson and Willkins [1995]).

4.3. Other phase imaging methods Image forming elements in the X-ray region are investigated including Fresnel zone plates and Wolter mirrors. With these element it may be possible to form a Schlieren system or phase contrast system. But they are limited to soft X-ray region at present. They will not be mentioned here. The most quantitative phase imaging method is interferometry, which will be stated in the next two sections.

§ 5. Reference type interferometers 5.1. Visible ray region 5.1.1. Mach–Zehnder interferometer In the visible spectrum Mach–Zehnder type of interferometer are widely used in the field of fluid dynamics. As shown in fig. 10, it consists of two half mirrors and two mirrors. Light from a light source is divided into two by a half mirror forming two arms. Flow to be tested is inserted in one arm of the interferometer. Light passing through the object is recombined at the other half mirror with the light through the other arm, which is used as a reference beam. The phase difference between the object beam and the reference beam is displayed as an interference pattern.

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Fig. 10. Mach–Zehnder interferometer.

When the phase of the object beam is denoted by φ(X, ξ ), intensity variation of the interfered beam is expressed as

I (X, ξ ) = Ia (X, ξ ) + Ib (X, ξ ) cos φ(X, ξ ) + Φ , (5.1) where Φ is phase difference caused by the path length difference between the two arms without the object. Ia (X, ξ ) and Ib (X, ξ ) are determined by the intensity of the two interfering beams and their coherence. From this expression, phase change of 2π can be detected by counting the peak of the intensity variation. 5.1.2. Phase detection methods In order to obtain smaller phase change, we use sub-fringe measuring techniques. They are phase shift method (Bruning, Herriott, Gallagher, Rosenfeld, White and Brangaccio [1974]) and Fourier transform method (Takeda and Mutoh [1983]). These methods are widely used in interferometry of visible spectrum (Creath [1988]). For the phase shift method Φ is changed by a known amount, for example, by translating one of the mirrors, and I (X, ξ ) is measured. In a frequently adopted case, the phase is changed four times as π Φn = n , n = 0, 1, 2, 3. (5.2) 2 If the corresponding intensity is expressed as In (X, ξ ), n = 0, 1, 2, 3, then φ(X, ξ ) = tan−1

I0 − I2 . I1 − I3

(5.3)

In the Fourier transform method Φ is changed linearly in space. For example, Φ = 2παX.

(5.4)

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This term can be introduced by tilting the two interfering beams. The resultant intensity is displayed as carrier fringes of frequency α modulated by the phase φ(X, ξ ). The intensity I (X, ξ ) is Fourier transformed, the spectrum around the frequency α is extracted and displaced to the origin of the spectrum space. The displaced spectrum is Fourier inverse transformed. If the obtained Fourier inversion data is expressed by f (X, ξ ), then φ(X, ξ ) = tan−1

Im(f (X, ξ )) . Re(f (X, ξ ))

(5.5)

As the Fourier transform method requires only one interferogram, it can be used for temporary changing objects such as flows but its spatial resolution is lower than that of the phase shift method.

5.2. X-ray region 5.2.1. Bonse–Hart type X-ray interferometer In the X-ray region, similar interferometer was devised about 40 years ago. It is a Bonse–Hart type interferometer, shown in fig. 11 (Bonse and Hart [1965]). As there is no adequate mirror in the X-ray region, Bragg diffraction by a single Si crystal is used for dividing the X-rays. The interferometer consists of three Si blades S, M, A and their base. Whole interferometer is made monolithically from a Si single crystal. The lattice plane diffracting the X-ray is perpendicular to the blade surface. The planes are shown in the upper part of the blades in the figure. If a X-ray

Fig. 11. Bonse–Hart type X-ray interferometer.

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is incident on the first blade S in the direction satisfying the Bragg condition, the ray is divided into diffracted ray and penetrating ray. Both rays reach the blade M, where they are diffracted again. The two diffracted rays coincide on the blade A, where one of them is diffracted and the other penetrate through the blade. They propagate in the same direction and interfere. As this interferometer is made monolithically from a single Si crystal, the X-ray satisfying the Bragg condition for S satisfies also the condition for other blades. As the wavelength of X-ray is smaller than the visible spectrum by the order of 10−4 , stability of the components of the interferometer is very important. This condition is assured by the monolithic structure. In this interferometer, X-rays in one arm is the reference beam and those in the other penetrate an object. The intensity variation is expressed in the same equation as eq. (5.1). If we use the two beams a–b1 –c1 and a–b2 –c1 shown in fig. 11, both beams are diffracted twice, resulting in the same intensity. In addition the distance between the first and the second blade is the same as that between the second and third. Path difference between the two interfering beams is almost zero and conventional X-ray source can be used with good contrast fringes. For a long time, this type of interferometer was used in the investigation for determining the lattice constant of Si and for measuring translational movement much smaller than 1 nm (Hart [1975]) and angular movement much smaller than 1 arcsec (Windisch and Becker [1992]). 5.2.2. Phase imaging It is rather recent that this interferometer began to be used for phase imaging and refractive index tomography (Momose and Fukuda [1995]). In this phase imaging, phase shift method is adopted to obtain phase smaller than 2π quantitatively. For the purpose, an acrylic plate is inserted in one of the arms as shown in fig. 11. By rotating the plate additional phase shifts are introduced. An example of the phase image obtained using this type of X-ray interferometer is shown in fig. 12(a). The object is an 1 mm sagittal slice of a rat cerebellum. The X-ray for this experiment is synchrotron radiation of wavelength 0.1 nm. For comparison, intensity image of the same object is shown in fig. 12(b). As clearly seen phase image has contrast much better than the intensity image. This type of interferometer is used in the investigation to detect soft tissues without using contrast media such as Iodine. These investigation proves the description in Section 2.1 that phase imaging is suitable to obtain materials of low atomic number in good contrast. As this type of interferometer has to made monolithically, large interferometers cannot be fabricated. The size is limited by the size of Si ingot. In order to separate the interferometer into two parts, a skew symmetric type of interferometer

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Fig. 12. X-ray images of a rat cerebellum (1 mm sagittal slice). (a) Interference pattern. (b) Intensity image (Momose and Fukuda [1995]).

was constructed (Becker and Bonse [1974], Bauspiess, Bonse and Graeff [1976]), which is shown in fig. 13. This interferometer can be divided along the broken line in the figure. According to the analysis of the interferometer, the allowable alignment angle θ of the two part of the interferometer is roughly expressed by θ

d , D

(5.6)

where D is the spacing between the two blades and d is the pitch of the lattice constant (Windisch and Becker [1992]). In the reference interferometer D should be large for incorporate a large object. For example, D = 10 mm and d = 0.3 nm, θ 3 × 10−7 . This interferometer is used also for phase imaging (Yoneyama and Momose [1999]), but the adjustment seems to be very difficult to be used in usual circumstances.

Fig. 13. Crystal lattice skew symmetric interferometer.

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§ 6. Shearing type interferometers 6.1. Visible ray region 6.1.1. General description A shearing type of interferometer shown in fig. 14 is used in visible spectrum region. In this interferometer, beams penetrating a object is displaced laterally by a shearing element. Then the two slightly displaced beams A–A and B–B are interfered after the shearing element (Murty [1978]). If the displacement of beams or amount of shear is denoted by s, the obtained interference fringe intensity is expressed by the expression

I (X, ξ ) = Ia (X, ξ ) + Ib (X, ξ ) cos φ(X, ξ ) + Φ , (6.1) where φ(X, ξ ) = φ(X + s, ξ ) − φ(X, ξ ).

(6.2)

In this review φ(X, ξ ) is named phase difference. Ia (X, ξ ) and Ib (X, ξ ) are determined by the intensity of the two interfering beams and their coherence. As eq. (6.1) is the same form as eq. (5.1), phase shift method and Fourier transform method can be used to obtain phase difference φ(X, ξ ) smaller than 2π. For obtaining phase shift we have to invent new methods which will be shown later. For performing Fourier transform method, we have to make carrier fringes, which is obtained by inclining the two interfering beams by an adequate angle. In the visible spectrum region, shearing interferometers are used in differential interference microscopes (Bryngdahl [1965]), where birefringent prisms are used

Fig. 14. Schematic diagram of a shearing interferometer.

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to produce a shear. Its aim is to obtain images of good contrast for phase object. Quantitative data is seldom utilized. Other shearing elements include diffraction grating, transparent parallel plate and so on. In the next section, shearing with gratings will be explained in consideration of the situation in X-ray region. 6.1.2. Shearing with double period gratings One way of making shear is the double period grating shown in fig. 15 (Wyant [1973]). In this interferometer, a grating with double period is used to diffract beams that penetrate the object. Beams incident on the grating are diffracted into two different directions as the first order. The difference of diffracted angles causes the shear as shown in the figure. When the grating period is d1 and d2 , the diffraction angle θ1 and θ2 of the first order diffraction is expressed as sin θi =

λ , di

i = 1, 2.

(6.3)

If the grating is put on the front focal plane of the lens with the focal length f , the amount of shear s at the image is obtained as s = f (θ1 − θ2 ).

(6.4)

In this optical system, the difference of the grating periods is much smaller than the periods themselves. Therefore, higher diffraction orders do not overlap with the images formed by the two first orders. The light source for this shearing interferometer is a laser. Therefore two beams interfere and good contrast fringes are obtained.

Fig. 15. Shearing interferometer using double period grating.

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6.1.3. Shearing with a single period grating Another way is using a single period grating with larger grating period d shown in fig. 16 (Yokozeki and Suzuki [1971]). Observation is made on a plane at a distance L from the object. In this case, many diffraction orders appear and many sheared images are superposed on the observation plane. Interference carrier fringes with the same period as the grating period appear. The carrier fringes are modulated by the phase difference φ of the object. As many diffracted beams are superposed, intensity at the observation plane is not expressed by eq. (6.1). In this interferometer, contrast of fringe changes according to the distance Ln from the grating. The positions of high contrast appears periodically. The high contrast images are called Fourier images and this phenomenon is called Talbot effect (Patorski [1989]). The nth positions of Fourier image appear at the distance Ln from the grating given by the expression for an absorption grating d2 . (6.5) λ To obtain a good contrast image, observation is made at the position of Fourier image. Thus this interferometer is called a Talbot interferometer. When the period of the carrier fringes are too small to detect by a detector, the second grating with the same period as the first grating is put at the plane of Fourier image before the observation plane. The fringe period is magnified by Moiré effect. By adjusting the relative angle between the two grating lines, we can obtain carrier fringes with adequate fringe period for Fourier transform method. If we want to adopt phase shift method, the second grating is displaced to obtain phase shifted images. Although the intensity variation of the fringes are not sinusoidal expressed as eq. (6.1), we can extract the first diffraction orders by image processing of the Ln = n

Fig. 16. Talbot interferometer.

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phase shift method or Fourier transform method. Therefore the amount of shear is expressed by Lλ (6.6) . d This interferometer can be considered as a strict shearing interferometer when only two diffraction orders are present. One way is to eliminate the zeroth order diffraction and leaving only the plus and minus first orders. A phase grating is suitable for the purpose, whose groove depth corresponds to phase shift of π and groove width is the same as land width. With this grating, the most strong diffraction order is the plus and minus first orders. The second orders disappear. The two first orders produce interference fringes of period d2 . However, in this interferometer, the image is not located at the position conjugate to the object. Therefore, in the visible ray region, diffraction or refraction effect will affect the image if high spatial resolution is required. To obtain the phase immediately after the object, image forming lens has to be used. s=

6.1.4. Shearing with four gratings A strict shearing interferometer using gratings was constructed with an extended source of broad spectrum. The basic configuration of the interferometer is shown in fig. 17 (Iwata, Kikuta, Kondo and Mizutani [2000]), where G1, G2, G3 and G4 are gratings with the same grating constant. The spacing between G1 and G2 is the same as the spacing between G3 and G4. The first pair of grating, the object and the second pair of grating are conjugate to each other. A light beam incident on the first grating pair converted into a pair of slightly displaced (sheared) light beams after passing through it. The sheared beams pass through the object and are recombined after the second pair of gratings. Interference fringes appear on the image plane.

Fig. 17. Optical system for shearing interferometer with four gratings.

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The interference fringes correspond to the phase difference between the two sheared beams. For using the phase shift method deliberate phase shifts between the two sheared beams are introduced by displacing one of the paired gratings. In this interferometer, the observation plane is conjugate with the object. Thus the phase difference immediate after the object are observed. With this interferometer high contrast fringes was formed using an extended light source of broad wavelength.

6.2. X-ray region 6.2.1. Crystal lattice grating Many X-ray interferometers use gratings for shearing elements. The first shearing interferometer in the X-ray region was a modification of the Bonse–Hart interferometer, which uses Bragg diffraction (Iwata, Tadano, Kikuta, Hagino and Nakano [1998], Iwata, Kikuta, Tandano, Hagino and Nakano [1999]). The system is shown in fig. 18. This system is similar to the skew symmetric interferometer shown in fig. 13, but the distance D between blades are smaller. The whole interferometer is cut from a silicon single crystal. The incident beam is diffracted by Bragg diffraction at the first pair of blades and two sheared parallel beams a and b are formed. Both beams are transmitted through the object and superposed after diffraction by the second pair of blades to interfere. The superposed beams are denoted by A.

Fig. 18. Crystal lattice shearing interferometer.

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This interferometer is similar to the interferometer for the visible spectrum mentioned in the previous section. In the X-ray region, diffraction and refraction effect is not large. As no optical element such as a lens is available, no image forming system is introduced. An X-ray line source perpendicular to the diffraction lattice planes are used as shown in fig. 18 (Iwata, Kawasaki and Kikuta [2000]). With this configuration direction of the interfering beams are limited to a small angle range determined by the breadth of the Bragg angle. The angle is in the order of 10−5 . Therefore the width of the image in the horizontal view is substantially the same as the width of the source. As no diffraction occurs at the blade in the plane perpendicular to the source, the image is magnified geometrically in this direction. For adopting a phase-shifting method an acrylic wedges is inserted as phase shifter. The wedge is rotated around the axis perpendicular to the apex line. The rotation brings different phase change to the two interfering beams because they penetrate the wedge at the position of different thickness. Because both of the two interfering beams are diffracted twice and transmitted twice through the blades, they have the same intensity at the image. Thus, in principle, the contrast of the fringes is unity as long as these two interfering beams are concerned. However, as the width of the beam is larger than the shearing distance, the interfering beams A are overlapped with the beams B and C which are diffracted by the other blade. As the optical paths of these beams are much different from those of A, they do not interfere. The two noninterfering beams B and C are also diffracted twice and transmitted twice. Hence their intensity is the same as the two interfering beams. Since the these four beams are superposed, the intensity of the image at the point X is expressed by eq. (6.1) with Ia (X, ξ ) = 2Ib (X, ξ ). This reduces the contrast of fringes but the theoretical contrast is 12 . In the constructed interferometer, the X-ray source with Mo target emits Kα line of wavelength 71 pm. The X-rays are diffracted by (111) crystal planes perpendicular to the blade surface. Lattice spacing is 313 pm and Bragg angle is 6.5◦ . The amount of shear s is 0.7 mm and length of the line source is 8 mm. An X-ray CCD camera is used to acquire the image intensity. Figure 19 shows an example of the interference fringes when an inclined acrylic pipe is used as an object. The degradation in contrast is not so large. In this interferometer shearing distance s is determined by the formula λ s = 2D sin θ = 2D , (6.7) d where D is the spacing between the pair of blade and θ is the Bragg angle. If we use silicon single silicon crystal, the Bragg angle is determined by its crystal lattice constants. Therefore to obtain smaller shearing distance, we have to make

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417

Fig. 19. Interference fringes by a crystal lattice shearing interferometer.

smaller spacing D. However, as far as machining fabrication technology is used, the smallest possible spacing is about 1 mm. Therefore the shearing distance is in the order of 0.1 mm. Large shearing distance restricts the spatial resolution of the image. 6.2.2. Ruled grating Recently a new type of shearing interferometer was reported which uses an artificially ruled grating for splitting the X-ray (David, Nöhammer, Solak and Ziegler [2001], David, Nöhammer, Solak and Ziegler [2003]). Its schematic diagram is shown in fig. 20. In this interferometer the X-rays transmitted through an object is introduced to a pair of gratings and it shears the X-rays. The period of the grating is about 1 µm and the angle between the two interfering beams is about 0.01◦ (2×10−4 rad). This angle is more than 10−2 times smaller than that for the crystal lattice.

Fig. 20. X-ray shearing interferometer using ruled gratings.

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Fig. 21. Interferogram of a shearing interferometer with ruled gratings. The objects are polystyrene spheres. Amount of shear is 14 µm in vertical direction. (Reprinted with permission from David, Nöhammer, Solak and Ziegler [2001]. Copyright 2001, American Institute of Physics.)

For obtaining only two sheared beams, a phase grating was used. The groove depth is deeper than 10 µm and the groove width is equal to the land width. In order to eliminate the 0th diffraction order, the gratings are inclined to increase the apparent groove depth. The grating is made of Si crystal and the grooves are fabricated by anisotropic wet etching. Two 1st order beams are used as the shearing beams. The period of the interference fringes is half the grating period. The period of the fringes are too small to be detected by the CCD camera. The analyzer grating broadens the fringe period by the Moiré method. There are at least four beams after the analyzer grating. Their diffraction orders at the first and second gratings are denoted by (1, −1), (−1, 1), (1, 1) and (−1, −1). The rays required for interferometry are the first two diffraction orders. A single Si crystal plate was used for selecting the required two beams. The width of the Bragg angle of the Si plate is about 10−5 while the angle between the beams of different orders are 2 × 10−4 rad. Therefore the Si crystal plate can be arranged to diffracted only the required beams. Interferogram obtained with this interferometer is shown in fig. 21. The object is 100 and 200 mm diameter polystyrene spheres. The distance between the two gratings is 80 mm and the amount of shear is 14 µm in the vertical direction. X-rays from synchrotron is used with photon energy of 12.4 keV (wavelength 0.1 nm). Because the X-ray of the synchrotron radiation has good coherency, the coherent requirement is satisfied for this small shear. A similar shearing interferometer was made by Momose, Kawamoto, Koyama, Hamaishi, Takai and Suzuki [2003] and Momose, Kawamoto, Koyama and Suzuki [2004]. It uses an absorption type gratings made of gold on a glass plate. The groove depth of the first grating is 1.25 µm which maximizes the contrast of Fourier image. The groove depth of the second grating is 8 µm to make absorp-

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Fig. 22. Shearing interferometer using twin zone plates.

tion as high as possible. No crystal plate is used for extracting the first orders. The second grating is put at the Fourier image position. Therefore this is a Talbot interferometer shown in fig. 16. The grating period is 8 µm and the distance between the gratings is 32 cm. Fringes of very good contrast was obtained using Moiré method. The fringe spacing was about 200 µm. With the shearing data three-dimensional reconstruction of refractive index was obtained as shown later. 6.2.3. Twin zone plate Another type of shearing interferometer was made using twin zone plates (Wilhein, Kaulich, Fabrizio, Romanato, Cabrini and Susini [2001], Kaulich, Wilhein, Fabrizio, Romanato, Altissimo, Cabrini, Fayard and Susini [2002]). Schematic diagram of this interferometer is shown in fig. 22. This interferometer has two zone plates made of Au on both side of a 1-µm-thick Si3 N4 substrate. Their radius is 37.5 µm and the outmost zone width is 200 nm. As the two zone plates are displaced in the direction perpendicular to the optical axis, two displaced images are formed. The two images interfere resulting in a shearing interferometer. As the interference occurs at the image of the object, obtained phase is that of the penetrating rays just after the object. However, large zone plate is difficult to fabricate because the pitch of the outer rings of the zone plate becomes very small.

§ 7. Refractive index tomography 7.1. Reference type interferometers X-ray computed tomography is widely used for reconstructing absorption distribution. Equation (3.3) is the equation from which we start for obtaining the re-

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construction algorithm. Based on this equation, many reconstruction algorithms were derived. On the other hand, the basic equation for reconstructing refractive index is eq. (3.4). The two equations are mathematically the same. Thus we can use mathematically the same reconstruction formula for refractive index as for absorption coefficient. As we can obtain phase with reference type interferometers quantitatively for X-rays and visible rays, refractive index in the interior of the object can be reconstructed with the data. The obtained data is the phase φ(X, ξ ) in eq. (5.1). The coordinates for this formula is shown in fig. 4. To reconstruct refractive index, we have to change the direction of incident rays. The direction is expressed by the angle ξ . The obtained phase φ(X, ξ ) is changed to path length data g(X, ξ ), λ (7.1) φ(X, ξ ). 2π There are many reconstruction algorithms. Among them filtered back projection formula is shown below. This algorithm is expressed as the refractive index reconstruction problem:  ∞ G(ζ, ξ ) = g(X, ξ ) exp(iXζ ) dX, −∞  ∞ 1 q(X, ξ ) = (7.2) G(ζ, ξ ) exp(−iXζ )|ς| dζ, 2π −∞  π 1 n(x, y) = q(x cos ξ + y sin ξ, ξ ) dξ. 2π 0 g(X, ξ ) =

In the visible wavelength region, the first refractive index reconstruction experiment was made by Matulka and Collins [1971] using holographic interferometry. This experiment was made before the first paper on X-ray absorption tomography was published (Hounsfield [1973]). The object was a jet stream. Measured value was phase difference between the two situations with flow absent and present. The phase difference corresponds to the reference type interferometer. There reconstruction algorithm they adopted was not the filtered back projection. One of other reconstruction experiments for a solid object was made later using a Mach–Zehnder interferometer (Maruyama, Iwata and Nagata [1977]). The object was a bundle of three acrylic cylinder. As the refractive index of the acrylic resin is much larger than that of air, rays are refracted and straight approximation is difficult to be applied. To reduce the refractive index difference between the object and the surrounding medium, matching liquid was used. In these experiments phase shift method was not used. Only integer fringe orders or fringe displacement was estimated graphically. The number of projection directions was not so

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Fig. 23. Tomographic reconstruction of a plastic sphere of diameter 1 mm. (a) Plane sectional view. (b) Whole object with an octant section showing inside (Momose, Takeda, Itai and Hirano [1996]).

large (36 directions in every 5◦ ) that the spatial resolution of the reconstructed refractive index distribution was not very good. First refractive index reconstruction for X-ray was made later by Momose, Takeda, Itai and Hirano [1996]. They used a Bonse–Hart type interferometer with synchrotron radiation source of wavelength 70 pm. Phase shift method was adopted using an acrylic plate shown in fig. 11. The object was a plastic sphere with small bubbles in it. In order to reduce the phase shift, the object was immersed in matching liquid. The number of projection was 200 over 180◦ . The reconstructed object is shown in fig. 23. The reconstructed image is piled up slices of 12 µm thickness in the vertical direction. It should be noticed that the bubbles in the acrylic resin is clearly seen although the refractive index difference between the resin and the air is in the order of 10−6 . The spatial resolution of this system was estimated as 30 µm.

7.2. Shearing type interferometers If we use a shearing interferometer, phase difference φ in eq. (6.1) can be obtained. We can convert it into phase φ by adding the phase difference data φ as follows: φ(N s, ξ ) =

N 

φ(ns, ξ ) + φ(0, ξ ).

(7.3)

n=0

With this phase φ we can use the reconstruction algorithm eq. (7.2) to reconstruct refractive index distribution.

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However, we can reconstruct directly from the data of a shearing interferometer. In the case of small s, we can obtain an approximation λφ(X, ξ ) ∂g(X, ξ ) = . (7.4) ∂X 2πs With this approximation, we obtain the following reconstruction formula (Iwata and Kikuta [2000]), under the assumption of continuity of φ(X, ξ ) with respect to X and φ = 0 outside the object,  ∞ ∂g(X, ξ ) H (ζ, ξ ) = exp(iXζ ) dX, ∂X −∞  1 ∞ q(X, ξ ) = (7.5) iH (ζ, ξ ) sign(ζ ) exp(−iXζ ) dζ, 2π −∞  1 π q(x cos ξ + y sin ξ, ξ ) dξ, n(x, y) = 2π 0 where

 (ζ > 0), 1 sign(ζ ) = 0 (ζ = 0),  −1 (ζ < 0).

(7.6)

The main difference between eq. (7.2) and eq. (7.5) is their spatial filters. The reconstruction from the phase uses a filter whose transmission is proportional to the frequency, while the reconstruction from the phase difference uses a filter whose transmission is constant with respect to frequency. The former filter transmits high frequency components of the data. In X-ray measurement, measured quantity often influenced by photon noise containing high frequency components. The high pass filter magnifies the noise. However, the latter filter does not magnify it. In this respect, measurement of phase difference is more suitable to X-ray measurement. A system of shearing interferometers was used to obtain phase difference data for many different directions of incident rays. The reconstruction was made by the formula eq. (7.5). It measured refractive index distribution in a flow using the light from a laser diode. The system is shown in fig. 24 (Iwata and Kikuta [2000]). The shearing interferometers are made of double period gratings shown in fig. 15 with diode lasers as light sources. As they are insensitive to noise such as vibration or temperature variation, stability of the whole system was good. The temporal variation of refractive index in a dynamic flow is measured tomographically. As the number of interferometers are limited to six, spatial resolution of the reconstructed refractive index distribution is not so good. But it demonstrate the usefulness of the shearing interferometers.

6, § 7]

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Fig. 24. System of shearing interferometers for obtaining temporary varying flow density.

A similar reconstruction was made with X-ray shearing interferometers of crystal grating type (Iwata, Takeda and Kikuta [2004]). In this case, one shearing interferometer is used and the object was rotated. Although its shearing distance is large (0.6 mm) compared with the beam width (about 6 mm) and numbers of rotation is small (6 directions), successful reconstruction of acrylic cylinder immersed in water was demonstrated using an X-ray shearing interferometer. It should be noted that this experiment was done with a conventional incoherent X-ray source. Another reconstruction for X-ray was made using a ruled grating Talbot interferometer shown in fig. 16 and explained in the last paragraph of Section 6.2.2.

Fig. 25. Reconstructed refractive index distribution on a section of a head of an ant in water. The density corresponds to refractive index (Momose, Kawamoto, Koyama and Suzuki [2004]).

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As the higher order diffractions are superposed, the interference intensity was not sinusoidal. To eliminate the higher orders, phase shift method with five steps was adopted. Phase maps were obtained every 0.72◦ over 180◦ . The reconstructed image of a head of an ant in water is shown in fig. 25 (Momose, Kawamoto, Koyama and Suzuki [2004]). The sensitivity of this image is comparable with the crystal lattice reference interferometer although the spatial resolution has to be compromised.

§ 8. Discussion on interferometers and refractive index tomography 8.1. Comparison between the interferometers In the preceding chapters, many types of interferometers are described. In this section their features will be mentioned in consideration of applications to X-ray region. The main classification is the reference type and shearing type. In the reference type interferometer an object beam is interfered with a reference beam which travels outside the object. In the shearing interferometer both of the two slightly displaced beams travel through the object and are interfered. This brings following difference. The phase difference between the two interfered beams are large for the reference type but small for the shearing type. Difference between the refractive index of air and acrylic resin is about 10−6 . Therefore an object with the dimension of 10 mm produces 100 fringes for the reference type when the wavelength of 100 pm. However, for the shearing type the fringe order depends on the shearing distance. It may be smaller than 1 fringe. When the fringe order increases, it becomes more difficult to assign the fringe order to a particular integer. This relates to the so-called phase unwrapping problem. In the flow measurement in the visible spectrum, phase difference is not so large for a reference interferometer because wavelength is large compared with X-ray. For the shearing interferometer, spatial resolution is determined by the shearing distance. It cannot be arbitrarily small, because the phase difference should be detectable. On the other hand, reference type of interferometer does not have this restriction. Therefore when we want high resolution, reference type is advantageous. The reference type of interferometer is less stable than the shearing interferometer. As the two beams travels along the similar path, the environmental disturbances such as air disturbance and vibration cause less effect on the shearing

6, § 8]

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type. As the X-rays have the wavelength in the order of 100 pm, small disturbance causes path length change larger than the wave length. In the visible spectrum, shearing type is used for a practical measurement, e.g., production line. Reference type is used in laboratories for research. As the wavelength is much smaller for the X-ray, the stability is more stringent. If we want to measure large objects, we have to divide the interferometer into two parts. It is practically impossible for the reference type due to the environmental disturbance, but for the shearing type it may be possible. The reference type interferometer have to be in an environment where temperature and vibration are controlled in a very strict way. On the contrary, the effect of disturbance depends on the shearing distance for the shearing type. In the reference type interferometer, measured phase is path length divided by wavelength as shown in eq. (3.5). If we assume that the refractive index does not depend on wavelength for small spectral width, the path length is constant with respect to wavelength. In this case phase changes inversely proportional to wavelength and the contrast of interference fringes degrades as the spectral width of the source becomes large. On the other side, in the grating type of shearing interferometer, the amount of shear is proportional to wavelength for a smaller diffraction angle as shown in eqs. (6.6) and (6.7). If we assume that the path difference between the two sheared beams is proportional to the amount of shear, phase difference between the two sheared beams does not depend on wavelength. This means that the fringes does not degrade even if we use a source of broad spectrum. Thus we can use rather intense X-ray source with wide spectrum width for this type of shearing interferometer. In conclusion, the reference type of interferometer is advantageous for the measurement of small objects with high resolution. Therefore, it is suitable to the laboratory measurement. The shearing type is advantageous for the measurement of large object with lower resolution. It may be used as clinical diagnosis. Another classification is an image forming type and nonimage forming type. Image forming type is in principle better for obtaining high resolution. However, in the present state of the art, image forming elements for X-rays are not so good. They are limited to long wavelength region. We have to wait for the improvement of the technique for fabricating zone plates and mirrors. If we want to use interferometers for clinical use, we have to use ordinary X-ray source, we have to use a grating shearing interferometer. The grating shearing interferometers are divided into crystal lattice type and ruled grating type. For a crystal lattice type interferometer small shearing distance is difficult and large interferometer is difficult to fabricate. In addition, the

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[6, § 8

direction of incident X-rays are restricted to parallel beams because it uses Bragg diffraction. The restriction is the same for the ruled grating shearing interferometer using crystal plate to select the diffraction orders. As a result if we use X-ray interferometers for the purpose of clinical diagnosis, ruled grating shearing interferometer without crystal plate is most suitable. However, the interferometer is now used synchrotron radiation as a source. We should note that the four grating type of interferometers can be used with conventional sources of large size and wide spectrum.

8.2. Consideration for interferometers with a conventional X-ray source Many X-ray interferometers used synchrotron radiation as X-ray source because it is bright and coherent. However, when these interferometers are to be used in usual place and time for diagnosis or nondestructive testing, the source restricts convenience. We want to use a conventional X-ray point source whose size is as large as possible and contains as wide spectrum as possible. We have to make a discussion on interferometers usable with such an X-ray source. In the visible spectrum, there are lasers as ideal light sources. Thus consideration of the source size and spectrum width is not so important. In visible region, fringe localization problem was analyzed before the advent of laser (Born and Wolf [2001a]). When we use a source with small area and narrow wavelength, interference fringes appear everywhere two rays are superposed. But when the source has a large area and broad wavelength, interference fringes appear at the restricted space. This discussion can be used also for X-ray region. At first we shall analyze the dependence of fringe contrast on an optical system (Tsuruta [1990, 2000]). Consider an observation point P where two interference beams are superposed. The beams come to this point through an interferometric optical system along different paths. Let us define two points P1 and P2 conjugate to the observation point in the source space as shown in fig. 26. The conjugate points are located in the direction in which the interfering beams leave the source center to the observation point and have the same path length from the source center to the observation point. As the two interfering beams travel through different paths, there are two conjugate points in the source space. The contrast of the interference fringes at the observation point corresponds to the degree of coherence for the two conjugate points. When the source is extended and incoherent quasi-monochromatic, degree of coherence for the two point P1 and P2 can be calculated by the van Cittert–Zernike theorem. This theorem states that the complex degree of coherence with respect

6, § 8]

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427

Fig. 26. Source space and conjugate points in an interferometer.

to P1 and P2 can be calculated by replacing the source by a diffracting aperture of the same size and shape as the source. If this aperture is illuminated by a spherical wave converging to the point P2 , the complex degree of coherence is the same as the complex amplitude of the diffracted wave at P1 . Based on this theorem, we can obtain a criterion for a circular source. In this case, the absolute value of degree of coherence becomes 0.88 at the point ρ from P2 expressed by ρ=

0.61λ , γ

(8.1)

where γ is the view angle subtended by the source at P2 . Here we assume the distances from the source to the conjugate point P1 and P2 is the same. This theorem can be applied to X-ray interferometer. In a Bonse–Hart interferometer, X-ray from the source meets at the analyzer crystal again and after diffracted by the analyzer the two rays travels along paths parallel to each other as long as the distances between the blades are the same. But if the distance is different by D, the distance ρ between the two interfering rays is expressed by the equation ρ = 2D sin θ.

(8.2)

As the view angle at the observation point is limited by the width of Bragg angle of the blades, then γ = 10−5 . With sin θ = 0.3 and λ = 0.1 nm, eq. (8.1) tells us D = 10 µm. This result agrees well with the result obtained by a different analysis of Bonse and te Kaat [1971]. Similar consideration leads to the similar result for crystal lattice shearing interferometer. On the other hand, the ruled grating shearing interferometer shown in figs. 16 and 20, the distance ρ is the same as the shearing distance. When the shearing distance is equal to 10 µm, γ < 0.5 × 10−5 . For synchrotron radiation this condition is satisfied. But if the incoherent source is considered, this restrict the source size.

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[6, § 8

For example, if the distance from the source to observation point is 2 m, source size must be smaller than 10 µm. This restrict the X-ray power. In order to use larger size of X-ray source, shearing distance must be compensated. The crystal lattice shearing interferometer and the four grating satisfies the condition. In these case, the conjugate points P1 and P2 coincide and larger size source can be used. However, the source size restrict the spatial resolution, because the observation point is at some distance from the object. In the visible spectrum, image forming lens reduces the distance but in X-ray spectrum, it is not the case. As the spatial resolution is in the same order as the source size, larger source can be used if the required spatial resolution is larger than 10 µm. However, four grating type of interferometer is not suitable to X-ray because the unnecessary diffractions are blocked by the spatial filtering using lenses. This scheme is difficult to adopt in X-ray region. In addition, it decreases the efficiency of the effective power of the source. Another suitable scheme has to be devised.

8.3. Diffraction tomography As shown in Section 2, diffraction and refraction produce no significant effect on the image for X-rays if the spatial resolution is lower than 0.1 mm. Under this condition, straight path approximation is valid. However, if we want higher spatial resolution, straight path approximation is not sufficient. We have to take diffraction and refraction into consideration. In the visible region, these effects are more significant. But we can use lenses for imaging. They can reduce the influence of diffraction caused after penetrating the object. However, in the X-ray region, no imaging element is available. In the X-ray interferometer the observation plane is located at some distance from the object. We cannot take the phase distribution at the observation plane as that immediately after the object. We have to take these effect in the consideration. One way to reduce this effects is to calculate the phase and amplitude immediately after the object on the basis of the phase and amplitude measured on the observation plane. This is accomplished by the inverse formula for Fresnel diffraction because the ray travels through a uniform medium after the object. With this calculation diffraction and refraction effect after the object can be reduced. There are some tomographic reconstruction algorithms which take these effect into consideration (Born and Wolf [2001b], Iwata and Nagata [1975], Iwata [1980]). Among them, Born approximation is a familiar one. It is used for X-ray

6, § 8]

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429

crystal structure analysis. However, it assumes small object and small refractive index variation. If we consider a sphere of radius a with refractive index n, Born approximation is valid under the condition 2π na 1. λ

(8.3)

If we put n = 10−6 and λ = 0.1 nm, a 0.016 mm. Thus this approximation is not suitable for the object considered in this review. Another approximation is Rytov’s approximation. The condition for the Rytov’s approximation is that the variation of refractive index is much smaller than the refractive index itself. This is satisfied by the object mentioned in this review for X-rays. In addition, Rytov’s approximation approaches to the straight approximation in the limit of small wavelength. Thus this approximation may be used for reconstructing refractive index distribution. But this problem has not been investigated as refractive index reconstruction problem for X-rays. It may be a subject of future research. Rytov’s approximation takes diffraction into consideration. But as shown in Section 2.2 refraction is also a problem when high resolution is required. Another approximation for reconstruction starts from geometrical approximation (Iwata and Nagata [1970]). In this approximation diffraction effect is not considered but refraction effect seems to be included. The appropriateness of the algorithm should be investigated in future. There are other approximated inversion formula in other fields such as seismology and quantum mechanics. These may be helpful for considering the inversion in X-ray region. With regard to shearing interferometer, no reconstruction algorithm is investigated other than straight pass approximation. It is important to find an appropriate algorithm if we want to reconstruct it with higher resolution. Reconstruction of refractive index and absorption coefficient from the diffracted rays or waves is just the same scheme as the crystal structure analysis, where electron density distribution in a unit lattice cell is calculated from the diffracted intensity in the Fraunhofer region. In the analysis, generally phase of the diffracted pattern is not known and the straightforward reconstruction is impossible. However in the phase imaging described in this review, phase is the main data. In the present review interferometric measurement is the main theme while in the crystal analysis interferometric measurement is difficult. In addition, the diffraction pattern obtained in the crystal analysis is in the Fraunhofer region, while in the present review the diffraction pattern is in the Fresnel region. The author hopes that there appears a research that connects the two fields.

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[6, § 9

§ 9. Conclusion This chapter reviews phase imaging and refractive index tomography for X-rays and visible rays. Research on phase imaging techniques for X-rays began about 10 years ago. It becomes a very active field now. On the other hand, research on phase imaging for visible rays are rather old field. These researches are made in rather distant fields. This review clarified that there are difference and similarity between the two fields. It will be beneficial to the improvement of one field to get information from the other field and learn form the other field.

Acknowledgement The author started writing this review on the advice of T. Asakura. The author really appreciates his suggestions and encouragement.

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Fitzgerald, R., 2000, Phase sensitive X-ray imaging, Phys. Today 53, 7. Hart, M., 1975, Ten years of X-ray interferometry, Proc. R. Soc. London Ser. A 346, 1. Hecht, E., 1987, Optics, second ed., Addison–Wesley, Boston, p. 576. Hendee, W.R., Ritenour, R., 1992, Medical Imaging Physics, third ed., Mosby Year Book, St. Lous, p. 1. Hounsfield, G.N., 1973, Computed transverse axial scanning (tomography) 1. Description of system, Brit. J. Radiol. 46, 1016. Ishisaka, A., Ohara, H., Honda, C., 2000, A new method of analyzing edge effect in phase contrast imaging incoherent X-rays, Opt. Rev. 7, 566. Iwata, K., 1980, Measurement of three-dimensional refractive index distribution, Oyo Buturi 48, 487. A monthly publication of The Japan Society of Applied Physics in Japanese. Iwata, K., Kawasaki, A., Kikuta, H., 2000, Phase imaging with a phase-shifting X-ray shearing interferometer using an X-Ray line source, Opt. Rev. 7, 561. Iwata, K., Kikuta, H., 2000, Measurement of dynamic flow field by optical computed tomography with shearing interferometers, Opt. Rev. 7, 415. Iwata, K., Kikuta, H., Kondo, T., Mizutani, A., 2000, Shearing interferometer using four gratings, in: Proc. of IMEKO – XVI World Congress II, Wien, 2000, International Measurement Confederation, p. 207. Iwata, K., Kikuta, H., Tandano, H., Hagino, H., Nakano, T., 1999, Phase-shifting X-ray shearing interferometer, Jpn. J. Appl. Phys. 38, 6535. Iwata, K., Nagata, R., 1970, Calculation of three-dimensional refractive-index distribution from interferograms, J. Opt. Soc. Am. 60, 134. Iwata, K., Nagata, R., 1975, Calculation of refractive index distribution from interferograms using the Born and Rytov’s approximation, Jpn. J. Appl. Phys. Suppl. 14 (14-1), 379. Iwata, K., Tadano, H., Kikuta, K., Hagino, H., Nakano, T., 1998, X-ray shearing interferometer for non-destructive testing in experimental mechanics, in: Allison, I.M. (Ed.), Advances in Design, Testing and Analysis, Balkema, UK, p. 741. Iwata, K., Takeda, Y., Kikuta, H., 2004, X-ray shearing interferometer and tomographic reconstruction of refractive index from its data, in: Developments in X-Ray Tomography IV, in: Proc. SPIE, vol. 5535, p. 392. Jenkins, F.A., 1980a, Fundamentals of Optics, fourth ed., McGraw-Hill Kogakusha, Tokyo, p. 602. Jenkins, F.A., 1980b, Fundamentals of Optics, fourth ed., McGraw-Hill Kogakusha, Tokyo, p. 395. Kaulich, B., Wilhein, T., Fabrizio, E.D., Romanato, F., Altissimo, M., Cabrini, S., Fayard, B., Susini, J., 2002, Differential interference contrast X-ray microscopy with twin plates, J. Opt. Soc. Am. A 19, 797. Malacara, D. (Ed.), 1978, Optical Shop Testing, Wiley, New York. Maruyama, Y., Iwata, K., Nagata, R., 1977, Measurement of the refractive index distribution in the interior of a solid object from multidirectional interferograms, Jpn. J. Appl. Phys. 16, 1171. Matulka, R.D., Collins, D.J., 1971, Determination of three-dimensional density fields from holographic interferograms, J. Appl. Phys. 42, 1109. Merzkirch, W., 1974, Optical method for compressible flows, in: Flow Visualization, Academic Press, New York, p. 62. Mewes, D., Herman, C., Renz, R., 1994, Tomographic measurement and reconstructing techniques, in: Mayinger, F. (Ed.), Optical Measurements, Springer-Verlag, Berlin, p. 369. Momose, A., 2003, Phase-sensitive imaging and phase tomography using X-ray interferometers, Opt. Exp. 11, 2303. Momose, A., Fukuda, J., 1995, Phase-contrast radiographs of nonstained rat cerebellar specimen, Med. Phys. 22, 375. Momose, A., Kawamoto, S., Koyama, I., Hamaishi, Y., Takai, K., Suzuki, Y., 2003, Demonstration of X-ray Talbot interferometry, Jpn. J. Appl. Phys. 42, L866.

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Author index for Volume 47 Anderson, D., 294, 312–314, 337 Anderson, D.G.M., 221 Anderson, D.Z., 312 Anderson, M.H., 149, 151, 294, 364 Anderson, P.W., 203 Andrews, M.R., 149–151, 167, 170, 199, 364 Andrews, R.A., 34 Angelsky, O.V., 218 Anglin, J.R., 197, 198 Anker, Th., 182 Aranson, I., 300 Arbore, M.A., 34 Arecchi, F.T., 361, 362 Arie, A., 14, 16, 19, 43, 46, 47 Arlt, J., 247, 254, 360 Arnaut, H.H., 368–370 Arnold, V.I., 88 Arrizón, V., 220, 256–258 Asobe, M., 21, 23, 45 Aspect, A., 151 Assa, M., 302 Assanto, G., 7, 21, 25, 26, 57, 324 Astilean, S., 227 Astinov, V., 8 Atai, J., 300, 305 Aubry, S., 355, 356 Avdeenkov, A.V., 178 Axelsson, A., 160 Aytur, O., 31, 34 Azimov, B.S., 54 Azzam, R.M.A., 261

A Abo-Shaeer, J.R., 170, 199, 209, 294, 367 Abramovitz, M., 105, 118 Abrikosov, A.A., 360, 366 Ackemann, T., 361, 362 Adachi, J., 238, 240 Adamczyk, O.H., 21, 23 Adams, R.M., 361 Adhikari, S.K., 336 Afanasjev, V.V., 305, 361 Agarwal, G.S., 313 Agrawal, G.P., 180, 181, 297, 302, 310, 325, 349, 354 Agullo-Lopez, F., 316, 346 Ahles, M., 316, 320 Aiello, A., 371 Akgun, G., 31 Akhmanov, S.A., 8, 28, 39, 40 Akhmediev, N., 314 Akhmediev, N.N., 301, 303, 307, 332 Al Khawaja, U., 368 Albert, O., 25, 27 Albiez, M., 182 Alekseev, K.N., 16 Aleksic, N.B., 338, 339 Alexander, T.J., 18, 29, 54, 55, 60, 294, 312, 313, 320, 335, 341, 355–359, 365 Alexandrescu, A., 369, 370 Allen, L., 198, 219, 230, 233, 247, 306, 327, 328, 360, 368, 369 Almeida, M.P., 370 Altissimo, M., 419 Altmeyer, A., 178, 200 Amiran, E.Y., 85 Ammann, E.O., 31, 34 An, K., 79 Anastassiou, C., 305, 323 Andersen, D.R., 320 Andersen, J.A., 313 Anderson, B.P., 181, 294, 365, 367

B Baardsen, E.L., 40 Baba, K., 239 Baba, N., 240 Babiˇc, V.M., 80 Babiker, M., 219, 230, 233, 247, 327, 328, 369 Baboiu, D.M., 343 433

434

Author index for Volume 47

Bache, M., 35 Backus, S., 8, 15 Baek, Y., 325 Bagnato, V.S., 154, 163 Baillargeon, J.N., 79, 126 Baizakov, B.B., 355, 367 Bakker, H.J., 34 Baksmaty, L.O., 367 Baldi, P., 14, 23, 25–27, 29, 37 Baliev, S.T., 195 Balles, S., 362 Balmer, J.E., 256 Baluschev, S., 302 Band, Y., 183, 188 Band, Y.B., 163, 183, 185 Banfi, G., 23 Banfi, G.P., 21, 23, 40 Bang, O., 43–45, 310, 326, 336, 340, 341 Banks, P.S., 12, 14, 15, 39, 40 Bapp, S.P., 152, 175 Barakat, R., 220, 221 Barashenkov, I.V., 361 Barbosa, G.A., 368–370 Bardeen, J., 153, 174 Barland, S., 362 Barnett, A.H., 100 Barnett, S., 369, 371 Barnett, S.M., 166, 194, 293, 306, 368, 369, 371 Barrelet, J.C., 151 Bartal, G., 355, 356, 358, 359 Bartels, A., 34 Bartels, R., 8 Bartenstein, M., 178, 200 Barthelemy, A., 23, 27, 58, 59, 311, 325, 337, 346, 347 Bartlett, S.D., 371 Basistiy, I.V., 327 Bassi, P., 23 Bates, D.R., 199 Bauspiess, W., 410 Bava, G.P., 362 Bazhenov, V.Y., 327 Beck, W.A., 219, 224 Becker, P., 409, 410 Beckwitt, K., 338, 341 Begley, R.F., 39, 40 Beijersbergen, M.W., 247, 368, 369 Belashenkov, N.R., 324

Beli´c, M.R. (Belic, M.R.), 316, 345, 346, 352, 354 Ben-Reuven, A., 199 Bender, C.M., 107 Benkler, E., 361 Berezhiani, V.I., 338, 339, 354 Berg-Sørensen, K., 182 Berge, L., 306, 337, 346, 365 Bergeman, T., 205 Berger, V., 39, 47 Bergman, K., 360 Bermudez, V., 36, 37 Berntson, A., 337 Berry, M., 293, 306, 371 Berry, M.V., 81, 86, 219, 239, 293–295 Bertolotti, M., 53 Beržanskis, A. (Berzanskis, A.), 34, 328, 329 Bethlem, H.L., 205, 206 Bezryadina, A., 357 Bezuhanov, K., 360 Bhandari, R., 240, 249 Bhatnagar, A., 256 Biaggio, I., 39, 40 Biellak, S.A., 126 Biener, G., 218, 219, 229–232, 238, 239, 241–243, 245–254, 257, 259, 260, 262–265, 270–272, 274, 275, 277, 283 Bierlein, J.D., 14, 29, 37 Bigelow, M.S., 309, 320 Bigelow, N.P., 367 Bigio, I.J., 284 Billings, B.H., 269 Binosi, D., 359 Birch, J., 47 Bird, G.R., 224 Birkhoff, G.D., 83 Bishop, A.R., 355, 358 Blair, S., 341 Blasi, G., 329 Bleistein, N., 107 Blit, S., 219, 235, 236 Bloch, I., 202 Bloembergen, N., 8, 40, 324 Bloemer, M.J., 53 Boardman, A.D., 7, 25, 26, 54, 55 Boccaletti, S., 361, 362 Bochinsky, J.R., 206 Bogoliubov, N.N., 167 Bohn, J.L., 178, 205 Boiron, D., 151

Author index for Volume 47 Bokor, N., 220 Boland, B.F., 51, 52 Boller, K.J., 35 Bomzon, Z., 218, 219, 227–229, 231, 232, 235, 236, 238, 239, 241–243, 245, 246, 249, 253, 254, 257, 259, 260, 262 Bongs, K., 180, 181 Bonse, U., 408, 410, 427 Bontemps, P., 25, 26, 55 Boreman, G.D., 224 Borghi, R., 218, 220, 285 Born, M., 224, 426, 428 Bortolozzo, U., 361, 363 Bosenberg, W.R., 14, 16, 29, 34 Bosshard, C., 8, 12, 39, 40 Bouchal, Z., 293 Bouchitte, G., 225 Boudebs, G., 337 Boulanger, B., 12, 14, 39, 40 Bourdel, T., 152, 175, 182 Bouwmeester, D., 368 Bowden, C.M., 53 Bowden, M.J., 225 Boyd, R.W., 6, 295, 309, 320, 325 Brachet, M.E., 298 Bracken, J.A., 23 Bradley, C.C., 149, 150, 364 Brady, D., 268 Brambilla, M., 362 Brangaccio, D.J., 407 Bratfalean, R.T., 14, 29, 49, 50 Brecha, R.J., 151 Breidis, D., 310, 336 Breitenbach, G., 34 Brener, I., 15, 20, 21, 23, 24, 43–45 Brennen, G.K., 202 Broderick, N.G.R., 14, 29, 39, 48–50 Brosseau, C., 217, 219–221, 271 Browaeys, A., 151 Brown, B.R., 284 Brown, J.B., 37 Bruder, C., 202 Brundrett, D.L., 224, 226 Brune, M., 78 Bruning, J.H., 407 Bryant, P.E., 247, 360 Bryngdahl, O., 411 Budde, M., 181 Buldyrev, V.S., 80 Burger, S., 180, 181

435

Burkett, W.H., 23 Burnett, K., 163, 166 Burr, K.C., 34 Burt, M.A., 151 Buryak, A.V., 7, 25, 54, 56, 57, 324–326, 330, 335, 338–341, 343–345 Butcher, P.N., 6, 295, 325 Butterworth, S.D., 34 Butts, D.A., 152, 366 Byer, R.L., 20, 34, 39, 40 C Cabrini, S., 419 Caetano, D.P., 370 Cai, T., 339 Callejo, D., 36, 37 Calvo, G.F., 316, 346 Campillo, A.K., 77 Campo-Taboas, J., 338, 339 Capasso, F., 79, 123, 124, 126 Capmany, J., 36, 37 Carbone, F., 23 Cardakli, M.C., 21, 23 Carlsson, A.H., 294, 312–314 Carminati, R., 281–283 Carmon, T., 316, 354, 355 Carr, L.D., 153, 176, 182 Carrasco, S., 45, 335, 369 Carrascosa, M., 346 Carretero-Gonzalez, R., 355, 367 Carvalho, M.I., 314 Case, S.K., 225 Castin, Y., 153, 167, 176, 182 Celechovsky, R., 293 Center for X-ray Optics at Lawrence Berkley National Laboratory, 397 Centini, M., 53 Chaban, E.E., 20, 21, 23, 24 Challener, W.A., 224 Chang, J.S., 79, 302 Chang, L., 368 Chang, R.K., 77–79, 118–120, 122, 123, 126, 135 Chang, S., 79, 122 Chang, S.J., 115 Chemla, D.S., 39, 40 Chen, B., 21, 23 Chen, G., 79 Chen, W., 182 Chen, W.Q., 368

436

Author index for Volume 47

Chen, X.F., 23 Chen, Y., 281–283 Chen, Y.B., 12, 14, 34 Chen, Y.F., 338 Chen, Y.J., 300, 305 Chen, Y.L., 23 Chen, Y.P., 23 Chen, Z., 297, 309, 355, 357 Chen, Z.G., 311, 317, 355–359 Cheng, B., 49 Cheng, C.C., 223 Chern, G.D., 119 Cherukulappurath, S., 337 Chervenkov, S., 302 Chevy, F., 294, 366 Chiao, R.Y., 239 Chikkatur, A.P., 167 Chikkatur, S.P., 167 Chin, C., 178, 200 Chin, J.K., 199, 209 Chinaglia, W., 335, 336 Ching, E.S.C., 91 Chiofalo, M.L., 176 Chipman, R.A., 221, 269, 270 Chirkin, A.S., 12, 14, 16, 31, 35 Cho, A.Y., 79, 123, 124, 126 Choi, D.L., 182 Chou, M.H., 15, 20, 21, 23, 24, 43–45 Chou, P.C., 261 Chowdhury, A., 48, 51, 52 Christ, H., 174, 191 Christiani, I., 23 Christiansen, P.L., 43–45, 304, 305, 341, 346, 355 Christman, S.B., 20, 21, 23 Christodoulides, D.N., 182, 305, 314–316, 318, 323, 354–356, 358 Christou, J., 301, 302, 309, 345 Chu, F.Y.F., 180 Chu, P.L., 340, 341, 361 Cincotti, G., 238, 239 Cipparrone, G., 238 Cirac, J.I., 202 Ciuchi, F., 238 Clairon, A., 35 Clark, C.W., 180, 181 Clausen, C.B., 43–45, 341 Claussen, N.R., 151, 199 Cline, D., 217, 260, 261 Coblentz, D.L., 78

Coddington, I., 367 Coerwinkel, R.P.C., 247 Coffey, V., 339 Cohen, O., 355, 356, 358, 359 Cohen-Tannoudji, C., 144, 151, 154, 306, 368 Cojocaru, C., 309, 332 Colet, P., 363 Collett, E., 217, 218, 221, 228, 259, 261, 263, 272 Collins, D.J., 420 Collins, L.A., 180, 181 Collins, R.W., 261 Collot, L., 78 Columbo, L., 362 Conti, C., 25, 26 Cooper, L.N., 153, 174 Cormack, A.M., 395 Cornell, E.A., 149, 151, 181, 294, 364, 365, 367 Cornish, S.L., 151 Corwin, K.L., 152, 175 Coskun, T.H., 323 Côté, R., 198 Cotter, D., 6, 295, 325 Cottrell, D.M., 218, 237, 277, 278 Couderc, V., 23, 27, 58, 59, 337, 346 Coullet, P., 361 Courtial, J., 293, 306, 368, 369, 371 Coutinho dos Santos, B., 370 Craighead, H.G., 226, 227 Crasovan, L.C., 338, 339, 342–344, 351, 353, 359, 363, 367 Creath, K., 407 Crespi, B., 115 Crespo, H., 8 Cristiani, I., 23 Crompvoets, F.M.H., 205, 206 Crosignani, B., 311 Cross, M.C., 360 Cubizolles, J., 152, 175, 180–182 Cui, D.F., 34 Cyrot, M., 360 D D’Aguanno, G., 53 Dahan, N., 283 Dalfovo, F., 364 Dalibard, J., 167, 294, 366 Dalton, R.B., 371

Author index for Volume 47 Dammann, H., 249 Danaila, I., 342 Dandridge, A., 269 Danielius, R., 34 Datta, P.K., 21, 23 d’Auria, L., 249 David, C., 417, 418 Davidson, N., 194–196, 217, 219, 220, 224, 235, 236, 247, 249, 255, 280, 281, 285 Davis, J.A., 218, 237, 238, 240, 277, 278 Davis, K.B., 149–151, 364 Davis, T.J., 405, 406 Davydova, T.A., 339 De Angelis, C., 23, 27, 58, 59 de Cordoba, P.F., 359 De Micheli, M.P., 23 de Mirandes, E., 152 de Sterke, C.M., 14, 29, 49–51, 182 de Sterke, M., 29, 30, 48–51 Dearborn, M.E., 31 Debernardi, P., 362 Debye, P., 220 Degiorgio, V., 21, 23, 40 Deguzman, P.C., 217, 226, 227, 261, 268 Delacourt, D., 14, 25–27, 29, 37 Delafuente, R., 311 Delarue, P., 14, 40 DeMarco, B., 152, 175, 176 Demicheli, M.P., 14, 25–27, 29, 37 DeMille, D., 205–207 Deng, L., 151, 180, 181, 183, 188 Dennis, M., 306, 371 Dennis, M.R., 218 Denschlag, J., 180, 181 Denz, C., 312, 315, 316, 320, 321, 324, 346, 351–354 DeRossi, A., 25, 26 DeSalvo, R., 7, 40, 324 Desyatnikov, A.S., 317–321, 342, 343, 346, 348, 349, 351–354, 356 Dettmer, S., 180, 181 Deutsch, I.H., 202 Deyanova, Y., 25, 26, 28, 369, 370 Dhal, B.B., 360 Dholakia, K., 247, 253, 254, 327, 360, 368 di Sopra, F.M., 362 Di Trapani, P., 7, 54, 324–326, 328–330, 335, 336, 341 Dieckmann, K., 152, 175 Dieguez, E., 36, 37

437

Diener, R.B., 178 Dietz, B., 97, 98, 104, 130 Dikmelik, Y., 31, 34 Dimitrevski, K., 337 Dinev, S., 300, 302 Ding, Y.J., 14 Dirac, P.A.M., 194 Dmitriev, V.G., 11, 16, 25, 26, 29, 36 Dominic, V., 34 Donelli, G., 21, 23 Dong, B.Z., 46, 47 Dong, L., 320, 339 Dong, P., 49 Donley, E.A., 199 Donnat, P., 347 Donnelly, R.J., 298 Dorn, R., 217, 220, 230, 260 Doron, E., 97, 98, 100, 108, 127, 130, 131 Dos Santos, A., 8 Douady, J., 39 Douillet, A., 35 Doumuki, T., 226 Dreischuh, A., 300–302, 360 Drits, V.V., 338 Drummond, P.D., 163, 164, 196–198, 313 Du, J., 36, 37 Du, J.X., 14, 42 Du, Y., 14, 34, 45 Dubietis, A., 34 Dubovik, A.N., 8, 28 Dubra, A., 239, 240 Ducci, S., 361 Duda, B.J., 346, 354 Duine, R.A., 178 Dupont-Roc, J., 144, 154, 306, 368 Duree, G., 311 Durfee, C.G., 8, 15 Durfee, D.S., 149–151, 167, 170, 364 Durnin, J., 253 Dürr, S., 199, 209 Dutra, S.M., 370 Dutton, Z., 181 E Eberler, M., 217, 220, 230, 260 Eberly, J.H., 198, 253, 320 Ebrahimzadeh, M., 34 Eckardt, R.C., 34 Eckmann, J.P., 97, 98, 104, 130 Edmundson, D., 310, 336

438

Author index for Volume 47

Edmundson, D.E., 303, 342, 345 Efremidis, N.K., 355, 356, 358 Eggleton, B.J., 182 Egorov, A.A., 358 Egorov, O.A., 12, 16 Eick, A.A., 284 Eiermann, B., 182 Eilbeck, J.C., 357 Einstein, A., 84, 149 Ekert, A., 368 Eliel, E.R., 370, 371 Ellingson, R.J., 34 Engels, P., 367 Enger, R.C., 225 Enns, R.H., 345 Ensher, J.R., 149, 151, 294, 364 Eriksen, R.L., 277, 278 Ertmer, W., 180, 181 Eschmann, A., 35 Esslinger, T., 202 Etchepare, J., 25, 27 Etrich, C., 7, 54, 325, 330 Eugenieva, E.D., 305, 323, 357 Everett, M.J., 261 F Fabrizio, E.D., 419 Fainman, Y., 223 Faist, J., 79, 123, 124, 126 Falco, G.M., 178 Falk, J., 31, 34 Fauve, S., 298 Fayard, B., 419 Feder, D.L., 180, 181 Fedichev, P.O., 163, 200 Fedorov, A.V., 362 Fedorov, S.V., 361–363 Feigelson, R.S., 311 Feit, M.D., 12, 14, 15, 39, 40 Fejer, M.M., 15, 20, 21, 23, 24, 34, 39, 41, 43–45 Felbacq, D., 224 Feldmann, M., 361, 362 Feringa, B.L., 360 Ferlaino, F., 152 Fernández-Pousa, C.R., 238, 240 Ferrando, A., 357, 359 Ferrari, G., 151, 152, 175, 182 Ferrari, J.A., 239, 240

Ferwerda, R., 175 Feshbach, H., 150 Fetter, A.L., 144, 171, 365 Feve, J.P., 12, 14, 39, 40 Ficek, Z., 313 Figen, Z.G., 34 Filipowicz, P., 202 Fini, J.M., 261 Firth, W.J., 304–309, 314, 315, 317, 330, 332, 346, 354, 362 Fischer, D., 256 Fischer, St., 39, 40 Fitzgerald, R., 395, 403 Fleischer, J.W., 355, 356, 358, 359 Fliesser, W., 302 Flytzanis, C., 8, 40, 324 Fokkema, J.T., 98 Follonier, S., 39, 40 Fonseca, R.A., 346, 354 Ford, D.H., 219, 234 Fortusini, D., 21, 23 Fradkin-Kashi, K., 14, 16, 19, 43, 46, 47 Franceschina, C.P., 40 Franke-Arnold, S., 293, 368, 369, 371 Frantzeskakis, D.J., 355, 357, 367 Freeland, R.S., 163, 199 Freund, I., 218, 295 Freyer, J.P., 284 Friberg, A.T., 282 Fried, D.G., 151, 364 Friedberg, R., 176 Friese, M.E.J., 294, 313, 360 Friesem, A.A., 217, 219, 224, 235, 236, 247, 249, 255, 280, 281, 285 Frins, E.M., 239, 240 Frischat, S.D., 100, 108, 127, 130 Froehly, C., 311, 347 Fu, J.S., 14 Fuerst, R., 325 Fuerst, R.A., 343 Fujimura, M., 23 Fukuda, J., 397, 409, 410 Fukushima, T., 126 Furfaro, L., 362 Furlan, W.D., 256, 257 Furuya, K., 176 G Gagarsky, S.V., 324 Gahagan, K.T., 294, 360

Author index for Volume 47 Gaididei, Y.B., 346, 355 Gajda, M., 198 Galajda, P., 284 Gallagher, J.E., 407 Gallo, K., 21, 49 Ganeev, R.A., 40 Gao, D., 403, 405, 406 Gao, S.M., 21, 23, 45 Gao, X., 219, 224 Garbusi, E., 239 Garcés-Chávez, V., 253 Garcia Sole, J., 36, 37 Garcia-Fernandez, R., 338, 339 Garcia-Ripoll, J.J., 314, 316, 317, 365 Gardiner, C.W., 202 Gauthier, D.J., 29, 30 Gawith, C.B.E., 49 Gaylord, T.K., 218, 224, 226, 227 Gbur, G., 321 Gehm, M.E., 153, 175, 176, 178 Geisser, M., 315, 354 Ghosh, S., 365 Ghrist, R.W., 151 Giannoni, M.J., 87 Gianordoli, S., 79 Gibbs, H.M., 361 Gibson, G., 293 Giessen, H., 31, 32, 34 Gil, L., 361 Gilchrist, A., 371 Ginzburg, V.L., 360 Giorgini, S., 364 Giudici, M., 362 Glas, P., 256 Glöckl, O., 217, 220, 230, 260 Glogower, J., 194 Glückstad, J., 217, 273, 277, 278 Glytsis, E.N., 224, 226, 227 Gmachl, C., 79, 123, 124, 126 Goldwin, J., 205 Goodman, J.W., 119, 132, 219, 224, 258 Góral, K., 198 Gori, F., 218–220, 240, 249, 253, 254, 261, 266, 285 Görlitz, A., 167, 175 Gornik, E., 79 Graeff, W., 410 Graham, R., 361 Granade, S.R., 175, 176 Grann, E.B., 223

Grasbon, F., 300–302 Grayson, T.P., 34 Grechin, S.G., 16, 25, 26, 29 Greffet, J.J., 281–283 Greiner, M., 178, 200, 202 Greytak, T.J., 151, 364 Grier, D.G., 360 Griffin, A., 149 Grimm, R., 151, 163, 178, 200 Gringoli, F., 23, 27, 58, 59 Gross, E.P., 166 Gross, P., 35 Grossman, H.L., 79 Grundkotter, W., 23 Grynberg, G., 144, 154, 306, 368 Gu, B.Y., 46, 47, 49, 235 Gu, X., 14 Gu, X.H., 14 Guattari, G., 218 Gubler, U., 8, 12, 40 Guido, L.J., 118 Guillien, Y., 12, 14, 40 Guizal, B., 224 Gunter, P., 39, 40 Guo, C.S., 14, 42 Guo, J., 268 Gupta, S., 152, 167, 175, 200 Gureyev, T.E., 403, 405, 406 Gurkan, D., 21 Gurzadyan, G.G., 11 Gutzwiller, M.C., 87 Guzman, A.M., 201 H Haake, F., 87 Hachair, X., 362 Hackenbroich, G., 98, 99, 115 Hadzibabic, Z., 152, 175, 200 Haelterman, M., 312, 314 Hagan, D.J., 3, 7, 40, 324, 325 Hagino, H., 415 Hagley, E.W., 151, 180, 181, 183, 188 Hagness, S.C., 48, 49 Haken, H., 361 Haljan, P.C., 181, 294, 365, 367 Hall, D.G., 219 Hall, D.S., 294, 365, 367 Hamaishi, Y., 418 Han, D.J., 163, 199 Handelsman, R.A., 107

439

440

Author index for Volume 47

Hanna, D.C., 34, 39, 48–50 Hänsch, T.W., 202 Harada, N., 360 Harel, R., 23 Harju, A., 359 Haroche, S., 78 Hart, M., 408, 409 Harvey, E., 360 Harvey, M.D., 371 Hashimoto, M., 279 Hasman, E., 217–219, 224, 227–232, 235, 236, 238, 239, 241–243, 245–255, 257, 259, 260, 262–265, 270–272, 274, 275, 277, 280, 281, 283, 285 Hau, L.V., 181 Haus, H.A., 261, 303 Havstad, S.A., 21 Hayes, J.P., 360 He, J., 49 He, J.L., 14, 19, 36, 37, 42, 45 He, P., 217, 260, 261 Heatley, D.R., 303, 332 Hebling, J., 31, 32, 34 Hecht, E., 395 Heckenberg, N.R., 294, 313, 360 Hecker Denschlag, J., 163, 178, 200 Heinzen, D.J., 163, 164, 196, 198, 199 Heismann, F., 269 Heller, E.J., 115, 126 Hellwig, M., 163 Helmerson, J.K., 151 Helmerson, K., 180, 181, 183, 188 Hemker, R.G., 346, 354 Hemmer, S.L., 153, 176, 178 Hendee, W.R., 395 Hendl, G., 200 Hentschel, M., 130 Herbig, J., 151 Herman, C., 400 Hernandez-Garcia, E., 363 Herriott, R., 407 Hertz, H., 224 Hewlett, S.J., 314 Hielscher, A.H., 284 Hinds, E.A., 205 Hirano, K., 395, 421 Ho, T.L., 178 Hohenberg, P.C., 360 Holland, M., 176 Holland, M.J., 151, 294, 365

Hollberg, L., 14, 16, 29 Holme, N.C.R., 238 Holmgren, S.J., 25, 27 Honda, C., 403, 404 Honda, K., 151 Honkanen, M., 219, 227, 239 Hooper, B.A., 29, 30 Hopkins, H.H., 220 Houbiers, M., 175 Hounsfield, G.N., 395, 420 Hoyuelos, M., 363 Hradil, Z., 371 Huang, C.P., 32, 36 Huang, G.X., 18, 54 Huang, J.Y., 34 Huang, K., 144, 149, 156, 157 Hudock, J., 355, 356, 358 Hudson, E.R., 206 Hudson, J.J., 205 Huguenin, J.A.O., 370 Huignard, J.P., 249 Hulet, R.G., 149, 150, 152, 175, 176, 182, 364 Husimi, K., 114 Hussein, M., 150, 160, 207, 209 Hvilsted, S., 238 Hvozdara, L., 79 I Ide, M., 274, 275 Ikegami, T., 35 Inguscio, M., 151, 152 Inochkin, M.V., 324 Inouye, S., 167, 170, 186, 199, 205 Iseler, G.W., 8, 29 Ishigaki, T., 240 Ishii, T., 239 Ishisaka, A., 403, 404 Ishizuki, H., 23 Itai, I., 395, 397 Itai, Y., 395, 421 Ito, H., 21, 23 Ivanov, R., 16 Iwata, K., 414–416, 420, 422, 423, 428, 429 J Jackson, J.D., 95 Jacquod, P., 98, 99 Jäger, R., 362

Author index for Volume 47 Jaksch, D., 202 James, D.F.V., 219 Jander, P., 352, 354 Jaque, D., 36, 37 Jarutis, V., 328 Javanainen, J., 167, 197, 198, 202 Javidi, B., 217, 273–275 Jellison Jr, G.E., 221, 261 Jeng, C.C., 321, 323 Jenkins, F.A., 395, 398 Jennewein, T., 371 Jeon, J.H., 302 Jessen, P.S., 202 Jia, Y.L., 14, 42 Jiang, J., 219, 224 Jin, D.S., 152, 175, 176, 178, 200, 205 Jin, G.F., 21, 23, 45 Jochim, S., 178, 200 Johansen, S.K., 45 Johansson, M., 355, 357 Johnson, B.R., 78 Johnson, N.M., 119 Jones, M.W., 217, 226, 227, 261, 268 Jones, R.C., 221 Jongma, R.T., 205 Jordan, R.H., 219 Joseph, J., 273 Joseph, P.I., 182 Joseph, R.I., 314 Joulain, K., 281–283 Julienne, P.S., 154, 163, 183, 185, 188, 198, 199, 205, 341 Jundt, D.H., 20, 34, 39 Junglen, T., 205 Juškaitis, R., 237, 238 K Kaatz, P., 8, 12, 40 Kagan, Yu., 163, 200 Kaino, T., 40 Kaiser, F., 312, 316, 320, 321, 324, 345, 346, 352, 354 Kaivola, M., 282 Kakichashvili, S.D., 238 Kamenov, V., 302 Kaminskii, A.A., 37 Kanashov, A.A., 324, 341 Kaneko, T., 268, 269 Kang, C.H., 49

441

Kang, J.U., 14 Kaplan, L., 126 Kapoor, R., 313 Kapteyn, H.C., 8, 15 Karamzin, Yu.N., 324 Karaulanov, T., 48, 50 Kartaloglu, T., 34 Kartashov, Y.V., 310, 351–353, 357–359 Kato, T., 261 Katz, D.P., 206 Katz, N., 194–196 Kaulich, B., 419 Kawamoto, S., 418, 423, 424 Kawano, K., 239 Kawasaki, A., 416 Keller, J.B., 80, 85, 86 Kerman, A., 150, 160, 207, 209 Kerman, A.J., 178, 205 Kerman, A.K., 176 Kersey, A.D., 269 Ketterle, W., 149–152, 167, 170, 175, 178, 186, 199, 200, 205, 209, 294, 364, 367, 368 Kettunen, V., 219, 227, 239 Kevrekidis, I.G., 367 Kevrekidis, P.G., 355, 357, 358, 367 Khaykovich, L., 152, 175, 182 Kheradmand, R., 362 Kheruntsyan, K.H., 197 Kheruntsyan, K.V., 163, 164, 196, 198 Khodova, G.V., 361, 362 Khokhlov, R.V., 8 Khoury, A.Z., 370 Khurgin, J.B., 14 Kielpinski, D., 368 Kikuchi, K., 21, 23 Kikuta, H., 414–416, 422, 423 Kikuta, K., 415 Kildal, H., 8, 29 Killian, T.C., 151, 364 Kim, G.H., 234, 302 Kim, J.U., 302 Kim, M.S., 11, 39 Kim, S.W., 79 Kimble, H.J., 239 Kimura, W.D., 219, 234 Kinast, J., 153, 178 Kinnunen, J., 178 Kip, D., 305, 323 Kitayama, K.I., 279

442

Author index for Volume 47

Kittel, C., 174 Kivshar, Y.S., 7, 16, 18, 25, 26, 28–30, 39, 42, 43, 48–56, 58, 60, 294, 296, 297, 299–304, 306, 310, 312–321, 323–325, 330, 335, 336, 341, 345, 348, 349, 351–359, 367 Klakow, D., 100, 101 Klein, M.E., 35 Kleiner, V., 218, 219, 227–232, 238, 239, 241–243, 245–254, 257, 259, 260, 262–265, 270–272, 274, 275, 277, 283 Kleppner, D., 151, 364 Kley, E.B., 227, 278, 279 Klyshko, D.N., 369 Kneissl, M., 119 Knödl, T., 362 Ko, K.H., 302 Kobayashi, N., 367 Kobayashi, Y., 35 Koch, K., 31 Koh, J., 261 Kokkelmans, S.J.J.M.F., 176 Kolokolov, A.A., 56, 307 Komatsu, K., 40 Komissarova, M.V., 16, 35, 54 Komori, K., 151 Kondo, T., 414 Konotop, V.V., 53 Koprulu, K.G., 34 Korotkov, R.Y., 14 Kosinski, S., 21, 23, 24 Kosterlitz, J.M., 294 Kostrun, M., 198 Kotochigova, S., 205 Koumura, N., 360 Kovachev, L.M., 320 Koyama, I., 395, 397, 418 Koyama, K., 418, 423, 424 Koynov, K., 16–18, 25, 26, 28 Kozuma, M., 151, 183, 188 Kozyreff, G., 363 Kravtsov, Y.A., 79–81 Kristensen, M., 247 Krolikowski, W., 310, 314–318, 321, 336, 345, 346, 354, 355 Krug, P.A., 182 Kruglov, V.I., 304, 305, 338, 339 Kubarych, K.J., 8 Kuech, T.F., 51, 52

Kuhl, J., 31, 32, 34 Kuipers, L., 34 Kuittinen, M., 227 Kulagin, I.A., 40 Kumakura, M., 151 Kung, H.L., 256 Kunimatsu, D., 21, 23 Küpper, J., 205, 206 Kurn, D.M., 149–151, 167, 170, 364 Kuroda, K., 274, 275 Kurz, J., 21 Kuzmiak, V., 53 Kwiat, P.G., 239 L Lagendijk, A., 34 Lago, E.L., 23, 27, 58, 59 Lai, B., 360 Lalanne, P., 218, 219, 224, 227 Landau, L.D., 360 Landhuis, D., 151, 364 Lange, W., 361, 362 Langford, N.K., 371 Laptev, G.D., 12, 14 Larionova, Y., 362 Laurell, F., 25, 27, 37 Lautanen, J., 227 Law, C.T., 294, 297, 301, 312, 314 Lawrence, B., 343 Lawrence, B.L., 337 Lazutkin, V.F., 86, 88, 90 Le Berre, M., 35 Leach, J., 368, 371 Leanhardt, A.E., 368 Leblond, H., 337 Lederer, F., 7, 54, 320, 325, 330, 343, 344, 353, 354 Leduc, M., 151 Lee, C.K., 34 Lee, D.H., 35 Lee, H., 311 Lee, H.J., 302 Lee, J., 261 Lee, J.H., 79, 302 Lee, S.B., 79 Lee, T.D., 176 Lee, T.W., 49 Lee, W.H., 228, 284 Lee, Y.L., 23

Author index for Volume 47 Lefevreseguin, V., 78 Leger, J.R., 240, 241 Leggett, A.J., 166, 177 Leitner, M., 256 Lenz, G., 21, 23, 24, 182 Leonard, J., 151 Leuchs, G., 217, 220, 230, 260 Leung, P.T., 91 Levenson, M.D., 14, 16, 29 Levi, A.F.J., 78 Lewandowski, H.R., 206 Lewenstein, M., 180, 181, 206 Li, D.H., 337 Li, J., 34 Li, Y.J., 337 Li, Y.P., 320, 339 Liang, X.Y., 34, 45 Liao, J., 14, 36, 37, 45 Liberale, C., 23 Lichtenberg, A.J., 88, 90 Lieberman, M.A., 88, 90 Lin, X.C., 34 Lisak, M., 294, 312–314, 337 Liu, H., 12, 36, 37 Liu, J., 14, 42, 235 Liu, S.Y., 91 Liu, W., 21, 23 Liu, X., 341 Liu, Y., 217, 260, 261 Liu, Y.G., 14 Liu, Z.W., 14, 19, 34, 45 Lobanov, V.E., 55 Lodahl, P., 35 Loewenthal, F., 256 Logan, R.A., 78 Logvin, Y.A., 305, 338, 339 Lohmann, A.W., 220, 256–258, 279, 280, 284, 285 Loiko, N.A., 363 Lomdahl, P.S., 304, 305, 357 London, F., 149 Longhi, S., 16, 35 Lopez, A.G., 226, 227 Lu, E.Y.C., 35 Lu, S.Y., 221 Lu, Z.Z., 337 Lugiato, L.A., 362 Luo, G.Z., 14, 19, 42 Luo, Z., 273

443

Luther-Davies, B., 296, 301, 302, 309, 311, 314, 315, 317, 318, 345, 346, 354 Lutwak, R., 183, 188 Lyot, B.F., 269 M Ma, B., 49 MacDonald, M.P., 247, 254, 360 Machida, K., 367 MacKay, R.S., 356 Mackie, M., 197, 198 Macleod, H.A., 225 MacVicar, I., 368 Madey, J.M.J., 29, 30 Madison, K.W., 294, 366 Magel, G.A., 20, 39 Maggipinto, T., 362 Mahajan, S.M., 354 Maimistov, A., 320, 342, 343, 353 Maimistov, A.I., 346, 353 Mainguy, S., 282 Mair, A., 247, 368–371 Mait, J.N., 219, 223, 224 Maitland, D.J., 261 Makarov, M., 14 Makarov, M.V., 14 Makasyuk, I., 355–359 Makdissi, A., 35 Maker, P.D., 297, 309 Makhankov, V.G., 368 Maki, K., 151 Malacara, D., 396 Maleev, I.D., 321, 324 Malmberg, J.N., 294, 312–314 Malomed, A., 353 Malomed, B., 7, 54, 320, 325, 342, 343, 353 Malomed, B.A., 25, 57, 61, 316, 320, 330, 336, 338–345, 353, 355, 357, 358, 361–363, 367 Mamaev, A.V., 297, 301, 312, 320 Manakov, S.V., 311, 314 Mancuso, A.P., 360 Mandel, L., 190, 220 Mandel, O., 202 Mandel, P., 361, 362 Manela, O., 355, 356, 358, 359 Manenkov, A.B., 130 Marathay, A.S., 321, 324 Marazov, O., 302

444

Author index for Volume 47

Mark, M., 151 Marnier, G., 14, 40 Marquier, F., 281, 283 Marrone, M.J., 269 Marte, A., 209 Marte, M.A.M., 35 Martin, H., 355–359 Martorell, J., 309, 332 Martynov, V.A., 8 Maruyama, Y., 420 Marzlin, K.-P., 182 Massoumian, F., 237, 238 Matijosius, A., 328 Matoba, O., 274, 275 Matsumura, K., 235 Matthews, M.R., 149, 151, 294, 364, 365, 367 Matulka, R.D., 420 Mayer, E.J., 34 Mazerant, W., 8, 12, 40 Mazilu, D., 320, 343, 344, 353 Mazurczyk, V.J., 269 Mazzulla, A., 238 McAlexander, W.I., 152, 175, 176 McCall, S.L., 78 McCarthy, G., 315–318, 321, 354 McCaughan, L., 48, 51, 52 McGloin, D., 253 McGowan, C., 34 McGuire Jr., J.P., 269, 270 McLaughlin, D., 80 McLaughlin, D.W., 180 McLeod, R., 341 McMahon, P.J., 360 McNamara, D.E., 218, 237, 277, 278 McNulty, I., 360 McSloy, J.M., 315, 346, 354, 362 Meier, J.T., 217, 226, 227, 261, 268 Meier, U., 8, 12, 40 Meijer, G., 205, 206 Meiser, D., 174 Meisner, L.B., 39, 40 Mekis, A., 79 Melville, H., 253 Mendonca, J.T., 8 Menyuk, C.R., 311, 325, 326 Meredith, G.R., 40 Merzbacher, E., 144, 146, 154 Merzkirch, W., 395, 396 Meschede, D., 202

Mewes, D., 400 Mewes, M.O., 149–152, 175, 364 Meyn, J.P., 35 Meystre, P., 141, 143, 152, 165, 172, 174, 182, 186, 191, 194, 201, 202, 204 Miceli Jr., J.J., 253 Michaelis, D., 362 Michinel, H., 336–339 Mies, F.H., 198 Miesner, H.-J., 151, 167, 170, 199 Miguel, M.S., 363 Migus, A., 347 Mihalache, D., 320, 338, 339, 342–344, 351, 353, 359, 363, 367 Milburn, G.J., 169, 197 Miller, D.A.B., 256 Miller, D.E., 199 Miller, M., 362 Miller, R.J.D., 8 Mills, D.L., 182 Milne, C.J., 8 Milonni, P.W., 176 Minabe, J., 239 Minardi, S., 329, 335, 336 Ming, L., 49 Ming, N.B., 11, 12, 14–16, 19, 31, 32, 34, 36, 37, 43, 45, 46 Mingaleev, S.F., 356 Minkovski, N., 25, 27 Mirotznik, M.S., 219, 223, 224 Mishra, S.R., 220 Misoguti, L., 8, 15 Mitchell, D.J., 312, 314, 345 Mitchell, M., 311, 314, 323 Miyakawa, T., 191, 201 Miyazawa, H., 21, 23, 45 Mizushima, T., 367 Mizutani, A., 414 Mlynek, J., 34 Modugno, G., 151, 152 Moerdijk, A.J., 160 Mogensen, P.C., 217, 273, 277, 278 Moharam, M.G., 218, 223, 224 Mohideen, U., 78 Mokhun, A.I., 218 Mokhun, I.I., 218 Molina-Terriza, G., 293, 328, 330–332, 335, 352, 368–371 Mølmer, K., 167, 182

Author index for Volume 47 Momose, A., 395, 397, 409, 410, 418, 421, 423, 424 Momose Lab, 398 Monro, T.M., 14, 29, 49, 50 Monsoriu, J.A., 359 Montesinos, G.D., 336 Moon, H.J., 79, 302 Moore, G.T., 31 Moore, M.G., 186 Moreno, I., 238, 240 Mori, W.B., 346, 354 Morin, M., 311 Morozov, E.Y., 12, 14, 31 Morris, G.M., 218, 219, 224 Moss, S.C., 151, 364 Motzek, K., 316, 320, 321, 323, 324, 352, 354 Mourant, J.R., 284 Mu, X., 14 Mu, X.D., 14 Mueller, E.J., 368 Mukaiyama, T., 199, 209 Mukohzaka, N., 279 Mulet, J.P., 281–283 Müller, G., 202 Muller, R.E., 297, 309 Murakami, N., 240 Muraki, S., 14 Murnane, M.M., 8, 15 Murty, M.V.R.K., 411 Mushiake, Y., 235 Musslimani, Z.H., 314–316, 318, 323, 354, 355 Mutoh, K., 407 Myatt, C.J., 151 Myers, L.E., 34 N Nagata, R., 420, 428, 429 Nägerl, H.C., 151 Nakajima, N., 235 Nakano, T., 415 Napartovich, A.P., 256 Narimanov, E.E., 79, 86, 98, 99, 122–124, 126 Nau, D., 34 Naumova, I.I., 14 Neil, A.T., 368 Neil, M.A.A., 237, 238

445

Nepomnyashchy, A., 316, 354 Neshev, D., 300, 302, 316–318, 355 Neshev, D.N., 355, 356, 358, 359 Nesterov, A.V., 217, 230, 235, 261 Neu, J.C., 299, 300 Ni, P., 49 Nie, Y.X., 337 Nieminen, R.M., 359 Nieminen, T.A., 360 Nienhuis, G., 368, 370, 371 Niitsu, T., 239 Nikandrov, A.V., 35 Nikogosyan, D.N., 11 Nikolova, L., 218, 238 Nishida, Y., 21, 23, 45 Nishihara, H., 23 Nishikata, Y., 239 Nitti, S., 23 Niu, Q., 182 Niv, A., 218, 219, 229–232, 238, 239, 241, 243, 247–254, 257, 259, 260, 262–265, 270–272, 274, 275, 277, 283 Niziev, V.G., 217, 230, 235, 261 Noack, F., 34 Nöckel, J.U., 78, 79, 86, 93, 107, 115, 121–124, 126, 130 Noh, Y.C., 302 Nöhammer, B., 417, 418 Nomura, T., 217 Nordin, G.P., 217, 219, 224, 226, 227, 261, 268 Nore, C., 298 Norton, A.H., 29, 49, 51 Nouh, S., 25–27 Nowak, S., 151 Nozières, P., 172, 173, 187 Nugent, K.A., 360 Nussenzveig, H.M., 105 Nye, J.F., 217, 218, 293–295 O Oberthaler, M.K., 182 O’Brien, J.L., 371 Oemrawsingh, S.S.R., 371 Offerhaus, H.L., 39, 48–50 Ohara, H., 403, 404 O’Hara, K.M., 175, 176 Ohgren, A., 337 Ohshima, S., 35 Oka, K., 261, 268, 269

446

Author index for Volume 47

Okada-Shudo, Y., 274, 275 Olivieri, D., 339 Olsen, M.L., 205 Olsen, O.H., 304, 305 Onofrio, R., 176 Onsager, L., 149 Oppo, G.L., 362, 363 Orenstein, M., 293, 362 Orlov, R.Yu., 9 Orlov, S., 328 Orlov, Y.I., 79–81 Ormos, P., 284 Oron, R., 219, 235, 236, 247 Orszag, S.A., 107 Ortiz-Gutierrez, M., 220, 256–258 Osorio, C.I., 328, 371 Ostrovskaya, E., 294, 313, 355 Ostrovskaya, E.A., 294, 299, 304, 312–318, 341, 354–356, 358, 359, 367 Ostrovskii, L.A., 324 Ostrowsky, D.B., 14, 25–27, 29, 37 Ott, H., 152 Ovchinnikov, Y.N., 360, 361 Ozeri, R., 194–196 P Pääkkönen, P., 220, 253, 254 Padgett, M., 306, 369, 371 Padgett, M.J., 219, 230, 233, 247, 293, 327, 328, 360, 368, 369, 371 Palacios, D.M., 321, 324 Palamaru, M., 227 Palfalvi, L., 31, 32, 34 Pan, C.L., 34 Pan, J.W., 371 Pancharatnam, S., 219, 231, 239 Papanicolau, G., 80 Papuchon, M., 14, 25–27, 29, 37 Parameswaran, K.R., 15, 21, 23, 24, 43–45 Parinov, S.T., 39, 40 Park, Q.H., 320 Parrish Jr., M., 224 Partridge, G.B., 152, 175, 176, 182 Paschotta, R., 34 Pasiskevicius, V., 25, 27 Pas’ko, V., 293 Pastur, L., 363 Paterson, D., 360 Paterson, L., 247, 254, 360

Pathria, R.K., 144 Patorski, K., 413 Paulus, G.G., 300–302, 360 Paye, J., 347 Paz-Alonso, M.J., 339 Peacock, A.C., 49 Pearton, S.J., 78 Peele, A.G., 360 Pegg, D.T., 194 Pego, R.L., 339 Pekker, M., 354 Pelinovsky, D.E., 56, 304, 306, 315, 324 Pelouch, W.S., 34 Pelusi, M.D., 21, 23 Peng, G.D., 340, 341 Penman, Z.E., 34 Penrose, O., 149 Penzkofer, A., 14, 40 Pepper, D.M., 191 Perales, F., 151 Perciante, D., 239, 240 Pereira Dos Santos, F., 151 Perez, G., 115 Perez-Garcia, V.M., 314, 316, 317, 336, 342, 351, 353, 354, 365, 367 Perry, M.D., 12, 14, 15, 39, 40 Peschel, T., 7, 54, 325 Peschel, U., 7, 54, 325, 330, 362 Petersen, D.E., 310, 336 Pethick, C.J., 150, 172, 173 Petit, R., 225 Petrov, D.S., 196, 200 Petrov, D.V., 305, 307, 309, 328, 330–335 Petrov, G.I., 25, 27 Petrov, V., 34 Petschek, R.G., 237 Petter, J., 312, 346 Pfister, O., 14, 16, 29 Philen, D., 21, 23, 24 Phillips, W.D., 151, 180, 181, 183, 188 Piccirillo, B., 369 Pierce, J.W., 34 Pigier, C., 305, 316, 354 Pillet, C.A., 97, 98, 104, 130 Pines, D., 172, 173, 187 Pinkse, P.W.H., 205 Piquero, G., 220, 285 Piskarskas, A., 34, 328, 329, 335, 336 Pismen, L.M., 293, 302, 360 Pitaevskii, L.P., 166, 195, 364

Author index for Volume 47 Planken, P.C.M., 34 Pogany, A., 403 Pohit, M., 274 Pohl, D., 235 Poincaré, H., 83 Poladian, L., 312, 345 Pommet, D.A., 223 Ponomarev, A.V., 16 Poritsky, H., 85 Pötting, S., 182 Poupard, J., 151 Powers, P.E., 34 Powles, R., 302 Prather, D.W., 219, 223, 224 Presilla, C., 176 Pritchard, D.E., 167, 368 Pryde, G.J., 371 Pu, H., 367 Puscasu, I., 224 Puska, M.J., 359 Q Qin, Y.Q., 11, 12, 14, 45, 49 Qiu, P., 14, 40 Quabis, S., 217, 220, 230, 260 Quemard, C., 337 Quiring, V., 23 Quiroga-Teixeiro, M., 338 Quiroga-Teixeiro, M.L., 337–339 R Raab, R.E., 221 Rabal, H.J., 256, 257 Rabin, H., 34 Radnoczi, G., 47 Raether, H., 281 Raimond, J.M., 78 Ralston, J.V., 86 Raman, C., 170, 294, 367 Ramanujam, P.S., 238 Ramazza, P.L., 361–363 Randeria, M., 177 Rangwala, S.A., 205 Rasel, E., 151 Rasmussen, J.J., 346 Rasmussen, K.O., 355 Raupach, S.M.F., 178, 200, 205 Rayleigh, L., 98

447

Recolons, J., 328 Refregier, P., 273 Regal, C., 200 Regal, C.A., 178, 181 Rehacek, J., 371 Reichl, L.E., 88 Reid, D.T., 34 Reimhult, E., 337 Reinhardt, W.P., 180, 181 Reintjes, J.F., 8, 15 Rempe, G., 199, 201, 205, 209 Ren, C., 346, 354 Rentzepis, P.M., 14, 40 Renz, R., 400 Ressayre, E., 35 Rex, N.B., 79, 118–120, 122, 123, 126, 135 Richards, B., 220 Richardson, D.J., 14, 29, 39, 48–50 Richter, K., 130 Ricken, R., 23 Riedl, S., 178, 200 Rieger, T., 205 Riklund, R., 47 Risk, W.P., 14 Ritenour, R., 395 Roati, G., 151, 152 Robert, A., 151 Roberts, J.L., 151 Robnik, M., 86 Rocca, F., 361 Rodriguez, M., 178 Rokhsar, D.S., 366 Rokshar, R.S., 152 Rolston, S.L., 151, 180, 181, 183, 188 Romagnoli, M., 23 Romanato, F., 419 Romero, J.J., 37 Rosanov, N.N., 362 Rosenblatt, C., 237 Rosenfeld, D.P., 407 Rosenman, G., 14, 16, 19, 43, 46 Ross, G.W., 39, 48–50 Rostovtseva, V.V., 16 Rousseau, I., 14, 40 Rowen, E., 194–196 Roy, A.M., 249 Rozanov, N.N., 361–363 Rozas, D., 247 Rubakov, Y.P., 368 Rubenchik, A.M., 324, 341

448

Author index for Volume 47

Rubinow, S.I., 85, 86 Rubinsztein-Dunlop, H., 294, 313, 360 Ruff, G., 163 Ruhle, W.W., 31, 32, 34 Ruostekoski, J., 368 Ryabov, V.L., 205 Ryasnyanskii, A.I., 40 Rytov, S.M., 225 Ryu, C., 163, 199 Ryzhov, V.A., 205 Rz¸az˙ ewski, K., 198 S Saarikoski, H., 359 Sackett, C.A., 149, 150, 175, 364 Sacks, Z.S., 247 Saffman, M., 35, 297, 301, 312, 320, 346 Sage, J.M., 205 Sahin, A.B., 21, 23 Sainis, J., 205 Sakaguchi, H., 367 Sakurai, J.J., 177 Salamo, G., 311, 345, 346 Salerno, M., 355, 367 Salgueiro, J.R., 294, 313, 318, 319, 324, 336, 338, 339 Salomon, C., 152, 175, 182, 196, 200 Saltiel, S., 16–18, 25, 26, 28, 29, 39, 48, 54, 55 Saltiel, S.M., 8, 16, 25–30, 39, 40, 42, 43, 48–53, 56, 58, 60, 325 Sammut, R., 25, 57 Sammut, R.A., 25, 57, 335, 338–341 Sampera, A., 180, 181 San Miguel, M., 363 Sanchez, F., 337 Sankaran, V., 261 Santagiustina, M., 363 Santamato, E., 369 Santarelli, G., 35 Santarsiero, M., 218, 220, 285 Santos, L., 206 Sanyuk, V.I., 368 Sasso, A., 369 Sauer, B.E., 205 Saunder, H., 403, 404 Savage, C.M., 368 Scalora, M., 53 Schadt, M., 236, 237

Schapers, B., 361, 362 Schatzel, M.G., 360 Scherer, A., 223 Scheuer, J., 293, 362 Schiek, R., 325, 326 Schiller, S., 34 Schjodt-Eriksen, J., 346 Schmidt, M.R., 346 Schmidt-Kaler, F., 201 Schnabel, B., 227, 278, 279 Schneider, B.I., 180, 181 Scholten, R.E., 360 Schouten, H.F., 321 Schreck, F., 152, 175, 182 Schreiber, G., 23 Schrenk, W., 79 Schrieffer, J.R., 153, 174 Schunck, C.H., 152, 178, 200, 205 Schwefel, H.G.L., 79, 86, 118–120, 122, 123, 126, 135 Schweikhard, V., 367 Scott, A.C., 180, 357 Scotti, R., 21, 23, 24 Scroggie, A.J., 362, 363 Scully, M.O., 201, 203 Search, C.P., 152, 174, 191, 194, 201, 202, 204 Sears, S., 319, 348, 358 Segev, M., 297, 303, 305, 309, 311, 314–319, 323, 345–349, 353–356, 358, 359 Sengstock, K., 180, 181 Seshadri, S.R., 219 Setälä, T., 282 Severin, M., 47 Shalaby, M., 347 Shapere, A., 239 Shapiro, J.H., 191 Shatsev, A.N., 362, 363 Shchesnovich, V.S., 361 Sheik-Bahae, M., 7, 40, 324 Shen, D., 284 Shen, Y.R., 6, 191, 295, 325 Shen, Z.X., 49 Sheppard, A., 314 Sheppard, A.P., 312 Sheppard, C.J.R., 220 Sherwood, J.N., 21, 23 Shi, B., 53 Shi, S., 219, 224

Author index for Volume 47 Shih, M.F., 309, 311, 321, 323, 345, 346 Shimura, T., 274, 275 Shin, Y., 368 Shlyapnikov, G.V., 153, 163, 180, 181, 196, 200, 206 Shmulovich, J., 21, 23, 24 Shum, P., 359 Sibbett, W., 34, 247, 253, 254, 360 Sibilia, C., 53 Sicre, E.E., 256, 257 Siegel, C., 256 Siegman, A.E., 77, 79, 93, 126 Sigal, I.M., 360, 361 Silberberg, Y., 341, 354 Simon, R., 239 Simoni, A., 151 Simos, C., 346 Simpson, N.B., 327, 360, 368 Simsarian, J.E., 180, 181 Simsarian, S.E., 183, 188 Singh, K., 273, 274 Singh, R., 36 Singh, S.R., 314 Sipe, J.E., 182 Sirjean, O., 151 Sivco, D.L., 79, 123, 124, 126 Skarka, V., 338, 339 Skeldon, K., 368 Skryabin, D.V., 7, 54, 304–309, 314, 315, 317, 324–326, 330, 332, 335, 346, 354, 362, 363 Slekys, G., 342, 361, 362, 364 Slowe, C., 181 Slusher, R.E., 77, 78, 182 Slyusar, V.V., 371 Slyusarev, S., 35 Smektala, F., 337 Smilansky, U., 97, 98, 100, 101, 104, 130, 131 Smilgeviˇcius, V. (Smilgevicius, V.), 328 Smith, H., 150, 172, 173 Smith, P.G.R., 34, 49 Smith, R.E., 225, 226 Smithers, M.E., 35 Snoke, D.W., 149 Snyder, A.W., 312, 314, 345 Sobolev, V.V., 337 Socci, L., 23 Sohler, W., 23 Solak, H.H., 417, 418

449

Sole, J.G., 36, 37 Soljacic, M., 314–316, 318, 319, 323, 347–349 Sols, F., 166 Sonehara, T., 218, 237, 277, 278 Soskin, M., 306 Soskin, M.S., 217, 218, 294, 295, 327, 371 Soto-Crespo, J.M., 303, 305, 307, 309, 330–334 Souto Ribeiro, P.H., 370 Spencer, D., 224 Spiegel, E.A., 298 Spinelli, L., 362 Spitz, E., 249 Spreeuw, R.J.C., 368, 369 Spreiter, R., 39 Stabinis, A., 328 Stalder, M., 236, 237 Staliunas, K., 294, 342, 361, 362, 364 Stamper-Kurn, D.M., 167, 170, 199 Stan, C.A., 152, 175, 178, 200, 205 Staus, C., 51, 52 Steblina, V.V., 300, 345 Steel, M.J., 182 Stefanov, I., 302 Stegeman, G., 7, 40, 324, 337 Stegeman, G.I., 3, 7, 14, 21, 23, 25–27, 29, 37, 54, 303, 324, 325, 343, 353 Stegun, I.A., 105, 118 Steinberg, V., 300 Steinhauer, J., 194–196 Stenger, J., 167, 199 Stepken, A., 312, 316, 320, 345, 346 Stevenson, A.W., 403, 405, 406 Stokes, G.G., 220 Stone, A.D., 78, 79, 86, 98, 99, 118–124, 126, 135 Stoof, H., 368 Stoof, H.T.C., 160, 175, 178, 202 Strasser, G., 79 Strecker, K.E., 152, 175, 176, 182 Stringari, S., 149, 195, 364 Stwalley, W.C., 163 Suche, H., 23 Sudarshan, E.C.G., 239 Suen, W.M., 91 Suhara, T., 23 Suk, H., 302 Sukhorukov, A.A., 25, 26, 28, 29, 56–58, 60, 320, 325, 356–358

450

Author index for Volume 47

Sukhorukov, A.P., 7, 9, 12, 16, 35, 54, 55, 324, 325 Sun, J., 23 Sun, J.Q., 21 Sun, P.C., 223 Sun, W.X., 49 Sun, Y., 126 Sundgren, J.E., 47 Sundheimer, M.L., 14, 29, 37 Susini, J., 419 Susskind, L., 194 Suzuki, A., 21, 23 Suzuki, H., 21, 23, 45 Suzuki, T., 413 Suzuki, Y., 418, 423, 424 Svensson, E., 337 Svidzinsky, A.A., 365 Swartzlander, G.A., 294, 297, 301, 312, 314, 321, 324, 360 Swartzlander Jr., G.A., 247, 323 Synakh, V.S., 337 Szipocs, R., 34 T Tadanaga, O., 21, 23, 45 Tadano, H., 415 Taglieber, M., 182 Taima, T., 40 Takagi, Y., 14 Takahashi, Y., 151 Takai, K., 418 Takano, T., 151 Takasu, Y., 151 Takeda, M., 407 Takeda, T., 395, 397, 421 Takeda, Y., 423 Talbot, F., 256 Tallet, A., 35 Tamada, H., 226 Tamm, C., 361 Tan, H., 21, 23 Tan, H.M., 23 Tan, X., 274, 275 Tanbunek, T., 78 Tandano, H., 415 Tang, C.L., 34 Tang, S.H., 49 Tang, X.H., 21 Taranenko, V.B., 342, 361, 362, 364

Tarbutt, M.R., 205 Tartara, L., 23, 40 Tartarini, G., 23 te Kaat, E., 427 Tepichin, E., 220, 256–258 Tereshkov, V.A., 16 Tervo, J., 219, 220, 239, 240, 253, 254 Thalhammer, G., 163 Theis, M., 163 Thomas, J.E., 153, 175, 176, 178 Thompson, S.T., 199 Thouless, D.J., 294 Ticknor, C., 205 Tidwell, S.C., 219, 234 Tiesinga, E., 160, 198, 205 Tikhonenko, V., 300–302, 309, 345 Timmermans, E., 150, 160, 176, 207, 209 Tissoni, G., 362 Tlidi, M., 362, 363 Todorov, T., 218, 238 Tokuda, K.L., 269 Tollett, J.J., 149, 150, 364 Tomaselli, A., 21, 23 Tommasini, P., 150, 160, 207, 209 Tomov, I.V., 8, 9, 14, 28, 40 Tomova, N., 218, 238 Tong, S.S., 91 Torelli, I., 25, 57 Torizuka, K., 35 Törmä, P., 178 Torner, L., 3, 7, 43–45, 54, 293, 305, 307, 309, 310, 324–326, 328, 330–335, 341–344, 351–353, 357–359, 367–371 Torres, J.P., 293, 307, 309, 323, 328, 330–335, 343, 344, 367–371 Torruellas, W., 337 Torruellas, W.E., 7, 54, 325, 343 Tovar, A.A., 235 Towers, I., 25, 57, 320, 336, 338–341, 343, 344 Towers, I.N., 61 Towghi, N., 273 Townsend, C.G., 151, 167, 170 Tran, C.Q., 360 Tredicce, J., 362 Tredicce, J.R., 362 Treutlein, P., 182 Trevino-Palacios, C.G., 14, 23, 25–27, 29, 37 Trillo, S., 7, 25, 54, 57, 324–326, 330, 335, 343

Author index for Volume 47 Trippenbach, M., 183, 185, 188 Troles, J., 337 Trukhov, D.V., 54 Truscott, A.G., 152, 175, 176, 182, 294, 313 Tschudi, T., 346 Tsuruta, T., 426 Tugushev, R.I., 40 Tung, S., 367 Tunkin, V.G., 8, 16, 28, 39, 40 Türeci, H.E., 79, 86, 93, 105, 107, 118–120, 122, 123, 126, 135 Turitsyn, S.K., 314 Turlapov, A., 153, 178 Turner, L.D., 360 Turunen, J., 219, 220, 227, 239, 240, 253, 254 Tyan, R.C., 223 U Unnikrishnan, C.S., 151 Unnikrishnan, G., 273, 274 Unsbo, P., 39 Urenski, P., 14, 16, 19, 43, 46 Ursin, R., 371 Usmanov, T., 40 Ussishkin, I., 97, 98, 104, 130 Uzdin, R., 316, 354 V Vaccaro, J.A., 166 Vahala, K.J., 362 Vahimaa, P., 220, 253, 254 Vaidyanathan, M., 31, 34 Vakhitov, N.G., 56, 307 Valiulis, G., 329, 335, 336 Van Baak, D.A., 14, 16, 29 Van Delden, J.S., 265, 266 van Delden, R.A., 360 van den Berg, P.M., 98 van den Brink, A.M., 91 van der Straten, P., 202 van Druten, N.J., 149–151 Van Enk, S.J., 368 van Oosten, D., 202 van Roij, A.J.A., 205, 206 Van Wonterghem, B., 14, 40 Vandruten, N.J., 364 Vanherzeele, H., 7, 40, 324

Vanstryland, E., 7, 324 Vanstryland, E.W., 7, 40, 324, 325 Varanavicius, A., 34, 329 Vardi, A., 197, 198 Vasnetsov, M., 293, 294 Vasnetsov, M.V., 217, 294, 295, 327, 371 Vaupel, M., 342, 361, 362, 364 Vawter, G.A., 225, 226 Vaziri, A., 247, 368–371 Vekslerchik, V., 354, 367 Velchev, I., 300, 302 Vella, A., 369 Veretenov, N.A., 362 Verhaar, B.J., 160 Vicalvi, S., 218 Vilaseca, R., 309, 332 Villeneuve, A., 14, 29, 37 Visser, J., 370 Visser, T.D., 321 Vladimirov, A.G., 354, 362, 363 Vlasov, R.A., 304, 305, 338 Vogels, J.M., 170, 294, 367 Volke-Sepulveda, K., 254, 360 Volkov, V.M., 305, 338, 339 Volkov, V.V., 12, 14, 16, 35 Volyar, A.V., 314 Volz, T., 199, 209 Voros, A., 87 Vujic, D., 346 Vysloukh, V.A., 310, 357–359 Vysotsky, D.V., 256 W Wabnitz, S., 343 Wagner, K., 341 Wahlstrom, U., 47 Walecka, J.D., 144, 171 Wallenberg, L.R., 47 Wallenstein, R., 35 Walls, D.F., 169, 182, 197 Walraven, J.T.M., 163, 200 Walser, R., 176 Walsh Jr., J.T., 261 Walther, H., 201, 202, 300–302, 360 Wang, C.C., 40 Wang, D.Y., 337 Wang, F.H., 47 Wang, H.T., 12, 14, 15, 19, 36, 37, 45 Wang, J., 151, 320, 339 Wang, J.Y., 14

451

452

Author index for Volume 47

Wang, S., 25, 27 Wang, X., 21, 23, 49, 53 Wang, X.H., 49 Wang, Z., 325 Warchall, H.A., 339 Warren, M.E., 225, 226 Weber, T., 151 Wei, H., 12, 14, 15, 19 Wei, I.Q., 14 Weigelt, J., 279, 280, 285 Weihs, G., 247, 368–371 Weilnau, C., 315, 316, 320, 321, 346, 354 Weiner, J., 154, 163 Weiss, C.O., 339, 342, 361, 362, 364 Wells, J.S., 14, 16, 29 Wen, B., 237 Wen, J., 151, 183, 188 Wendt, J.R., 225, 226 Westbrook, C., 151 White, A.D., 407 White, A.G., 371 Wieman, C., 149, 151 Wieman, C.E., 151, 199, 294, 364, 365, 367 Wigner, E., 114 Wilczek, F., 239 Wilde, J.P., 311 Wilhein, T., 419 Wilkins, S.W., 403, 405, 406 Willemsen, M.B., 362 Williams, C.J., 199 Williams, J.E., 294, 365 Willmann, L., 151, 364 Willner, A.E., 21, 23 Wilson, D.W., 297, 309 Wilson, T., 220, 237, 238 Windholz, L., 302 Windisch, D., 409, 410 Winkler, K., 163 Wise, F., 341, 342 Wise, F.W., 338 Woerdman, J.P., 247, 368–371 Wohlleben, W., 294, 366 Wolf, E., 190, 218, 220, 224, 283, 321, 426, 428 Wong, G.K.L., 14 Wrage, M., 256 Wright, E.M., 182, 201, 303, 307, 332, 337 Wu, L.A., 34 Wyant, J.C., 412 Wynar, R., 163, 196, 198, 199

Wynar, R.H., 164 Wynne, J.J., 235 Wyrowski, F., 227, 278, 279 X Xia, Y.X., 23 Xiao, R.F., 14 Xiaoping, Y., 311 Xie, K., 25, 26, 55 Xie, S.W., 11 Xu, C.Q., 21, 23 Xu, F., 36, 37, 223 Xu, J.J., 357 Xu, K., 170, 199, 209, 294, 367 Xu, P., 34 Xu, Z., 359, 368 Xu, Z.Y., 34, 45 Y Yablonovitch, E., 8, 40 Yabuzaki, T., 151 Yakimenko, A., 310, 336 Yakimenko, A.I., 339 Yakunin, V.P., 235 Yamamoto, Y., 47, 77 Yan, M., 359 Yanauskas, Z.K., 303 Yang, C.X., 21, 23, 45 Yang, G.Z., 47, 235 Yang, J., 355 Yang, J.K., 315, 324, 357 Yang, Q.M., 360 Yang, S.X., 12 Yang, X., 311 Yang, X.L., 11 Yang, X.P., 302 Yao, A.Y., 34 Yao, E., 371 Yarborough, J.M., 31, 34 Yariv, A., 191, 311 Ye, F., 320, 339 Ye, J., 206 Yeh, P., 225 Yi, S., 367 Yokozeki, S., 413 Yoneyama, A., 395, 397, 410 Yoo, S.M., 167 Yoon, C.S., 11, 39 Yoshida, Z., 354

Author index for Volume 47 Young, K., 91 Yu, J., 329 Yuen, H.P., 191 Yur’ev, Yu.V., 25, 26 Yurke, B., 194 Yurovsky, V., 197, 198 Yurovsky, V.I., 199 Z Zacares, M., 359 Zacher, F., 300, 301 Zakharov, V.E., 337 Zaliznyak, Yu., 310, 336 Zeilinger, A., 247, 368–371 Zeitner, U.D., 278, 279 Zeng, X.L., 23 Zerom, P., 309 Zgonik, M., 39 Zhai, H., 368 Zhan, C.L., 337 Zhan, Q., 240, 241 Zhang, C., 11, 12, 14, 15, 19, 31, 34, 45 Zhang, D., 49 Zhang, D.Q., 337 Zhang, F.F., 34 Zhang, H.Z., 49 Zhang, J.Y., 34 Zhang, S.Y., 34

453

Zhang, W., 182, 201, 202, 204 Zhang, X., 31, 32, 34, 294, 312, 314 Zhang, X.P., 34 Zhang, X.R., 34 Zhang, Y., 46, 47 Zhao, L.M., 47 Zhao, L.Z., 337 Zhou, B., 21, 23 Zhou, Y.S., 47 Zhu, D.B., 337 Zhu, S., 14 Zhu, S.N., 12, 14, 15, 19, 31, 32, 34, 36, 37, 42, 45 Zhu, Y.Y., 11, 12, 14–16, 19, 31, 32, 34, 36, 37, 43, 45, 46 Ziegler, E., 417, 418 Zijlstra, R.W.J., 360 Zilio, S., 154, 163 Zink, L., 14, 16, 29 Zinn-Justin, J., 87 Zobay, O., 182 Zoller, P., 202, 206 Zondy, J.J., 35 Zozulya, A.A., 297, 300, 301, 312, 320 Zubairy, M.S., 201, 203 Zwierlein, M.W., 152, 175, 178, 200, 205 Zyskind, J.L., 269 Zyss, J., 119, 122, 123, 135

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Subject index for Volume 47 A

Brewster window 235

Abrikosov lattice 366 adiabatic rapid passage 198, 199 anisotropic crystal 3 atom interferometer 206 – optics 141 – – , nonlinear 142 – trapping 217, 229 axicon 254, 255

C caustic 81 centrosymmetric medium 296 Cooper pair 174, 175, 177, 179, 192 critical temperature 175 D de Broglie optics 202 – – wave 142 – – wavelength 141, 192 – – – , thermal 149, 156 Debye integral 220 degree of coherence 426 dielectric resonator 80, 87, 90 – – , deformed 107–113 – – , optical 113 dipole–dipole interaction 182, 206, 207 Dirichlet boundary condition 81, 82 dye-doped polymer micro laser 122

B Babinet compensator 269 Baliev damping 195 Bardeen–Cooper–Schrieffer ground state 175 – transition 153, 174, 175 Bell inequalities 371 Bessel beam 358 – – , vectorial 254 – function 105 Bessel–Gauss beam 219 birefringence 224 Bloch mode 357 Bogoliubov–DeGennes equations 172 Bogoliubov quasiparticle 193 – sound velocity 170 – transformation 169, 170, 192 Bose–Einstein condensation 142, 149–152, 177, 178, 182, 199, 200, 201, 294, 320, 354, 360, 364 – – , mean-field theory of 164–172 – quantum statistics 144 Bose gas 189 – grating 190 Bragg diffraction 183, 404, 405, 408, 409, 415 – scattering 173, 190 – – of condensate atoms 167 – spectroscopy 170

E eikonal approximation 84 – equation 81 – method 79, 80, 84, 86, 129 Einstein–Brillouin–Keller quantization 84 electro-optic effect 39 ellipsometer 261 encryption, polarization 272–275 entangled matter wave 142 – state 16, 368, 369 evaporative cooling 175 F Fermi–Dirac quantum statistics 144 Fermi energy 152 455

456

Subject index for Volume 47

– gas 153, 154, 173, 175 – – , degenerate 172–179 – grating 190 – liquid 172 – system, quantum degenerate 151–153 – – , superfluid 174–179 – temperature 152, 175 fermionic phase conjugation 191–194 – superfluid 191 Feshbach resonance 142, 143, 150, 158, 160, 161, 176, 178, 197, 200, 205, 207–210 Fibonacci structure 16 field quantization 143–148 Fock state 188, 201 four-wave mixing 143, 182–194, 320 – – , atomic 190 – – , bosonic 182–186 – – , fermionic 186–191 Franck–Condon factor 164 Fresnel approximation 258 – law of refraction 93 – propagation operator 254 – scattering 107 G

Husimi distribution 115, 134 Husimi–Poincare distribution 119, 120, 134 – plot 126 – projection 80, 87, 108, 109, 113, 116, 117 I imaging polarimetry 217 interferometer, shearing type 411, 421–424 – , Talbot 413, 423 J Jaynes–Cummings model 202, 203, 204 Jones calculus 242 – matrix 231 Jones–Mueller polarization-transfer matrix 218 K Kerr medium 183, 336, 347, 348 – – , self-defocusing 312 – nonlinearity 298, 303, 313 Kirchhoff’s law 282 Kolmogorov–Arnold–Moser theory 86, 87

Galilean transformation 304 Gaussian optics 77, 79, 86, 122 Ginzburg–Landau model 360 Gladstone–Dale constant 396 gradient force 141 grating, Lee-type binary 245, 271 – , polarization 262 – , sub-wavelength 223, 224, 226, 227, 229, 230, 255 Gross–Pitaevskii equation 143, 164–167, 172, 179, 183, 184, 360, 364

L

H

M

Hankel function 116, 118 Hartree variational principle 165 – wave function 167 Heisenberg equation of motion 145, 146, 167, 169 Helmholtz equation 80, 91, 94, 95–98, 114, 157 hologram, polarization-selective computergenerated 223 holographic recording 238 holography, polarization 238

Mach–Zehnder interferometer 234, 240, 406, 420 Madelung transformation 298, 299 Maxwell–Bloch equations 361 Maxwell boundary conditions 95 Maxwell–Dirac equations 320 Maxwell equations 80, 91, 94, 218, 320 Michelson atom interferometer 194 micromaser 143, 200, 201, 204 modulational instability 304, 306, 332, 343 Moiré effect 413

Laguerre–Gaussian beam 347, 369 – mode 309, 314, 327, 347 Landau damping 195 Landau–Zener model 198 Legendre polynomial 156 lens transform 132 liquid crystal 222 lithography 228 Lorentz model 141 Lyapunov exponent 126

Subject index for Volume 47 – method 418 Mott-insulator transition 202 Mueller calculus 218 – matrix 221, 263, 269 N neural network 217 neutron star 149 O optical bistability 361 – computing 217, 222 – encryption 217 – parametric amplifier 31 – – oscillator 31 – rectification 39 – resonator, high-Q 77 – superlattice 16 – – , aperiodic 46 – – , quasi-periodic 46 – tweezer 217, 254, 360 P Pancharatnam–Berry phase 219, 221, 239, 241–243, 246, 247, 249–251, 253, 284 Pancharatnam charge 232, 233 – phase 219, 232, 233, 243, 244 – ’s theorem 239 parametric down-conversion 35, 369 – instability dynamics 35 – interaction 29, 54 – – , multistep 25 – – , two-color 26 – oscillation 197 – process 7, 24, 42 – – in quadratic medium 325 – – , phase matched 325 – – , multistep 4, 5 paraxial approximation 340 – beam 327 – light field 217 Pauli’s exclusion principle 148, 151, 186, 189, 193, 196 phase image 401 – imaging methods 402–406 phase-matched interaction, multistep 8–39 – process, single 5–7 phase-matching 3–5, 11, 15, 21, 50

457

– , birefringence 21 – for multistep cascading 39–53 phase transition, Kosterlitz–Thouless 294 phonon excitation 170 photoassociation 162 – of fermionic atoms 202 – , stimulated Raman 203 – , two-photon Raman 163 photonic bandgap 52 – crystal 354 – – , nonlinear 47 photorefractive crystal 311 – medium 320, 323, 346 Poincaré sphere 219, 241, 243, 271 – surface of section 88 polarimetry, far-field 266–268 – , imaging 268 polariton, surface 281 polarization, degree of 262 – logic 279 – state, space variant 217, 221, 222, 227 Q quantum billiard 97, 127 – chaos 87 – computer 207 – degenerate atomic system 149–153 – dot 359 quantum-Hall effect 368 quantum tunneling 202 R Rabi frequency 163, 164 – oscillation 204 radiography, contact 395 Ramsey fringe 199 Rayleigh hypothesis 98 Riemann zeta function 150 Rytov approximation 429 S Schlieren method 396, 404 Schrödinger field operator 147 – equation 145, 155, 208, 364 – – , nonlinear 165, 166, 179, 180, 302 second harmonic generation 3–5, 7, 17, 48, 197 self-focusing 3, 179, 296, 297, 309, 323, 342 shadowgraphy 395, 402

458

Subject index for Volume 47

sine-Gordon model 361 skyrmion 367, 368 slowly-varying-envelope approximation 6 S-matrix 97, 99, 108 Snell’s law 399 soliton, atomic 179–182 – , bright 181, 182 – , dark 180, 299 – , – vortex 297 – , gap 180, 182 – gluon 317 – induced waveguide 310–313 – instability 57 – , multi-color 18, 53, 54, 60, 329 – , – vortex 324–336 – , necklace-ring vector 320 – , parametric 24, 54, 55 – , spatial optical 303 – , spatiotemporal spinning 341–344 – , vortex 298, 301, 306–310, 314–317, 321, 325, 328, 330, 335, 336, 338–340, 348 squeezed matter wave 142 – polarized light 35 – quantum state 16 Stern–Gerlach effect 199 Stokes–Mueller calculus 269 Stokes parameters 220, 221, 228, 259, 264, 277 – vector 262, 263 sub-Poissonian photon statistics 201 superconductivity, high-TC 176 superconductor 175 s-wave scattering 153–158, 164 symmetry point group 26 synchrotron radiation 409 T Talbot effect 221, 256–259 – – , fractional 258 – laser resonator 256

– self imaging 256 third-harmonic multistep process 10 – wave 17 – – , efficient generation of 11 three-wave mixing 5, 194–205 tomography, diffraction 428 – , refractive index 419, 424 transport equation 81 U uniaxial crystal 222 V Vakhitov–Kolokolov stability criterion 307, 316 van Cittert–Zernike theorem 220, 426 vortex 218, 305 – optical 293–295, 323 – , partially coherent 321 W walk-off effect, spatial 38 – – , temporal 38 Wannier state 203 Weigert effect 238 whispering gallery mode 121, 122 Wigner distribution 114, 115 WKB quantization condition 154 Wolf’s theory of coherence 218 X X-ray imaging 395, 396 – interferometer 409, 428 – – , Bonse–Hart type 408, 427 Z Zeeman effect 160

Contents of previous volumes*

VOLUME 1 (1961) 1 2 3 4 5 6 7 8

The modern development of Hamiltonian optics, R.J. Pegis Wave optics and geometrical optics in optical design, K. Miyamoto The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat Light and information, D. Gabor On basic analogies and principal differences between optical and electronic information, H. Wolter Interference color, H. Kubota Dynamic characteristics of visual processes, A. Fiorentini Modern alignment devices, A.C.S. Van Heel

1– 29 31– 66 67–108 109–153 155–210 211–251 253–288 289–329

VOLUME 2 (1963) 1 2 3 4 5 6

Ruling, testing and use of optical gratings for high-resolution spectroscopy, G.W. Stroke The metrological applications of diffraction gratings, J.M. Burch Diffusion through non-uniform media, R.G. Giovanelli Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi Fluctuations of light beams, L. Mandel Methods for determining optical parameters of thin films, F. Abelès

1– 72 73–108 109–129 131–180 181–248 249–288

VOLUME 3 (1964) 1 2 3

The elements of radiative transfer, F. Kottler Apodisation, P. Jacquinot, B. Roizen-Dossier Matrix treatment of partial coherence, H. Gamo

1 2 3 4

Higher order aberration theory, J. Focke Applications of shearing interferometry, O. Bryngdahl Surface deterioration of optical glasses, K. Kinosita Optical constants of thin films, P. Rouard, P. Bousquet

1– 28 29–186 187–332

VOLUME 4 (1965)

* Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

459

1– 36 37– 83 85–143 145–197

460 5 6 7

Contents of previous volumes The Miyamoto–Wolf diffraction wave, A. Rubinowicz Aberration theory of gratings and grating mountings, W.T. Welford Diffraction at a black screen, Part I: Kirchhoff’s theory, F. Kottler

199–240 241–280 281–314

VOLUME 5 (1966) 1 2 3 4 5 6

Optical pumping, C. Cohen-Tannoudji, A. Kastler Non-linear optics, P.S. Pershan Two-beam interferometry, W.H. Steel Instruments for the measuring of optical transfer functions, K. Murata Light reflection from films of continuously varying refractive index, R. Jacobsson X-ray crystal-structure determination as a branch of physical optics, H. Lipson, C.A. Taylor 7 The wave of a moving classical electron, J. Picht

1– 81 83–144 145–197 199–245 247–286 287–350 351–370

VOLUME 6 (1967) 1 2 3 4 5 6 7 8

Recent advances in holography, E.N. Leith, J. Upatnieks Scattering of light by rough surfaces, P. Beckmann Measurement of the second order degree of coherence, M. Françon, S. Mallick Design of zoom lenses, K. Yamaji Some applications of lasers to interferometry, D.R. Herriot Experimental studies of intensity fluctuations in lasers, J.A. Armstrong, A.W. Smith Fourier spectroscopy, G.A. Vanasse, H. Sakai Diffraction at a black screen, Part II: electromagnetic theory, F. Kottler

1– 52 53– 69 71–104 105–170 171–209 211–257 259–330 331–377

VOLUME 7 (1969) 1 2 3 4 5 6 7

Multiple-beam interference and natural modes in open resonators, G. Koppelman Methods of synthesis for dielectric multilayer filters, E. Delano, R.J. Pegis Echoes at optical frequencies, I.D. Abella Image formation with partially coherent light, B.J. Thompson Quasi-classical theory of laser radiation, A.L. Mikaelian, M.L. Ter-Mikaelian The photographic image, S. Ooue Interaction of very intense light with free electrons, J.H. Eberly

1– 66 67–137 139–168 169–230 231–297 299–358 359–415

VOLUME 8 (1970) 1 2 3 4 5 6

Synthetic-aperture optics, J.W. Goodman The optical performance of the human eye, G.A. Fry Light beating spectroscopy, H.Z. Cummins, H.L. Swinney Multilayer antireflection coatings, A. Musset, A. Thelen Statistical properties of laser light, H. Risken Coherence theory of source-size compensation in interference microscopy, T. Yamamoto 7 Vision in communication, L. Levi 8 Theory of photoelectron counting, C.L. Mehta

1– 50 51–131 133–200 201–237 239–294 295–341 343–372 373–440

Contents of previous volumes

461

VOLUME 9 (1971) 1 2 3 4 5 6 7

Gas lasers and their application to precise length measurements, A.L. Bloom Picosecond laser pulses, A.J. Demaria Optical propagation through the turbulent atmosphere, J.W. Strohbehn Synthesis of optical birefringent networks, E.O. Ammann Mode locking in gas lasers, L. Allen, D.G.C. Jones Crystal optics with spatial dispersion, V.M. Agranovich, V.L. Ginzburg Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, J. Petykiewicz 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden

1– 30 31– 71 73–122 123–177 179–234 235–280 281–310 311–407

VOLUME 10 (1972) 1 2 3 4 5 6 7

Bandwidth compression of optical images, T.S. Huang The use of image tubes as shutters, R.W. Smith Tools of theoretical quantum optics, M.O. Scully, K.G. Whitney Field correctors for astronomical telescopes, C.G. Wynne Optical absorption strength of defects in insulators, D.Y. Smith, D.L. Dexter Elastooptic light modulation and deflection, E.K. Sittig Quantum detection theory, C.W. Helstrom

1– 44 45– 87 89–135 137–164 165–228 229–288 289–369

VOLUME 11 (1973) 1 2 3 4 5 6 7

Master equation methods in quantum optics, G.S. Agarwal Recent developments in far infrared spectroscopic techniques, H. Yoshinaga Interaction of light and acoustic surface waves, E.G. Lean Evanescent waves in optical imaging, O. Bryngdahl Production of electron probes using a field emission source, A.V. Crewe Hamiltonian theory of beam mode propagation, J.A. Arnaud Gradient index lenses, E.W. Marchand

1– 76 77–122 123–166 167–221 223–246 247–304 305–337

VOLUME 12 (1974) 1 2 3 4 5 6

Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto Self-induced transparency, R.E. Slusher Modulation techniques in spectrometry, M. Harwit, J.A. Decker Jr Interaction of light with monomolecular dye layers, K.H. Drexhage The phase transition concept and coherence in atomic emission, R. Graham Beam-foil spectroscopy, S. Bashkin

1– 51 53–100 101–162 163–232 233–286 287–344

VOLUME 13 (1976) 1

On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment, H.P. Baltes 2 The case for and against semiclassical radiation theory, L. Mandel 3 Objective and subjective spherical aberration measurements of the human eye, W.M. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G. Schulz, J. Schwider

1– 25 27– 68 69– 91 93–167

462

Contents of previous volumes

5

Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi 6 Aplanatism and isoplanatism, W.T. Welford

169–265 267–292

VOLUME 14 (1976) 1 2 3 4 5 6 7

The statistics of speckle patterns, J.C. Dainty High-resolution techniques in optical astronomy, A. Labeyrie Relaxation phenomena in rare-earth luminescence, L.A. Riseberg, M.J. Weber The ultrafast optical Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, P.J. Vernier Optical fibre waveguides – a review, P.J.B. Clarricoats

1– 46 47– 87 89–159 161–193 195–244 245–325 327–402

VOLUME 15 (1977) 1 2 3 4 5

Theory of optical parametric amplification and oscillation, W. Brunner, H. Paul Optical properties of thin metal films, P. Rouard, A. Meessen Projection-type holography, T. Okoshi Quasi-optical techniques of radio astronomy, T.W. Cole Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, J.E. Sipe

1– 75 77–137 139–185 187–244 245–350

VOLUME 16 (1978) 1 2 3 4 5

Laser selective photophysics and photochemistry, V.S. Letokhov Recent advances in phase profiles generation, J.J. Clair, C.I. Abitbol Computer-generated holograms: techniques and applications, W.-H. Lee Speckle interferometry, A.E. Ennos Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis 6 Light emission from high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanical framework, I.R. Senitzky

1– 69 71–117 119–232 233–288 289–356 357–411 413–448

VOLUME 17 (1980) 1 2 3

Heterodyne holographic interferometry, R. Dändliker Doppler-free multiphoton spectroscopy, E. Giacobino, B. Cagnac The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wilhelmi 4 Michelson stellar interferometry, W.J. Tango, R.Q. Twiss 5 Self-focusing media with variable index of refraction, A.L. Mikaelian

1– 84 85–161 163–238 239–277 279–345

VOLUME 18 (1980) 1 2

Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan Photocount statistics of radiation propagating through random and nonlinear media, J. Pe˘rina

1–126 127–203

Contents of previous volumes Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, V.U. Zavorotnyi 4 Catastrophe optics: morphologies of caustics and their diffraction patterns, M.V. Berry, C. Upstill

463

3

204–256 257–346

VOLUME 19 (1981) 1 2 3 4 5

Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy Light scattering spectroscopy of surface electromagnetic waves in solids, S. Ushioda Principles of optical data-processing, H.J. Butterweck The effects of atmospheric turbulence in optical astronomy, F. Roddier

1– 43 45–137 139–210 211–280 281–376

VOLUME 20 (1983) 1 2 3 4 5

Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtès, P. Cruvellier, M. Detaille Shaping and analysis of picosecond light pulses, C. Froehly, B. Colombeau, M. Vampouille Multi-photon scattering molecular spectroscopy, S. Kielich Colour holography, P. Hariharan Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P. Stoicheff

1– 61 63–153 155–261 263–324 325–380

VOLUME 21 (1984) 1 2 3 4 5

Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, L.A. Lugiato The Radon transform and its applications, H.H. Barrett Zone plate coded imaging: theory and applications, N.M. Ceglio, D.W. Sweeney Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J.C. Englund, R.R. Snapp, W.C. Schieve

1– 67 69–216 217–286 287–354 355–428

VOLUME 22 (1985) 1 2 3 4 5

Optical and electronic processing of medical images, D. Malacara Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema Spectral and temporal fluctuations of broad-band laser radiation, A.V. Masalov Holographic methods of plasma diagnostics, G.V. Ostrovskaya, Yu.I. Ostrovsky Fringe formations in deformation and vibration measurements using laser light, I. Yamaguchi 6 Wave propagation in random media: a systems approach, R.L. Fante

1– 76 77–144 145–196 197–270 271–340 341–398

VOLUME 23 (1986) 1

Analytical techniques for multiple scattering from rough surfaces, J.A. DeSanto, G.S. Brown 2 Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka 3 Optical films produced by ion-based techniques, P.J. Martin, R.P. Netterfield

1– 62 63–111 113–182

464 4 5

Contents of previous volumes Electron holography, A. Tonomura Principles of optical processing with partially coherent light, F.T.S. Yu

183–220 221–275

VOLUME 24 (1987) 1 2 3 4 5

Micro Fresnel lenses, H. Nishihara, T. Suhara Dephasing-induced coherent phenomena, L. Rothberg Interferometry with lasers, P. Hariharan Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, I. Glaser

1– 37 39–101 103–164 165–387 389–509

VOLUME 25 (1988) 1

Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, L.M. Narducci 2 Coherence in semiconductor lasers, M. Ohtsu, T. Tako 3 Principles and design of optical arrays, Wang Shaomin, L. Ronchi 4 Aspheric surfaces, G. Schulz

1–190 191–278 279–348 349–415

VOLUME 26 (1988) 1 2 3 4 5

Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals, I.C. Khoo Single-longitudinal-mode semiconductor lasers, G.P. Agrawal Rays and caustics as physical objects, Yu.A. Kravtsov Phase-measurement interferometry techniques, K. Creath

1–104 105–161 163–225 227–348 349–393

VOLUME 27 (1989) 1 2 3 4

The self-imaging phenomenon and its applications, K. Patorski Axicons and meso-optical imaging devices, L.M. Soroko Nonimaging optics for flux concentration, I.M. Bassett, W.T. Welford, R. Winston Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C. Sibilia 5 Generalized holography with application to inverse scattering and inverse source problems, R.P. Porter

1–108 109–160 161–226 227–313 315–397

VOLUME 28 (1990) 1 2

Digital holography – computer-generated holograms, O. Bryngdahl, F. Wyrowski Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa 3 The quantum coherence properties of stimulated Raman scattering, M.G. Raymer, I.A. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, R.J. Cook

1– 86 87–179 181–270 271–359 361–416

Contents of previous volumes

465

VOLUME 29 (1991) 1 2

Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev 3 Generation and propagation of ultrashort optical pulses, I.P. Christov 4 Triple-correlation imaging in optical astronomy, G. Weigelt 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol

1– 63 65–197 199–291 293–319 321–411

VOLUME 30 (1992) 1

4 5

Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Fabre Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P. Shchepinov Localization of waves in media with one-dimensional disorder, V.D. Freilikher, S.A. Gredeskul Theoretical foundation of optical-soliton concept in fibers, Y. Kodama, A. Hasegawa Cavity quantum optics and the quantum measurement process, P. Meystre

1 2 3 4 5 6

Atoms in strong fields: photoionization and chaos, P.W. Milonni, B. Sundaram Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov Optical amplifiers, N.K. Dutta, J.R. Simpson Adaptive multilayer optical networks, D. Psaltis, Y. Qiao Optical atoms, R.J.C. Spreeuw, J.P. Woerdman Theory of Compton free electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre

2 3

1– 85 87–135 137–203 205–259 261–355

VOLUME 31 (1993) 1–137 139–187 189–226 227–261 263–319 321–412

VOLUME 32 (1993) 1 2 3 4

Guided-wave optics on silicon: physics, technology and status, B.P. Pal Optical neural networks: architecture, design and models, F.T.S. Yu The theory of optimal methods for localization of objects in pictures, L.P. Yaroslavsky Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, V.U. Zavorotny 5 Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus

1– 59 61–144 145–201 203–266 267–312 313–361

VOLUME 33 (1994) 1 2 3 4

The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin Quantum statistics of dissipative nonlinear oscillators, V. Peˇrinová, A. Lukš Gap solitons, C.M. De Sterke, J.E. Sipe Direct spatial reconstruction of optical phase from phase-modulated images, V.I. Vlad, D. Malacara 5 Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt 6 Digital halftoning: synthesis of binary images, O. Bryngdahl, T. Scheermesser, F. Wyrowski

1–127 129–202 203–260 261–317 319–388 389–463

466

Contents of previous volumes VOLUME 34 (1995)

1 2 3 4 5

Quantum interference, superposition states of light, and nonclassical effects, V. Bužek, P.L. Knight Wave propagation in inhomogeneous media: phase-shift approach, L.P. Presnyakov The statistics of dynamic speckles, T. Okamoto, T. Asakura Scattering of light from multilayer systems with rough boundaries, I. Ohlídal, K. Navrátil, M. Ohlídal Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss

1–158 159–181 183–248 249–331 333–402

VOLUME 35 (1996) 1 2 3 4 5 6

Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis Interferometric multispectral imaging, K. Itoh Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo Coherent population trapping in laser spectroscopy, E. Arimondo Quantum phase properties of nonlinear optical phenomena, R. Tana´s, A. Miranowicz, Ts. Gantsog

1– 60 61–144 145–196 197–255 257–354 355–446

VOLUME 36 (1996) 1 2 3 4 5

Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders Super-resolution by data inversion, M. Bertero, C. De Mol Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, L.A. Apresyan Photon wave function, I. Bialynicki-Birula

1– 47 49–128 129–178 179–244 245–294

VOLUME 37 (1997) 1 2 3 4 5 6

The Wigner distribution function in optics and optoelectronics, D. Dragoman Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura Spectra of molecular scattering of light, I.L. Fabelinskii Soliton communication systems, R.-J. Essiambre, G.P. Agrawal Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller Tunneling times and superluminality, R.Y. Chiao, A.M. Steinberg

1– 56 57– 94 95–184 185–256 257–343 345–405

VOLUME 38 (1998) 1 2 3

Nonlinear optics of stratified media, S. Dutta Gupta Optical aspects of interferometric gravitational-wave detectors, P. Hello Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W. Nakwaski, M. Osi´nski 4 Fractional transformations in optics, A.W. Lohmann, D. Mendlovic, Z. Zalevsky 5 Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L. Horner 6 Free-space optical digital computing and interconnection, J. Jahns

1– 84 85–164 165–262 263–342 343–418 419–513

Contents of previous volumes

467

VOLUME 39 (1999) 1 2

Theory and applications of complex rays, Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T. Opatrný 3 Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford 4 The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs

1– 62 63–211 213–290 291–372 373–469

VOLUME 40 (2000) 1 2 3 4

Polarimetric optical fibers and sensors, T.R. Woli´nski Digital optical computing, J. Tanida, Y. Ichioka Continuous measurements in quantum optics, V. Peˇrinová, A. Lukš Optical systems with improved resolving power, Z. Zalevsky, D. Mendlovic, A.W. Lohmann 5 Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, F. Wyrowski 6 Spectroscopy in polychromatic fields, Z. Ficek, H.S. Freedhoff

1– 75 77–114 115–269 271–341 343–388 389–441

VOLUME 41 (2000) 1 2 3 4 5 6 7

Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur Ellipsometry of thin film systems, I. Ohlídal, D. Franta Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu Quantum statistics of nonlinear optical couplers, J. Peˇrina Jr, J. Peˇrina Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. Sánchez-Soto Optical solitons in media with a quadratic nonlinearity, C. Etrich, F. Lederer, B.A. Malomed, T. Peschel, U. Peschel

1– 95 97–179 181–282 283–358 359–417 419–479 483–567

VOLUME 42 (2001) 1 2 3 4 5 6

Quanta and information, S.Ya. Kilin Optical solitons in periodic media with resonant and off-resonant nonlinearities, G. Kurizki, A.E. Kozhekin, T. Opatrný, B.A. Malomed Quantum Zeno and inverse quantum Zeno effects, P. Facchi, S. Pascazio Singular optics, M.S. Soskin, M.V. Vasnetsov Multi-photon quantum interferometry, G. Jaeger, A.V. Sergienko Transverse mode shaping and selection in laser resonators, R. Oron, N. Davidson, A.A. Friesem, E. Hasman

1– 91 93–146 147–217 219–276 277–324 325–386

VOLUME 43 (2002) 1 2 3

Active optics in modern large optical telescopes, L. Noethe Variational methods in nonlinear fiber optics and related fields, B.A. Malomed Optical works of L.V. Lorenz, O. Keller

1– 69 71–193 195–294

468

Contents of previous volumes

4

Canonical quantum description of light propagation in dielectric media, A. Lukš, V. Peˇrinová 5 Phase space correspondence between classical optics and quantum mechanics, D. Dragoman 6 “Slow” and “fast” light, R.W. Boyd, D.J. Gauthier 7 The fractional Fourier transform and some of its applications to optics, A. Torre

295–431 433–496 497–530 531–596

VOLUME 44 (2002) 1 2 3

Chaotic dynamics in semiconductor lasers with optical feedback, J. Ohtsubo Femtosecond pulses in optical fibers, F.G. Omenetto Instantaneous optics of ultrashort broadband pulses and rapidly varying media, A.B. Shvartsburg, G. Petite 4 Optical coherence tomography, A.F. Fercher, C.K. Hitzenberger 5 Modulational instability of electromagnetic waves in inhomogeneous and in discrete media, F.Kh. Abdullaev, S.A. Darmanyan, J. Garnier

1– 84 85–141 143–214 215–301 303–366

VOLUME 45 (2003) 1 2 3 4 5 6

Anamorphic beam shaping for laser and diffuse light, N. Davidson, N. Bokor Ultra-fast all-optical switching in optical networks, I. Glesk, B.C. Wang, L. Xu, V. Baby, P.R. Prucnal Generation of dark hollow beams and their applications, J. Yin, W. Gao, Y. Zhu Two-photon lasers, D.J. Gauthier Nonradiating sources and other “invisible” objects, G. Gbur Lasing in disordered media, H. Cao

1– 51 53–117 119–204 205–272 273–315 317–370

VOLUME 46 (2004) 1 2

Ultrafast solid-state lasers, U. Keller Multiple scattering of light from randomly rough surfaces, A.V. Shchegrov, A.A. Maradudin, E.R. Méndez 3 Laser-diode interferometry, Y. Ishii 4 Optical realizations of quantum teleportation, J. Gea-Banacloche 5 Intensity-field correlations of non-classical light, H.J. Carmichael, G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice

1–115 117–241 243–309 311–353 355–404

Cumulative index – Volumes 1–47* Abdullaev, F.Kh., S.A. Darmanyan, 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, L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, V.M., V.L. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.P., see Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett, M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefringent networks Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A., A.W. Smith: Experimental studies of intensity fluctuations in lasers 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, A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-free diffraction images Barrett, H.H.: The Radon transform and its applications Bashkin, S.: Beam-foil spectroscopy Bassett, I.M., W.T. Welford, R. Winston: Nonimaging optics for flux concentration * Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

469

44, 303 2, 249 7, 139 16, 71 25, 1 11, 1 9, 235 26, 163 37, 185 9, 179 39, 291 9, 123 41, 97 36, 179 35, 257 6, 211 11, 247 34, 183 37, 57 39, 1 39, 291 45, 53 13,

1

29, 65 1, 67 21, 217 12, 287 27, 161

470

Cumulative index – Volumes 1–47

Beckmann, P.: Scattering of light by rough surfaces Beran, M.J., J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M. Berry, M.V., C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M., C. De Mol: Super-resolution by data inversion Bertolotti, M., see 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. Bloom, A.L.: Gas lasers and their application to precise length measurements Bokor, N., see Davidson, N. Bouman, M.A., W.A. Van De Grind, P. Zuidema: Quantum fluctuations in vision Bousquet, P., see Rouard, P. Boyd, R.W., D.J. Gauthier: “Slow” and “fast” light Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W., H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: Applications of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., T. Scheermesser, F. Wyrowski: Digital halftoning: synthesis of binary images Bryngdahl, O., F. Wyrowski: Digital holography – computer-generated holograms Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Bužek, V., 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, 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, M. Mansuripur: Principles of optical disk data storage Casasent, D., D. Psaltis: Deformation invariant, space-variant optical pattern recognition Ceglio, N.M., D.W. Sweeney: Zone plate coded imaging: theory and applications Chang, R.K., see Fields, M.H. Charnotskii, M.I., J. Gozani, V.I. Tatarskii, V.U. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T., Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y., A.M. Steinberg: Tunneling times and superluminality Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, W.M.

6, 53 33, 319 35, 61 18, 257 36, 129 36, 1 27, 227 16, 357 36, 245 47, 215 9, 1 45, 1 22, 77 4, 145 43, 497 23, 1 35, 61 15, 1 4, 37 11, 167 33, 389 28, 1 2, 73 19, 211 34,

1

17, 85 45, 317 46, 355 41, 97 16, 289 21, 287 41, 1 32, 203 41, 283 37, 345 41, 97 13, 69

Cumulative index – Volumes 1–47 Christov, I.P.: Generation and propagation of ultrashort optical pulses Chumash, V., I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.J., C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.J.B.: Optical fibre waveguides – a review Cohen-Tannoudji, C., A. Kastler: Optical pumping Cojocaru, I., see Chumash, V. Cole, T.W.: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.J.: Quantum jumps Courtès, G., P. Cruvellier, 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 Cruvellier, P., see Courtès, G. Cummins, H.Z., 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. Dattoli, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton free electron lasers Davidson, N., N. Bokor: Anamorphic beam shaping for laser and diffuse light Davidson, N., see Oron, R. Decker Jr, J.A., see Harwit, M. Delano, E., R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses De Mol, C., see Bertero, M. DeSanto, J.A., G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Desyatnikov, A.S., Y.S. Kivshar, L. Torner: Optical vortices and vortex solitons De Sterke, C.M., J.E. Sipe: Gap solitons Detaille, M., see Courtès, G. Dexter, D.L., see Smith, D.Y. Dragoman, D.: The Wigner distribution function in optics and optoelectronics Dragoman, D.: Phase space correspondence between classical optics and quantum mechanics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dutta, N.K., J.R. Simpson: Optical amplifiers Dutta Gupta, S.: Nonlinear optics of stratified media Eberly, J.H.: Interaction of very intense light with free electrons Englund, J.C., R.R. Snapp, W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity Ennos, A.E.: Speckle interferometry

471 29, 199 36, 1 16, 71 14, 327 5, 1 36, 1 15, 187 20, 63 28, 361 20, 1 26, 349 11, 223 20, 1 8, 133 14, 1 17, 1 44, 303 31, 321 45, 1 42, 325 12, 101 7, 67 9, 31 36, 129 23, 1 47, 291 33, 203 20, 1 10, 165 37, 1 43, 433 12, 163 14, 161 31, 189 38, 1 7, 359 21, 355 16, 233

472

Cumulative index – Volumes 1–47

Essiambre, R.-J., G.P. Agrawal: Soliton communication systems Etrich, C., F. Lederer, B.A. Malomed, T. Peschel, U. Peschel: Optical solitons in media with a quadratic nonlinearity Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C., see Reynaud, S. Facchi, P., S. Pascazio: Quantum Zeno and inverse quantum Zeno effects Fante, R.L.: Wave propagation in random media: a systems approach Fazio, E., see Chumash, V. Fercher, A.F., C.K. Hitzenberger: Optical coherence tomography Ficek, Z., H.S. Freedhoff: Spectroscopy in polychromatic fields Fields, M.H., J. Popp, R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Flytzanis, C., F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics Focke, J.: Higher order aberration theory Forbes, G.W., see Kravtsov, Yu.A. Foster, G.A., see Carmichael, H.J. Françon, M., S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlídal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, V.D., S.A. Gredeskul: Localization of waves in media with one-dimensional disorder Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions Friesem, A.A., see Oron, R. Froehly, C., B. Colombeau, M. Vampouille: Shaping and analysis of picosecond light pulses Fry, G.A.: The optical performance of the human eye Fu, Z., see Chen, R.T. Gabor, D.: Light and information Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H., G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media Gantsog, Ts., see Tana´s, R. Gao, W., see Yin, J. Garnier, J., see Abdullaev, F.Kh. Gauthier, D.J.: Two-photon lasers 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., K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E., B. Cagnac: Doppler-free multiphoton spectroscopy

37, 185 41, 483 37, 95 30, 1 42, 147 22, 341 36, 1 44, 215 40, 389 41, 1 1, 253 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 1, 109 3, 187 34, 333 35, 355 45, 119 44, 303 45, 205 43, 497 45, 273 46, 311 18, 1 13, 169 17, 85

Cumulative index – Volumes 1–47 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, P.R. Prucnal: Ultra-fast all-optical switching in optical networks Gniadek, K., J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, J., see Charnotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, V.D.

473 30, 1 31, 321 32, 267 9, 235 2, 109 24, 389 45, 53 9, 281 8, 1 32, 203 12, 233 30, 137

Hache, F., see Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, P.: Colour holography Hariharan, P.: Interferometry with lasers Hariharan, P., B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M., J.A. Decker Jr: Modulation techniques in spectrometry Hasegawa, A., see Kodama, Y. Hasman, E., G. Biener, A. Niv, V. Kleiner: Space-variant polarization manipulation Hasman, E., see Oron, R. Heidmann, A., see Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Herriot, D.R.: Some applications of lasers to interferometry Hitzenberger, C.K., see Fercher, A.F. Horner, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images

29, 321 29, 1 20, 263 24, 103 36, 49 12, 101 30, 205 47, 215 42, 325 30, 1 38, 85 10, 289 6, 171 44, 215 38, 343 10, 1

Ichioka, Y., see Tanida, J. Imoto, N., see Yamamoto, Y. 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 47, 393

Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, P., B. Roizen-Dossier: Apodisation Jacquod, Ph., see Türeci, H.E. Jaeger, G., A.V. Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection

5, 247 3, 29 47, 75 42, 277 38, 419

474

Cumulative index – Volumes 1–47

Jamroz, W., B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation Javidi, B., J.L. Horner: Pattern recognition with nonlinear techniques in the Fourier domain Jones, D.G.C., see Allen, L.

20, 325 38, 343 9, 179

Kastler, A., see Cohen-Tannoudji, C. Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Keller, O.: Optical works of L.V. Lorenz 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. Klein, M.C., see Flytzanis, C. Kleiner, V., see Hasman, E. Klyatskin, V.I.: The imbedding method in statistical boundary-value wave problems Knight, P.L., see Bužek, V. Kodama, Y., A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, F.: The elements of radiative transfer Kottler, F.: Diffraction at a black screen, Part I: Kirchhoff’s theory Kottler, F.: Diffraction at a black screen, Part II: electromagnetic theory Kozhekin, A.E., see Kurizki, G. Kravtsov, Yu.A.: Rays and caustics as physical objects Kravtsov, Yu.A., L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Forbes, A.A. Asatryan: Theory and applications of complex rays Kravtsov, Yu.A., see Barabanenkov, Yu.N. Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrný, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities

5, 1 37, 257 43, 195 46, 1 26, 105 41, 97 20, 155 42, 1 4, 85 28, 87 47, 291 47, 1 29, 321 47, 215 33, 1 34, 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

Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves Lederer, F., see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N., J. Upatnieks: Recent advances in holography Letokhov, V.S.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication

14, 47 11, 123 41, 483 16, 119 6, 1 16, 1 39, 373 8, 343

42, 93

Cumulative index – Volumes 1–47 Li, L., see Carriere, J. Lipson, H., C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Lohmann, A.W., D. Mendlovic, Z. Zalevsky: Fractional transformations in optics Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L.A.: Theory of optical bistability Luis, A., L.L. Sánchez-Soto: Quantum phase difference, phase measurements and Stokes operators Lukš, A., V. Peˇrinová: Canonical quantum description of light propagation in dielectric media Lukš, A., see Peˇrinová, V. Lukš, A., see Peˇrinová, V. Machida, S., see Yamamoto, Y. Mainfray, G., C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas Malacara, D.: Optical and electronic processing of medical images Malacara, D., see Vlad, 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 Martin, P.J., R.P. Netterfield: Optical films produced by ion-based techniques Masalov, A.V.: Spectral and temporal fluctuations of broad-band laser radiation Maystre, D.: Rigorous vector theories of diffraction gratings Meessen, A., see Rouard, P. Mehta, C.L.: Theory of photoelectron counting Mendez, E.R., see Shchegrov, A.V. Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Meystre, P., see Search, C.P. Michelotti, F., see Chumash, V. Mihalache, D., M. Bertolotti, C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L.: Self-focusing media with variable index of refraction Mikaelian, A.L., M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mills, D.L., K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids

475 41, 97 5, 287 38, 263 40, 271 35, 61 21, 69 41, 419 43, 295 33, 129 40, 117 28, 87 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 23, 113 22, 145 21, 1 15, 77 8, 373 46, 117 38, 263 40, 271 30, 261 47, 139 36, 1 27, 227 17, 279 7, 231 19, 45

476

Cumulative index – Volumes 1–47

Milonni, P.W., 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., A. Thelen: Multilayer antireflection coatings Nakwaski, W., M. Osi´nski: 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. Nishihara, H., T. Suhara: Micro Fresnel lenses Niv, A., see Hasman, E. Noethe, L.: Active optics in modern large optical telescopes

31, 1 35, 355 1, 31 19, 1 5, 199 8, 201

38, 165 41, 97 25, 1 34, 249 23, 113 24, 1 47, 215 43, 1

Ohlídal, I., D. Franta: Ellipsometry of thin film systems Ohlídal, I., K. Navrátil, M. Ohlídal: Scattering of light from multilayer systems with rough boundaries Ohlídal, M., see Ohlídal, I. Ohtsu, M., T. Tako: Coherence in semiconductor lasers Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback Okamoto, T., T. Asakura: The statistics of dynamic speckles Okoshi, T.: Projection-type holography Omenetto, F.G.: Femtosecond pulses in optical fibers Ooue, S.: The photographic image Opatrný, T., see Kurizki, G. Opatrný, T., see Welsch, D.-G. Oron, R., N. Davidson, A.A. Friesem, E. Hasman: Transverse mode shaping and selection in laser resonators Orozco, L.A., see Carmichael, H.J. Orrit, M., J. Bernard, R. Brown, B. Lounis: Optical spectroscopy of single molecules in solids Osi´nski, M., see Nakwaski, W. Ostrovskaya, G.V., Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.I., 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.

41, 181

35, 61 38, 165 22, 197 30, 87 22, 197 24, 165 33, 319 29, 65

Padgett, M.J., see Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status

39, 291 32, 1

34, 249 34, 249 25, 191 44, 1 34, 183 15, 139 44, 85 7, 299 42, 93 39, 63 42, 325 46, 355

Cumulative index – Volumes 1–47 Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., see Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., see Brunner, W. Pegis, R.J.: The modern development of Hamiltonian optics Pegis, R.J., see Delano, E. Peiponen, K.-E., E.M. Vartiainen, T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C., see Carriere, J. Pe˘rina, J.: Photocount statistics of radiation propagating through random and nonlinear media Peˇrina, J., see Peˇrina Jr, J. Peˇrina Jr, J., J. Peˇrina: Quantum statistics of nonlinear optical couplers Peˇrinová, V., A. Lukš: Quantum statistics of dissipative nonlinear oscillators Peˇrinová, V., 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. Picht, J.: The wave of a moving classical electron Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view Popp, J., see Fields, M.H. Porter, R.P.: Generalized holography with application to inverse scattering and inverse source problems Presnyakov, L.P.: Wave propagation in inhomogeneous media: phase-shift approach Prucnal, P.R., see Glesk, I. Psaltis, D., Y. Qiao: Adaptive multilayer optical networks Psaltis, D., see Casasent, D. Qiao, Y., see Psaltis, D. Raymer, M.G., 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, C. Fabre: Quantum fluctuations in optical systems Ricard, D., see Flytzanis, C. Rice, P.R., see Carmichael, H.J. Riseberg, L.A., 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 Roizen-Dossier, B., see Jacquinot, P.

477 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67 37, 57 41, 97 18, 127 41, 359 41, 359 33, 129 40, 115 43, 295 5, 83 41, 483 41, 483 44, 143 9, 281 5, 351 31, 139 41, 1 27, 315 34, 159 45, 53 31, 227 16, 289 31, 227

28, 181 46, 355 31, 321 30, 1 29, 321 46, 355 14, 89 8, 239 19, 281 3, 29

478

Cumulative index – Volumes 1–47

Ronchi, L., see Wang Shaomin Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems Rosenblum, W.M., J.L. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P., P. Bousquet: Optical constants of thin films Rouard, P., A. Meessen: Optical properties of thin metal films Roussignol, Ph., see Flytzanis, C. Rubinowicz, A.: The Miyamoto–Wolf diffraction wave Rudolph, D., see Schmahl, G. Saichev, A.I., see Barabanenkov, Yu.N. Saito, S., see Yamamoto, Y. Sakai, H., see Vanasse, G.A. Saleh, B.E.A., see Teich, M.C. Saltiel, S.M., A.A. Sukhorukov, Y.S. Kivshar: Multistep parametric processes in nonlinear optics 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., D. Rudolph: Holographic diffraction gratings Schubert, M., B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G.: Aspheric surfaces Schulz, G., 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., K.G. Whitney: Tools of theoretical quantum optics Search, C.P., 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. Sharma, S.K., D.J. Somerford: Scattering of light in the eikonal approximation Shchegrov, A.V., A.A. Maradudin, E.R. Méndez: Multiple scattering of light from randomly rough surfaces Shchepinov, V.P., see Ostrovsky, Yu.I. Shvartsburg, A.B., G. Petite: Instantaneous optics of ultrashort broadband pulses and rapidly varying media Sibilia, C., see Mihalache, D. Simpson, J.R., see Dutta, N.K. Sipe, J.E., see De Sterke, C.M. Sipe, J.E., see Van Kranendonk, J. Sittig, E.K.: Elastooptic light modulation and deflection

25, 279 35, 1 13, 69 24, 39 4, 145 15, 77 29, 321 4, 199 14, 195 29, 65 28, 87 6, 259 26, 1 47, 1 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 47, 139 16, 413 42, 277 39, 213 46, 117 30, 87 44, 143 27, 227 31, 189 33, 203 15, 245 10, 229

Cumulative index – Volumes 1–47 Sizmann, A., G. Leuchs: The optical Kerr effect and quantum optics in fibers Slusher, R.E.: Self-induced transparency Smith, A.W., see Armstrong, J.A. Smith, D.Y., D.L. Dexter: Optical absorption strength of defects in insulators Smith, R.W.: The use of image tubes as shutters Snapp, R.R., see Englund, J.C. Sodha, M.S., A.K. Ghatak, V.K. Tripathi: Self-focusing of laser beams in plasmas and semiconductors Somerford, D.J., see Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices Soskin, M.S., M.V. Vasnetsov: Singular optics Spreeuw, R.J.C., J.P. Woerdman: Optical atoms Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. 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 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, Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J., R.Q. Twiss: Michelson stellar interferometry Tanida, J., Y. Ichioka: Digital optical computing Tatarskii, V.I., 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., 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 Desyatnikov, A.S. Torre, A.: The fractional Fourier transform and some of its applications to optics

479 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355 13, 169 39, 213 27, 109 42, 219 31, 263 5, 145 37, 345 20, 325 47, 75 9, 73 2, 1 19, 45 24, 1 47, 1 31, 1 12, 1 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 47, 291 43, 531

480

Cumulative index – Volumes 1–47

Torre, A., see Dattoli, G. Tripathi, V.K., see Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering Türeci, H.E., H.G.L. Schwefel, Ph. Jacquod, A.D. Stone: Modes of wave-chaotic dielectric resonators Turunen, J., M. Kuittinen, F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.

47, 75 40, 343 17, 239

Upatnieks, J., see Leith, E.N. Upstill, C., see Berry, M.V. Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids

6, 1 18, 257 19, 139

Vampouille, M., see Froehly, C. Vanasse, G.A., H. Sakai: Fourier spectroscopy Van De Grind, W.A., see Bouman, M.A. Van Heel, A.C.S.: Modern alignment devices Van Kranendonk, J., J.E. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vartiainen, E.M., see Peiponen, K.-E. Vasnetsov, M.V., see Soskin, M.S. Vernier, P.J.: Photoemission Vlad, V.I., D. Malacara: Direct spatial reconstruction of optical phase from phasemodulated images Vogel, W., see Welsch, D.-G.

20, 63 6, 259 22, 77 1, 289

Walmsley, I.A., see Raymer, M.G. Wang, B.C., see Glesk, I. Wang Shaomin, L. Ronchi: Principles and design of optical arrays Weber, M.J., see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weiss, G.H., see Gandjbakhche, A.H. Welford, W.T.: Aberration theory of gratings and grating mountings Welford, W.T.: Aplanatism and isoplanatism Welford, W.T., see Bassett, I.M. Welsch, D.-G., W. Vogel, 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. Woerdman, J.P., see Spreeuw, R.J.C. Woli´nski, 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

31, 321 13, 169 2, 131

15, 245 37, 57 42, 219 14, 245 33, 261 39, 63 28, 181 45, 53 25, 279 14, 89 29, 293 34, 333 4, 241 13, 267 27, 161 39, 63 10, 89 17, 163 27, 161 31, 263 40, 1 1, 155 10, 137

Cumulative index – Volumes 1–47

481

Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Turunen, J.

28, 1 33, 389 40, 343

Xu, L., see Glesk, I.

45, 53

Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y., S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa: 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, 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 Zalevsky, Z., D. Mendlovic, 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. Zuidema, P., see Bouman, M.A.

22, 271 6, 105 8, 295 28, 87 28, 87 32, 145 41, 97 45, 119 11, 77 23, 221 32, 61

40, 271 38, 263 32, 203 18, 204 45, 119 22, 77

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