Progress in Optics, Volume 46

Preface This volume presents five review articles of areas of current research interest in several branches of modern optics. The first article, by U. Keller, discusses the progress which has been made in recent years in ultrafast pulse generation, using solid-state lasers. In the last decade, the performance of compact ultrafast solid-state lasers has been improved by several orders of magnitude in pulse duration, in average power, in pulse energies and in pulse repetition rates. In this article these and other breakthroughs are discussed. The article provides both the expert and the non-expert with an overview of this rapidly advancing field. The second article, by A.V. Shchegrov, A.A. Maradudin and E.R. Méndez, reviews research on the scattering of electromagnetic waves from randomly rough surfaces. Unlike earlier investigations in this field, the article covers the more difficult subject of multiple scattering from such surfaces and discusses more recently discovered effects such as enhanced backscattering and enhanced transmission. In the next article, by Y. Ishii, an interesting advance in interferometry is described, namely a phase-measuring technique which uses direct frequency modulation of a laser diode source by changing the current. This technique has been implemented in holographic interferometry and in phase conjugate interferometry, for example. The fourth article, by J. Gea-Banacloche, reviews the theory of quantum teleportation and the ways in which this intriguing quantum phenomenon has been experimentally demonstrated in optical systems. Among the topics covered are the relationship between teleportation of discrete and continuous variables, entanglement and teleportation, the difficulty of Bell measurements, and schemes for near-deterministic teleportation with linear optics. The limitations of current experiments and some questions of interpretation are also discussed. The concluding article, by H.J. Carmichael, G.T. Foster, L.A. Orozco, J.E. Reiner and P.R. Rice, uses intensity–field correlation functions of the electromagnetic field as a tool for studying quantum fluctuations of light. The relationship between the correlation functions and quadrature squeezing is noted and conditions are developed to distinguish between classical and non-classical field fluctuations. The v

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Preface

theoretical analysis is illustrated by examples such as the optical parametric oscillator, a cavity QED system and the composite system of a single atom coupled to an optical parametric oscillator. The results of experimental measurements on a cavity QED system are also reviewed.

Emil Wolf Department of Physics and Astronomy and the Institute of Optics University of Rochester Rochester, NY 14627, USA February 2004

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

v

Chapter 1. Ultrafast solid-state lasers, Ursula Keller (Zürich, Switzerland) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Definition of Q-switching and mode locking . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Q-switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 3. Overview of ultrafast solid-state lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Overview of solid-state laser materials . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Design guidelines of diode-pumped solid-state lasers . . . . . . . . . . . . . . . . . § 4. Loss modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Optical modulators: acousto-optic and electrooptic modulators . . . . . . . . . . . 4.2. Saturable absorber: self-amplitude modulation (SAM) . . . . . . . . . . . . . . . . 4.3. Semiconductor saturable absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Effective saturable absorbers using the Kerr effect . . . . . . . . . . . . . . . . . . § 5. Pulse propagation in dispersive media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Dispersive pulse broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Dispersion compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 6. Mode-locking techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Haus’s master equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Active mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Passive mode locking with a slow saturable absorber and dynamic gain saturation . 6.5. Passive mode locking with a fast saturable absorber . . . . . . . . . . . . . . . . . 6.6. Passive mode locking with a slow saturable absorber without gain saturation and soliton formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. Soliton mode locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8. Design guidelines to prevent Q-switching instability . . . . . . . . . . . . . . . . . 6.9. External pulse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 7. Pulse characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Electronic techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Optical autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. FROG and SPIDER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 8. Carrier envelope offset (CEO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 9. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

3 7 10 10 11 15 15 39 43 43 44 49 52 54 54 56 65 65 66 70 74 77 80 81 87 89 90 90 91 92 93 95 98

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Chapter 2. Multiple scattering of light from randomly rough surfaces, Andrei V. Shchegrov (Rochester, NY, USA), Alexei A. Maradudin (Irvine, CA, USA) and Eugenio R. Méndez (Ensenada, Mexico) . . . . . . . . . . .

117

§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Characterization of randomly rough surfaces . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Two-dimensional random surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. One-dimensional random surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . § 3. Equations for electromagnetic fields and observable quantities . . . . . . . . . . . . . . 3.1. Physical quantities studied in rough surface scattering problems . . . . . . . . . . . 3.2. Rayleigh method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Surface integral equations for electromagnetic fields . . . . . . . . . . . . . . . . . § 4. Weak localization effects in the multiple scattering of light from randomly rough surfaces. Enhanced backscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The nature of enhanced backscattering effect . . . . . . . . . . . . . . . . . . . . . 4.2. Theoretical methods employed in the study of multiple-scattering phenomena . . . 4.3. Experimental techniques used in the study of multiple scattering effects, including the enhanced backscattering effect . . . . . . . . . . . . . . . . . . . . . . . . . . . § 5. Angular intensity correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Memory and reciprocal memory effects: theory and experiment . . . . . . . . . . . 5.2. The correlation function C (10) (q, k|q
, k
): theory and experiment . . . . . . . . . 5.3. Correlations in film systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 6. Multiple-scattering effects in the scattering of light from complex media bounded by a rough surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Coherent effects associated with the interference of nonreciprocal optical paths . . 6.2. Scattering from the random surface of an amplifying medium . . . . . . . . . . . . 6.3. Scattering from a nonlinear medium bounded by a rough surface . . . . . . . . . . § 7. Near-field effects: localization phenomena for surface waves . . . . . . . . . . . . . . . § 8. Spectral changes induced by multiple scattering . . . . . . . . . . . . . . . . . . . . . . § 9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3. Laser-diode interferometry, Yukihiro Ishii (Sagamihara, Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Laser-diode operation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Single-mode laser diodes . . . . . . . . . . . . . . . . . . . . . 2.2. Wavelength tunability in laser diodes . . . . . . . . . . . . . . . § 3. Modulation methods in laser-diode interferometers . . . . . . . . . . § 4. Laser-diode phase-shifting interferometers . . . . . . . . . . . . . . 4.1. Single-wavelength phase-shifting interferometry . . . . . . . . 4.2. Phase-shift calibration . . . . . . . . . . . . . . . . . . . . . . . 4.3. Laser-diode Fizeau interferometry . . . . . . . . . . . . . . . . 4.4. Phase-extraction algorithm insensitive to changes in LD power 4.5. Two-wavelength phase-shifting interferometry . . . . . . . . . § 5. Sinusoidal phase-modulating interferometry . . . . . . . . . . . . . § 6. Feedback interferometry . . . . . . . . . . . . . . . . . . . . . . . .

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ix

§ 7. Heterodyne interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Single-wavelength heterodyne interferometry . . . . . . . . . . . . . . . . 7.2. Two-wavelength heterodyne interferometry . . . . . . . . . . . . . . . . . § 8. Optical coherence function synthesized by tunable LD . . . . . . . . . . . . . . § 9. Holographic interferometer and phase-conjugate interferometer by tunable LD § 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. Optical realizations of quantum teleportation, Julio GeaBanacloche (Fayetteville, AR, USA) . . . . . . . . . . . . . . . . . . . . . .

311

§ 1. A brief primer on quantum teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Teleportation of a two-state system (qubit) . . . . . . . . . . . . . . . . . . . . . . 1.3. Generalization to an N -state system . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Continuous-variable teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Imperfect teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Possible applications and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Optical teleportation of discrete variables . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The difficulties of Bell-state measurements . . . . . . . . . . . . . . . . . . . . . . 2.2. Teleportation using entanglement between different degrees of freedom of the same photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Teleportation with less than a full Bell-state measurement (conditional teleportation) 2.4. Low-efficiency teleportation with complete Bell-state measurement . . . . . . . . . 2.5. Near-deterministic teleportation of photonic qubits using only linear optics . . . . § 3. Optical teleportation of continuous variables . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Squeezed states as EPR states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Teleportation fidelity for coherent states . . . . . . . . . . . . . . . . . . . . . . . . 3.3. The role of “classical” fields, and the nature of laser fields . . . . . . . . . . . . . . § 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5. Intensity-field correlations of non-classical light, H.J. Carmichael (Auckland, New Zealand), G.T. Foster (New York, NY, USA), L.A. Orozco (Stony Brook, NY, USA), J.E. Reiner (Stony Brook, NY, USA) and P.R. Rice (Oxford, OH, USA) . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . § 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The intensity-field correlation function hθ (τ ) . . . 2.2. Classical bounds for hθ (τ ) . . . . . . . . . . . . . 2.3. Time reversal properties of hθ (τ ) . . . . . . . . . 2.4. Intensity-field correlations in classical optics . . . § 3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Optical parametric oscillator . . . . . . . . . . . . 3.2. Cavity QED . . . . . . . . . . . . . . . . . . . . . 3.3. Two-level atom in an optical parametric oscillator

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§ 4. Experiment in cavity QED . . . . . . . . . . . . . . . . . . . 4.1. Cavity QED apparatus . . . . . . . . . . . . . . . . . . . 4.2. Conditional homodyne detector . . . . . . . . . . . . . . 4.3. Measurements . . . . . . . . . . . . . . . . . . . . . . . § 5. Equal-time cross- and auto-correlations . . . . . . . . . . . . 5.1. Cross-correlations . . . . . . . . . . . . . . . . . . . . . § 6. Quantum measurements and quantum feedback . . . . . . . . 6.1. Weak measurements . . . . . . . . . . . . . . . . . . . . 6.2. Vacuum state squeezing versus squeezed classical noise 6.3. Application of hθ (τ ) to quantum feedback . . . . . . . § 7. Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Author index for Volume 46 . . . . Subject index for Volume 46 . . . . Contents of previous volumes . . . Cumulative index – Volumes 1–46

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E. Wolf, Progress in Optics 46 © 2004 Elsevier B.V. All rights reserved

Chapter 1

Ultrafast solid-state lasers by

Ursula Keller Swiss Federal Institute of Technology (ETH), Physics Department, Zürich, Switzerland

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(03)46001-0 1

Contents

Page List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

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

7

§ 2. Definition of Q-switching and mode locking . . . . . . . . . . . . . .

10

§ 3. Overview of ultrafast solid-state lasers . . . . . . . . . . . . . . . . .

15

§ 4. Loss modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

§ 5. Pulse propagation in dispersive media . . . . . . . . . . . . . . . . . .

54

§ 6. Mode-locking techniques . . . . . . . . . . . . . . . . . . . . . . . .

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§ 7. Pulse characterization . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 8. Carrier envelope offset (CEO) . . . . . . . . . . . . . . . . . . . . . .

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§ 9. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

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

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2

List of symbols

pulse envelope normalized such that |A(z, t)|2
= P (z, t) (eq. (6.2)) A˜ is the Fourier transformation of A, i.e. A˜ = A(t)e−i
ωt dt (eqs. (6.8) and (6.9)) laser mode area on saturable absorber AA AL laser mode area in laser gain media Ap pump mode area b depth of focus or confocal parameter of a Gaussian beam D dispersion parameter (eq. (6.18) and Table 7), i.e. half of the total group delay dispersion per cavity roundtrip gain dispersion (eq. (6.10)) and Table 7) Dg Dp width of the pump source (i.e. approximately the stripe width of a diode array or bar) DR(t) impulse response of a saturable absorber mirror measured with standard pump probe (fig. 4) E electric field of the electromagnetic wave
  is the Fourier transformation of E, i.e. E  = E(t)e−iωt dt (eqs. (6.6) E E and (6.7)) intracavity pulse energy Ep Ep,out output pulse energy Esat,A absorber saturation energy (Table 3) Esat,L laser saturation energy Esat,L = Fsat,L · AL
P (t ) f (t) f (t) = Ep , with f (t) dt = 1 (eq. (4.7)) frep pulse repetition
frequency F fluence, F = I (t) dt, in units of cmJ 2 Fsat,A absorber saturation fluence (Table 3) Fsat,L laser saturation fluence (eqs. (3.1) and (3.2)) Fp,A incident pulse fluence on saturable absorber (Table 3) g saturated amplitude laser gain coefficient small signal amplitude laser gain g0

A A˜

3

4

hν I IA Isat,A k kn l

lout ls La Lg M Ms M2 2 Mslow 2 Mfast n n2 P P0 Pabs Pav,out q q0 qp qs R S t tD T

Ultrafast solid-state lasers

[1

photon energy intensity incident intensity on saturable absorber absorber saturation intensity (Table 3) vacuum wave number, i.e. k = 2π/λ wave number in a dispersive media, i.e. kn = nk total saturated amplitude loss coefficient. l includes the output coupler, all the residual cavity losses and the unsaturated loss of the saturable absorber amplitude loss coefficient of output coupler amplitude loss coefficient of soliton due to gain filtering and absorber saturation (eq. (6.53)) absorption length length of laser gain material modulation depth of loss modulator (eq. (6.11)) curvature of loss modulation (eq. (6.11) and Table 7) M 2 factor defining the laser beam quality (eq. (3.3)) M 2 factor in the “slow” axis, parallel to the pn junction of the diode laser M 2 factor in the “fast” axis, perpendicular to the pn junction of the diode laser refractive index of a dispersive media nonlinear refractive index (eq. (6.20)) power peak power of pulse absorbed pump power average output power saturable amplitude loss coefficient (i.e. nonsaturable losses not included) (eq. (4.2)) unsaturated amplitude loss coefficient or maximal saturable amplitude loss coefficient (eq. (4.1)) total absorber loss coefficient which results from the fact that part of the excitation pulse needs to be absorbed to saturate the absorber residual saturable absorber amplitude loss coefficient for a fully saturated ideal fast absorber with soliton pulses (eq. (6.46)) reflectivity for intensity Fp,A (Section 4.2.1) saturation parameter S = Fsat,A time time shift (eq. (6.37)) time that develops on a time scale of TR (eq. (6.1))

1]

List of symbols

5

Tout TR Vp W0,G W0,opt x z

intensity transmission of the laser output coupler cavity roundtrip time pump volume beam waist of a Gaussian beam (eq. (3.3)) optimized beam waist for efficient diode pumping (eq. (3.5)) chirp parameter (eqs. (6.30) and (6.31)) pulse propagation distance

z0
A
R
Rns
T
Tns
λg
νg δL φ φnl φs (z) φs φnl (z)

Rayleigh range of a Gaussian beam, i.e. z0 = λ 0 change in the pulse envelope modulation depth of a saturable absorber mirror (fig. 5) nonsaturable reflection loss of saturable absorber mirror (fig. 5) modulation depth of saturable absorber in transmission nonsaturable transmission loss of saturable absorber FWHM gain bandwidth
ν
λ FWHM gain bandwidth, i.e. ν0g = λ0g SPM coefficient (eq. (6.22) and Table 7) phase shift, φ = kn · z, z: propagation distance nonlinear phase shift per cavity roundtrip (eq. (6.33)) phase shift of the soliton during propagation along the z-axis (eq. (6.51)) phase shift of the soliton per cavity round trip (eq. (6.54)) nonlinear phase shift of a pulse with peak intensity I0 during propagation through a Kerr media along the z-axis, i.e. φnl (z) = kn2 I0 z phase shift (eq. (6.28)) absorber coefficient (eq. (4.13) and Table 7) vacuum wavelength of light center vacuum wavelength wavelength in a dispersive media with refractive index n, i.e. λn = λ/n effective wavelength (eq. (3.4)) frequency pump photon frequency radian frequency center radian frequency modulation frequency in radians/second half-width-half-maximum (HWHM) gain bandwidth of laser in radians/seconds, i.e. Ωg = π
νg (eq. (6.3)) divergence angle of a pump source (i.e. the beam radius increases approximately linearly with propagation distance, defining a cone with half-angle θ ) divergence angle of a Gaussian beam, i.e. θG = πWλ0,G (eq. (3.3))

ψ γA λ λ0 λn λeff ν νpump ω ω0 ωm Ωg θ

θG

πW 2

6

σA σL σLabs τA τAu τc τL τp τp,min

Ultrafast solid-state lasers

absorber cross section gain cross section absorption cross section recovery time of saturable absorber FWHM of intensity autocorrelation pulse photon cavity lifetime upper state lifetime of laser gain material FWHM intensity pulse duration minimal τp

[1

§ 1. Introduction Since 1990 we have observed tremendous progress in ultrashort pulse generation using solid-state lasers (fig. 1). Until the end of the 1980s, ultrashort pulse generation was dominated by dye lasers which produced pulses as short as 27 fs with a typical average output power of about 20 mW (Valdmanis, Fork and Gordon [1985]). Shorter pulse durations, down to 6 fs, were only achieved through additional amplification and fiber-grating pulse compression at much lower repetition rates (Fork, Cruz, Becker and Shank [1987]). The tremendous success of ultrashort dye lasers in the 1970s and 1980s diverted research interest away from solid-state lasers. In 1974 the first sub-picosecond passively mode-locked dye lasers (Shank and Ippen [1974], Ruddock and Bradley [1976], Diels, Stryland and Benedict [1978]) and in 1981 the first sub-100-fs colliding pulse mode-locked (CPM) dye lasers (Fork, Greene and Shank [1981]) have been demonstrated. The CPM dye laser was the “work horse” all through the 1980s for ultrafast laser spectroscopy in physics and chemistry.

Fig. 1. Development of shortest reported pulse duration over the last three decades. Circles, dye-laser technology; triangles, Ti:sapphire laser systems; solid symbols, results directly obtained from an oscillator; open symbols, results achieved with additional external pulse compression. 7

8

Ultrafast solid-state lasers

[1, § 1

Fig. 2. Different modes of operation of a laser, cw: continuous-wave. Single and multi mode refers to longitudinal modes.

The development of higher average-power diode lasers in the 1980s stimulated a strong interest in solid-state lasers again. Diode laser pumping provides dramatic improvements in efficiency, lifetime, size and other important laser characteristics. For example, actively mode-locked diode-pumped Nd:YAG and Nd:YLF lasers generated 7–12 ps pulse durations for the first time (Maker and Ferguson [1989a], Maker and Ferguson [1989b], Keller, Li, Khuri-Yakub, Bloom, Weingarten and Gerstenberger [1990], Weingarten, Shannon, Wallace and Keller [1990], Juhasz, Lai and Pessot [1990]). In comparison, flashlamp-pumped Nd:YAG and Nd:YLF lasers typically produced pulse durations of ≈ 100 ps and ≈ 30 ps, respectively. Before 1990, all attempts to passively mode-lock solidstate lasers with long upper state lifetimes (i.e. > 100 µs) resulted however in Q-switching instabilities which at best produced stable mode-locked pulses within longer Q-switched macropulses (i.e. Q-switched mode locking) (fig. 2). In Q-switched mode locking, the mode-locked pico- or femtosecond pulses are inside much longer Q-switched pulse envelopes (typically in the µs-regime) at much lower repetition rates (typically in the kHz-regime). The strong interest in an all-solid-state ultrafast laser technology was the driving force and formed the basis for many new inventions and discoveries. Today, reliable self-starting passive mode locking for all types of solid-state lasers is obtained with semiconductor saturable absorbers, first demonstrated in 1992 (Keller, Miller, Boyd, Chiu, Ferguson and Asom [1992]). Since then, more than a decade later, the performance of compact ultrafast solid-state lasers has been improved by several orders of magnitude in pulse duration, average power, pulse energy and pulse repetition rate, based on semiconductor saturable ab-

1, § 1]

Introduction

9

sorbers that are integrated into a cavity laser mirror (i.e. generally referred to as semiconductor saturable absorber mirrors – SESAMs) (Keller, Weingarten, Kärtner, Kopf, Braun, Jung, Fluck, Hönninger, Matuschek and Aus der Au [1996], Keller [1999]). Diode-pumped SESAM-mode-locked solid-state lasers offer very reliable and compact ultrafast lasers with unsurpassed performance (Paschotta and Keller [2003], Keller [2003]). Currently, pulse duration can range from picoseconds to a few femtoseconds depending on the different laser materials and saturable absorber parameters. The average power has been increased to 60 W directly from a mode-locked diode-pumped laser with pulse energies larger than 1 µJ (Innerhofer, Südmeyer, Brunner, Häring, Aschwanden, Paschotta, Hönninger, Kumkar and Keller [2003]). The pulse repetition rate has been increased to more than 100 GHz (Krainer, Paschotta, Lecomte, Moser, Weingarten and Keller [2002]). The breakthrough of femtosecond solid-state lasers came about with the discovery of the Ti:sapphire laser (Moulton [1986]), the first laser able to support sub-10-femtosecond pulses. The existing passive mode-locking techniques, primarily developed for dye lasers, were inadequate due to the much longer upper state lifetime (i.e. in the µs regime) and smaller gain cross-section (i.e. in the 10−19 cm2 regime) of Ti:sapphire compared to dyes (i.e. in the ns and 10−16 cm2 regime). Therefore, passive pulse generation techniques had to be re-evaluated with new laser material properties in mind. Kerr lens mode locking (KLM) (Spence, Kean and Sibbett [1991]) of Ti:sapphire lasers was discovered in 1991 and is still the only successful technique to push the frontier in ultrashort pulse duration into the few femtosecond regime. Today, the shortest laser pulses with less than 6 fs duration (Sutter, Steinmeyer, Gallmann, Matuschek, Morier-Genoud, Keller, Scheuer, Angelow and Tschudi [1999], Morgner, Kärtner, Cho, Chen, Haus, Fujimoto, Ippen, Scheuer, Angelow and Tschudi [1999]) are generated directly from the laser cavity without any additional external cavity pulse compression. Slightly shorter pulses have been demonstrated with external pulse compression (Baltuska, Wei, Pshenichnikov, Wiersma and Szipöcs [1997], Nisoli, Stagira, Silvestri, Svelto, Sartania, Cheng, Lenzner, Spielmann and Krausz [1997]) and continuum generation together with parametric optical amplification (Shirakawa, Sakane, Takasaka and Kobayashi [1999]). Adaptive pulse compression finally resulted in a new world record of only 3.8 fs pulse duration (Schenkel, Biegert, Keller, Vozzi, Nisoli, Sansone, Stagira, De Silvestri and Svelto [2003]). Although very different in technical detail, all of these sub-6-fs pulse generation techniques rely on the same three main components: nonlinear Kerr effect, higher-order dispersion control and ultrabroadband amplification (Steinmeyer, Sutter, Gallmann, Matuschek and Keller [1999]).

10

Ultrafast solid-state lasers

[1, § 2

KLM however has serious limitations because the mode-locking process is generally not self-starting and critical cavity alignment is required to obtain stable pulse generation. Thus, the laser cavity has to be optimized for best KLM and not necessarily for best efficiency and output power – this sets serious constraints on the cavity design, which becomes even more severe at higher average output powers and more compact monolithic cavities. Thus, passively mode-locked solid-state lasers using intracavity SESAMs have become a very attractive alternative to KLM and are more widely used today. The purpose of this chapter is to give an updated review of the progress in ultrafast solid-state lasers over the past ten years. The goal is also to give the non-expert an efficient starting position to enter into this field without providing all the detailed derivations. Relevant and useful references for further information are provided and a brief historic perspective is given throughout this paper. A basic knowledge in lasers is required. We put the emphasis on solid-state lasers because they will dominate the field in the future. More extended reviews and books summarize the dye laser era (Shank [1988], Diels [1990]). Fiber and semiconductor lasers will not be treated, but some useful references to recent review articles and book chapters will be provided.

§ 2. Definition of Q-switching and mode locking 2.1. Q-switching The history of Q-switching goes back to 1961, when Hellwarth [1961] predicted that a laser could emit short pulses if the loss of an optical resonator was rapidly switched from a high to a low value. Experimental proof was produced a year later (McClung and Hellwarth [1962], Collins and Kisliuk [1962]). The technique of Q-switching allows the generation of laser pulses of short duration (from the nanosecond to the picosecond range) and high peak power. The principle of the technique is as follows. Suppose a shutter is introduced into the laser cavity. If the shutter is closed, laser action cannot occur and the population inversion can reach a value far in excess of the threshold population that would have occurred if the shutter were not present. If the shutter is now opened suddenly, the laser will have a gain that greatly exceeds the losses, and the stored energy will be released in the form of a short and intense light pulse. Since this technique involves switching the cavity Q-factor from a low to a high value, it is known as Q-switching. Ideally, Q-switched lasers operate with only one axial mode because strong intensity noise is observed in a multi-mode Q-switched laser.

1, § 2]

Definition of Q-switching and mode locking

11

In passive Q-switching the shutter is replaced by an intracavity saturable absorber. The saturable absorber starts to bleach as the intensity inside the laser continues to grow from noise due to spontaneous emission. Thus, the laser intensity continues to increase which in turn results in stronger bleaching of the absorber, and so on. If the saturation intensity of the absorber is comparatively small compaired to the gain, the inversion still left in the laser medium after the absorber is bleached is essentially the same as the initial inversion. Therefore, after bleaching of the saturable absorber the laser will have a gain well in excess of the losses and if the gain cannot saturate fast enough, the intensity will continue to grow and stable Q-switching can occur. A large modulation depth of the saturable absorber then results in a high Q-switched pulse energy. Typically, the pulse repetition rate in Q-switched solid-state lasers is in the hertz to few megahertz regime, always much lower than the cavity round-trip frequency. Today, picosecond pulse durations can be obtained with Q-switched diodepumped microchip lasers (Zayhowski and Mooradian [1989], Zayhowski and Harrison [1997]) with pulses as short as 115 ps for active Q-switching using electrooptic light modulators (Zayhowski and Dill [1995]) and 37 ps for passive Q-switching using SESAMs (Spühler, Paschotta, Fluck, Braun, Moser, Zhang, Gini and Keller [1999]). For LIDAR applications passively Q-switched Er:Yb:glass microchip lasers around 1.5 µm are particularly interesting (Fluck, Häring, Paschotta, Gini, Melchior and Keller [1998], Häring, Paschotta, Fluck, Gini, Melchior and Keller [2001]). The performance of Q-switched microchip lasers bridges the gap between Q-switching and mode locking both in terms of pulse duration (nanoseconds to a few tens of picoseconds) and pulse repetition rates (kilohertz to a few tens of megahertz).

2.2. Mode locking Mode locking is a technique to generate ultrashort pulses from lasers. In cw mode locking the pulses are typically much shorter than the cavity round trip and the pulse repetition rate (from few tens of megahertz to a few hundreds of gigahertz) is determined by the cavity round trip time. Typically, an intracavity loss modulator (i.e. a loss modulator inside a laser cavity) is used to collect the laser light in short pulses around the minimum of the loss modulation with a period given by the cavity round trip time TR = 2L/νg , where L is the laser cavity length and νg the group velocity (i.e. the propagation velocity of the peak of the pulse intensity). Under certain conditions, the pulse repetition rate can be some integer multiple of the fundamental repetition rate (i.e. harmonic mode locking)

12

Ultrafast solid-state lasers

[1, § 2

(Becker, Kuizenga and Siegman [1972]). We distinguish between active and passive mode locking. For active mode locking, an external signal is applied to an optical loss modulator typically using the acousto-optic or electro-optic effect. Such an electronically driven loss modulation produces a sinusoidal loss modulation with a period given by the cavity round trip time TR . For passive mode locking, a saturable absorber is used to obtain a selfamplitude modulation (SAM) of the light inside the laser cavity. Such an absorber introduces some loss to the intracavity laser radiation, which is relatively large for low intensities but significantly smaller for a short pulse with high intensity. Thus, a short pulse then produces a loss modulation because the high intensity at the peak of the pulse saturates the absorber more strongly than its low intensity wings. This results in a loss modulation with a fast initial loss saturation (i.e. reduction of the loss) determined by the pulse duration and typically a somewhat slower recovery which depends on the detailed mechanism of the absorption process in the saturable absorber. A homogeneously broadened laser normally lases at one axial mode at the peak of the gain. However, the periodic loss modulation transfers additional energy phase-locked to adjacent modes separated by the modulation frequency. This modulation frequency is normally adapted to the cavity round trip frequency. The resulting frequency comb with equidistant axial modes locked together in phase forms a short pulse in the time domain. Mode locking was first demonstrated in the mid-1960s using a HeNe-laser (Hargrove, Fork and Pollack [1964]), ruby laser (Mocker and Collins [1965]) and Nd:glass laser (De Maria, Stetser and Heynau [1966]). The passively mode-locked lasers were also Q-switched, which means that the mode-locked pulse train was strongly modulated (fig. 2). This continued to be a problem for passively modelocked solid-state lasers until the first intracavity saturable absorber was designed correctly to prevent self-Q-switching instabilities in solid-state lasers with microsecond or even millisecond upper state lifetimes (Keller, Miller, Boyd, Chiu, Ferguson and Asom [1992]). Passive mode locking starts from normal noise fluctuations in a laser. One noise spike is strong enough to sufficiently start to saturate the absorber, which results in lower loss and therefore more gain. Thus, this noise spike begins to grow and becomes shorter until a stable pulse duration is obtained. For reliable passive cw mode locking, the pulse generation starts from normal laser noise within less than about 1 ms. Q-switching instabilities are a serious issue with passively mode-locked solidstate lasers. The parameters of the saturable absorber have to be chosen such that

1, § 2]

Definition of Q-switching and mode locking

13

the mode locking is self-starting (i.e. starting from the normal intensity noise of the laser) and stable, i.e. without any Q-switching instabilities. For example, if the loss modulation becomes too large it can drive the laser unstable: The loss saturation increases the intensity inside the laser cavity. The gain then needs to saturate more strongly to compensate for the reduced loss and to keep the intensity inside the laser cavity constant. If the gain cannot respond fast enough, the intensity continues to increase as the absorber is bleached which leads to selfQ-switching instabilities or in the best case to stable Q-switched mode locking. In the latter case, the mode-locked pulse train is strongly modulated at close to the relaxation oscillation frequency of the laser (typically in the kHz rate) (fig. 2). A large modulation depth of the saturable absorber results in shorter pulses but an upper limit is set by the onset of Q-switching instabilities which is a serious challenge in solid-state lasers (Hönninger, Paschotta, Morier-Genoud, Moser and Keller [1999]). Passive mode-locking mechanisms are well-explained by three fundamental models: slow saturable absorber mode locking with dynamic gain saturation (New [1972, 1974]) (fig. 3a), fast saturable absorber mode locking (Haus [1975a], Haus, Fujimoto and Ippen [1992]) (fig. 3b), and slow saturable absorber mode locking without dynamic gain saturation stable both in the picosecond domain (Paschotta and Keller [2001]) and in the femtosecond domain described by soliton mode locking (Kärtner and Keller [1995], Kärtner, Jung and Keller [1996])

Fig. 3. Passive modelocking mechanisms explained by three fundamental models: (a) slow saturable absorber modelocking with dynamic gain saturation; (b) fast saturable absorber modelocking; (c) slow saturable absorber modelocking without dynamic gain saturation, which in the femtosecond regime is described by soliton modelocking.

14

Ultrafast solid-state lasers

[1, § 2

(fig. 3c). In the first two cases, a short net-gain window forms and stabilizes an ultrashort pulse. In fig. 3a, an ultrashort net-gain window can be formed by the combined saturation of absorber and gain for which the absorber has to saturate and recover faster than the gain, while the recovery time of the saturable absorber can be much longer than the pulse duration. Dynamic gain saturation means that the gain experiences a fast, pulse-induced saturation that then recovers again between consecutive pulses (fig. 3a). For solid-state lasers we cannot apply slow saturable absorber mode locking as shown in fig. 3a, because no significant dynamic gain saturation is taking place due to the small gain cross-section and the long upper state lifetime of the laser. The upper state lifetime of solid-state lasers is typically in the µs to ms regime, much longer than the pulse repetition period, which is typically in the nanosecond regime. In addition, the gain cross-section is 1000 or even more times smaller than for dye lasers. We therefore do not observe any significant dynamic gain saturation and the gain is only saturated to a constant level by the average intracavity intensity. This is not the case for dye, semiconductor and color center lasers for which fig. 3a describes most mode-locking processes. Therefore it was assumed that without other pulse forming mechanisms (such as soliton pulse shaping) a fast saturable absorber is required for solid-state lasers. Kerr lens mode locking is an almost ideal example of fast saturable absorber mode locking. However, during the last 10 years we have seen that a slow saturable absorber can support significantly shorter pulses even though a net gain window remains open after the short pulse (fig. 3c). At first this seems surprising, because on the trailing edge of the pulse there is no shaping action of the absorber and, even worse, one would expect that the net gain after the pulse would destabilize the pulse. However, we have shown that in the femtosecond regime soliton formation is actually the dominant pulse forming mechanism and the slow saturable absorber needs only to be fast enough to stabilize this soliton – which is referred to as soliton mode locking (Kärtner and Keller [1995], Jung, Kärtner, Brovelli, Kamp and Keller [1995], Kärtner, Jung and Keller [1996]). In the picosecond regime without soliton formation, a more strongly saturated slow saturable absorber can stabilize a shorter pulse because the pulse is constantly delayed by the absorber and therefore swallows any noise growing behind it (Paschotta and Keller [2001]). This means that even with solid-state lasers we can work with relatively slow saturable absorbers that have approximately a recovery time in the range of 10 to 30 times the absorber recovery time.

1, § 3]

Overview of ultrafast solid-state lasers

15

§ 3. Overview of ultrafast solid-state lasers 3.1. Overview of solid-state laser materials Solid-state lasers can be grouped in two types: transition-metal doped Cr3+ , Cr4+ , Ti3+ , Ni2+ , Co2+ and rare-earth doped (Nd3+ , Tm3+ , Ho3+ , Er3+ , Yb3+ ) solidstate lasers (Table 1). Color center lasers have also supported ultrashort pulse durations, but they require cryogenic cooling (Mitschke and Mollenauer [1987], Islam, Sunderman, Soccolich, Bar-Joseph, Sauer, Chang and Miller [1989]). A similar wavelength range can be covered with Cr:YAG lasers, for example. Tables 1 and 2 summarize the laser parameters of these solid-state lasers and the performances that have been demonstrated with these lasers up to date. Because of their complexity we do not include results that are based on coupled-cavity modelocking techniques. For further information see the detailed summary in Table 2. The rare-earth-doped solid-state lasers have 4f-electrons responsible for the laser transition, which are shielded from the crystal host. Thus, the gain bandwidth is normally not very large and pulse durations are limited to a few 100 fs (Tables 1 and 2). However, inhomogeneously broadened rare-earth-doped lasers in glass hosts, for example, can produce significantly shorter pulses in the sub100-fs regime (Tables 1 and 2). Transition-metal-doped solid-state lasers have 3d-electrons responsible for the laser transition, which are not very well shielded from the host material. Thus, these lasers are strongly phonon broadened and can support much shorter pulses than the rare-earth-doped crystal. Presently, the shortest pulses generated from a laser are based on Ti:sapphire, a transition-metal-doped solid-state laser (Table 1). Mode-locked and Q-switched ion-doped fiber lasers also showed a lot of progress during the last ten years. More recent reviews on mode-locked fiber lasers are given in book chapters and review articles by Duling et al. (Duling [1995]), by Fermann (Fermann [1994], Fermann [1995], Fermann, Galvanauskas, Sucha and Harter [1997], Fermann [2003]), and by Haus (Haus, Tamura, Nelson and Ippen [1995], Haus [1995]). Generally, mode-locked fiber lasers generate significantly lower pulse energies and longer pulse durations than bulk lasers. However, recent progress in mode-locked fiber lasers resulted in Er/Yb-doped fiber lasers that generate 2.7 nJ pulses at 32 MHz with 100-fs pulse duration (Nelson, Fleischer, Lenz and Ippen [1996]). Much shorter pulses but also at much lower pulse energies have been obtained in Nd-doped fiber lasers with pulse durations as short as 38 fs (Hofer, Ober, Haberl and Fermann [1992]) and in erbium fiber lasers with pulses as short as 84 fs (Fermann, Andrejco, Stock, Silberberg and Weiner [1993]) and 63 fs (Tamura, Ippen and Haus [1995]). Fiber lasers require

Laser material

Ref.

λ0 [nm]

σL [10−20 cm2 ]

τL [µs]


λg [nm]

τp,min

16

Table 1 Relevant laser materials for short and ultrashort pulse generation. λ0 : center lasing wavelength. σL : gain cross-section (σem –σabs for 3 level lasers). τL : upper state lifetime.
λg : FWHM gain bandwidth. τp,min : minimal pulse duration that has been demonstrated so far (coupled cavity mode-locking schemes are not considered because of their increased complexity, for further information with regards to coupled cavity mode-locking results see Table 2) Ref.

Transition metal doped solid-state lasers (Cr3+ , Cr4+ , Ti3+ , Ni2+ , Co2+ ) [86Mou], [96Koe] [89Pay], [91Sch], [96Koe] [88Pay] [92Smi] [92Cha] [79Wal], [89Sam], [96Koe] [88Pet], [93Car], [96Koe] [88Ang], [95Küc]

790 850 758 835 ≈ 800 750 1240 1378

41 5 1.23 3.3 0.76 14.4 33

3.2 67 175 88 80 260 2.7 4.1

≈ 230 180 115 190 ≈ 155 ≈ 100 170 ≈ 250

5–6 fs 9.9 fs 9 fs 14 fs 90 fs 3 ps 14 fs 20 fs

[99Sut], [01Ell] [03Uem] [02Wag] [97Sor] [94Wan] [84Pes] [01Chu] [02Rip]

6.7 ps/6.8 ps 1.5 ps 5.7 ps 2.7 ps/2.8 ps 4.6 ps 1.9 ps 1.6 ps 150 fs

[97Hen]/[94Kel] [90Mal] [96Flu2] [00Kra]/[99Cou] [96Flu] [98Kel] [96Bra] [95Kop]

Rare earth doped solid-state lasers (Nd3+ , Tm3+ , Ho3+ , Er3+ ) Nd:YAG Nd:YLF

[64Geu], [97Zay] [69Har], [97Zay], [96Koe]

Nd:YVO4

[97Zay]

Nd:YAlO3 Nd:LSB (10 at.%) Nd:phosphate glass (LG-760) Nd:silicate glass (LG-680)

[91Kut], [94Mey] [99Sch]

1064 1047 1314 1064 1342 930 1064 1054

[99Sch]

1059.7

33 18 300 60 4.1 13 4.5 2.54

230 480 480 100 100 160 118 323

0.6 1.2

2.5 4 24.3

361

35.9

0.8

68 fs

Ultrafast solid-state lasers

Ti3+ :sapphire Cr3+ :LiSAF Cr3+ :LiCAF Cr3+ :LiSGAF Cr3+ :LiSCAF Cr3+ :Alexandrite Cr4+ :Forsterite Cr4+ :YAG

[97Aus] [1, § 3

(continued on next page)

λ0 [nm]

σL [10−20 cm2 ]

τL [µs]

[78Sch]

1054

2.6

495

26.1

60 fs

[93Fan] [97Kul1], [97Kul2], [03Kul] [97Kul1], [97Kul2], [02Puj], [00Dem] [99Mou] [02Dru]

1030 1026 1025

2.0 2.8 3.0

950 ≈ 250 ≈ 250

6.3 ≈ 25 ≈ 25

340 fs 112 fs 71 fs

[96Kel], [99Hön] [00Bru] [01Liu]

1045 1062 1025−1060 1025−1060 1025−1060 1535

0.36 0.2 0.049 0.093 0.158 0.8

2600 1100 1300 1100 1300 7900

44 60 62 77 81 55

90 fs 69 fs 58 fs 61 fs 60 fs 380 fs

[00Dru] [02Dru] [98Hön], [99Hön] [98Hön], [99Hön] [98Hön], [99Hön] [01Was]

Laser material

Ref.

Nd:fluorophosphate glass (LG-812) Yb:YAG Yb:KGW Yb:KYW

[96Kig]


λg [nm]

τp,min

Ref. [97Aus]

[64Geu] Geusic, Marcos and Uitert [1964], [69Har] Harmer, Linz and Gabbe [1969], [78Sch] Schott Glass Technologies [1978], [79Wal] Walling, Jenssen, Morris, O’Dell and Peterson [1979], [84Pes] Pestryakov, Trunov, Matrosov and Razvalyaev [1984], [86Mou] Moulton [1986], [88Ang] Angert, Borodin, Garmash, Zhitnyuk, Okhrimchuk, Siyuchenko and Shestakov [1988], [88Pay] Payne, Chase, Newkirk, Smith and Krupke [1988], [88Pet] Petricevic, Gayen and Alfano [1988], [89Pay] Payne, Chase, Smith, Kway and Newkirk [1989], [89Sam] Sam [1989], [90Mal] Malcolm, Curley and Ferguson [1990], [91Kut] Kutovoi, Laptev and Matsnev [1991], [91Sch] Scheps, Myers, Serreze, Rosenberg, Morris and Long [1991],

17

(continued on next page)

Overview of ultrafast solid-state lasers

Yb:GdCOB Yb:BOYS Yb:phosphate glass Yb:silicate glass Yb:fluoride phosphate Er,Yb:glass

1, § 3]

Table 1 (Continued)

18

Table 1 (Continued)

[1, § 3

(continued on next page)

Ultrafast solid-state lasers

[92Cha] Chai, Lefaucheur, Stalder and Bass [1992], [92Smi] Smith, Payne, Kway, Chase and Chai [1992], [93Car] Carrig and Pollock [1993], [93Fan] Fan [1993], [94Kel] Keller [1994], [94Mey] Meyn, Jensen and Huber [1994], [94Wan] Wang, Kam Wa, Lefaucheur, Chai and Miller [1994], [95Kop] Kopf, Kärtner, Keller and Weingarten [1995], [95Küc] Kück, Petermann, Pohlmann and Huber [1995], [96Bra] Braun, Hönninger, Zhang, Keller, Heine, Kellner and Huber [1996], [96Flu] Fluck, Zhang, Keller, Weingarten and Moser [1996], [96Kel] Keller, Weingarten, Kärtner, Kopf, Braun, Jung, Fluck, Hönninger, Matuschek and Aus der Au [1996], [96Kig] Kigre Inc. [1996], [96Koe] Koechner [1996], [97Aus] Aus der Au, Kopf, Morier-Genoud, Moser and Keller [1997], [97Hen] Henrich and Beigang [1997], [97Kul1] Kuleshov, Lagatsky, Podlipensky, Mikhailov and Huber [1997], [97Kul2] Kuleshov, Lagatsky, Shcherbitsky, Mikhailov, Heumann, Jensen, Diening and Huber [1997], [97Sor] Sorokina, Sorokin, Wintner, Cassanho, Jenssen and Szipöcs [1997], [97Zay] Zayhowski and Harrison [1997], [98Hön] Hönninger, Morier-Genoud, Moser, Keller, Brovelli and Harder [1998], [98Kel] Kellner, Heine, Huber, Hönninger, Braun, Morier-Genoud and Keller [1998], [99Cou] Couderc, Louradour and Barthelemy [1999], [99Hön] Hönninger, Paschotta, Graf, Morier-Genoud, Zhang, Moser, Biswal, Nees, Braun, Mourou, Johannsen, Giesen, Seeber and Keller [1999], [99Mou] Mougel, Dardenne, Aka, Kahn-Harari and Vivien [1999], [99Sch] Schott Glass Technologies [1999], [99Sut] Sutter, Steinmeyer, Gallmann, Matuschek, Morier-Genoud, Keller, Scheuer, Angelow and Tschudi [1999],

1, § 3]

Table 1 (Continued)

Overview of ultrafast solid-state lasers

[00Bru] Brunner, Spühler, Aus der Au, Krainer, Morier-Genoud, Paschotta, Lichtenstein, Weiss, Harder, Lagatsky, Abdolvand, Kuleshov and Keller [2000], [00Dem] Demidovich, Kuzmin, Ryabtsev, Danailov, Strek and Titov [2000], [00Dru] Druon, Balembois, Georges, Brun, Courjaud, Hönninger, Salin, Aron, Mougel, Aka and Vivien [2000], [00Kra] Krainer, Paschotta, Moser and Keller [2000b], [01Chu] Chudoba, Fujimoto, Ippen, Haus, Morgner, Kärtner, Scheuer, Angelow and Tschudi [2001], [01Ell] Ell, Morgner, Kärtner, Fujimoto, Ippen, Scheuer, Angelow, Tschudi, Lederer, Boiko and Luther-Davis [2001], [01Liu] Liu, Nees and Mourou [2001], [01Was] Wasik, Helbing, König, Sizmann and Leuchs [2001], [02Dru] Druon, Chenais, Raybaut, Balembois, Georges, Gaume, Aka, Viana, Mohr and Kopf [2002], [02Puj] Pujol, Bursukova, Güell, Mateos, Sole, Gavalda, Aguilo, Massons, Diaz, Klopp, Griebner and Petrov [2002], [02Rip] Ripin, Chudoba, Gopinath, Fujimoto, Ippen, Morgner, Kärtner, Scheuer, Angelow and Tschudi [2002], [02Wag] Wagenblast, Morgner, Grawert, Schibli, Kärtner, Scheuer, Angelow and Lederer [2002], [03Kul] Kuleshov, Private communication [2003], [03Uem] Uemura and Torizuka [2003]

19

Laser material

ML technique

λ0

τp

Pav,out

frep

Active-AOM CCM-APM

814 nm

150 fs 1.4 ps

600 mW 300 mW

80.5 MHz

CCM-RPM Dye sat. absorber

860 nm 750 nm

2 ps 140 fs

90 mW

250 MHz

KLM

880 nm

60 fs

300 mW

852 nm

73 fs

240 mW

Remarks

20

Table 2 CW mode-locked solid-state lasers using different mode-locking techniques. “Best” means in terms of pulse duration, highest average output power, highest pulse repetition rate, etc. – the result for which “best” applies is in bold letters. The lasers are assumed to be diode-pumped, if not stated otherwise (except Ti:sapphire laser). ML: mode locking. CCM: coupled cavity mode locking. APM: additive pulse mode locking. RPM: resonant passive mode locking. KLM: Kerr lens mode locking. SESAM: passive mode locking using semiconductor saturable absorber mirrors (SESAMs). Soliton-SESAM: Soliton mode locking with a SESAM. AOM: acousto-optic modulator. EOM: electro-optic phase modulator. λ0 : center lasing wavelength. τp : measured pulse duration. Pav,out : average output power. frep : pulse repetition rate Ref.

Ti:sapphire

47 fs 32 fs

110 mW 320 mW

134 MHz

≈ 100 MHz

KLM started with dye saturable absorber (not understood – assumed to have a CPM Ti:sapphire laser) First demonstration of KLM (but KLM not understood) First experimental evidence for KLM, self-starting due to RPM KLM, a Gaussian approximation LaFN28-glass prisms, 2 cm Ti:sapphire crystal thickness

[91Cur] [89Goo]

[90Kel2] [91Sar]

Ultrafast solid-state lasers

813 nm

Highly chirped 1.4 ps output pulses externally compressed down to 200 fs

[91Spe] [91Kel]

[91Sal] [92Riz1] [92Hua1] [1, § 3

(continued on next page)

Laser material

ML technique

λ0

τp

Pav,out

1, § 3]

Table 2 (Continued) frep

33 fs 22 fs

950 mW

100 MHz

817 nm

17 fs

500 mW

80 MHz

775 nm

12.3 fs

780 nm

11 fs

500 mW

11 fs 8.5 fs

300 mW ≈ 1 mW

100 MHz

850 nm 800 nm 800 nm 800 nm

8.2 fs 7.5 fs 6.5 fs

100 mW 150 mW 200 mW

80 MHz 86 MHz

≈ 800 nm

5.8 fs

300 mW

85 MHz

≈ 800 nm

< 6 fs

200 mW

90 MHz

Ref.

Schott F2 prisms, 8 mm Ti:Sapphire crystal thickness Schott LaK31 prisms, 2 cm Ti:Sapphire crystal thickness Schott LaKL21 prisms, 9 mm Ti:Sapphire crystal thickness Fused silica prisms, 4 mm Ti:Sapphire crystal thickness Fused silica prisms, 4.5 mm Ti:Sapphire crystal thickness Chirped mirrors, no prisms Metal mirrors and fused silica prisms Chirped mirrors only Chirped mirrors, ring cavity Fused silica prisms and double-chirped mirrors, KLM is self-starting with SESAM Fused silica prisms and double-chirped mirrors, KLM is self-starting with SESAM, pulse duration measured with SPIDER CaF2 prisms, double chirped mirrors, pulse duration estimated with fit to IAC (not very accurate)

[92Kra] [92Lem] [92Hua2] [93Cur] [93Asa] [94Sti] [94Zho] [95Sti] [96Xu] [97Jun]

[99Sut], [00Mat]

Overview of ultrafast solid-state lasers

804 nm

Remarks

[99Mor]

21

(continued on next page)

Laser material

λ0

ML technique

τp

Pav,out

frep

≈ 800 nm

≈ 5 fs

120 mW

65 MHz

780 nm

8.5 fs

1W

75 MHz

850 nm 800 nm ≈ 780 nm 782 nm

13 fs 16.5 fs 80 fs 23 fs

1.5 W 170 mW 360 mW 300 mW

110 MHz 15 MHz 4 MHz 2 GHz

≈ 800 nm

9.5 fs

180 mW

85 MHz

840 nm

34 fs

140 mW

98.9 MHz

90 fs 810 nm

13 fs

80 mW

85 MHz

800–880 nm

150 fs

50 mW

82 MHz

≈ 840 nm

50 fs

135 mW

Remarks

Ref.

CaF2 prisms, double chirped mirrors, pulse duration estimated with fit to IAC 1.5 MW peak, focused intensity 5 × 1013 W/cm2 1 MW peak, 13 nJ out 0.7 MW peak 90 nJ pulse energy Ring laser, ML is self-starting due to feedback from external mirror KLM self-starting with novel broadband fluoride SESAM First SESAM design with a single quantum well absorber in a Bragg reflector New acronym for special SESAM design: SBR Shortest pulse with soliton mode locking and no KLM

[01Ell]

[98Xu] [99Bed] [99Cho] [01Cho] [99Bar]

[02Sch] [95Bro]

Ultrafast solid-state lasers

Soliton-SESAM

22

Table 2 (Continued)

[95Tsu] [96Kär]

Cr:LiSAF KLM

[92Mil]

[92Riz2]

(continued on next page)

[1, § 3

First mode-locked femtosecond Cr:LiSAF laser, Kr-pumped Ar-ion pumped, intracavity dye absorber for starting KLM

Laser material

ML technique

λ0 880–920 nm

860 nm 880 nm 850 nm 875 nm 880 nm 850 nm ≈ 850 nm Soliton-SESAM 840 nm 850 nm 865 nm

Pav,out

50 fs

150 mW

33 fs

25 mW

93 fs

5 mW

300 ps

≈ 1 mW

220 fs 90 fs

≈ 10 mW < 20 mW

frep 85 MHz

90 MHz

97 fs 34 fs 55 fs 40 fs (24 fs) 18 fs 14.8 fs 12 fs 9.9 fs 98 fs

2.7 mW 80 MHz 42 mW 80 MHz 10 mW 70 mW (≈ 1 mW) ≈ 1 mW 70 mW 70 MHz 23 mW 200 MHz 50 mW

120 MHz

45 fs 100 fs

105 mW 11 mW

176 MHz 178 MHz

Remarks

Ref.

Ar-ion pumped, regenerative AOM for starting KLM Ar-ion pumped, intracavity dye absorber for starting KLM Ar-ion pumped, KLM started with intracavity SESAM First diode-pumped mode-locked Cr:LiSAF laser, AOM or RPM for starting KLM KLM started by SESAM Pumped by a SHG diode-pumped actively mode-locked Nd:YLF laser, prism mode locker to start KLM KLM started by regen. AOM KLM started by regen. AOM KLM self-starting

[92Eva]

Kr-ion-laser pumped ML not self-starting Strong wings in IAC First soliton mode locking, no KLM required

[92Riz3] [93Riz] [93Fre]

[94Mel] [94Lin1]

[94Dym] [95Dym] [95Fal] [95Mel] [97Dym] [97Sor] [99Uem] [03Uem] [94Kop1]

Overview of ultrafast solid-state lasers

870 nm

τp

1, § 3]

Table 2 (Continued)

[95Kop2], [97Kop1] [95Tsu] 23

(continued on next page)

Laser material

ML technique

Pav,out

24

Table 2 (Continued) τp

frep

842 nm

160 fs

25 mW

868 mW

70 fs

100 mW

860 nm 875 nm

60 fs 50 fs

125 mW 340 mW

176 MHz 150 MHz

875 nm

110 fs

500 mW

150 MHz

857 nm

146 fs

3 mW

1 GHz

800 nm 793 nm 820 nm ≈ 850 nm

170 fs 52 fs 20 fs 9 fs

100 mW 75 mW 13 mW 220 mW

90 MHz 95 MHz 95 MHz 97 MHz

35 mW

71 MHz

Remarks

Ref.

Compact cavity design with no prisms for dispersion compensation (dispersive SESAM) 0.5 W diffraction limited MOPA pump

[96Kop1]

Low-brightness 0.9-cm wide, 15 W diode laser array Low-brightness 0.9-cm wide, 15 W diode laser array

[96Tsu] [97Kop1] [97Kop2] [97Kop2] [01Kem]

Cr:LiCAF KLM

Kr-pumped

Ti:sapphire laser pumped, only fit to IAC measurement

[92LiK] [98Gab] [98Gab] [02Wag]

Ultrafast solid-state lasers

λ0

Cr:LiSGaF KLM

830 nm 835 nm 842 nm 895 nm

100 fs (50 fs) 64 fs 44 fs 14 fs

200 mW 100 mW

80 MHz 70 MHz

[95Yan] [96Sor1] [96Sor2] [97Sor]

(continued on next page)

[1, § 3

Kr-ion-laser pumped Kr-ion-laser pumped, GTI Kr-ion-laser pumped, chirped mirror

Laser material

ML technique

λ0

τp

Pav,out

1, § 3]

Table 2 (Continued) frep

Soliton-SESAM

839 nm

61 fs

78 mW

119 MHz

KLM

860 nm

90 fs

100 mW

140 MHz

Remarks

Ref. [97Loe]

Cr:LiSCAF Kr-ion-laser pumped

[94Wan]

Nd:YAG laser pumped, KLM self-starting with AOM Nd:YAG laser pumped Nd:YAG laser pumped Nd:YVO4 laser pumped, not self-starting Double chirped mirrors KLM selfstarting due to Semiconductor-doped silica films Nd:YVO4 laser pumped Nd:YVO4 laser pumped Double-clad fiber pumped Nd:YAG laser pumped

[93Sen]

Cr:Forsterite

Soliton-SESAM

1.23 µm (1.21–1.27 µm) 1.24–1.27 µm 1.27 µm 1.28 µm

48 fs

380 mW

81 MHz

50 fs 25 fs 20 fs

45 mW 300 mW

82 MHz 80 MHz

1.3 µm 1.3 µm

14 fs 25 fs

80 mW

100 MHz

1.29 µm 1.29 µm 1.27 µm 1.26 mW

40 fs 36 fs 80 fs 78 fs

60 mW 60 mW 68 mW 800 mW

90 MHz 83 MHz

1.52 µm

120 fs

360 mW

81 MHz

1.51 µm 1.55 µm

70 fs 60 fs

50 mW 50 mW

235 MHz

[93Sea] [93Yan] [97Zha2] [01Chu] [02Pra]

[97Zha1] [97Zha2] [98Liu] [98Pet]

Overview of ultrafast solid-state lasers

KLM

Cr:YAG KLM

Nd:YAG laser pumped, regen. AOM for starting KLM Nd:YAG laser pumped Nd:YAG laser pumped

[94Sen] [94Con] [94Ish] 25

(continued on next page)

Laser material

ML technique

λ0

τp

Pav,out

26

Table 2 (Continued) frep

53 fs 43 fs 75 fs

250 mW 200 mW 280 mW

1.52 µm 1.54 µm 1.45 µm Soliton-SESAM 1.541 µm 1.5 µm 1.52 µm

55 fs 115 fs 20 fs 110 fs 114 fs 200 fs

150 mW 400 mW 70 mW 94 mW 82 mW

1.46 µm

400 fs

230 mW

1.52 µm

44 fs

65 mW

Active AOM

1.064 µm

25 ps

Active FM Active EOM Active AOM

1.064 µm 1.32 µm 1.32 µm

12 ps 8 ps 53 ps

65 mW 240 mW 1.5 W

350 MHz 1 GHz 200 MHz

CCM-APM

1.064 µm 1.064 µm

1.7 ps 2 ps

25 mW 110 mW

136 MHz 125 MHz

70 MHz 1 GHz 1.2 GHz 2.64 GHz 110 MHz 185 MHz 0.9, 1.8, 2.7 GHz

152 MHz

Ref.

Nd:YAG laser pumped Nd:YVO4 laser pumped Nd:YVO4 laser or Yb:fiber laser pumped Nd:YVO4 laser pumped Nd:YVO4 laser pumped Nd:YVO4 laser pumped Nd:YVO4 laser pumped Nd:YVO4 laser pumped Nd:YVO4 laser pumped, 0.9 GHz fundamental, then double and triple harmonics Nd:YAG laser pumped, single prism for dispersion comp. Nd:YVO4 laser pumped

[96Ton] [97Ton] [98Mel]

Lamp-pumped: Pulse shortening due to intracavity etalon

[86Ros]

[00Tom] [01Tom] [02Rip] [96Col] [97Spä] [97Col]

[98Cha] [99Zha]

Ultrafast solid-state lasers

1.54 µm 1.54 µm 1.52 µm

Remarks

Nd:YAG

Lamp-pumped and harmonic mode-locked

[89Mak2] [91Zho] [88Kel]

(continued on next page)

[1, § 3

[90Goo] [91McC]

Laser material

ML technique

KLM

Polarization switching in nonlinear crystal

τp

Pav,out

frep

1.064 µm 1.32 µm 1.064 µm 1.064 µm

6 ps 10 ps 8.5 ps

2.4 W 700 mW 1W

1.064 µm 1.064 µm 1.064 µm 1.064 µm 1.064 µm 1.064 µm

6.7 ps 8.7 ps 6.8 ps 16 ps 19 ps 23 ps

675 mW 100 mW 400 mW 10.7 W 27 W 4W

106 MHz 100 MHz 217 MHz 88 MHz 55 MHz 150 MHz

1.053 µm 1.053 µm 1.047 µm 1.047 µm 1.047 µm 1.053 µm 1.053 µm 1.3 µm 1.047 µm

37 ps 18 ps 9 ps 7 ps 6.2 ps 4 ps 4.5 ps 8 ps 200 ps

6.5 W 12 mW 150 mW 135 mW 20 mW 400 mW 400 mW 240 mW 27 mW

100 MHz 230 MHz 500 MHz 2 GHz 1 GHz 237.5 MHz 2.85 GHz 1 GHz 100 MHz

100 MHz 100 MHz 100 MHz

Remarks

Ref.

Lamp-pumped Lamp-pumped

[90Liu1] [90Liu1] [92Liu] [95Chu]

Lamp-pumped, severe instabilities Ti:sapphire laser pumped

Multiple laser heads Lamp-pumped, KTP crystal

[97Hen] [93Wei] [94Kel] [99Spü] [00Spü] [99Kub]

Nd:YLF Active AOM

Active EOM

Active MQW

Lamp-pumped

Ti:sapphire laser pumped

Semiconductor multiple quantum well (MQW) modulator

[87Bad] [89Mak1] [90Kel1] [90Wei] [90Wal] [92Wei] [92Wei] [91Zho] [95Bro2]

27

(continued on next page)

Overview of ultrafast solid-state lasers

SESAM

λ0

1, § 3]

Table 2 (Continued)

Laser material

ML technique KLM

SESAM

CCM-RPM

τp

Pav,out

1.047 µm 1.047 µm

3 ps 2.3 ps

250 mW 800 mW

1.047 µm 1.047 µm 1.047 µm 1.3 µm 1.053 µm 1.053 µm 1.047 µm 1.047 µm 1.047

3.3 ps 5.1 ps 2.8 ps 5.7 ps 3.7 ps 1.7 ps 1.5 ps 4 ps 3.7 ps

700 mW 225 mW 460 mW 130 mW 7W

1.064 µm 1.064 µm 1.064 µm 1.064 µm 1.064 µm 1.064 µm 1.064 µm 1.064 µm

7 ps 8.3 ps 6.8 ps 33 ps 21 ps 4.8 ps 2.7 ps 2.7 ps 13.7 ps 4.6 ps 2.7 ps 7.9 ps

4.5 W 198 mW 81 mW 4.4 W 20 W 80 mW 65 mW 45 mW 2.1 W 50 mW 1.1 W 1.35 W

20 mW 1W 550 mW

frep 82 MHz 220 MHz 100 MHz 220 MHz 98 MHz 76 MHz 103 MHz 123 MHz 120 MHz 250 MHz

Remarks

Ref.

Lamp-pumped, microdot mirror Ti:sapphire laser pumped

Lamp-pumped

Pump laser for PPLN OPO Ti:sapphire laser pumped

[94Lin2] [93Ram] [92Kel2] [93Wei] [94Kel] [96Flu] [90Liu2] [99Jeo] [90Mal] [99Lef] [92Kel1]

Nd:YVO4 SESAM

Ti:sapphire laser pumped

Ti:sapphire laser pumped Ti:sapphire laser pumped Ti:sapphire laser pumped

[97Ruf] [99Kra1] [99Kra2] [99Gra] [00Bur] [00Kra1] [00Kra2] [02Kra2] [02Kra2] [96Flu] [02McC] [97Agn]

(continued on next page)

[1, § 3

1.3 µm CCM-APM 1.064 µm Nonlinear mirror 1.064 µm ML

84 MHz 13 GHz 29 GHz 235 and 440 MHz 90 MHz 39, 49, 59 GHz 77 GHz 157 GHz 10 GHz 93 MHz 90.7 MHz 150 MHz

Ultrafast solid-state lasers

CCM-APM

λ0

28

Table 2 (Continued)

Laser material

τp

Pav,out

Intensity-dependent 1.064 µm polarization rotation

2.8 ps

670 mW

130 MHz

SESAM

1.062 µm 1.062 µm

1.6 ps 2.8 ps

210 mW 400 mW

240 MHz 177 MHz

Active-AOM Active-FM

1.070 µm 1.070 µm 1.070 µm

7.5 ps 2.9 ps 3.9 ps

230 mW 30 mW 30 mW

250 MHz 238 MHz 20 GHz

Active-FM CCM-APM SESAM

1.067 µm 1.067 µm 1.067 µm

12 ps 2.3 ps 6.3 ps

850 mW 1W

200 MHz 76.5 MHz 64.5 MHz

Active-AOM

1.054 µm 1.054 µm 1.054 µm 1.063 µm

7 ps ≈ 10 ps 9 ps 310 fs

20 mW 30 mW 30 mW 70 mW

240 MHz 240 MHz

1.054 µm

9 ps

14 mW

235 MHz

ML technique

λ0

1, § 3]

Table 2 (Continued) frep

Remarks

Ref. [99Cou]

Nd:LSB Ti:sapphire laser pumped

Nd:BEL Harmonic mode locking Harmonic mode locking

[91Li] [91God] [91God]

Nd:KGW [95Flo] [02Maj1] [02Maj2]

Nd:glass Nd:phosphate

Nd:phosphate

Active-FM

Ar-ion laser pumped

Ti:sapphire laser pumped, regeneratively actively mode-locked

Overview of ultrafast solid-state lasers

[96Bra] [96Bra]

[86Yan] [88Bas] [92Hug] [94Kop2]

[91Hug] 29

(continued on next page)

30

Table 2 (Continued) Pav,out

CCM-APM 1.054 µm Soliton-SESAM 1.054 µm 1.054 µm 1.054 µm

122 fs 130 fs 150 fs 120 fs

200 mW 160 mW 110 mW 30 mW

180 MHz 150 MHz

1.057 µm 1.054 µm 1.065 µm 1.064 µm

175 fs 275 fs 60 fs 130 fs

1W 1.4 W 84 mW 80 mW

117 MHz 74 MHz 114 MHz 180 MHz

1.03 µm

540 fs

100 mW

81 MHz

1.03 µm 1.03 µm 1.03 µm

340 fs 2.2 ps 730 fs

170 mW 8.1 W 16.2 W

63 MHz 35 MHz

1.03 µm

3.3–89 ps 0.83–1.57 ps 810 fs

12 W

28 MHz

60 W

34.3 MHz

ML technique

Nd:phosphate Nd:phosphate

Nd:fluorophosphate Nd:silicate

λ0

frep

Remarks

Ref.

Kr-ion laser pumped Ti: sapphire laser pumped

[91Spi] [93Kel] [95Kop1] [96Kop2]

Single-prism for dispersion compensation

[98Aus] [00Pas] [97Aus] [95Kop1]

Yb:YAG Soliton-SESAM

1.03 µm

First passively mode-locked Yb:YAG laser

First passively mode-locked thin disk laser Tunable pulse duration Thin disk laser, 1.75 µJ pulse energy, 1.9 MW peak power

[95Hön] [99Hön] [99Aus] [00Aus]

Ultrafast solid-state lasers

τp

Laser material

[01Bru] [03Inn]

Yb:KGW Soliton-SESAM 1.037 µm 1.046 µm

176 fs 112 fs

1.1 W 200 mW

86 MHz 86 MHz

[00Bru] [00Bru] [1, § 3

(continued on next page)

1, § 3]

Table 2 (Continued) Laser material

ML technique

λ0

τp

Pav,out

frep

Remarks

Ref.

Thin disk laser, 0.9 µJ pulse energy, 3.3 MW peak power

[02Bru]

Yb:KYW Soliton-SESAM

1.028 µm

240 fs

22 W

25 MHz

KLM

1.057 µm

71 fs

120 mW

110 MHz

[01Liu]

Soliton-SESAM

1.045 µm

90 fs

40 mW

100 MHz

[00Dru]

Soliton-SESAM

1.062 µm

69 fs

80 mW

113 MHz

[02Dru]

58 fs 61 fs

65 mW 53 mW

112 MHz 112 MHz

[98Hön] [98Hön]

Yb:GdCOB

Yb:glass Yb:phosphate glass Yb:silicate glass

Soliton-SESAM

1.025–1.065 µm 1.03–1.082 µm

Active AOM Active FM

≈ 1.53 µm 1.53 µm ≈ 1.5 µm 1.533 µm

90 ps 9.6 ps 9.6–30 ps 48 ps

7 mW 3 mW 3 mW 1 mW

1.535 µm 1.534 µm

2.5 ps 380 fs 3.8 ps

4 mW 12 mW

Er:Yb:glass

SESAM

100 MHz 2.5 GHz 2.5 and 5 GHz 5 GHz

100 MHz 10 GHz

Dual wavelength with 165 GHz separation

[94Cer] [94Lon] [95Lap] [98Lon]

Overview of ultrafast solid-state lasers

Yb:BOYS

[99Spü2] [01Was] [02Kra1] 31

(continued on next page)

Laser material

ML technique

λ0 1.535 µm

1.528–1.561 µm

τp

Pav,out

frep

2.5 ps 2.5 ps

50 mW > 5 dBm

1.9 ps

25 mW

25 GHz

4.3 ps

18 mW

40 GHz

10 GHz 40 GHz MUX

Remarks

Ref.

Timing jitter 80 fs rms over 10 Hz and 1 MHz, 40 GHz multiplexed (MUX) Gain equalized frequency comb. with 36 discrete channels, tunable over C-band

[02Spü]

[03Spü]

[03Zel]

(continued on next page)

[1, § 3

[86Ros] Roskos, Robl and Seilmeier [1986], [86Yan] Yan, Ling, Ho and Lee [1986], [87Bad] Bado, Bouvier and Coe [1987], [88Bas] Basu and Byer [1988], [88Kel] Keller, Valdmanis, Nuss and Johnson [1988], [89Goo] Goodberlet, Wang, Fujimoto and Schulz [1989], [89Mak1] Maker and Ferguson [1989a], [89Mak2] Maker and Ferguson [1989b], [90Goo] Goodberlet, Jacobson, Fujimoto, Schulz and Fan [1990], [90Kel1] Keller, Li, Khuri-Yakub, Bloom, Weingarten and Gerstenberger [1990], [90Kel2] Keller, Knox and Roskos [1990], [90Liu1] Liu, Huxley, Ippen and Haus [1990], [90Liu2] Liu and Chee [1990], [90Mal] Malcolm, Curley and Ferguson [1990], [90Wal] Walker, Avramopoulos and Sizer II [1990], [90Wei] Weingarten, Shannon, Wallace and Keller [1990], [91Cur] Curley and Ferguson [1991], [91God] Godil, Hou, Auld and Bloom [1991], [91Hug] Hughes, Barr and Hanna [1991],

Ultrafast solid-state lasers

1.534 µm

32

Table 2 (Continued)

1, § 3]

Table 2 (Continued)

Overview of ultrafast solid-state lasers

[91Kel1] Keller, ’t Hooft, Knox and Cunningham [1991], [91Li] Li, Sheridan and Bloom [1991], [91McC] McCarthy, Maker and Hanna [1991], [91Sal] Salin, Squier and Piché [1991], [91Sar] Sarukura, Ishida and Nakano [1991], [91Spe] Spence, Kean and Sibbett [1991], [91Spi] Spielmann, Krausz, Brabec, Wintner and Schmidt [1991], [91Zho] Zhou, Malcolm and Ferguson [1991], [92Eva] Evans, Spence, Sibbett, Chai and Miller [1992], [92Hua1] Huang, Kapteyn, McIntosh and Murnane [1992], [92Hua2] Huang, Asaki, Backus, Murnane, Kapteyn and Nathel [1992], [92Hug] Hughes, Phillips, Barr and Hanna [1992], [92Kel1] Keller and Chiu [1992], [92Kel2] Keller, Miller, Boyd, Chiu, Ferguson and Asom [1992], [92Kra] Krausz, Spielmann, Brabec, Wintner and Schmidt [1992], [92Lem] Lemoff and Barty [1992], [92LiK] LiKamWa, Chai and Miller [1992], [92Liu] Liu, Flood, Walker and van Driel [1992], [92Mil] Miller, LiKamWa, Chai and Van Stryland [1992], [92Riz1] Rizvi, French and Taylor [1992a], [92Riz2] Rizvi, French and Taylor [1992b], [92Riz3] Rizvi, French and Taylor [1992c], [92Wei] Weingarten, Godil and Gifford [1992], [93Asa] Asaki, Huang, Garvey, Zhou, Kapteyn and Murnane [1993], [93Cur] Curley, Spielmann, Brabec, Krausz, Wintner and Schmidt [1993], [93Fre] French, Mellish, Taylor, Delfyett and Florez [1993], [93Kel] Keller, Chiu and Ferguson [1993], [93Ram] Ramaswamy, Gouveia-Neto, Negus, Izatt and Fujimoto [1993],

33

(continued on next page)

34

Table 2 (Continued)

Ultrafast solid-state lasers

[93Riz] Rizvi, French, Taylor, Delfyett and Florez [1993], [93Sea] Seas, Petricevic and Alfano [1993], [93Sen] Sennaroglu, Pollock and Nathel [1993], [93Wei] Weingarten, Keller, Chiu and Ferguson [1993], [93Yan] Yanovsky, Pang, Wise and Minkov [1993], [94Cer] Cerullo, De Silvestri, Laporta, Longhi, Magni, Taccheo and Svelto [1994], [94Con] Conlon, Tong, French, Taylor and Shestakov [1994], [94Dym] Dymott and Ferguson [1994], [94Ish] Ishida and Naganuma [1994], [94Kel] Keller [1994], [94Kop1] Kopf, Weingarten, Brovelli, Kamp and Keller [1994], [94Kop2] Kopf, Kärtner, Weingarten and Keller [1994], [94Lin1] Lincoln, Dymott and Ferguson [1994], [94Lin2] Lincoln and Ferguson [1994], [94Lon] Longhi, Laporta, Taccheo and Svelto [1994], [94Mel] Mellish, French, Taylor, Delfyett and Florez [1994], [94Sen] Sennaroglu, Pollock and Nathel [1994], [94Sti] Stingl, Spielmann, Krausz and Szipöcs [1994], [94Wan] Wang, Kam Wa, Lefaucheur, Chai and Miller [1994], [94Zho] Zhou, Taft, Huang, Murnane, Kapteyn and Christov [1994], [95Bro1] Brovelli, Jung, Kopf, Kamp, Moser, Kärtner and Keller [1995], [95Bro2] Brovelli, Lanker, Keller, Goossen, Walker and Cunningham [1995], [95Chu] Chung and Siegman [1995], [95Dym] Dymott and Ferguson [1995], [95Fal] Falcoz, Balembois, Georges and Brun [1995], [95Flo] Flood, Walker and van Driel [1995], [95Hön] Hönninger, Zhang, Keller and Giesen [1995], [95Kop1] Kopf, Kärtner, Keller and Weingarten [1995],

[1, § 3

(continued on next page)

1, § 3]

Table 2 (Continued)

Overview of ultrafast solid-state lasers

[95Kop2] Kopf, Weingarten, Brovelli, Kamp and Keller [1995], [95Lap] Laporta, Longhi, Marchesi, Taccheo and Svelto [1995], [95Mel] Mellish, Barry, Hyde, Jones, French, Taylor, van der Poel and Valster [1995], [95Sti] Stingl, Lenzner, Spielmann, Krausz and Szipöcs [1995], [95Tsu] Tsuda, Knox, de Souza, Jan and Cunningham [1995], [95Yan] Yanovsky, Wise, Cassanho and Jenssen [1995], [96Bra] Braun, Hönninger, Zhang, Keller, Heine, Kellner and Huber [1996], [96Col] Collings, Stark, Tsuda, Knox, Cunningham, Jan, Pathak and Bergman [1996], [96Flu] Fluck, Zhang, Keller, Weingarten and Moser [1996], [96Kär] Kärtner, Jung and Keller [1996], [96Kop1] Kopf, Zhang, Fluck, Moser and Keller [1996], [96Kop2] Kopf, Spühler, Weingarten and Keller [1996], [96Sor1] Sorokina, Sorokin, Wintner, Cassanho, Jenssen and Noginov [1996], [96Sor2] Sorokina, Sorokin, Wintner, Cassanho, Jenssen and Szipöcs [1996], [96Ton] Tong, Sutherland, French, Taylor, Shestakov and Chai [1996], [96Tsu] Tsuda, Knox and Cundiff [1996], [96Xu] Xu, Spielmann, Krausz and Szipöcs [1996], [97Agn] Agnesi, Pennacchio, Reali and Kubecek [1997], [97Aus] Aus der Au, Kopf, Morier-Genoud, Moser and Keller [1997], [97Col] Collings, Bergman and Knox [1997], [97Dym] Dymott and Ferguson [1997], [97Hen] Henrich and Beigang [1997], [97Jun] Jung, Kärtner, Matuschek, Sutter, Morier-Genoud, Zhang, Keller, Scheuer, Tilsch and Tschudi [1997], [97Kop1] Kopf, Prasad, Zhang, Moser and Keller [1997] [97Kop2] Kopf, Weingarten, Zhang, Moser, Emanuel, Beach, Skidmore and Keller [1997], [97Loe] Loesel, Horvath, Grasbon, Jost and Niemz [1997], [97Ruf] Ruffing, Nebel and Wallenstein [1997], [97Sor] Sorokina, Sorokin, Wintner, Cassanho, Jenssen and Szipöcs [1997],

35

(continued on next page)

36

Table 2 (Continued)

[1, § 3

(continued on next page)

Ultrafast solid-state lasers

[97Spä] Spälter, Böhm, Burk, Mikulla, Fluck, Jung, Zhang, Keller, Sizmann and Leuchs [1997], [97Ton] Tong, French, Taylor and Fujimoto [1997], [97Zha1] Zhang, Torizuka, Itatani, Kobayashi, Sugaya and Nakagawa [1997a], [97Zha2] Zhang, Torizuka, Itatani, Kobayashi, Sugaya and Nakagawa [1997b], [98Aus] Aus der Au, Loesel, Morier-Genoud, Moser and Keller [1998], [98Cha] Chang, Maciejko, Leonelli and Thorpe [1998], [98Gab] Gäbel, Rußbüldt, Lebert and Valster [1998], [98Hön] Hönninger, Morier-Genoud, Moser, Keller, Brovelli and Harder [1998], [98Liu] Liu, Qian, Wise, Zhang, Itatani, Sugaya, Nakagawa and Torizuka [1998], [98Lon] Longhi, Sorbello, Taccheo and Laporta [1998], [98Mel] Mellish, Chernikov, French and Taylor [1998], [98Pet] Petrov, Shcheslavskiy, Mirtchev, Noack, Itatani, Sugaya and Nakagawa [1998], [98Xu] Xu, Tempea, Spielmann, Krausz, Stingl, Ferencz and Takano [1998], [99Aus] Aus der Au, Schaer, Paschotta, Hönninger, Keller and Moser [1999], [99Bar] Bartels, Dekorsky and Kurz [1999], [99Bed] Beddard, Sibbett, Reid, Garduno-Mejia, Jamasbi and Mohebi [1999], [99Cho] Cho, Bouma, Ippen and Fujimoto [1999], [99Cou] Couderc, Louradour and Barthelemy [1999], [99Gra] Graf, Ferguson, Bente, Burns and Dawson [1999], [99Hön] Hönninger, Paschotta, Graf, Morier-Genoud, Zhang, Moser, Biswal, Nees, Braun, Mourou, Johannsen, Giesen, Seeber and Keller [1999], [99Jeo] Jeong, Kang and Nam [1999] [99Kra1] Krainer, Paschotta, Aus der Au, Hönninger, Keller, Moser, Kopf and Weingarten [1999], [99Kra2] Krainer, Paschotta, Spühler, Moser and Keller [1999], [99Kub] Kubecek, Couderc, Bourliaguet, Louradour and Barthelemy [1999], [99Lef] Lefort, Puech, Butterworth, Svirko and Hanna [1999], [99Mor] Morgner, Kärtner, Cho, Chen, Haus, Fujimoto, Ippen, Scheuer, Angelow and Tschudi [1999], [99Spü1] Spühler, Paschotta, Keller, Moser, Dymott, Kopf, Meyer, Weingarten, Kmetec, Alexander and Truong [1999], [99Spü2] Spühler, Gallmann, Fluck, Zhang, Brovelli, Harder, Laporta and Keller [1999],

1, § 3]

Table 2 (Continued)

37

(continued on next page)

Overview of ultrafast solid-state lasers

[99Sut] Sutter, Steinmeyer, Gallmann, Matuschek, Morier-Genoud, Keller, Scheuer, Angelow and Tschudi [1999], [99Uem] Uemura and Torizuka [1999], [99Zha] Zhang, Nakagawa, Torizuka, Sugaya and Kobayashi [1999], [00Aus] Aus der Au, Spühler, Südmeyer, Paschotta, Hövel, Moser, Erhard, Karszewski, Giesen and Keller [2000], [00Bru] Brunner, Spühler, Aus der Au, Krainer, Morier-Genoud, Paschotta, Lichtenstein, Weiss, Harder, Lagatsky, Abdolvand, Kuleshov and Keller [2000], [00Bur] Burns, Hetterich, Ferguson, Bente, Dawson, Davies and Bland [2000], [00Dru] Druon, Balembois, Georges, Brun, Courjaud, Hönninger, Salin, Aron, Mougel, Aka and Vivien [2000], [00Kra1] Krainer, Paschotta, Moser and Keller [2000a], [00Kra2] Krainer, Paschotta, Moser and Keller [2000b], [00Mat] Matuschek, Gallmann, Sutter, Steinmeyer and Keller [2000], [00Pas] Paschotta, Aus der Au, Spühler, Morier-Genoud, Hövel, Moser, Erhard, Karszewski, Giesen and Keller [2000], [00Spü] Spühler, Südmeyer, Paschotta, Moser, Weingarten and Keller [2000], [00Tom] Tomaru and Petek [2000], [01Bru] Brunner, Paschotta, Aus der Au, Spühler, Morier-Genoud, Hövel, Moser, Erhard, Karszewski, Giesen and Keller [2001], [01Cho] Cho, Kärter, Morgner, Ippen, Fujimoto, Cunningham and Knox [2001], [01Chu] Chudoba, Fujimoto, Ippen, Haus, Morgner, Kärtner, Scheuer, Angelow and Tschudi [2001], [01Ell] Ell, Morgner, Kärtner, Fujimoto, Ippen, Scheuer, Angelow, Tschudi, Lederer, Boiko and Luther-Davis [2001], [01Kem] Kemp, Stormont, Agate, Brown, Keller and Sibbett [2001], [01Liu] Liu, Nees and Mourou [2001], [01Tom] Tomaru [2001], [01Was] Wasik, Helbing, König, Sizmann and Leuchs [2001], [02Bru] Brunner, Südmeyer, Innerhofer, Morier-Genoud, Paschotta, Kisel, Shcherbitsky, Kuleshov, Gao, Contag, Giesen and Keller [2002], [02Dru] Druon, Chenais, Raybaut, Balembois, Georges, Gaume, Aka, Viana, Mohr and Kopf [2002], [02Kra1] Krainer, Paschotta, Spühler, Klimov, Teisset, Weingarten and Keller [2002], [02Kra2] Krainer, Paschotta, Lecomte, Moser, Weingarten and Keller [2002], [02Maj1] Major, Langford, Graf and Ferguson [2002], [02Maj2] Major, Langford, Graf, Burns and Ferguson [2002], [02McC] McConnell, Ferguson and Langfrod [2002],

38

[02Pra] [02Rip] [02Sch] [02Spü]

Prasankumar, Chudoba, Fujimoto, Mak and Ruane [2002], Ripin, Chudoba, Gopinath, Fujimoto, Ippen, Morgner, Kärtner, Scheuer, Angelow and Tschudi [2002], Schön, Haiml, Gallmann and Keller [2002], Spühler, Dymott, Klimov, Luntz, Baraldi, Kilburn, Crosby, Thomas, Zehnder, Teisset, Brownell, Weingarten, Dangel, Offrein, Bona, Buccafusca, Kaneko, Krainer, Paschotta and Keller [2002], [02Wag] Wagenblast, Morgner, Grawert, Schibli, Kärtner, Scheuer, Angelow and Lederer [2002], [03Inn] Innerhofer, Südmeyer, Brunner, Häring, Aschwanden, Paschotta, Hönninger, Kumkar and Keller [2003], [03Spü] Spühler, Golding, Krainer, Kilburn, Crosby, Brownell, Weingarten, Paschotta, Haiml, Grange and Keller [2003], [03Uem] Uemura and Torizuka [2003], [03Zel] Zeller, Krainer, Spühler, Weingarten, Paschotta and Keller [2003]

Ultrafast solid-state lasers

Table 2 (Continued)

[1, § 3

1, § 3]

Overview of ultrafast solid-state lasers

39

somewhat different saturable absorber parameters than bulk lasers. However, as has been demonstrated early on, the SESAM parameters can be adjusted for stable cw mode locking of fiber lasers (Zirngibl, Stulz, Stone, Hugi, DiGiovanni and Hansen [1991], Ober, Hofer, Keller and Chiu [1993]). Interested readers are referred to the review articles given above. Mode-locked and gain-switched semiconductor lasers compared to solid-state lasers are still low in output power (milli-watt regime) and also have significantly longer pulse durations (a few hundreds of femtosecond at best). They have, however, the advantage of being electrically pumped. Mode-locked semiconductor lasers are compact, easy to use and cost effective. They can be used in applications that do not require high powers. Higher output power can be achieved from semiconductor optical amplifiers (Delfyett, Florez, Stoffel, Gmitter, Andreadakis, Silberberg, Heritage and Alphonse [1992], Delfyett [1995]). A more extensive recent review of ultrashort pulse generation with semiconductor lasers is given in (Jiang and Bowers [1995], Vail’ev [1995], Delfyett [2003]). More recently, diode-pumped vertical external cavity semiconductor lasers have been passively mode-locked (Hoogland, Dhanjal, Roberts, Tropper, Häring, Paschotta, MorierGenoud and Keller [2000]) with an intracavity saturable absorber for the first time. Average power scaling is not limited and 950 mW of average power in 15ps pulses with a 6-GHz repetition rate (Häring, Paschotta, Aschwanden, Gini, Morier-Genoud and Keller [2002]) has been generated, with the promise of even higher powers in the multi-GHz regime.

3.2. Design guidelines of diode-pumped solid-state lasers An all-solid-state ultrafast laser technology is based on diode-pumped solid-state lasers. These lasers have to be optimized to support stable pulse generation. The discussion in the following sections will show that a small saturation energy of the laser medium results in a lower tendency of self-Q-switching instabilities. The saturation fluence of a four-level laser system is hν mσL and for a three-level system Fsat,L =

Fsat,L =

hν m(σL + σLabs )

(3.1)

,

(3.2)

where hν is the lasing photon energy, σL is the gain cross section, σLabs is the absorption cross section of the three-level gain medium and m is the number

40

Ultrafast solid-state lasers

[1, § 3

of passes through the gain medium per cavity round-trip. In case of a standing wave laser cavity this factor is m = 2, in a unidirectional ring laser cavity it is m = 1. A small saturation energy, low pump threshold and good beam quality is obtained with a small pump and cavity mode volume while maintaining good spatial overlap of the pump laser and laser mode. This can be easily obtained when a diffraction-limited pump laser is used, as, for example, in a Ti:sapphire laser. The lower limit of the pump volume is then set by diffraction, and ultimately pump-induced damage to the crystal. However, diode laser arrays or bars do not generate diffraction limited pump beams, which makes the situation a bit more complicated and is therefore explained next. The propagation of diffraction-limited Gaussian laser beams is extensively described in many text books, see for example Saleh and Teich [1991] and Siegman [1986]. A beam quality factor M 2 was introduced to describe the propagation of non-diffraction limited beams (Siegman [1986], Sasnett [1989]). The objective was to provide propagation equations for non-diffraction limited beams that retain the simplicity of the fundamental Gaussian mode beam equations. The M 2 -factor is given by M2 ≡

θ , θG

with θG ≡

λ , πW0,G

(3.3)

where θ is the actual far-field divergence angle of any beam with any mixtures of modes, θG the far-field Gaussian beam divergence angle, W0,G the beam waist of a Gaussian beam which is set equal to the beam waist of the non-diffraction limited beam. The quantity M 2 is then a numerical expression of (inverse) beam quality with 1 being a perfect Gaussian beam and higher values indicating “poorer” quality. This is entirely equivalent to the “number of times diffraction limit” quantity introduced by Siegman [1986]. The beam quality does not give any information about the details of higher-order mode content in the beam. The propagation of laser beams with M 2 larger than 1, can be reduced to standard Gaussian beam propagation after substituting the wavelength λn with a new “effective wavelength” λeff given by λeff = M 2 · λn ,

(3.4)

where λn is the wavelength in the dispersive medium (i.e. λn = λ/n) in which the beam is propagating. Physically, this means that non-diffraction-limited beams propagate like an ideal diffraction-limited Gaussian beam but with the new, longer “effective wavelength”. Beams with larger M 2 have larger “effective wavelengths”, and therefore a smaller depth of focus for a given beam waist. The output beam of a laser diode array or broad-stripe diode suffers from poor beam quality. In the so-called “fast” axis, perpendicular to the pn-junction of the

1, § 3]

Overview of ultrafast solid-state lasers

41

diode laser, the light diverges with a large angle (25 to 40◦ , typically) from a narrow aperture of ≈ 1 µm. However, in this direction the light is nearly diffraction2 ≈ 1. Thus, even though the light in the fast axis is highly dilimited with Mfast vergent, it can be efficiently collected with a “fast” high-numerical aperture lens and tightly focussed due to its diffraction-limited nature. In the “slow” axis, parallel to the pn-junction of the diode laser, the beam typically has a divergence of ≈ 10◦ . For single-stripe diodes, the emitting aperture is ≈ 3 µm, resulting in a beam close to diffraction-limited. For higher-power “arrays” of such apertures, the divergence is also ≈ 10◦ , but the total aperture has increased from typically 50 µm to more than 200 µm, or in case of “arrays of arrays” (i.e. bars) to a width of approximately 1 cm. The diode laser light in the slow axis is therefore many times worse than diffraction limited. High brightness diode arrays with 5 W out2 of ≈ 20. More recently put power and a 100 µm stripe width can have an Mslow diode arrays of 1.5 W and a 50 µm strip width can have an M 2 ≈ 12. Low bright2 ness bars with ≈ 20 W and ≈ 1 cm stripe width have an Mslow > 1000. The slow axis ultimately limits the spot size of focussed pump due to the requirements of mode matching to the laser mode. The good news is that the brightness of diode pump lasers continues to improve. With such pump lasers, the lowest pump threshold can be achieved with the following optimized mode matching (OMM) design guidelines applied to both the fast and slow axis of the diode pump laser (Fan and Sanchez [1990], Kopf, Weingarten, Zhang, Moser, Emanuel, Beach, Skidmore and Keller [1997]): (i) Determine M 2 for the pump beam (eq. (3.3)) where 2W0,G is set equal to the width of the pump source Dp . The width of the pump source is approximately given by the stripe width of a diode array or bar or more accurately by the second-order intensity moment. (ii) Determine the “effective wavelength” λeff (eq. (3.4)). (iii) Set the depth of focus or confocal parameter b of the pump beam approximately equal to the absorption length La of the pump beam in the laser medium, i.e. b ≈ La . (iv) Determine the smallest pump beam waist W0,opt for which a good mode overlap over the absorption length of the pump and the cavity mode can be obtained. This is the minimum pump spot size in the gain medium that still guarantees good laser beam quality and therefore determines the lowest pump threshold: Calculate optimum beam waist radius W0,opt using Rayleigh range formula for an ideal Gaussian beam (i.e. b = 2z0 , where z0 is the Rayleigh range of a Gaussian beam) with the “effective wavelength” given in eq. (3.3) and a confocal parameter b given in (iii):

42

Ultrafast solid-state lasers

 W0,opt =

[1, § 3

 λeff b = 2π

M 2 λn La . 2π

(3.5)

From eq. (3.5) it becomes clear that for a small spot size the absorption length La in the gain medium should be as short as possible. The absorption length, however, limits the maximum pump power at which some thermal effects will start to degrade the laser’s performance. This will be more severe for “thermally challenged” lasers which exhibit a low thermal heat conductivity and/or upper state lifetime quenching. Low thermal conductivity results in large thermal lenses and distortions, which limit the maximum pump power. Such a thermally challenged laser material is Cr:LiSAF which is interesting for an all-solid-state femtosecond laser. Upper state lifetime quenching as observed in Cr:LiSAF results in the following: As the temperature in the laser medium increases, the upper state lifetime of the laser drops, and the pump threshold increases. Beyond a critical temperature, the laser actually switches off. If the absorption length is too short for these materials, this critical temperature occurs at relatively low pump powers. There is an optimum doping level for best mode matching to the available pump diodes and for minimizing pump-induced upper-state lifetime quenching. In standard diode pumping, we use high-brightness diode arrays (i.e. brightness as high as possible) and apply OMM (eqs. (3.1)–(3.5)) only in the slow axis of the diodes and weaker focussing in the fast axis. This results in an approximately circular pump beam that becomes slightly elliptical when the laser crystal is pumped at a Brewster angle. Standard diode pumping is explained in more detail by the example of a diode-pumped Cr:LiSAF laser (Kopf, Weingarten, Brovelli, Kamp and Keller [1994]). With this standard pumping approach, the average output power was limited by the above-mentioned thermal problems to 230 mW cw and 125 mW mode-locked with 60 fs pulses (Kopf, Prasad, Zhang, Moser and Keller [1997]). Standard diode pumping has also been successfully used with most other solid-state lasers such as Nd:YAG. Such lasers are not “thermally challenged”, and much higher average output power has been achieved with this approach. Significantly more output power can be obtained with a diode-pumped Cr:LiSAF laser for which OMM (eqs. (3.1)–(3.5)) is applied to both axes in combination with a long absorption length and efficient cooling (Kopf, Keller, Emanuel, Beach and Skidmore [1997], Kopf, Aus der Au, Keller, Bona and Roentgen [1995]). Optimized mode matching in both axes results in a highly elliptical laser mode in the crystal, because the pump beam can be focused to a much smaller beam radius in the diffraction limited fast axis compared to the slow axis. Additionally, we can extract the heat very efficiently with a thin crystal of ≈ 1 mm height and obtain approximately a one-dimensional heat flow. With a cylindrical

1, § 4]

Loss modulation

43

cavity mirror we still obtained nearly ideal TEM00 output beams. Using a 15 W, 2 2 = 1 (Skidmore, = 1200 and Mfast 0.9 cm wide diode pump array with an Mslow Emanuel, Beach, Benett, Freitas, Carlson and Solarz [1995]), the average output power of such a diode-pumped Cr:LiSAF laser was 1.4 W cw, 500 mW modelocked with 110 fs pulses, and 340 mW mode-locked with 50 fs pulses (Kopf, Prasad, Zhang, Moser and Keller [1997]). Combined with a relatively long absorption length, we pumped a thin sheet volume with a width of approximately 1 mm, a length of La ≈ 4 mm and a thickness of ≈ 80 µm in the laser crystal. This approach was also applied to a diode-pumped Nd:glass laser, resulting in an average output power of about 2 W cw and 1 W mode-locked with pulses as short as 175 fs (Aus der Au, Loesel, Morier-Genoud, Moser and Keller [1998]) and more recently 1.4 W with pulses as short as 275 fs (Paschotta, Aus der Au, Spühler, Morier-Genoud, Hövel, Moser, Erhard, Karszewski, Giesen and Keller [2000]). In addition, a diode-pumped Yb:YAG laser with the same approach produced 3.5 W average power with 1-ps pulses and 8.1 W with 2.2-ps pulses (Aus der Au, Schaer, Paschotta, Hönninger, Keller and Moser [1999]).

§ 4. Loss modulation 4.1. Optical modulators: acousto-optic and electrooptic modulators Many textbooks, for example, Siegman [1986], Yariv and Yeh [1984], Svelto [1998], have reviewed optical modulators for pulse generation. Today the most important optical modulators for short pulse generation are the acousto-optic and electrooptic modulators. The acousto-optic modulators have the advantage of low optical insertion loss and can readily be driven at high repetition rates. They are typically used for cw mode locking. However, for Q-switching their loss modulation is limited and the switching time is rather slow. Therefore, acousto-optic modulators are primarily used for repetitive Q-switching of cw-pumped lasers (e.g., Nd:YAG) and electrooptic modulators are used for Q-switching in general. For mode locking, the acousto-optic modulator typically consists of an acoustooptic substrate (typically fused quartz) and a transducer that launches an acoustic wave into the substrate. An acoustic resonator is formed when opposite to the transducer the crystal substrate is air backed. Then the acoustic wave is reflected and an acoustic standing wave is formed which produces a light modulator at twice the microwave drive frequency. At higher frequencies (a few hundreds of megahertz to a few gigahertz) the loss modulation is strongly reduced by the acoustic attenuation in the substrate. Thus, at higher modulation frequencies a

44

Ultrafast solid-state lasers

[1, § 4

sapphire (Keller, Li, Khuri-Yakub, Bloom, Weingarten and Gerstenberger [1990], Weingarten, Shannon, Wallace and Keller [1990]), or a GaP (Walker, Avramopoulos and Sizer II [1990]) substrate has been used successfully.

4.2. Saturable absorber: self-amplitude modulation (SAM) Saturable absorbers have been used to passively Q-switch and mode-lock many lasers. Different saturable absorbers, such as organic dyes, color filter glasses, dye-doped solids, ion-doped crystals and semiconductors have been used. Independent of the specific saturable absorber material, we can define a few macroscopic absorber parameters that will determine the pulse generation process. The relevant macroscopic properties of a saturable absorber are the modulation depth, the nonsaturable loss, the saturation fluence, the saturation intensity and the impulse response or recovery times (Table 3). These parameters determine the operation of a passively mode-locked or Q-switched laser. In our notation we assume that the saturable absorber is integrated within a mirror structure, thus, we are interested in the nonlinear reflectivity change R(t) as a function of time and R(Fp,A ) as a function of the incident pulse energy fluence on the saturable absorber. If the saturable absorber is used in transmission, we simply characterize the absorber by nonlinear transmission measurements, i.e. T (t) and T (Fp,A ). Both the saturation fluence Fsat,A and the absorber recovery time τA are determined experimentally without the need to determine the microscopic properties of the nonlinearities. The saturation fluence of the absorber does not only depend on material properties but also on the specific device structure in which the absorber is integrated, which gives significantly more design freedom. Table 3 Saturable absorber quantities, their defining equations and units Quantity

Symbol

Defining equation or measurement

Unit

Saturation fluence Recovery time Incident beam area Saturation energy Saturation intensity Modulation depth Nonsaturable loss Incident pulse energy Incident pulse fluence Incident intensity

Fsat,A τA AA Esat,A Isat,A
R or
T
Rns or
Tns Ep Fp,A IA (t)

Measurement R(Fp,A) or T (Fp,A ) (fig. 5) Measurement R(t) or T (t) (fig. 4) Measurement Esat,A = AA Fsat,A Isat,A = Fsat,A /τA Measurement R(Fp,A) or T (Fp,A ) (fig. 5) Measurement R(Fp,A) or T (Fp,A ) (fig. 5) Measurement Fp,A = Ep /AA
Fp,A = IA (t) dt

J/cm2 s cm2 J W/cm2

J J/cm2 W/cm2

1, § 4]

Loss modulation

45

Fig. 4. Typical measured impulse response of a SESAM measured with standard degenerate pump-probe measurements using two different excitation pulse durations. The saturable absorber was grown at low temperature, which reduced the recovery time to about 20 ps. The short intraband thermalization recovery time results in negligible modulation depth with a 4-ps excitation pulse. Thus, only the slower recovery time due to carrier trapping is important in the picosecond regime.

Standard pump-probe techniques determine the impulse response R(t) and therefore τA . In the picosecond regime we normally only have to consider one picosecond recovery time, because much faster femtosecond nonlinearities in the saturable absorber give negligible modulation depth. This is shown with a semiconductor saturable absorber in fig. 4, where the differential impulse response DR(t) was measured for two different excitation pulse durations. For excitation with a picosecond pulse the pump-probe trace clearly shows no significant modulation depth with a fast time constant. In the femtosecond pulse regime we normally have to consider more than one absorber recovery time. In this case the slow component normally helps to start the initial pulse formation process. The modulation depth of the fast component then determines the pulse duration at steady state. Further improvements of the saturable absorber require a better understanding of the underlying physics of the nonlinearities which is very interesting and rather complex. A more detailed review about the microscopic properties of ultrafast semiconductor nonlinearities for saturable absorber applications is provided by Keller [1999] and Siegner and Keller [2000]. Ultrafast semiconductor dynamics in general are discussed in more detail by Shah [1996]. However, a basic knowledge of the macroscopic properties of the absorber and how set them to a certain value is normally sufficient for stable pulse generation. The saturation fluence Fsat,A is determined and defined by measuring the nonlinear change in reflectivity R(Fp,A ) as a function of increased incident pulse fluence (fig. 5). The common travelling wave rate equations (Agrawal and Olsson [1989]) in the slow absorber approximation usually give a very good fit and deter-

46

Ultrafast solid-state lasers

[1, § 4

Fig. 5. Measured nonlinear reflectivity as a function of incident pulse fluence on a SESAM. Theoretical fit determines the macroscopic saturable absorber parameters: saturation fluence Fsat,A , modulation depth
R and nonsaturable loss
Rns .

mine the saturation fluence Fsat,A , the modulation depth
R and the nonsaturable losses
Rns of the absorber (Brovelli, Keller and Chiu [1995], Keller [1999]). The modulation depth is typically small to prevent Q-switching instabilities in passively mode-locked solid-state lasers (Hönninger, Paschotta, Morier-Genoud, Moser and Keller [1999]). Thus it is reasonable to make the following approximation:
R = 1 − e−2q0 ≈ 2q0 ,

q0  1,

(4.1)

where q0 is the unsaturated amplitude loss coefficient. The saturation of an absorber can be described with the following differential equation (Agrawal and Olsson [1989]): q(t) − q0 q(t)P (t) dq(t) =− − , dt τA Esat,A

(4.2)

where q(t) is the saturable amplitude loss coefficient that does not include any nonsaturable losses. Note that eq. (4.2) may not be precise for strongly saturated real absorbers. For example, strongly saturated semiconductor absorbers can exhibit additional effects such as two-photon absorptions, free carrier absorption, thermal and even various damage effects. Two-photon-absorption only starts to become significant in the femtosecond pulse-width regime and results in an earlier roll-off of the nonlinear reflectivity at high incident pulse fluences (Thoen, Koontz. M. Joschko, Langlois, Schibli, Kärtner, Ippen and Kolodziejski [1999]). This well-known effect has been used in power-limiting devices before (Walker, Kar, Ji, Keller and Smith [1986]). In this regime, however, the absorber is operated more closely to the damage threshold which needs to be evaluated separately. Our

1, § 4]

Loss modulation

47

experience is that damage in semiconductor saturable absorbers typically occurs at around 100 times the saturation fluence of the absorber with long-term degradation observed close below this damage threshold. Therefore, we normally operate the device at least an order of magnitude below this damage threshold, ideally at an incident pulse fluence of 3 to 5 times the saturation fluence of the absorber. We therefore neglect these very high-fluence effects in the following discussion. At any time t the reflected (or transmitted) intensity Iout (t) from the saturable absorber is given by Iout (t) = R(t)Iin (t) = e−2q(t )Iin (t). Then the total net reflectivity is given by
 Iout (t) dt 2 Fout =1− = q(t)Iin (t) dt. Rtot =
Fin Fin Iin (t) dt

(4.3)

(4.4)

This determines the total absorber loss coefficient qp , which results from the fact that part of the excitation pulse needs to be absorbed to saturate the absorber: Rtot = e−2qp ≈ 1 − 2qp . From eqs. (4.4) and (4.5) it then follows   1 qp = q(t)Iin (t) dt = q(t)f (t) dt, Fin where   1 Iin (t) Pin (t) = , with f (t) dt = f (t) ≡ Iin (t) dt = 1. Fin Ep,in Fin

(4.5)

(4.6)

(4.7)

We then distinguish between two typical cases: a slow and a fast saturable absorber. 4.2.1. Slow saturable absorber In the case of a slow saturable absorber, we assume that the excitation pulse duration is much shorter than the recovery time of the absorber (i.e. τp  τA ). Thus, we can neglect the recovery of the absorber during pulse excitation and eq. (4.2) reduces to: q(t)P (t) dq(t) ≈− . (4.8) dt Esat,A This differential equation can be solved and we obtain for the self-amplitude modulation (SAM):    t Ep f (t  ) dt  . q(t) = q0 exp − (4.9) Esat,A 0

48

Ultrafast solid-state lasers

[1, § 4

Equation (4.6) then determines the total absorber loss coefficient for a given incident pulse fluence Fp,A :  Fsat,A  1 − e−Fp,A /Fsat,A . qp (Fp,A ) = q(t)f (t) dt = q0 (4.10) Fp,A It is not surprising that qp does not depend on any specific pulse form because τp  τA . It is useful to introduce a saturation parameter S ≡ Fp,A /Fsat,A . For strong saturation (S > 3), we have qp (Fp,A ) ≈ q0 /S (eq. (4.10)) and the absorbed pulse fluence is about ≈ Fsat,A
R. 4.2.2. Fast saturable absorber In the case of a fast saturable absorber, the absorber recovery time is much faster than the pulse duration (i.e. τp  τA ). Thus, we can assume that the absorption instantaneously follows the absorption of a certain power P (t) and eq. (4.2) reduces to 0=−

q(t) − q0 q(t)P (t) − . τA Esat,A

(4.11)

The saturation of the fast absorber then follows directly from eq. (4.11): q(t) =

q0 1+

IA (t ) Isat,A

,

(4.12)

where we used the fact that Psat,A = Esat,A /τA and P (t)/Psat,A = IA (t)/Isat,A . In the linear regime we can make the following approximation in eq. (4.12): q(t) ≈ q0 − γA P (t),

with γA ≡

q0 . Isat,A AA

(4.13)

The total absorber loss coefficient qp (eqs. (4.5)–(4.7)) now depends on the pulse form and for a sech2 -pulse shape we obtain for an incident pulse fluence Fp,A and the linear approximation of q(t) for weak absorber saturation (eq. (4.13)):

  1 1 Fp,A q(t)IA (t) dt = q0 1 − . qp (Fp,A ) = (4.14) Fp,A 3 τ Isat,A We will later see that we only obtain an analytic solution for fast saturable absorber mode locking, if we assume an ideal fast absorber that saturates linearly with pulse intensity (eq. (4.13)) – which in principle only applies for weak absorber saturation in a real absorber. For a maximum modulation depth, we then can assume that q0 = γA P0 (assuming an ideal fast absorber over the full modulation depth). We then obtain with eq. (4.14) a residual saturable absorber loss of q0 /3 suffered by the pulse to fully saturate the ideal fast saturable absorber.

1, § 4]

Loss modulation

49

4.3. Semiconductor saturable absorbers Semiconductor saturable absorbers have been used as early as 1974 in CO2 lasers (Gibson, Kimmitt and Norris [1974]) and 1980 for semiconductor diode lasers (Ippen, Eichenberger and Dixon [1980]). A color center laser was the first solidstate laser that was cw mode-locked with an intracavity semiconductor saturable absorber (Islam, Sunderman, Soccolich, Bar-Joseph, Sauer, Chang and Miller [1989]). However, for both the diode and color center laser, dynamic gain saturation supported pulse formation and the recovery time of the slow saturable absorber was not relevant for pulse generation (fig. 3a). In addition, because of the large gain cross section (i.e. approximately 10−14 cm2 for diode lasers and 10−16 cm2 for color center lasers) Q-switching instabilities were not a problem. This is not the case for most other solid-state lasers, such as ion-doped solid-state lasers, which have typically 1000 or even more times smaller gain cross sections. Thus, the semiconductor saturable absorber parameters (figs. 4 and 5) have to be chosen much more carefully for stable cw mode locking. We typically integrate the semiconductor saturable absorber into a mirror structure, which results in a device whose reflectivity increases as the incident optical intensity increases. This general class of devices is called semiconductor saturable absorber mirrors (SESAMs) (Keller, Miller, Boyd, Chiu, Ferguson and Asom [1992], Keller, Weingarten, Kärtner, Kopf, Braun, Jung, Fluck, Hönninger, Matuschek and Aus der Au [1996]). Detailed descriptions and guidelines for designing a SESAM for either passive mode locking or Q-switching for different laser parameters have been given by Keller [1999] and Paschotta and Keller [2003]. SESAMs are well-established as useful devices for passive mode locking and Q-switching in many kinds of solid-state lasers. The main reason for its utility is that both the linear and nonlinear optical properties can be engineered over a wide range, allowing for more freedom in the specific laser cavity design. In addition, semiconductor saturable absorbers are ideally suited for passive modelocking solid-state lasers because the large absorber cross section (in the range of 10−14 cm2 ) and therefore small saturation fluence is ideally suited for suppressing Q-switching instabilities (Hönninger, Paschotta, Morier-Genoud, Moser and Keller [1999]). Initially, semiconductor saturable absorber mirrors for solid-state lasers were used in coupled cavities (Keller, Knox and Roskos [1990], Keller and Chiu [1992]), because these early SESAM designs introduced too much loss for the laser cavity (fig. 6a). In 1992, this work resulted in a new type of intracavity saturable absorber mirror, the antiresonant Fabry–Perot saturable absorber (A-FPSA) (Keller, Miller, Boyd, Chiu, Ferguson and Asom [1992], Keller

50

Ultrafast solid-state lasers

[1, § 4

Fig. 6. Historical evolution of different SESAM designs: (a) Initially the semiconductor saturable absorber was used inside a nonlinear coupled cavity, termed resonant passive modelocking (RPM) (Keller, Knox and Roskos [1990]). (b) First intracavity saturable absorber to passively modelock diode-pumped solid-state lasers without Q-switching instabilities: antiresonant Fabry–Perot saturable absorber (A-FPSA) (Keller, Miller, Boyd, Chiu, Ferguson and Asom [1992]). (c) Scaling of the A-FPSA resulted in a single quantum well saturable absorber integrated into a Bragg mirror (Brovelli, Jung, Kopf, Kamp, Moser, Kärtner and Keller [1995]) – later also referred to as saturable Bragg reflector (SBR) (Tsuda, Knox, de Souza, Jan and Cunningham [1995]). (d) General concept of semiconductor saturable absorber mirror (SESAM) without any restrictions on the mirror design (Keller and Kopf [1995], Keller, Weingarten, Kärtner, Kopf, Braun, Jung, Fluck, Hönninger, Matuschek and Aus der Au [1996]).

[1994]), where the absorber was integrated inside a Fabry–Perot structure of which the bottom reflector was a high reflector (i.e. approximately 100%) (fig. 6b). This was the first intracavity saturable absorber design that allowed for passive mode-locking of diode-pumped solid-state lasers without Q-switching instabilities. The Fabry–Perot was operated at antiresonance to obtain broad bandwidth and low loss. The A-FPSA mirror was mainly based on absorber layers sandwiched between the lower semiconductor and the higher SiO2 /TiO2 dielectric Bragg mirrors. The top reflector of the A-FPSA provides an adjustable parameter that determines the intensity entering the semiconductor saturable absorber and therefore the saturation fluence of the saturable absorber device. Therefore, this design allowed for a large variation of absorber parameters by simply changing absorber thickness and top reflectors (Brovelli, Keller and Chiu [1995], Jung, Brovelli, Kamp, Keller and Moser [1995]). This resulted in an even simpler SESAM design with a single quantum well absorber layer integrated into a Bragg mirror (Brovelli, Jung, Kopf, Kamp, Moser, Kärtner and Keller

1, § 4]

Loss modulation

51

[1995], Tsuda, Knox, de Souza, Jan and Cunningham [1995]) (fig. 6c) – this was also referred to as saturable Bragg reflectors (SBRs) (Tsuda, Knox, de Souza, Jan and Cunningham [1995]). In the 10-femtosecond regime with Ti:sapphire lasers we have typically replaced the lower semiconductor Bragg mirror with a metal mirror to support the required large reflection bandwidth (Fluck, Jung, Zhang, Kärtner and Keller [1996], Jung, Kärtner, Matuschek, Sutter, MorierGenoud, Shi, Scheuer, Tilsch, Tschudi and Keller [1997]). However, more recently an ultrabroadband monolithically-grown fluoride semiconductor saturable absorber mirror was demonstrated that covers nearly the entire gain spectrum of the Ti:sapphire laser. Using this SESAM inside a Ti:sapphire laser resulted in 9.5fs pulses (Schön, Haiml, Gallmann and Keller [2002]). The reflection bandwidth was achieved with a AlGaAs/CaF2 semiconductor Bragg mirror (Schön, Haiml and Keller [2000]). Keller and Kopf [1995] further realized that the intracavity saturable absorber can be integrated in a more general mirror structure that allows for both saturable absorption and negative dispersion control, which is now generally referred to as a semiconductor saturable absorber mirror (SESAM) (fig. 6d). In a general sense we can then reduce the design problem of a SESAM to the analysis of multilayered interference filters for a given desired nonlinear reflectivity response for both the amplitude and phase. The A-FPSA (Keller, Miller, Boyd, Chiu, Ferguson and Asom [1992]), the saturable Bragg reflector (SBR) (Brovelli, Jung, Kopf, Kamp, Moser, Kärtner and Keller [1995], Tsuda, Knox, de Souza, Jan and Cunningham [1995], Knox [1995]), and the dispersive saturable absorber mirror (DSAM) (Kopf, Zhang, Fluck, Moser and Keller [1996]) are examples of SESAM designs. In this more general class of design we do not restrict ourselves to Bragg mirror structures, which are defined by a stack of quarter-wave layers with alternating high and low refractive indices (e.g., Knox [1995], Kim, Garmire, Hummel and Dapkus [1989]). For example, we have demonstrated with many examples that non-quarter-wave layers in mirrors give more design freedom for integrating the absorber layers into the mirror structure. Furthermore, double-chirped semiconductor mirror structures can provide very broadband negative dispersion (Paschotta, Spühler, Sutter, Matuschek, Keller, Moser, Hövel, Scheuer, Angelow and Tschudi [1999]). In addition, low-field enhancement (LFR) SESAM designs (Weingarten, Spühler, Keller and Krainer [2001]) can be used to further reduce saturation fluence without the detrimental effects of strongly resonant structures such as bistability and narrow bandwidth. Such a LFR-SESAM design has a lowfinesse resonant structure such that the field strength is substantially higher in the spacer layer containing the absorber and therefore reducing the saturation fluence further. Thus, different absorber materials and parameters have been used

52

Ultrafast solid-state lasers

[1, § 4

in many passively mode-locked solid-state lasers which resulted in a much better understanding of the absorber and laser design necessary to obtain stable passive mode locking (Hönninger, Paschotta, Morier-Genoud, Moser and Keller [1999]) or Q-switching (Spühler, Paschotta, Fluck, Braun, Moser, Zhang, Gini and Keller [1999]) of many different solid-state lasers. With a few exceptions (Braun and Keller [1995]), the SESAM is mostly used as an end mirror of a standing-wave cavity. Very compact cavity designs have been achieved, for example, in passively Q-switched microchip lasers (Braun, Kärtner, Keller, Meyn and Huber [1996], Spühler, Paschotta, Fluck, Braun, Moser, Zhang, Gini and Keller [1999], Häring, Paschotta, Fluck, Gini, Melchior and Keller [2001], Häring, Paschotta, Gini, Morier-Genoud, Melchior, Martin and Keller [2001]) and passively mode-locked miniature lasers (Krainer, Paschotta, Spühler, Moser and Keller [1999], Krainer, Paschotta, Moser and Keller [2000b]), where a short laser crystal defines a simple monolithic cavity. The SESAM attached directly to the laser crystal then formed one end-mirror of this laser cavity. As the laser can not be pumped through the SESAM, the laser output needs to be separated from the collinear pump by a dichroic mirror. These examples suggest that there is need for a device which combines the nonlinear properties of the SESAM with an output coupler. This has been demonstrated before for a passively mode-locked fiber laser (Sharp, Spock, Pan and Elliot [1996]) and more recently for solid-state lasers (Spühler, Reffert, Haiml, Moser and Keller [2001]). Semiconductor-doped dielectric films have been demonstrated for saturable absorber applications (Bilinsky, Fujimoto, Walpole and Missaggia [1999]). In this case, InAs-doped thin-film rf-sputtering technology was used which offers similar advantages as SESAMs, i.e. the integration of the absorber into a mirror structure. However, the saturation fluence of ≈ 10 mJ/cm2 is rather high for stable solidstate laser mode locking. In comparison, epitaxially grown SESAMs typically have a saturation fluence in the range of 10 µJ/cm2 depending on the specific device structure.

4.4. Effective saturable absorbers using the Kerr effect The extremely rapid response and the broad bandwidth of the Kerr nonlinearity are very attractive for a mode-locking process. For high intensities, the polarization inside a dielectric medium does not proportionally follow the electric field anymore. This gives rise to an index change proportional to intensity. Offresonance, this nonlinear optical effect is extremely fast, with estimated response times in the few-femtosecond range. The transverse and longitudinal effects resulting from the intensity dependence are shown schematically in fig. 7. The

1, § 4]

Loss modulation

53

Fig. 7. The Kerr effect gives rise to an increase of the refractive index with intensity, causing a retardation of the most intense parts of a pulse (i.e. for n2 > 0). In its longitudinal form, the Kerr effect causes self-phase modulation and in its transverse form, a nonlinear lens is formed in the central part of the beam profiles (i.e. Kerr lens).

transverse Kerr effect retards the central and most intense part of a plane wavefront and thus acts as a focusing lens, referred to as the Kerr lens. Along the axis of propagation, the longitudinal Kerr effect retards the center of an optical pulse, producing a red shift of the leading part of the pulse, and a blue shift in the trailing part. Consequently, the longitudinal Kerr effect has been named self-phase modulation (SPM). The longitudinal Kerr effect can also be used to produce the same effect as a fast saturable absorber. To do this, the phase nonlinearity provided by the longitudinal Kerr effect has to be converted into an effective amplitude nonlinearity. The earliest mode-locking schemes based only on SPM used a coupled cavity to convert SPM into SAM. In the soliton laser (Mollenauer and Stolen [1984]), pulses compressed by SPM and anomalous dispersion in the coupled cavity are directly coupled back into the main laser cavity. This provides more gain for the center of the pulse. Pulses as short as 19 fs have been demonstrated with color center lasers (Mitschke and Mollenauer [1987]). Later, the SPM-to-SAM conversion with a coupled cavity was demonstrated even when the pulses inside the coupled cavity were broadened due to positive group velocity dispersion (Kean, Zhu, Crust, Grant, Landford and Sibbett [1989]). In this case, no compressed pulse was fed back into the main cavity. An effective SAM was obtained because SPM inside the coupled cavity generates a phase modulation on the pulse that adds constructively at the peak of the pulse in the main cavity and destructively in the wings, thus shortening the pulse duration inside the main cavity. This was also later referred to as additive pulse mode locking (APM) (Ippen, Haus and Liu [1989], Haus, Fujimoto and Ippen [1991]). Although very powerful in principle, these coupled-cavity schemes have the severe disadvantage that the auxiliary cavity has

54

Ultrafast solid-state lasers

[1, § 5

to be stabilized interferometrically. An alternative method for converting the reactive Kerr nonlinearity into an effective saturable absorber was discovered in 1991: Kerr-lens mode locking (KLM) (Spence, Kean and Sibbett [1991]). The discovery of Kerr lens mode locking has been a breakthrough in ultrashort pulse generation (Spence, Kean and Sibbett [1991]). Initially, the mode-locking mechanism was not understood and was something of a mystery. But a short time after the initial discovery it became clear that the transverse Kerr effect provides a fast saturable absorber. In KLM, the transverse Kerr effect produces a Kerr lens (fig. 7) that focuses the high intensity part of the beam more strongly than the low intensity part. Thus, combined with an intracavity aperture, the Kerr lens produces less loss for high intensity and forms an effective fast saturable absorber (Keller, ’t Hooft, Knox and Cunningham [1991], Salin, Squier and Piché [1991], Negus, Spinelli, Goldblatt and Feugnet [1991]). A similar mode-locking effect can be obtained without a hard aperture when the Kerr lens produces an increased overlap of the laser mode with the pump profile in the gain medium (Piché and Salin [1993]). The Kerr lens provides the strongest advantage for the pulsed operation when the cavity is operated close to the stability limit. Optimization guidelines for SAM produced by the Kerr lens in different cavities can be found in (Magni, Cerullo, de Silvestri and Monguzzi [1995]). Unfortunately, the transverse Kerr effect couples the mode-locking process with the laser cavity mode. In contrast, the use of only the longitudinal Kerr effect in mode locking totally decouples the mode-locking process from the laser mode. This allows optimum cavity design for scaling the laser to higher powers and to higher pulse repetition rates without being constrained by the Kerr lens. In fiber lasers a different Kerr-effect-based effective saturable absorber has been used to generate pulses as short as 38 fs (Hofer, Ober, Haberl and Fermann [1992]) – the shortest pulses generated directly from a fiber laser so far. An effective fast saturable absorber is obtained with a Kerr-induced nonlinear polarization rotation in a weakly birefringent fiber combined with a polarization dependent loss. Previously, a similar idea has been used to “clean up” high intensity pulses by reducing the low-intensity pulse pedestals (Tapié and Mourou [1992], Beaudoin, Chien, Coe, Tapié and Mourou [1992]).

§ 5. Pulse propagation in dispersive media 5.1. Dispersive pulse broadening Dispersion compensation is important in ultrashort pulse generation. The laser material and other elements inside the laser cavity normally produce positive dis-

1, § 5]

Pulse propagation in dispersive media

55

persion which broadens a transform-limited pulse. Dispersion is a linear effect and therefore is additive and can be treated in the spectral domain. A very good summary of linear pulse propagation in dispersive media is given in Chapter 9 of (Siegman [1986]). Assuming a Gaussian pulse shape, an analytical solution can be determined for the pulse broadening due to second-order dispersion (Siegman [1986]):

 τp (z) 4 ln 2d2 φ/dω2 2 = 1 + , (5.1) τp (0) τp2 (0) where it is assumed that the incident pulse is transform limited, i.e. the timebandwidth product of the Gaussian pulse is τp (0)
νp = 0.4413, where
νp is the FWHM (full width half maximum) spectral width of the pulse intensity. Only second-order dispersion (i.e. d2 φ/dω2 , Table 5) and higher orders are broadening the pulse. The first-order dispersion gives the group delay, i.e. the delay of the peak of the pulse envelope (Table 5). It is important to note that in the linear pulse propagation regime the spectrum of the pulse remains unchanged, only the spectral content of the pulse is redistributed in time. With positive dispersion the long wavelength part of the spectrum is in the leading edge of the pulse and the short wavelength part in the trailing edge of the pulse, i.e. “red is faster than blue” d2 φ 2 (fig. 8). In the regime of strong pulse broadening, i.e. dω 2  τp (0), we can reduce eq. (5.1) to d2 φ (5.2)
ωp , dω2 where
ωp = 2π
νp is the FWHM spectral width (in radians/second) of the pulse intensity. τp (z) ≈

Fig. 8. Dispersive pulse broadening through a material with positive dispersion for the pulse envelope A(t) and/or intensity I (t) and the electric field E(t).

56

Ultrafast solid-state lasers

[1, § 5

Table 4a Sellmeier equations for different materials. The wavelength λ is given in units of µm Material

Defining Sellmeier equation Aλ2 λ2 −λ21

Fused quartz

n2 = 1 +

+

SF10 glass

n2 = a0 + a1 λ2 +

Sapphire

n2 = 1 +

Bλ2 λ2 −λ22

+

Constants Cλ2 λ2 −λ23

a2 a a a + 34 + 46 + 58 λ2 λ λ λ

a λ2 a1 λ2 a λ2 + 22 + 23 λ2 −b1 λ −b2 λ −b3

A = 0.6961663 λ1 = 0.0684043 B = 0.4079426 λ2 = 0.1162414 C = 0.8974794 λ3 = 9.896161 a0 = 2.8784725 a1 = −0.010565453 a2 = 3.327942 × 10−2 a3 = 2.0551378 × 10−3 a4 = −1.1396226 × 10−4 a5 = 1.6340021 × 10−5 a1 = 1.023798 a2 = 1.058264 a3 = 5.280792 b1 = 0.00377588 b2 = 0.0122544 b3 = 321.3616

Material dispersion is normally described with Sellmeier equations for the refractive index as a function of the wavelength, i.e. n(λ). With the Sellmeier equations (Table 4a) all the necessary dispersive quantities can be calculated (Table 5).

5.2. Dispersion compensation The challenge in ultrashort pulse generation is dispersion compensation over a large bandwidth, that is, to compensate for the dispersive pulse broadening that occurs in the gain material and other elements inside the laser cavity. Dispersion compensation is important because, for example, a 10 fs (1 fs) Gaussian pulse at the center wavelength of 800 nm is broadened to 100 fs (1 ps) after only 1 cm of fused quartz due to second-order dispersion. This follows from eq. (5.2) for the regime of strong pulse broadening and Table 4. In addition, in femtosecond lasers the pulses are ideally soliton pulses for which a constant negative dispersion over the full spectral width of the pulse balances the chirp of the self-phase modulation. The negative dispersion required for a certain pulse duration follows from eq. (6.52). It is necessary that all higher-order dispersion terms are negligibly small.

1, § 5]

Pulse propagation in dispersive media

57

Table 4b Examples of material dispersions calculated from the Sellmeier equations given in Table 4a and the equations given in Table 5 Material

Refractive index n at a center wavelength of 800 nm

Fused quartz

n(0.8 µm) = 1.45332  ∂n  1 ∂λ 800 nm = −0.017 µm  ∂2n  = 0.04 1 2 ∂λ2 800 nm µm  ∂3n  1 3 800 nm = −0.24 3 µm

∂λ

SF10 glass

n (0.8 µm) = 1.71125  ∂n  1 ∂λ 800 nm = −0.0496 µm  2 ∂ n = 0.176 1 2 ∂λ2 800 nm µm  1 ∂3n  = −0.997 3 800 nm 3 µm

∂λ

Sapphire

n(0.8 µm) = 1.76019  ∂n  1 ∂λ 800 nm = −0.0268 µm  ∂2n  = 0.064 1 2 ∂λ2 800 nm µm  ∂3n  1 = −0.377 3 800 nm 3 µm

∂λ

Propagation constant kn at a center wavelength of 800 nm ∂kn  −9 s = 4.84 ns m m ∂ω 800 nm = 4.84 × 10  2 ∂ kn  −26 s2 = 36.1 fs2 = 3.61 × 10 m mm ∂ω2 800 nm 3 ∂ 3 kn  fs3 = 2.74 × 10−41 sm = 27.4 mm ∂ω3 800 nm

∂kn  −9 s = 5.70 ns m m ∂ω 800 nm = 5.70 × 10  2 2 ∂ kn  fs2 s −25 = 1.59 × 10 m = 159 mm ∂ω2 800 nm 3 ∂ 3 kn  fs3 = 1.04 × 10−40 sm = 104 mm ∂ω3 800 nm ∂kn  −9 s = 5.87 ns m m ∂ω 800 nm = 5.87 × 10 ∂ 2 kn  −26 s2 = 58 fs2 = 5.80 × 10 m mm ∂ω2 800 nm

3 ∂ 3 kn  fs3 = 4.21 × 10−41 sm = 42.1 mm ∂ω3 800 nm

Table 5 Dispersion quantities, their defining equations and units. kn : wavevector in the dispersive media, i.e. kn = kn = n2π /λ, where λ is the vacuum wavelength. z: a certain propagation distance. c: vacuum light velocity. ω: frequency in radians/second Quantity

Symbol

Defining equation

Defining equation using n(λ)

Phase velocity

υp

Group velocity

υg

ω kn dω dkn

Group delay

Tg

dφ Tg = υzg = dω , φ ≡ kn z

c n 1 c n 1− dn λ dλ n  nz 1 − dn c dλ nz 1 − dn c dλ λ3 z d2 n 2π c2 dλ2

Dispersion: 1st order Dispersion: 2nd order Dispersion: 3rd order

dφ dω d2 φ dω2 d3 φ dω3

λ n λ n

−λ4 z 3 d2 n + λ d3 n 4π 2 c3 dλ2 dλ3

For optimum soliton formation of a sub-10 fs pulse inside the laser, only a very small amount of a constant negative total intracavity dispersion is necessary to form a stable soliton pulse (eq. (6.52)). For example, we estimated the neces-

58

Ultrafast solid-state lasers

[1, § 5

sary dispersion to be only −10 fs2 for a Ti:sapphire laser producing 6.5 fs pulses (Jung, Kärtner, Matuschek, Sutter, Morier-Genoud, Zhang, Keller, Scheuer, Tilsch and Tschudi [1997], Sutter, Jung, Kärtner, Matuschek, Morier-Genoud, Scheuer, Tilsch, Tschudi and Keller [1998]). Here we assumed an estimated self-phase modulation coefficient of about 0.07/mW, an average output power of 200 mW using a 3% output coupler and a pulse repetition rate of 86 MHz. This results in an intracavity pulse energy of 77.5 nJ. With eq. (6.52) then follows an estimated negative group delay dispersion of −10 fs2 in a cavity round trip. There are different methods for dispersion compensation, such as Gires– Tournois Interferometer (GTI) grating pairs, prism pairs and chirped mirrors. The dispersion compensation is summarized in Table 6 with the symbols defined in fig. 9.

(a)

(b)

(c) (continued on next page) Fig. 9. Dispersion compensation techniques: (a) Gires–Tournois Interferometer (GTI); (b) grating pair; (c) prism pair; (d) chirped mirror (i.e. shown here a double chirped mirror).

1, § 5]

Pulse propagation in dispersive media

59

(d) Fig. 9. (Continued)

5.2.1. Gires–Tournois Interferometer (GTI) A GTI (Gires and Tournois [1964], Kafka, Watts and Pieterse [1992]) is a very compact dispersion compensation element, which basically replaces one flat laser cavity mirror. The negative dispersion is obtained due to the Fabry–Perot structure (fig. 9a). Normally, in a GTI the bottom mirror is a 100% reflector and the top mirror is a relatively low reflector, typically with a reflectivity of a few percent. The spacer layer in the Fabry–Perot should contain a non-absorbing material and is very often air, such that the thickness can be easily changed. In Table 7 the material dispersion in the GTI spacer layer was neglected. The bandwidth compared to the other techniques is limited, thus a GTI is typically used for pulse durations above 100 fs. There is a trade-off: a broader bandwidth is obtained with a smaller Fabry–Perot thickness but then the amount of negative dispersion is strongly reduced: d2 φ ∝ d 2, dω2

bandwidth of

d2 φ 1 ∝ . dω2 d

(5.3)

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Ultrafast solid-state lasers

[1, § 5

Table 6 Dispersion compensation, their defining equations and figures. c: light velocity in vacuum, λ: wavelength in vacuum, λ0 : center wavelength of pulse spectrum, ω: frequency in radians/second Quantity

Defining equation

Gires–Tournois Interferometer (GTI) (fig. 9a)

d: thickness of Fabry–Perot n: refractive index of material inside Fabry–Perot (airspaced n = 1). Note: material dispersion is neglected. t0 = 2nd c : roundtrip time of the Fabry–Perot Rt : intensity reflectivity of top reflector of Fabry–Perot (bottom reflector is assumed to have a 100%-reflectivity)

Dispersion: 2nd order Four-grating compressor (fig. 9b)

Dispersion: 2nd order Dispersion: 3rd order Four-prism compressor (fig. 9c)

Dispersion: 2nd order

Dispersion: 3rd order

√ −2t02 (1−Rt ) Rt sin ωt0 d2 φ √ = dω2 (1+Rt −2 Rt cos ωt0 )2

Lg : grating pair spacing Λ: grating period θi : angle of incidence at grating λ 2 −3/2 λ 3 Lg  d2 φ 2 =− 2 2 1 − Λ − sin θi dω

πc Λ

λ sin θ −sin2 θ 2 1+ Λ d3 φ i i = − d φ2 6πc λ 2 λ dω3 dω 1− Λ −sin θi

n: refractive index of prisms θB : angle of incident of prism is at Brewster angle θB = arctan[n(λ0 )] α: apex angle of prism, α = π − 2θB   sin θ  θ2 (λ) = arcsin n(λ) sin π − 2θB − arcsin n(λ)B L: apex-to-apex prism distance h: beam insertion into second prism h cos θ2 sin β = L cos(α/2) 3 2 d2 φ = λ 2 d P2 dω2 2π c dλ d2 P = 2 ∂ 2 n  ∂θ2 +  ∂ 2 θ2  ∂n 2 L sin β − 2 ∂θ2 ∂n 2 L cos β ∂λ ∂n ∂λ dλ2 ∂λ2 ∂n ∂n2  ∂2n   ∂n 2   ∂n 2 1 ≈ 4 2 + 2n − 3 ∂λ L sin β − 8 ∂λ L cos β ∂λ n 4  2 3 d3 φ = −λ2 3 3 d P2 + λ d P3 dω3 4π c dλ dλ

d3 P ≈ 4 d3 n L sin β − 24 dn d2 n L cos β dλ dλ2 dλ3 dλ3

For example, an air-spaced GTI with 80 µm thickness and a top reflectivity of 4% produces a negative dispersion of about −0.13 ps2 at a wavelength of 799 nm. In comparison a 2.25 µm thick air space results in about −100 fs2 at a wavelength of 870 nm.

Laser cavity element

Linearized operator

New constants

Constants

Gain

(6.10)

 2 
A ≈ g + Dg ∂ 2 A

Dg ≡ g2

Dg : gain dispersion (eq. (6.10)) g: saturated amplitude gain coefficient Ωg : HWHM of gain bandwidth in radians/second
νg : FWHM of gain bandwidth, i.e.
νg = Ωg /π

Loss modulator

(6.12)


A ≈ −Ms t 2 A

Ms ≡

∂t

Ωg

2 Mωm 2


A ≈ −lA

Constant loss

l: amplitude loss coefficient


A ≈ iψA

Constant phase shift

Ms : curvature of loss modulation (eq. (6.11)) ωm : loss modulation frequency in radians/second 2M: peak-to-peak modulation depth for amplitude loss coefficient

ψ: phase shift q

Fast saturable absorber

(6.14)


A ≈ γA |A|2 A

γA ≡ I 0 A sat,A A

γA : absorber coefficient (eq. (4.13)) q0 : maximum saturable amplitude loss coefficient Isat,A : saturation intensity AA : laser mode area in saturable absorber

Dispersion: 2nd order

(6.19)

2
A ≈ iD ∂ 2 A

D ≡ 12 kn z

D: dispersion parameter (half of the total group delay dispersion per cavity roundtrip (eq. (6.18)))

∂t

Pulse propagation in dispersive media

Eq.

1, § 5]

Table 7 Linearized operators that model the change in the pulse envelope A(t) for each element in the laser cavity and their defining equations. The pulse envelope is normalized such that |A(z, t)|2 is the pulse power P (z, t) (eq. (6.2)). kn : wavevector in the dispersive media, i.e. kn = kn = n2π/λ, where λ is the vacuum wavelength. z: the relevant propagation distance for negative dispersion or SPM, respectively. c: vacuum light velocity. ω: frequency in radians/second

2 kn = d k2n

dω

SPM

(6.25)


A ≈ −iδL |A|2 A

kn z

δL ≡ A 2 L

δL : SPM coefficient (eq. (6.22)) n2 : nonlinear refractive index AL : laser mode area inside laser material (Note: here we assume that the dominant SPM occurs in the laser material. Then z is equal to 2 times the length of the laser crystal in a standing wave cavity) 61

62

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[1, § 5

5.2.2. Grating pairs Grating pairs (Treacy [1969]) produce negative dispersion due to the wavelengthdependent diffraction (fig. 9b). To obtain a spatially coherent beam 2 paths through the grating pair are required. Normally gratings introduce too much loss inside a solid-state laser resonator. Thus, they have only been used for external pulse compression schemes (Tomlinson, Stolen and Shank [1984]). The grating pairs alone can only be used to compensate for second-order dispersion. Higherorder dispersion limits pulse compression in the ultrashort pulse-width regime. Therefore, a combination of a grating and prism compressor was used to generate the long-standing world record of 6-fs pulses with dye lasers (Fork, Cruz, Becker and Shank [1987]). There are mainly two types of gratings: ruled or holographic. Removing material from a master substrate with a precise instrument called a ruling engine produces ruled gratings. Replicas of the ruled grating are then pressed and the pressings are coated. Holographic gratings are produced by interfering two laser beams on a substrate coated with a photoresist, which is subsequently processed to reproduce the sinusoidal interference pattern. Generally, replicas cannot be produced from holographic masters, i.e. they are more expensive. Higher damage threshold can be obtained with gold coated ruled gratings (i.e. > 500 mJ/cm2 at 1 ps). The diffraction efficiency in this case is from around 88% to 92% depending on the grating. Because 4 paths through the gratings are required for dispersion compensation and a spatially coherent beam, this amounts to at least about 30% loss. 5.2.3. Prism pairs Prism pairs (Fork, Martinez and Gordon [1984]) are well established for intracavity dispersion compensation. Negative dispersion is obtained with the wavelength dependent refraction (fig. 9c). To obtain a spatially coherent beam 2 paths through the prism pairs are required. The insertion loss is very small because the angle of incidence is the Brewster angle. The prism apex angle is chosen such that the incident beam at Brewster angle is also at minimum deviation. Prism pairs offer two advantages. First, the pulse width can be varied by simply moving one of the prisms (see fig. 9c) and second, the laser can be tuned in wavelength by simply moving a knife edge at a position where the beam is spectrally broadened. Both properties are often desired for spectroscopic applications, for example. However, the prism pair suffers from higher-order dispersion, which is the main limitation in ultrashort pulse generation in the sub-10-fs regime. Different prism materials introduce different amounts of higher-order dispersion. For compact lasers with pulse durations of few tens to hundreds of femtoseconds the more dispersive

1, § 5]

Pulse propagation in dispersive media

63

SF10-prisms are better because they require a smaller prism separation than fused quartz prism, for example. But a smaller prism separation comes at the expense of a larger higher-order dispersion. Fused quartz is one of the best materials for ultrashort pulse generation with minimal higher-order dispersion. The higher-order dispersion of the prism pairs is dominated by the prism spacing, which is not changed significantly when we adjust the dispersion by inserting the prisms into the laser beam. For example, at a fused quartz prism spacing of 40 cm, the total dispersion produced by a double pass through the prism pairs amounts to −862 fs2 and a thirdorder dispersion of −970 fs3 at a center wavelength of 800 nm, assuming zero prism insertion into the beam (Table 6). We can reduce the negative dispersion by moving the prisms into the laser beam: Each additional millimeter of prism insertion produces a positive dispersion of 101 fs2 but only a third-order dispersion of 78 fs3 per cavity round trip. Thus, the prism pairs can generally only be used to compensate for either second or third-order dispersion. With fused quartz prisms alone pulses as short as 10 fs can be produced with a Ti:sapphire laser (Asaki, Huang, Garvey, Zhou, Kapteyn and Murnane [1993], Fluck, Jung, Zhang, Kärtner and Keller [1996]). Slightly shorter pulses can be achieved if the Ti:sapphire laser is operated at a center wavelength of approximately 850 nm, where nearzero second and third-order dispersion can be obtained for a Ti:sapphire crystal of about 2 mm thickness. In this case 8.5 fs have been generated (Zhou, Taft, Huang, Murnane, Kapteyn and Christov [1994]). Another dispersion compensation is then required for shorter pulse generation. Therefore, the chirped mirrors were designed to show the inverse higher-order group delay dispersion of the dispersion of the prism pair plus laser crystal to eliminate the higher-order dispersion and obtain a slightly negative but constant group delay dispersion required for an ideal soliton pulse. It is interesting to note that under special cavity design a single prism can also be used for dispersion compensation (Kopf, Spühler, Weingarten and Keller [1996]). 5.2.4. Chirped mirror Chirped mirrors (Szipöcs, Ferencz, Spielmann and Krausz [1994]) have been established in different set-ups for the generation of ultrashort laser pulses. Figure 9d shows a typical chirped mirror structure which schematically shows the path of a long-wavelength (i.e. 1000 nm) and short-wavelength (i.e. 650 nm) beam. This results in negative dispersion, because the long wavelengths are made to penetrate deeper into the mirror structure than short wavelengths. Figure 9d also shows the standing wave electric field patterns in a chirped mirror structure

64

Ultrafast solid-state lasers

[1, § 5

versus wavelength. The negative dispersion of the mirrors is clearly illustrated by the dependence of penetration depth on wavelength for the range of 650 to 1050 nm. The highly transmissive region around 500 nm is used for pumping the Ti:sapphire laser through these chirped mirrors. According to (Szipöcs, Ferencz, Spielmann and Krausz [1994]), chirping means that the Bragg wavelength λB is gradually increased along the mirror, producing a negative group delay dispersion (GDD). However, no analytical explanation of the unwanted oscillations typically observed in the group delay and GDD of such a simple-chirped mirror was given. These oscillations were mostly minimized purely by computer optimization. An exact coupled-mode analysis (Matuschek, Kärtner and Keller [1997]) was used to develop a double-chirp technique. The strong periodic variation in the group delay of the original chirped mirrors occurs due to impedance mismatch between the incident medium (i.e. typically air) and the mirror stack and also within the mirror stack (Kärtner, Matuschek, Schibli, Keller, Haus, Heine, Morf, Scheuer, Tilsch and Tschudi [1997], Matuschek, Kärtner and Keller [1999]). Using the accurate analytical expressions for phase, group delay, and GDD (Matuschek, Kärtner and Keller [1999]), double-chirped mirrors (DCM) could be designed and fabricated with a smooth and custom tailored GDD suitable for generating pulses in the two-cycle regime directly from a Ti:sapphire laser (Sutter, Steinmeyer, Gallmann, Matuschek, Morier-Genoud, Keller, Scheuer, Angelow and Tschudi [1999], Morgner, Kärtner, Cho, Chen, Haus, Fujimoto, Ippen, Scheuer, Angelow and Tschudi [1999]). A double chirped mirror (DCM) (Kärtner, Matuschek, Schibli, Keller, Haus, Heine, Morf, Scheuer, Tilsch and Tschudi [1997], Matuschek, Kärtner and Keller [1999]) is a multilayer interference coating that can be considered as a composition of at least two sections, each with a different task. The layer materials are typically SiO2 and TiO2 . The first section is the AR coating, typically composed of 10 to 14 layers. It is necessary because the theory is derived assuming an ideal matching to air. The other section represents the actual DCM structure, as derived from theory. The double-chirp section is responsible for the elimination of the oscillations in the GDD. Double-chirping means that in addition to the local Bragg wavelength λB the local coupling of the incident wave to the reflected wave is independently chirped as well. The local coupling is adjusted by slowly increasing the high-index layer thickness in every pair so that the total optical thickness remains λB /2. This corresponds to adiabatic matching of the impedance. The AR-coating, together with the rest of the mirror, is used as a starting design for a numerical optimization program. Since theoretical design is close to the desired design goal, a local optimization using a standard gradient algorithm is sufficient. At this point, only the broadband AR-coating sets a limitation on the reduction of

1, § 6]

Mode-locking techniques

65

the GDD oscillations. An AR coating with a residual reflectivity of less than 10−4 is required for a DCM at a center wavelength of around 800 nm, which results in a bandwidth of only 250 nm. This bandwidth limitation cannot be removed with more layers in the mirror structure. To overcome this AR-coating limitation, we demonstrated a new technique for the design of chirped mirrors with extremely smooth dispersion characteristics over an extended ultrabroadband wavelength range: back-side coated (BASIC) chirped mirrors (Matuschek, Gallmann, Sutter, Steinmeyer and Keller [2000]). Dispersion oscillations are significantly reduced by coating the chirped mirror structure on the back side of a substrate, providing ideal impedance matching between coating and ambient medium. The chirped mirror structure and the AR coating are non-interfering due to slightly wedged substrates or substrates with differently curved surfaces (e.g., focusing mirror) for example. Thus, chirped mirror and AR-coating multilayer structures and can be independently designed and optimized. The separation of both coating sections provides a much better solution for the impedance-matching problems than previous approaches to chirped mirror design. The AR-coating may be added on the front side of the substrate, geometrically separated from the chirped mirror, to simply reduce the intracavity losses. Thus, the AR coating does not need to be as good and we can easily obtain a much broader bandwidth. We showed by a theoretical analysis and numerical simulations that minimum dispersion oscillations are achieved if the index of the substrate is identical to the index of one of the coating materials and if double chirping is used for the chirped mirror structure. Based on this analysis, we designed a back-side coated double-chirped mirror (BASIC DCM) that supports a bandwidth of 220 THz with group delay dispersion oscillations of about 2 fs2 (rms), an order-of magnitude improvement compared to previous designs of similar bandwidth.

§ 6. Mode-locking techniques 6.1. Overview Passive mode-locking mechanisms are well-explained by three fundamental models: slow saturable absorber mode locking with dynamic gain saturation (New [1972, 1974]) (fig. 3a), fast saturable absorber mode locking (Haus [1975a], Haus, Fujimoto and Ippen [1992]) (fig. 3b), and slow saturable absorber mode locking without dynamic gain saturation which in the femtosecond regime is described by soliton mode locking (Kärtner and Keller [1995], Jung, Kärtner, Brovelli, Kamp

66

Ultrafast solid-state lasers

[1, § 6

and Keller [1995], Kärtner, Jung and Keller [1996]) and in the picosecond regime by Paschotta and Keller [2001] (fig. 3c). The physics of most of these techniques can be well explained with Haus’s master equation formalism as long as at steady state the changes in the pulse envelope during the propagation inside the cavity are small. At steady state the pulse envelope has to be unchanged after one roundtrip through the cavity. Passive mode locking, however, can only be analytically modelled in the weak saturation regime, which is typically not the case in SESAM mode-locked solidstate lasers. However, this formalism still provides a useful approach to describe mode-locking techniques in a unified fashion. Recent numerical simulations show that analytical results with fast saturable absorbers only slightly underestimate numerical solutions and correctly describe the dependence on saturated gain, gain bandwidth and absorber modulation, taking into account more strongly saturated absorbers and somewhat longer saturation recovery times in SESAM modelocked solid-state lasers (Paschotta and Keller [2001]). A short introduction to Haus’s formalism is given in Section 6.2 and Table 7. Afterwards we will describe all mode-locking techniques using this formalism and summarize the theoretical prediction for pulse shape and pulse duration (Table 9). For solid-state lasers self-Q-switching instabilities in passive mode locking is a serious challenge. Simple guidelines to prevent those instabilities and obtain stable cw mode locking are presented in Section 6.8.

6.2. Haus’s master equations Haus’s master equation formalism (Haus [1995], Haus [2000]) is based on linearized differential operators that describe the temporal evolution of a pulse envelope inside the laser cavity. At steady-state we then obtain the differential equation: ∂A(T , t)  TR (6.1) =
Ai = 0, ∂T i

where A is the pulse envelope, TR is the cavity round-trip time, T is the time that develops on a time scale of the order of TR , t is the fast time of the order of the pulse duration, and
Ai are the changes of the pulse envelope due to different elements in the cavity, such as gain, loss modulator or saturable absorber, dispersion, etc. (Table 7). Equation (6.1) basically means that at steady state after one laser roundtrip the pulse envelope cannot change and all the small changes due to the different elements in the cavity have to sum up to zero. Each element is modeled as a linearized operator, which will be discussed in more detail below.

1, § 6]

Mode-locking techniques

67

The pulse envelope is normalized such that |A(z, t)|2 is the pulse power P (z, t):  2 E(z, t) ∝ A(z, t)ei[ω0 t −kn (ω0 )z] , with A(z, t) ≡ P (z, t), (6.2) where E(z, t) is the electric field, ω0 the center frequency in radians/second of the pulse spectrum and kn = nk with k = 2π/λ the wave number with λ the vacuum wavelength and n the refractive index. Before we discuss the different mode-locking models we briefly discuss the linearized operators for the differential equations. Gain. A homogeneously broadened gain medium is described by a Lorentzian lineshape for which the frequency dependent gain coefficient g(ω) is given by

 g
ω2 g(ω) = (6.3) , for (ω − ω0 /Ωg )2  1, ≈g 1−  0 2 Ωg2 1 + ω−ω Ωg where
ω = ω − ω0 , g is the saturated gain coefficient for a cavity round trip and Ωg is the HWHM (half width half maximum) of the gain bandwidth in radians/seconds. In the frequency domain the pulse envelope after the gain medium is given by   A˜ out (ω) = eg(ω) A˜ in (ω) ≈ 1 + g(ω) A˜ in(ω), for g  1, (6.4) ˜ where A(ω) is the Fourier transformation of A(t). Equations (6.3) and (6.4) then give   g A˜ out (ω) = 1 + g − 2
ω2 A˜ in (ω) ⇒ Ωg (6.5)   g ∂2 Aout (t) = 1 + g + 2 2 Ain (t), Ωg ∂t where we used the fact that a factor of
ω in the frequency domain produces a time derivative in the time domain. For example, for the electric field we obtain:     ∂ ∂ 1 1 iωt iωt   (6.6) E(t) = dω = dω E(ω)e E(ω)iωe ∂t ∂t 2π 2π and

    ∂2 1 1 ∂2 iωt 2 iωt   E(t) = 2 ]e dω E(ω)e dω = E(ω)[−ω 2π ∂t 2 ∂t 2π

(6.7)

˜ ˜  0 +
ω) and similarly for the pulse envelope. Note that A(ω) ≡ A(
ω) = E(ω     ∂ ∂ 1 1 i
ωt i
ωt ˜ ˜ (6.8) A(t) = dω = dω A(ω)e A(ω)i
ωe ∂t ∂t 2π 2π

68

Ultrafast solid-state lasers

[1, § 6

and

    ∂2 ∂2 1 1 i
ωt 2 i
ωt ˜ ˜ A(t) = dω = ]e dω. A(ω)e A(ω)[−
ω 2π ∂t 2 ∂t 2 2π (6.9) For the change in the pulse envelope
A = Aout − Ain after the gain medium we then obtain:   ∂2 g
A ≈ g + Dg 2 A, Dg ≡ 2 , (6.10) ∂t Ωg

where Dg is the gain dispersion. Loss modulator. A loss modulator inside a laser cavity is typically an acoustooptic modulator and produces a sinusoidal loss modulation given by a time dependent loss coefficient: 2 Mωm (6.11) , 2 where Ms is the curvature of the loss modulation, 2M is the peak-to-peak modulation depth and ωm the modulation frequency which corresponds to the axial mode spacing in fundamental mode locking. In fundamental mode locking we only have one pulse per cavity round trip. The change in the pulse envelope is then given by   Aout (t) = e−l(t ) Ain (t) ≈ 1 − l(t) Ain (t) ⇒
A ≈ −Ms t 2 A. (6.12)

l(t) = M(1 − cos ωm t) ≈ Ms t 2 ,

Ms ≡

Fast saturable absorber. In case of an ideal fast saturable absorber we assume that the loss recovers instantaneously and therefore shows the same time dependence as the pulse envelope (eqs. (4.12) and (4.13)): q0 q0 q(t) = (6.13) ≈ q0 − γA P (t), γA ≡ . 1 + I (t)/Isat,A Isat,A AA The change in the pulse envelope is then given by   Aout (t) = e−q(t ) Ain (t) ≈ 1 − q(t) Ain (t) ⇒


A ≈ γA |A|2A.

(6.14)

Group velocity dispersion (GVD). The wave number kn (ω) in a dispersive material depends on the frequency and can be approximately written as: 1 kn (ω) ≈ kn (ω0 ) + kn
ω + kn
ω2 + · · · , 2 2

(6.15)

∂ kn  n where
ω = ω −ω0 , kn = ∂k ∂ω |ω=ω0 and kn = ∂ω2 |ω=ω0 . In the frequency domain the pulse envelope in a dispersive medium after a propagation distance of z is

1, § 6]

Mode-locking techniques

69

given by     ˜ ω) = e−i[kn (ω)−kn (ω0 )]z A(0, ˜ ω) ≈ 1 − i kn (ω) − kn (ω0 ) z A(0, ˜ ω), A(z, (6.16) where we used the slowly varying envelope approximation (which is applicable for pulse durations of more than 10 fs in the near infrared wavelength regime). Taking into account only the first- and second-order dispersion terms we then obtain:   1   2 ˜ ω). ˜ A(z, ω) ≈ 1 − ikn
ωz − i kn
ω z A(0, (6.17) 2 The linear term in
ω determines the propagation velocity of the pulse envelope (i.e. the group velocity νg ) and the quadratic term in
ω determines how the pulse envelope gets deformed due to second-order dispersion. The influence of higher-order dispersion can be considered with more terms in the expansion of kn (ω) (eq. (6.15)). However, higher-order dispersion only becomes important for ultrashort pulse generation with pulse durations below approximately 30 fs depending how much material is inside the cavity. Normally we are only interested in changes of the pulse envelope and therefore it is useful to restrict our observation to a reference system that is moving with the pulse envelope. In this reference system we only need to consider second- and higher-order dispersion. In the time domain we then obtain for second-order dispersion:   1 1 d2 φ ∂2 , A(z, t) ≈ 1 + iD 2 A(0, t), D ≡ kn z = (6.18) 2 2 dω2 ∂t where D is the dispersion parameter which is half of the total group delay dispersion per cavity roundtrip. Therefore we obtain for the change in the pulse envelope:
A ≈ iD

∂2 A. ∂t 2

(6.19)

Self-phase modulation. The Kerr effect introduces a space and time dependent refractive index: n(r, t) = n + n2 I (r, t),

(6.20)

where n is the linear refractive index, n2 the nonlinear refractive index and I (r, t) the intensity of the laser beam, typically a Gaussian beam profile. For laser host materials, n2 is typically of the order of 10−16 cm2 /W and does not change very much for different materials. For example, for sapphire n2 =

70

Ultrafast solid-state lasers

[1, § 6

3 × 10−16 cm2 /W, fused quartz n2 = 2.46 × 10−16 cm2 /W, Schott glass LG760 n2 = 2.9 × 10−16 cm2 /W, YAG n2 = 6.2 × 10−16 cm2 /W, and YLF n2 = 1.72 × 10−16 cm2 /W. The nonlinear refractive index produces a nonlinear phase shift during pulse propagation: 2    φ(z, r, t) = −kn(r, t)z = −k n + n2 I (r, t) z = −knz − δL A(r, t) , (6.21) where δL is the self-phase modulation coefficient (SPM coefficient): δL ≡ kn2 z/AL ,

(6.22)

where AL is the laser mode area inside the laser medium. Here we assume that the dominant SPM inside the laser occurs in the gain medium. In this case, z is equal to twice the laser material length. Of course, the mode area can also be very small in other materials. In this case, we will have to add up all the SPM contributions inside the laser resonator. The laser mode area AL is an “averaged value” in case the mode is changing within the gain medium. The electric field during propagation is changing due to SPM: E(z, t) = eiφ E(0, t) ∝ e−iδL |A(t )| A(0, t)eiω0 t −ikn (ω0 )z . 2

(6.23)

For δL |A|2  1 we obtain:

2   2 A(z, t) = e−iδL |A(t )| A(0, t)e−ikn (ω0 )z ≈ 1 − iδL A(t) A(0, t)e−ikn (ω0 )z . (6.24)

After one cavity roundtrip we then obtain
A ≈ −iδL |A|2 A.

(6.25)

6.3. Active mode locking Short pulses from a laser can be generated with a loss or phase modulator inside the resonator. For example, the laser beam is amplitude-modulated when it passes through an acousto-optic modulator. Such a modulator can modulate the loss of the resonator at a period equal to the round-trip time TR of the resonator (i.e. fundamental mode locking). The pulse evolution in an actively mode-locked laser without self-phase modulation (SPM) and group-velocity dispersion (GVD) can be described by the master equation of Haus (Haus [1975b]). Taking into account gain dispersion and loss modulation we obtain with eqs. (6.10) and (6.12) (Table 7) the following differential equation: 

   1 ∂2 ωm t 2 (6.26)
Ai = g 1 + 2 2 − l − M A(T , t) = 0. Ωg ∂t 2 i

1, § 6]

Mode-locking techniques

71

Typically, we obtain pulses which are much shorter than the roundtrip time in the cavity and which are placed in time at the position where the modulator introduces the least amount of loss. Therefore, we were able to approximate the cosine modulation by a parabola (eq. (6.11)). The only stable solution to this differential equation is a Gaussian pulse shape with a pulse duration:  τp = 1.66 × 4 Dg /Ms , (6.27) where Dg is the gain dispersion (eq. (6.10) and Table 7) and Ms is the curvature of the loss modulation (eq. (6.11) and Table 7). Therefore, in active mode locking the pulse duration becomes shorter until the pulse shortening of the loss modulator balances the pulse broadening of the gain filter. Basically, the curvature of the gain is given by the gain dispersion Dg and the curvature of the loss modulation is given by Ms . The pulse duration is only scaling with the fourth root of √ √ the saturated gain (i.e. τp ∝ 4 g ) and the modulation depth (i.e. τp ∝ 4 1/M ) √ and with the square root of the modulation frequency (i.e. τp ∝ 1/ωm ) and the  gain bandwidth (i.e. τp ∝ 1/
ωg ). A higher modulation frequency or a higher modulation depth increases the curvature of the loss modulation and a larger gain bandwidth decreases gain dispersion. Therefore, we obtain shorter pulse durations in all cases. At steady-state the saturated gain is equal to the total cavity losses, therefore a larger output coupler will result in longer pulses. Thus, higher average output power is generally obtained at the expense of longer pulses (Table 2). The results obtained in actively mode-locked flashlamp pumped Nd:YAG and Nd:YLF lasers can be very well explained by this result. For example, with Nd:YAG at a lasing wavelength of 1.064 µm we have a gain bandwidth of
λg = 0.45 nm. With a modulation frequency of 100 MHz (i.e. ωm = 2π · 100 MHz), a 10% output coupler (i.e. 2g ≈ Tout = 0.1) and a modulation depth M = 0.2, we obtain a FWHM pulse duration of 93 ps (eq. (6.27)). For example, with Nd:YLF at a lasing wavelength of 1.047 µm and a gain bandwidth of
λg = 1.3 nm we obtain with the same mode-locking parameters a pulse duration of 53 ps (eq. (6.27)). The same result for the pulse duration (eq. (6.27)) has been previously derived by Kuizenga and Siegman [1970a, 1970b] where they assumed a Gaussian pulse shape and then followed the pulse once around the laser cavity, through the gain and the modulator. They then obtain a self-consistent solution when no net change occurs in the complete round trip. The advantage of Haus’s theory is that no prior assumption has to be made for the pulse shape. His theory then predicts a Gaussian pulse shape for actively mode-locked lasers, which then in principle justifies Kuizenga and Siegman’s assumption. Equation (6.27) shows that increasing the modulation frequency is an effective method to shorten the pulses. Harmonic mode locking is a technique in which the

72

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[1, § 6

cavity is modulated at a frequency that is some integer multiple of the fundamental pulse repetition rate with a period given by the cavity round trip time. This technique was first introduced and analyzed by Becker, Kuizenga and Siegman [1972] in a phase mode-locked Nd:YAG laser. Second harmonic mode locking of a flashlamp pumped Nd:YAG laser at 1.06 µm resulted in less than 50 ps (Johnson and Simpson [1983], Johnson and Simpson [1985]) and at 1.32 µm in 53 ps (Keller, Valdmanis, Nuss and Johnson [1988]). It has been well known for quite some time that the addition of a nonlinear index medium to a passively (Martinez, Fork and Gordon [1984], Martinez, Fork and Gordon [1985]) or actively (Haus and Silberberg [1986]) mode-locked laser system can lead to shorter pulses. The bandwidth limitation that results from gain dispersion can be partially overcome by the spectral broadening caused by the nonlinearity. We can extend the differential equation (eq. (6.26)) with the additional terms for SPM (eq. (6.25) and Table 7): 

   1 ∂2 ωm t 2 − iδL |A|2 + iψ A(T , t) = 0.
Ai = g 1 + 2 2 − l − M Ωg ∂t 2 i (6.28) In this case, however, we have to include an additional phase shift ψ to obtain a self-consistent solution. So far, we always assumed ψ = 0. This phase shift is an additional degree of freedom because the boundary condition for intracavity pulses only requires that the pulse envelope is unchanged after one cavity round trip. The electric field, however, can have an arbitrary phase shift ψ after one round trip. This is normally the case because the phase and the group velocity are not the same. It is important to note, that for ultrashort pulses in the one to two cycle regime this becomes important and stabilization of the electric field with respect to the peak of the pulse envelope is required (Telle, Steinmeyer, Dunlop, Stenger, Sutter and Keller [1999]). We can obtain an analytic solution of eq. (6.28) if we assume a parabolic approximation for |A|2 :

 t2 2 2 |A|2 = |A0 |2 e−t /τ ≈ P0 1 − 2 . (6.29) τ The solution of eq. (6.28) is then a chirped Gaussian pulse   1 t2 A(t) = A0 exp − 2 (1 − ix) 2τ

(6.30)

with the chirp parameter x=

τ 2 φnl , 2Dg

(6.31)

1, § 6]

Mode-locking techniques

and the FWHM pulse duration  τp = 1.66 · τ = 1.66 ·

4

Dg

2 /4D Ms + φnl g

,

73

(6.32)

where φnl is the nonlinear phase shift per cavity roundtrip at peak intensity. Typically, the beam diameter is very small in the laser gain material, thus the dominant SPM contribution comes from propagation through the gain material: φnl = 2kLg n2 I0,L

(6.33)

and Lg is the length of the laser gain material (assuming standing wave cavity) and I0,L the peak intensity inside the SPM medium, i.e. the laser gain medium. This analytical result can explain the much shorter 7–12 ps pulses in actively mode-locked diode-pumped Nd:YAG (Maker and Ferguson [1989a]) and Nd:YLF (Maker and Ferguson [1989b], Keller, Li, Khuri-Yakub, Bloom, Weingarten and Gerstenberger [1990], Weingarten, Shannon, Wallace and Keller [1990], Juhasz, Lai and Pessot [1990]) lasers because the laser mode area in those diodepumped lasers is very small which results in significant SPM pulse shortening (eq. (6.32)). For example, our experiments with an actively mode-locked diodepumped Nd:YLF laser (Braun, Weingarten, Kärtner and Keller [1995]) are very well explained with eq. (6.32). In this case the lasing wavelength is 1.047 µm, the gain bandwidth is
λg = 1.3 nm, the pulse repetition rate is 250 MHz, the output coupler is 2.5%, the average output power is 620 mW, the mode radius inside the 5 mm long Nd:YLF crystal is 127 µm × 87 µm and the loss modulation of the acousto-optic mode locker is about 20%. We then obtain a FWHM pulse duration of 17.8 ps (eq. (6.32)) which agrees well with the experimentally observed pulse duration of 17 ps. Without SPM we would predict a pulse duration of 33 ps (eq. (6.27)). Equation (6.32) would predict that more SPM continues to reduce the pulse duration. However, too much SPM will ultimately drive the laser unstable. This has been shown by the numerical simulations of Haus and Silberberg [1986] which predict that pulse shortening in an actively mode-locked system is limited by roughly a factor of 2 in the case of SPM only. They also showed that the addition of negative GVD can undo the chirp introduced by SPM, and therefore both effects together may lead to stable pulse shortening by a factor of 2.5. However, experimental results with fiber lasers and solid-state lasers indicate that soliton shaping in the negative GVD regime can lead to pulse stabilization and considerable more pulse shortening. We have extended the analysis of Haus and Silberberg by investigating the possible reduction in pulse width of an actively mode-locked laser as a result of soliton-like pulse formation, i.e. the pres-

74

Ultrafast solid-state lasers

[1, § 6

ence of SPM and an excessive amount of negative GVD (Kärtner, Kopf and Keller [1995]). We show, by means of soliton perturbation theory, that beyond a critical amount of negative GVD a soliton-like pulse is formed and kept stable by an active mode locker. If the bandwidth of the gain is large enough, the width of this solitary pulse can be much less than the width of a Gaussian pulse generated by the active mode locker and gain dispersion alone. We established analytically that the pulse shortening possible by addition of SPM and GVD does not have a firm limit of 2.5. Numerical simulations and experiments with a regeneratively actively mode-locked Nd:glass laser (Kopf, Kärtner, Weingarten and Keller [1994]) confirm these analytical results. The pulse-width reduction achievable depends on the amount of negative GVD available. For an actively mode-locked Nd:glass laser a pulse shortening up to a factor of 6 may result, until instabilities arise. 6.4. Passive mode locking with a slow saturable absorber and dynamic gain saturation Dynamic gain saturation can strongly support pulse formation in passive mode locking and has allowed pulses with a duration much shorter than the absorber recovery time. Dynamic gain saturation means that the gain undergoes a fast pulseinduced saturation that recovers between consecutive pulses. This technique has been used to produce sub-100-fs pulses with dye lasers and dye saturable absorbers even though the absorber recovery time was in the nanosecond regime. Dynamic gain saturation can only help if the following conditions are fulfilled (fig. 3a): (i) The loss needs to be larger than the gain before the pulse: q0 > g0 , (6.34) where q0 is the unsaturated loss coefficient (eq. (4.1)) and g0 is the small signal gain coefficient in the laser (Weingarten, Braun and Keller [1994]). (ii) The absorber needs to saturate faster than the gain. From eq. (4.9) (i.e. slow saturable absorber and gain) then follows that AA AL Esat,A < Esat,L ⇔ (6.35) < , σA σL where Esat,A and Esat,L are the saturation energy of the absorber and the gain; σA and σL the absorber and gain cross section; and AA and AL the laser mode area in the absorber and gain. (iii) The absorber has to recover faster than the gain: τA < τL ,

(6.36)

where τA is the absorber recovery time and τL the upper state lifetime of the gain medium (Table 1).

1, § 6]

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75

Table 8 Femtosecond Rhodamine 6G dye laser with its saturable absorber DODCI at a wavelength of 620 nm Rhodamine 6G DODCI Photoisomer

σL = 1.36 × 10−16 cm2 σA = 0.52 × 10−16 cm2 σ˜ A = 1.08 × 10−16 cm2

τL = 4 ns τA = 2.2 ns τ˜L = 1 ns

Passively mode-locked dye lasers are based on this mode-locking technique and a more extensive review of ultrashort pulse generation with dye lasers is given in Shank [1988] and Diels [1990]. The most important dye laser for sub-100-fs pulse generation is the colliding pulse mode-locked (CPM) laser (Fork, Greene and Shank [1981]). The design considerations of such a laser are very well described in Valdmanis and Fork [1986]. This laser is based on Rhodamine 6G as the gain medium and on DODCI as the absorber (Ippen, Shank and Dienes [1972]). From Table 8 follows that the conditions (eqs. (6.34)–(6.36)) are only fulfilled if the mode area in the absorber jet is smaller than in the gain jet. In addition, a sixmirror ring cavity design where the absorber and the gain jet are separated by 1/4 of the resonator round trip gives the two counter propagating pulses the same gain and the absorber is more strongly saturated because of the two pulses colliding inside the saturable absorber jet (i.e. colliding pulse mode locking – CPM) (Fork, Greene and Shank [1981]). This effectively shortens the pulses and increases the stability. The best performance of this laser was only obtained after both the gain and the absorber dyes have been freshly prepared, because both photodegrade within a relatively short time (i.e. 1–3 weeks). The best result was 27 fs pulses at a center wavelength of 620 nm with an average output power of 20 mW each in two output beams at a pulse repetition rate of 100 MHz (Valdmanis, Fork and Gordon [1985], Valdmanis and Fork [1986]). More typically, this laser produced slightly below 100 fs pulses. This laser was the “work horse” for all the pioneering work on sub-100-fs spectroscopy for nearly 10 years in the 1980s. However, the rather large maintenance of this laser explains the success of the Ti:sapphire laser at the beginning of the 1990s. Today, solid-state lasers with much shorter pulses, higher output powers and better stability have replaced practically all ultrafast dye laser systems. Another important application of this mode-locking technique is the semiconductor laser, which also has an upper state lifetime in the nanosecond regime and a large gain cross section in the range of 10−14 cm2 . More extensive reviews of ultrashort pulse generation with semiconductor lasers have been provided by (Jiang and Bowers [1995]) and (Vail’ev [1995]). The CPM technique, as developed for dye lasers, has also been used for passively mode-locked semiconductor

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[1, § 6

lasers (van der Ziel, Logan and Mikulyak [1981], Vasil’ev, Morzov, Popov and Sergeev [1986]). Wu, Chen, Tanbun-Ek, Logan, Chin and Raybon [1990] and Chen and Wu [1992] monolithically incorporated the CPM technique in quantum well lasers. Pulses of 0.64 ps at a repetition rate of 710 MHz were generated (Chen, Wu, Tanbun-Ek, Logan and Chin [1991]). Compared with other modelocked solid-state lasers and mode-locked dye lasers, semiconductor lasers still have low output power (milli-watt regime) and long pulse width (a few hundreds of femtoseconds). The master equation for this mode-locking mechanism (fig. 3a) is given by (Haus [1975c], Ippen [1994]):    d g0 d2 A(T , t) = 0.
Ai = g(t) − q(t) + 2 2 + tD (6.37) Ωg dt dt i

For a self-consistent solution we will have to include a time shift tD of the pulse envelope due to the saturation of the absorber. g(t) and q(t) are given by the slow saturation approximation (eq. (4.9)). For an analytical solution we have to expand the exponential function up to the second order, i.e. ex ≈ 1 + x + x 2 /2:    t   2  σA   A(t ) dt q(t) = q0 exp − AA hν −∞  ≈ q0

σA 1− AA hν



t −∞

  2  A(t ) dt +

σA2 2(AA hν)2



t −∞

  2  A(t ) dt

2  ,

(6.38) and analogous for g(t). In this case we can obtain an analytical solution with a sech2 -pulse shape:

 t A(t) = A0 sech (6.39) , τ and a FWHM pulse duration τp ≈ 1.76 ×

4 1 , π
νg

(6.40)

where
νg is the FWHM gain bandwidth of the laser. In eq. (6.40) the conditions (eqs. (6.34)–(6.36)) are assumed and in addition Esat,L  Esat,A and Ep  Esat,A (i.e. a fully saturated absorber). For the example of Rhodamine 6G and DODCI (Table 8) we then obtain for a gain bandwidth of
νg ≈ 4 × 1013 Hz a pulse duration of about 56 fs (eq. (6.40)). In contrast, pulses as short as 27 fs have been demonstrated (Valdmanis, Fork

1, § 6]

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and Gordon [1985]). However, it was recognized early on that SPM together with negative dispersion results in soliton formation and further reduces pulse duration by about a factor of 2 in dye lasers (Martinez, Fork and Gordon [1984, 1985]). This would explain the difference in the theoretical prediction of eq. (6.40) from the experimentally demonstrated 27 fs. However, at that time an analytic solution for the pulse shortening effect was not presented. For semiconductor lasers we typically observe strongly chirped pulses (Jiang and Bowers [1995]). Therefore, we would have to include dispersion and selfphase modulation in the rate equation. This is not so easy because we would also have to include the refractive index change that occurs during gain saturation.

6.5. Passive mode locking with a fast saturable absorber In passive mode locking the loss modulation is obtained by self-amplitude modulation (SAM), where the pulse saturates an absorber. In the ideal case, the SAM follows the intensity profile of the pulse. This is the case of an ideal fast saturable absorber. In this case, SAM produces a much larger curvature of loss modulation than in the sinusoidal loss modulation of active mode locking, because the modelocked pulse duration is much shorter than the cavity roundtrip time. Therefore, we would expect from the previous discussion of active mode locking, that we obtain much shorter pulses with passive mode locking. This is indeed observed. In the fast saturable absorber model no dynamic gain saturation is required and the short net-gain window is formed by a fast recovering saturable absorber alone (fig. 3b). This was initially believed to be the only stable approach to passively mode-locked solid-state lasers with long upper state lifetimes. Additive pulse mode locking (APM) was the first fast saturable absorber for such solidstate lasers (Section 4.4). However, APM required interferometric cavity length stabilization. Kerr lens mode locking (KLM) (Spence, Kean and Sibbett [1991]) (Section 4.4) was the first useful demonstration of an intracavity fast saturable absorber for a solid-state laser and because of its simplicity replaced coupled cavity mode-locking techniques. KLM is very close to an ideal fast saturable absorber, where the modulation depth is produced either by the decreased losses because of self-focusing through a hard aperture (Keller, ’t Hooft, Knox and Cunningham [1991], Salin, Squier and Piché [1991], Negus, Spinelli, Goldblatt and Feugnet [1991]) or by increased gain in the laser as a result of an increased overlap of the laser mode with the pump mode in the laser crystal (Piché and Salin [1993]). Only in the ultrashort pulse regime of a few optical cycles more complicated space– time coupling occurs and wavelength dependent effects start to limit further pulse reduction.

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Besides the tremendous success of KLM, there are some significant limitations for practical or “real-world” ultrafast lasers. First, the cavity is typically operated near one end of its stability range, where the Kerr-lens-induced change of the beam diameter is large enough to sustain mode locking. This results in a requirement for critical cavity alignment where mirrors and the laser crystal have to be positioned to an accuracy of several hundred microns. Once the cavity is correctly aligned, KLM can be very stable and under certain conditions even selfstarting. However, self-starting KLM lasers in the sub-50-fs regime have not yet been demonstrated without any additional starting mechanisms. This is not surprising, since in a 10 fs Ti:sapphire laser with a 100 MHz repetition rate, the peak power changes by 6 orders of magnitude when the laser switches from cw to pulsed operation. Therefore, nonlinear effects that are still effective in the sub10 fs regime are typically too small to initiate mode locking in the cw-operation regime. In contrast, if self-starting is optimized, KLM tends to saturate in the ultrashort pulse regime or the large self-phase modulation (SPM) will drive the laser unstable. For an ideal fast saturable absorber, Haus, Fujimoto and Ippen [1992] developed an analytic solution for the pulse width starting with the following master equation: 

   1 ∂2
Ai = g 1 + 2 2 − l + γA |A|2 A(T , t) = 0, (6.41) Ωg ∂t i

not taking into account GVD and SPM. The solution is an unchirped sech2 -pulse shape

 t P (t) = P0 sech2 (6.42) τ with a FWHM pulse width: τp = 1.7627 · τ = 1.7627

4Dg , γ A Ep

(6.43)

where P0 is the peak power of the pulse, Ep is the intracavity pulse energy and Dg is the gain dispersion of the laser medium (eq. (6.10)). The shortest possible pulses can be obtained when we use the full modulation depth of the fast saturable absorber. We only obtain an analytic solution if we assume an ideal fast absorber that saturates linearly with the pulse intensity (eq. (4.13)) over the full modulation depth. This is clearly a strong approximation because eq. (4.13) only holds for weak absorber saturation. For a maximum modulation depth and this linear approximation we then can assume that γA P0 = q0 .

1, § 6]

Mode-locking techniques

For a sech2 -shaped pulse (eq. (6.42)), the pulse energy is given by  Ep = P (t) dt = 2τ P0 .

79

(6.44)

The minimal pulse width for a fully saturated ideal fast absorber then follows from eq. (6.43):  1.7627 2g . τp,min = (6.45) Ωg q0 This occurs right at the stability limit when the filter loss due to gain dispersion is equal to the residual loss a soliton undergoes in an ideal fast saturable absorber (eq. (4.14)): Dg q0 = qs = residual saturable absorber loss. = (6.46) 3 3τ 2 The residual saturable absorber loss qs results from the fact that the soliton pulse initially experiences loss to fully saturate the absorber (see Section 4.2.2). This residual loss is exactly q0 /3 for a sech2 -pulse shape and a fully saturated ideal fast saturable absorber. This condition results in a minimal FWHM pulse duration given by eq. (6.45). Including GVD and SPM, i.e. soliton formation, in the fast saturable absorber model, an additional pulse shortening of a factor of 2 was predicted. However, unchirped soliton pulses (i.e. ideal sech2 -shaped pulses) are only obtained for a certain negative dispersion value given by Filter loss =

|D| Dg = . δL γA

(6.47)

This is also where we obtain the shortest pulses with a fast saturable absorber. Here we assume that higher-order dispersion is fully compensated or negligibly small. In addition, computer simulations show that too much self-phase modulation drives the laser unstable. KLM is well described by the fast absorber mode-locking model discussed above even though it is not so easy to determine the exact saturable absorber parameters such as the effective saturation fluence. However, the linearized model does not describe the pulse generation with Ti:sapphire lasers in the sub-10fs regime very well. Pulse-shaping processes in these lasers are more complex (Brabec, Spielmann and Krausz [1991], Krausz, Fermann, Brabec, Curley, Hofer, Ober, Spielmann, Wintner and Schmidt [1992]). Under the influence of the different linear and nonlinear pulse shaping mechanisms, the pulse is significantly broadened and recompressed, giving rise to a “breathing” of the pulse width. The

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[1, § 6

order of the pulse shaping elements in the laser cavity becomes relevant and the spectrum of the mode-locked pulses becomes more complex. In this case, an analytical solution can no longer be obtained. As a rough approximation, the pulses still behave like solitons and consequently these lasers are also called solitary lasers (Brabec, Spielmann and Krausz [1991]).

6.6. Passive mode locking with a slow saturable absorber without gain saturation and soliton formation Over many years we consistently observed in experiments that even without soliton effects the pulse duration can be much shorter than the absorber recovery time in SESAM mode-locked solid-state lasers. It has always been postulated that without soliton pulse shaping, we need to have a fast saturable absorber for stable mode locking, which is in disagreement with our experimental observations. More recently, we therefore performed some more detailed numerical investigations (Paschotta and Keller [2001]). For a strongly saturated slow absorber with a saturation parameter S > 3 and an absorber recovery time smaller than 10 to 30 times the pulse duration, we found a useful guideline for the predicted pulse duration  g 1.5 τp,min ≈ (6.48) .
νg
R We neglected, similar to the ideal fast saturable absorber mode-locking model, the effects of Kerr nonlinearity and dispersion in the cavity, phase changes on the absorber and spatial hole burning in the gain medium. Compared to the analytical solution of a fully saturated absorber (eq. (6.45)) we would predict a slightly longer pulse duration given by a factor of about 1.3. Otherwise, the dependence with regards to gain saturation, gain bandwidth and absorber modulation depth has been explained very well with the analytical solution. Numerical simulations show that the pulse duration in eq. (6.48) can be significantly shorter than the absorber recovery time and has little influence on the pulse duration as long as τA < 10τp to 30τp . Numerical simulations show that this is a reasonable estimate and that with too long recovery time, the pulse does not simply become longer but unstable. At first this long recovery time might be surprising, because on the trailing edge of the pulse there is no shaping action of the absorber. There is even net gain, because the loss caused by the absorber is very small for the trailing edge (always assuming a fully saturated absorber), while the total loss for the pulse is larger and is balanced by the saturated gain in steady-state. Thus, one might expect that this

1, § 6]

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81

net gain after the pulse would destabilize the pulse – but this is not the case. The reason is that the pulses experience a temporal shift by the absorber, which limits the time in which noise behind the pulse can be amplified. The absorber attenuates mostly the leading wing of the pulse, thus shifting the pulse center backwards in each cavity round-trip. This means that the pulse is constantly moving backward and swallows any noise growing behind itself. An upper limit on the recovery time then follows from the condition that this noise cannot grow too much. Note that weak reflections in the laser cavity could generate weak satellite pulses behind the main pulse. These satellite pulses could be stronger than the noise level and thus significantly reduce maximum tolerable recovery time of the absorber. So far we have not included any additional pulse shaping effects such as SPM or solitons. Further simulations show that SPM alone in the positive dispersion regime should always be kept small because it always makes pulses longer and even destabilizes them, particularly for absorbers with small modulation depth (Paschotta and Keller [2001]). This result might be surprising at first – but again the temporal delay caused by the absorber gives a simple explanation. SPM (with positive n2 , as is the usual case) decreases the instantaneous frequency in the leading wing and increases the frequency in the trailing wing. The absorber always attenuates the leading wing, thus removing the lower frequency components, which results in an increased center frequency and broader pulses due to the decrease in pulse bandwidth. For strong SPM the pulses become unstable. A rule of thumb is that the nonlinear phase shift for the peak should be at most a few mrad per 1% of modulation depth. It is clear that SPM could hardly be made weak enough in the sub-picosecond regime. For this reason, soliton mode locking is usually required in the sub-picosecond domain, because there the nonlinear phase changes can be much larger.

6.7. Soliton mode locking In soliton mode locking, the pulse shaping is done solely by soliton formation, i.e. the balance of group velocity dispersion (GVD) and self-phase modulation (SPM) at steady state, with no additional requirements on the cavity stability regime as, for example, for KLM. In contrast to KLM we use only the time-dependent part of the Kerr effect at the peak intensity (i.e. n(t) = n + n2 I0 (t), eq. (6.20)) and not the radially dependent part as well (i.e. n(r, t) = n + n2 I (r, t), eq. (6.20)). The radially dependent part of the Kerr effect is responsible for KLM because it forms the nonlinear lens that reduces the beam diameter at an intracavity aperture inside the gain medium. Thus, this nonlinear lens forms an effective fast saturable

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[1, § 6

absorber because the intensity dependent beam diameter reduction at an aperture introduces less loss at high intensity and more loss at low intensity. Such a radially dependent effective saturable absorber however couples the mode-locking mechanism with the cavity mode. In contrast, soliton mode locking does not depend on the transverse Kerr effect and has therefore the advantage that the mode-locking mechanism is decoupled from the cavity design and no critical cavity stability regime is required – it basically works over the full cavity stability range. In soliton mode locking, an additional loss mechanism, such as a saturable absorber (Kärtner and Keller [1995]), or an acousto-optic modulator (Kärtner, Kopf and Keller [1995]), is necessary to start the mode-locking process and to stabilize the soliton pulse forming process. In soliton mode locking, we have shown that the net-gain window (fig. 3c) can remain open for significantly more than 10 times the width of the ultrashort pulse, depending on the specific laser parameters (Jung, Kärtner, Brovelli, Kamp and Keller [1995], Kärtner, Jung and Keller [1996]). This strongly relaxed the requirements on the saturable absorber and we can obtain ultrashort pulses even in the 10 fs regime with semiconductor saturable absorbers that have much longer recovery times. With the same modulation depth, one can obtain almost the same minimal pulse duration as with a fast saturable absorber, as long as the absorber recovery time is roughly less than ten times the final pulse width. In addition, high dynamic range autocorrelation measurements showed excellent pulse pedestal suppression over more than seven orders of magnitude in 130 fs pulses of a Nd:glass laser (Kopf, Kärtner, Keller and Weingarten [1995]) and very similar to or even better than KLM pulses in the 10-fs pulse-width regime (Jung, Kärtner, Henkmann, Zhang and Keller [1997]). Even better performance can be expected if the saturable absorber also shows a negative refractive index change coupled with the absorption change as is the case for semiconductor materials (Kärtner, Jung and Keller [1996]). With a slow saturable absorber as a starting and stabilizing mechanism for soliton mode locking, there remains a time window with net round-trip gain behind the pulse, where the loss of the still saturated absorber is smaller than the total loss for the pulse that is balancing the saturated gain. At first glance it may seem that the discussion in the last section about slow saturable absorbers would apply here as well. However, there is another limiting effect which usually becomes more effective in soliton mode-locked lasers: the dispersion causes the background to temporally broaden and thus permanently loosing the energy in those parts which drift into the time regions with net loss. We can describe soliton mode locking by the Haus’s master equation formalism, where we take into account GVD, SPM and a slow saturable absorber q(T , t) that recovers slower than the pulse duration (see fig. 3c) (Kärtner and Keller

1, § 6]

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[1995], Kärtner, Jung and Keller [1996]):     2 ∂2  
Ai = −iD 2 + iδL A(T , t) A(T , t) ∂t i

  ∂2 + g − l + Dg 2 − q(T , t) A(T , t) = 0. ∂t

(6.49)

This differential equation can be solved analytically using the soliton perturbation theory. The first bracket term of this differential equation determines the nonlinear Schrödinger equation for which the soliton pulse is a stable solution for negative GVD (i.e. D < 0) and positive SPM (i.e. n2 > 0):

 t −iφs (z) e A(z, t) = A0 sech (6.50) . τ This soliton pulse propagates without distortion through a medium with negative GVD and positive SPM because the effect of SPM exactly cancels that due to dispersion. The FWHM soliton pulse duration is given by τp = 1.7627 · τ and the time-bandwidth product by
νp τp = 0.3148. φs (z) is the phase shift of the soliton during propagation along the z-axis 1 1 φs (z) = kn2 I0 z ≡ φnl (z), (6.51) 2 2 where I0 is the peak intensity inside the SPM medium (i.e. typically the gain medium). Thus, the soliton pulse experiences during propagation a constant phase shift over the full beam profile. For a given negative dispersion and an intracavity pulse energy Ep we obtain a pulse duration τp = 1.7627

2|D| δ L Ep

(6.52)

for which the effects of SPM and GVD are balanced with a stable soliton pulse. This soliton pulse loses energy due to gain dispersion and losses in the cavity. Gain dispersion and losses can be treated as perturbation to the nonlinear Schrödinger equation for which a soliton is a stable solution (i.e. second bracket term in eq. (6.50)). This lost energy, called continuum in soliton perturbation theory, is initially contained in a low intensity background pulse, which experiences negligible SPM, but spreads in time due to GDD (fig. 10a). In soliton mode locking a stable soliton pulse is formed for all group delay dispersion values as long as the continuum loss is larger than the soliton loss (eq. (6.53)) or the pulses break up in two or more pulses (Aus der Au, Kopf, Morier-Genoud, Moser and Keller [1997]). Thus, for smaller negative dispersion the pulse duration becomes shorter

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[1, § 6

(a)

(b) Fig. 10. Soliton modelocking in (a) time and (b) frequency domain. The continuum pulse spreads in time due to group velocity dispersion and thus undergoes more loss in the relatively slow absorber, which is saturated by the shorter soliton pulse. However, the longer continuum pulse has a narrower spectrum and thus undergoes more gain than the spectrally broader soliton pulse.

(eq. (6.52)) until minimal pulse duration is reached. With no pulse-break-up, the minimal pulse duration is given when the loss for the continuum pulse becomes equal to the loss of the soliton pulse. When the soliton pulse is stable, then the saturated gain is equal to the loss: 

 Dg Ep Esat,A 1 − exp − , g = l + ls , with ls = 2 + q0 (6.53) 3τ Ep Esat,A where l is the total saturated amplitude loss coefficient per cavity roundtrip (i.e. output coupler, residual cavity losses and nonsaturable absorber loss of the saturable absorber) and ls the additional loss experienced by the soliton as a result of gain filtering (eq. (6.46)) and the amplitude loss coefficient for saturation of the slow absorber (eq. (4.10)). Soliton perturbation theory then determines the roundtrip loss of the continuum pulse (Kärtner, Jung and Keller [1996]). The continuum is spread in time because of dispersion and therefore experiences enhanced

1, § 6]

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85

loss in the recovering absorber that has been saturated by the much shorter soliton pulse. Solitons alone in the cavity are not stable. The continuum pulse is much longer and therefore experiences only the gain at line center, while the soliton exhibits an effectively lower average gain due to its larger bandwidth. Thus, the continuum exhibits a higher gain than the soliton. After a sufficient build-up time, the continuum would actually grow until it reached lasing threshold, destabilizing the soliton. However, we can stabilize the soliton by introducing a relatively slow saturable absorber into the cavity. This absorber is fast enough to add sufficient additional loss for the growing continuum that spreads in time during its build-up phase so that it no longer reaches lasing threshold. We then predict a minimum pulse duration for a soliton pulse (Kärtner and Keller [1995], Kärtner, Jung and Keller [1996]): 3/4



3/2 1/4 1 −1/8 τA g φs τp,min = 1.7627 √ q0 6Ωg



 1 3/4 τA 1/4 g 3/8 ≈ 0.45 , (6.54) 1/8
νg
R φs where φs is the phase shift of the soliton per cavity round trip (assuming that the dominant SPM occurs in the laser gain medium),
νg the FWHM gain bandwidth and
R ≈ 2q0 (eq. (4.1)). With eq. (6.51) then follows φs = φs (z = 2Lg ) = kn2 Lg I0,L = 12 δL P0 , where δL is the SPM coefficient (eq. (6.25)) and P0 the peak power inside the laser cavity. In eq. (6.54) we assume a fully saturated slow saturable absorber and a linear approximation for the exponential decay of the slow saturable absorber. The analytical solution for soliton mode locking (eq. (6.54)) has been experimentally confirmed with a Ti:sapphire laser where a Fabry–Perot filter has been inserted to give a well-defined gain bandwidth (Jung, Kärtner, Brovelli, Kamp and Keller [1995]). However, this equation still does not tell us what soliton phase shift would be ideal. Equation (6.54) would suggest that very high values are preferred, which actually leads to instabilities. Also this equation is not taking into account that the soliton pulse may break up into two solitons which occurs more easily if the absorber is too strongly saturated. Numerical simulations can give better estimates for these open questions (Paschotta and Keller [2001]). In soliton mode locking, the dominant pulse formation process is assumed to be soliton formation. Therefore, the pulse has to be a soliton for which the negative GVD is balanced with the SPM inside the laser cavity. The pulse duration is then given by the simple soliton solution (eq. (6.52)). This means that the pulse duration scales linearly with the negative group delay dispersion inside the laser cavity

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Ultrafast solid-state lasers

[1, § 6

(i.e. τp ∝ |D|). In the case of an ideal fast saturable absorber, an unchirped soliton pulse is only obtained at a very specific dispersion setting (eq. (6.47)), whereas for soliton mode locking an unchirped transform-limited soliton is obtained for all dispersion levels as long as the stability requirement against the continuum is fulfilled. This fact has been also used to experimentally confirm that soliton mode locking is the dominant pulse formation process and not a fast saturable absorber such as KLM (Aus der Au, Kopf, Morier-Genoud, Moser and Keller [1997]). Higher-order dispersion only increases the pulse duration, therefore is undesirable and is assumed to be compensated. The break-up into two or even three pulses can be explained as follows: Beyond a certain pulse energy, two soliton pulses with lower power, longer duration, and narrower spectrum will be preferred, since their loss introduced by the limited gain bandwidth decreases so much that the slightly increased residual loss of the less saturated saturable absorber cannot compensate for it. This results in a lower total roundtrip loss and thus a reduced saturated or average gain for two pulses compared to one pulse. The threshold for multiple pulsing is lower for shorter pulses, i.e. with spectra which are broad compared to the gain bandwidth of the laser. A more detailed description of multiple pulsing is given elsewhere (Kärtner, Aus der Au and Keller [1998]). Numerical simulations show, however, that the tendency for pulse break-up in cases with strong absorber saturation is found to be significantly weaker than expected from simple gain/loss arguments (Paschotta and Keller [2001]). To conclude, soliton shaping effects can allow for the generation of significantly shorter pulses, compared to cases without SPM and dispersion. The improvement is particularly large for absorbers with a relatively low modulation depth and when the absorber recovery is not too slow. In this regime, the pulse shaping is mainly done by the soliton effects, and the absorber is only needed to stabilize the solitons against growth of the continuum. The absorber parameters are generally not very critical. It is important not only to adjust the ratio of dispersion and SPM to obtain the desired soliton pulse duration (eq. (6.52)), but also to keep their absolute values in a reasonable range where the nonlinear phase change is in the order of a few hundred mrad per round-trip (i.e. significantly larger than acceptable in cases without negative dispersion). Soliton formation is generally very important in femtosecond lasers, which has been already recognized in colliding pulse-mode-locked dye lasers. However, no analytic solution was presented for soliton pulse shortening. It was always assumed that for a stable solution the mode-locking mechanism without soliton effects has to generate a net gain window as short as the pulse (fig. 3a and b). In contrast to these cases, in soliton mode locking we present an analytic solution based on soliton pertur-

1, § 6]

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87

bation theory, where soliton pulse shaping is clearly assumed to be the dominant pulse formation process, and the saturable absorber required for a stable solution is treated as a perturbation. Then, the net gain window can be much longer than the pulse (fig. 3c). Stability of the soliton against the continuum then determines the shortest possible pulse duration. This is a fundamentally different mode-locking model than previously described. We therefore refer to it as soliton mode locking, emphasizing the fact that soliton pulse shaping is the dominant factor.

6.8. Design guidelines to prevent Q-switching instability For picosecond solid-state lasers the self-amplitude modulation of a saturable absorber with a picosecond or tens of picoseconds recovery time is sufficient for stable pulse generation. A picosecond recovery time can be achieved with low-temperature grown semiconductor saturable absorbers where mid-gap defect states form very efficient traps for the photoexcited electrons in the conduction band (Haiml, Siegner, Morier-Genoud, Keller, Luysberg, Lutz, Specht and Weber [1999]). In the picosecond regime, we developed a very simple stability criteria for stable passive mode locking without Q-switching instabilities (Hönninger, Paschotta, Morier-Genoud, Moser and Keller [1999]): 2 Ep2 > Ep,c = Esat,L Esat,A
R.

(6.55)

The critical intracavity pulse energy Ep,c is the minimum intracavity pulse energy, which is required to obtain stable cw mode locking, that is, for Ep > Ep,c we obtain stable cw mode locking and for Ep < Ep,c we obtain Q-switched mode locking. For good stability of a mode-locked laser against unwanted fluctuations of pulse energy, operation close to the stability limit is not recommended. Thus, a large modulation depth supports shorter pulses (Table 9), but an upper limit is given by the onset of self-Q-switching instabilities (eq. (6.55)). In the femtosecond regime, we observe a significant reduction of the tendency of Q-switching instabilities compared to pure saturable absorber mode-locked picosecond lasers (eq. (6.55). This can be explained as follows: if the energy of an ultrashort pulse rises slightly due to relaxation oscillations, SPM and/or SAM broadens the pulse spectrum. A broader pulse spectrum, however, increases the loss due to the finite gain bandwidth (eq. (6.46)), which provides some negative feedback, thus decreasing the critical pulse energy which is necessary for stable cw mode locking. The simple stability requirement of eq. (6.55) then has to

88

Ultrafast solid-state lasers

[1, § 6

Table 9 Predicted pulse duration for the different mode-locking (ML) techniques. The parameters used here are summarized and defined in Table 7. The pulse is transform-limited for proper dispersion compensation and is either an unchirped Gaussian or soliton pulse ML technique

Eq.

Active ML: Amplitude loss modulation

(6.27) Gauss

τp = 1.66 ×

Passive ML: Slow saturable absorber and dynamic gain saturation

(6.40) Soliton

1 τp ≈ 1.76 × π4
ν g

Slow saturable absorber for solid-state lasers and strongly saturated absorbers (S > 3)

1.5 (6.48) Numerical τp,min ≈
ν g simulations

Passive ML: Fast saturable absorber (FSA)

(6.43) Soliton

Fully saturated ideal fast saturable absorber

(6.45)

only transform-limited soliton pulses for a well-defined intracavity group delay dispersion (assuming negligible higher order dispersion): |D|/δL = Dg /γA   Dg 2g g |D| 1.12 τp,min = 1.76 q ≈
νg
R , for δ = γ Ωg

Passive ML: Soliton mode locking

(6.52) Soliton

τp = 1.76 δ2|D| Ep

(6.54)

Pulse shape Pulse duration (FWHM)  4 Dg Ms



= 1.66 ×



4 2g



M

1 ωm Ω g

g
R , for τA
30τp

4D

τp = 1.76 γ Egp A

0

L

A

L

transform-limited soliton pulses for all intracavity group delay dispersion (assuming negligible higher order dispersion)  3/4 −1/8  τA g3/2 1/4 φs τp,min = 1.7627 √ 1 q0 6Ωg  1 3/4  τA 1/4 g3/8 ≈ 0.45
ν 1/8 ,
R g

φs

be modified as follows (Hönninger, Paschotta, Morier-Genoud, Moser and Keller [1999]): Esat,L gK 2 Ep2 + Ep2 > Esat,L Esat,A
R,

(6.56)

where K is given by K≡

0.315 2πn2 Lg . 1.76 DAL λ0
νg

(6.57)

1, § 6]

Mode-locking techniques

89

Here we assume a standing wave cavity and that the dominant SPM is produced in the laser medium. In other cases we have to add all other contributions as well.

6.9. External pulse compression SPM generates extra frequency components. The superposition with the other frequency components in the pulse can spectrally broaden the pulse (i.e. for positively chirped pulses assuming n2 > 0). SPM alone does not modify the pulse width, but with proper dispersion compensation a much shorter pulse can be generated with the extra bandwidth (Mollenauer, Stolen and Gordon [1980]). The positively chirped spectrally broadened pulse then can be compressed with appropriate negative dispersion compensation, where the “blue” spectral components have to be advanced relative to the “red” ones. A careful balance between a nonlinear spectral broadening process and negative dispersion is needed for efficient compression of a pulse. Typically, self-phase modulation in a single-mode fiber with optimized length was used to linearly chirp the pulse, which is then compressed with a grating pair compressor (Tomlinson, Stolen and Shank [1984]). Ultimately, uncompensated higher-order dispersion and higher-order nonlinearities limit compression schemes. For pulses shorter than 100 fs, compression is typically limited to factors of less than 10. Compression of amplified CPM dye laser pulses with 50 fs duration produced the long-standing world record of 6 fs (Fork, Cruz, Becker and Shank [1987]). Similar concepts have been recently used for external pulse compression of 13-fs pulses from a cavity dumped Ti:sapphire laser (Baltuska, Wei, Pshenichnikov, Wiersma and Szipöcs [1997]) and of 20-fs pulses from a Ti:sapphire laser amplifier (Nisoli, Stagira, Silvestri, Svelto, Sartania, Cheng, Lenzner, Spielmann and Krausz [1997]) resulting, in both cases, in approximately 4.5-fs pulses. In the latter case, the use of a noble-gas-filled hollow fiber resulted in pulse energies of about 0.5 mJ with 5.2 fs pulses and a peak power of 0.1 TW (Sartania, Cheng, Lenzner, Tempea, Spielmann, Krausz and Ferencz [1997]). More recently, adaptive pulse compression of a super-continuum produced in two cascaded hollow fibers generated 3.8 fs pulses with energies of up to 15 µJ at 1 kHz (Schenkel, Biegert, Keller, Vozzi, Nisoli, Sansone, Stagira, De Silvestri and Svelto [2003]). The pulse duration was fully characterized by spectral-phase interferometry for direct electric-field reconstruction (SPIDER). This is currently the world record in the visible and near-infrared spectral regime. The extra bandwidth obtained with SPM can be extremely large producing a white-light continuum (Fork, Shank, Hirlimann, Yen and Tomlinson [1983]), which can be used as a seed for broadband parametric amplification. Parametric processes can provide amplification with an even broader bandwidth than

90

Ultrafast solid-state lasers

[1, § 7

can typically be achieved in laser amplifiers. Noncollinear phase-matching at a crossing angle of 3.8◦ in Barium beta borate (BBO) provides more than 150 THz amplification bandwidth (Gale, Cavallari, Driscoll and Hache [1995]). With this type of set-up, parametric amplification has been successfully demonstrated with pulse durations of less than 5 fs (Shirakawa, Sakane, Takasaka and Kobayashi [1999], Zavelani-Rossi, Polli, Cerullo, De Silvestri, Gallmann, Steinmeyer and Keller [2002]). Much higher average power in compressed pulses can be achieved in combination with a passively mode-locked Yb:YAG thin-disk laser (Innerhofer, Südmeyer, Brunner, Häring, Aschwanden, Paschotta, Hönninger, Kumkar and Keller [2003]). Pulse compression after this laser resulted in 33-fs pulses with an average power of 18 W at a pulse repetition rate of 34 MHz. This gives a peak power of 12 MW (Südmeyer, Brunner, Innerhofer, Paschotta, Furusawa, Baggett, Monro, Richardson and Keller [2003]).

§ 7. Pulse characterization 7.1. Electronic techniques Electronic techniques for pulse-width measurements are limited to the picosecond regime. Today, photo detectors and sampling heads with bandwidths up to 60 GHz are commercially available. This means that the measured pulse duration is limited to about 7 ps. This follows from simple linear system analysis for which the impulse response of a photo detector or a sampling head can be approximated by a Gauss-function. The impulse response for a given system bandwidth B has a FWHM τFWHM in units of ps: τFWHM [ps] ≈

312 GHz . B[GHz]

(7.1)

The impulse response of a measurement system can be determined from the impulse response of each element in the detection chain:  2 τFWHM (7.2) = τ12 + τ22 + τ32 + · · ·, where, for example, τ1 is the FWHM of the impulse response of the photo detector, τ2 of the sampling head and so on. Thus, with a 40 GHz sampling head and a 60 GHz photo detector we only measure a impulse response with FWHM of 9.4 ps.

1, § 7]

Pulse characterization

91

7.2. Optical autocorrelation Optical autocorrelation techniques with second harmonic generation of two identical pulses that are delayed with respect to each other are typically used to measure shorter pulses (Sala, Kenney-Wallace and Hall [1980], Weiner [1983]). We distinguish between collinear and non-collinear intensity autocorrelation measurements for which the two pulse beams in the nonlinear crystal are either collinear or non-collinear. In the non-collinear case the second harmonic signal only depends on the pulse overlap and is given by  I2ω (
t) ∝ I (t)I (t −
t) dt. (7.3) The FWHM of the autocorrelation signal I2ω (
t) is given by τAu and determines the FWHM pulse duration τp of the incoming pulse I (t). However, τAu depends on the specific pulse shape (Table 10) and any phase information is lost in this measurement. So normally, transform-limited pulses are assumed. This assumption is only justified as long as the measured spectrum also agrees with the assumption of pulse shape and constant phase (i.e. the time bandwidth product corresponds to a transform-limited pulse, Table 10). For passively mode-locked lasers, for example, that allow for parabolic approximation of pulse formation mechanisms one expects a sech2 temporal and spectral pulse shape (Section 6). Passively mode-locked solid-state lasers with pulse durations well above 10 fs normally generate pulses close to this ideal sech2 -shape. Therefore, this noncollinear autocorrelation technique is a good standard diagnostic for such laser sources. For ultrashort pulse generation in the sub-10-fs regime this is generally not the case anymore. Experimentally, this is clearly indicated by more complex pulse spectra. Interferometric autocorrelation (IAC) techniques (Mindl, Hefferle, Schneider and Dörr [1983]) have been used to get more information. In IAC a collinear inTable 10 Optical pulses: defining equations for Gaussian and soliton pulse shapes, FWHM (full width half maximum) intensity pulse duration τp , time-bandwidth products
νp τp for which
νp is the FWHM of the spectral intensity, FWHM intensity autocorrelation pulse duration τAu τp /τ


νp τp

τp /τAu

 2 I (t) ∝ exp t 2

√ 2 ln 2

0.4413

0.7071

Soliton  I (t) ∝ sech2 τt

1.7627

0.3148

0.6482

Pulse shape Gauss

τ

92

Ultrafast solid-state lasers

[1, § 7

tensity autocorrelation is fringe resolved and gives some indication how well the pulse is transform-limited. However, we still do not obtain full phase information about the pulse. The temporal parameters have usually been obtained by fitting an analytical pulse shape with constant phase to the autocorrelation measurement. Theoretical models of the pulse formation process motivate the particular fitting function. For lasers obeying such a model the a priori assumption of a theoretically predicted pulse shape is well-motivated and leads to good estimates of the pulse duration as long as the measured spectrum also agrees with the theoretical prediction. However, we have seen that fits to an IAC trace with a more complex pulse spectrum tend to underestimate the true duration of the pulses.

7.3. FROG and SPIDER For a more precise measurement, a variety of methods have been proposed to fully reconstruct both pulse amplitude and phase from measured data only (Chilla and Martinez [1991], Kane and Trebino [1993], Chu, Heritage, Grant, Liu, Dienes, White and Sullivan [1995], Rhee, Sosnowski, Tien and Norris [1996], Iaconis and Walmsley [1998]). Of these methods, especially frequency-resolved optical gating (FROG, Kane and Trebino [1993] and Trebino, DeLong, Fittinghoff, Sweetser, Krumbügel and Richman [1997]) and spectral phase interferometry for direct electric-field reconstruction (SPIDER, Iaconis and Walmsley [1998] and Iaconis and Walmsley [1999]) have found widespread use. FROG is a characterization method based on the measurement of a spectrally resolved autocorrelation signal followed by an iterative phase-retrieval algorithm to extract the intensity and phase of the laser pulse. SPIDER is a self-referencing variant of spectral interferometry. Conventional spectral interferometry measures the spectral phase differences between two pulses. To access the spectral phase of a single pulse, SPIDER generates a spectral shear between the carrier frequencies of two replicas of this pulse. The spectral shear is generated by upconversion of the two replicas with a strongly chirped pulse using sum-frequency generation in a nonlinear optical crystal. The phase information of the resulting interferogram allows the direct reconstruction of the spectral phase of the input pulse. In the sub-10-femtosecond regime non-collinear FROG (Baltuska, Pshenichnikov and Wiersma [1998], Cheng, Fürbach, Sartania, Lenzner, Spielmann and Krausz [1999], Durfee, Backus, Kapteyn and Murnane [1999]), collinear FROG (Gallmann, Steinmeyer, Sutter, Matuschek and Keller [2000]), and SPIDER (Gallmann, Sutter, Matuschek, Steinmeyer, Keller, Iaconis and Walmsley [1999]) have been used. The applicability of a particular measure-

1, § 8]

Carrier envelope offset (CEO)

93

ment technique in this pulse duration regime is not solely determined by possible bandwidth limitations but also by the requirements on the nonlinear optical process or the beam geometry involved (Gallmann, Sutter, Matuschek, Steinmeyer and Keller [2000]). SPIDER offers more bandwidth than any other technique and still gives valid results even if the beam profile is not spatially coherent anymore (Gallmann, Steinmeyer, Sutter, Rupp, Iaconis, Walmsley and Keller [2001]). Thus, we can measure pulses even in the single-cycle regime (Schenkel, Biegert, Keller, Vozzi, Nisoli, Sansone, Stagira, De Silvestri and Svelto [2003]). SPIDER even allows for kHz single shot characterization (Kornelis, Biegert, Tisch, Nisoli, Sansone, De Silvestri and Keller [2003]). Only fully characterized pulses in phase and amplitude will provide reliable information about pulse shape and pulse duration – this becomes even more important for pulses with very broad and complex spectra. Any other technique, such as fitting attempts to IAC traces, is very erroneous and generally underestimates the pulse duration.

§ 8. Carrier envelope offset (CEO) Progress in ultrashort pulse generation (Steinmeyer, Sutter, Gallmann, Matuschek and Keller [1999]) has reached a level where the slowly varying envelope approximation starts to fail. Pulse durations of Ti:sapphire laser oscillators have reached around 5 fs which is so short that only about two optical cycles of the underlying electric field fit into the full-width half maximum of the pulse envelope. More recently, even shorter pulses of 3.8 fs have been achieved with adaptive pulse compression of a cascaded hollow fiber supercontinuum (Schenkel, Biegert, Keller, Vozzi, Nisoli, Sansone, Stagira, De Silvestri and Svelto [2003]). These are currently the shortest pulses generated in the visible and near infrared spectral regime. For such short pulses the maximum electric field strength depends strongly on the exact position of the electric field with regards to the pulse envelope (fig. 11), i.e. the carrier envelope offset (CEO) (Telle, Steinmeyer, Dunlop, Stenger, Sutter and Keller [1999]). In passively mode-locked lasers this carrier envelope offset is a freely varying parameter because the steady-state boundary condition only requires that the pulse envelope is the same after one round-trip. Therefore the CEO phase may exhibit large fluctuations, even when all other laser parameters are stabilized. We have discussed the physical origin of these fluctuations before (Helbing, Steinmeyer, Keller, Windeler, Stenger and Telle [2001], Helbing, Steinmeyer, Stenger, Telle and Keller [2002]). Because nonlinear laser– matter interaction depends strongly on the strength of the electric field, these

94

Ultrafast solid-state lasers

[1, § 8

Fig. 11. Electric field of a few-cycle pulse for different values of the carrier-envelope-offset (CEO) phase of 0 (solid line) and π/2 (dashed line).

CEO fluctuations cause strong signal fluctuations in nonlinear experiments such as high harmonic generation (Brabec and Krausz [2000]), attosecond pulse generation (Drescher, Hentschel, Kienberger, Tempea, Spielmann, Reider, Corkum and Krausz [2001]), photoelectron emission (Paulus, Grasborn, Walther, Villoresi, Nisoli, Stagira, Priori and Silvestri [2001]), etc. Different techniques have been proposed to stabilize these CEO fluctuations in the time domain (Xu, Spielmann, Poppe, Brabec, Krausz and Haensch [1996]) and in the frequency domain (Telle, Steinmeyer, Dunlop, Stenger, Sutter and Keller [1999]). The frequency domain technique is much more sensitive and is the technique that is being used today. Using the f -to-2f heterodyne technique with additional external spectral broadening (Jones, Diddams, Ranka, Stentz, Windeler, Hall and Cundiff [2000], Apolonski, Poppe, Tempea, Spielmann, Udem, Holzwarth, Hänsch and Krausz [2000]) we achieved a long-term CEO stabilization with residual 10 attosecond timing jitter which corresponds to 0.025 rad rms CEO phase noise in a (0.01 Hz–100 kHz) bandwidth (Helbing, Steinmeyer, Stenger, Telle and Keller [2002]). The f -to-2f interference technique requires an octave-spanning spectrum. So far, all attempts to generate this spectrum directly from a laser source have only reached unsatisfactory control of the CEO frequency, which was mainly caused by poor signal strength. Therefore most experiments required additional spectral broadening, e.g., in an external microstructure fiber. The continuum generation process with its strong nonlinearity, however, introduces additional CEO noise. It is important to note that CEO stabilization is achieved for the pulses after the microstructure fiber, which means that the pulses directly from the Ti:sapphire laser will exhibit excess CEO phase noise even with a perfectly working CEO stabilization. In our experiments we observed

1, § 9]

Outlook

95

typical phase drift values of about 1.2 rad/mW oscillator power fluctuations (Helbing, Steinmeyer and Keller [2003]). The relative phase between the carrier and the envelope of an optical pulse is the key parameter linking the fields of precision frequency metrology and ultrafast laser physics (Udem, Holzwarth and Hänsch [2002]). As we have discussed, the spectrum of a mode-locked laser consists of a comb of precisely equally spaced frequencies. The uniformity of this frequency comb has been demonstrated to a relative uncertainty below 10−15 (Udem, Reichert, Holzwarth and Hänsch [1999]). Knowledge of only two parameters, the comb spacing and a common offset frequency of all modes, provides one with a set of reference frequencies, similar to the tick marks on a ruler. The beat (difference-frequency) signal of this unknown frequency with the closest frequency in the ruler always gives a beat signal smaller than half the comb period. Thus, an optical frequency in the 100 THz regime can be translated down to a microwave frequency in the range of 100 MHz, which can then be measured very accurately. This can be used for an all-optical atomic clock that is expected to outperform today’s state-of-the-art caesium clocks (Udem, Holzwarth and Hänsch [2002]). The mode-locked laser provides a phase-locked clockwork mediating between radio frequencies and the terahertz frequencies of the lines in the optical comb, effectively rendering optical frequencies countable. Details on precision frequency measurements with mode-locked laser combs can be found in Reichert, Holzwarth, Udem and Hänsch [1999], Holzwarth, Zimmermann, Udem and Hänsch [2001], Cundiff, Ye and Hall [2001] and Bauch and Telle [2002].

§ 9. Outlook We have shown that the technology of ultrafast lasers has become very refined and is now suitable for application in many areas. Points of particular importance in this respect are: • The transition from dye lasers to solid-state lasers, which can be compact, powerful, efficient and long-lived. It has been shown that solid-state lasers can generate pulses which are even shorter than those generated in dye lasers with much more average output power. • The development of diode lasers for direct pumping of solid-state lasers. This has lead not only to very efficient and compact lasers, but also to mode-locked lasers with tens of watts of output power. • The development of novel saturable absorbers such as KLM which has pushed the frontier in ultrashort pulse duration into the two-optical-cycle regime.

96

Ultrafast solid-state lasers

[1, § 9

• The development of semiconductor saturable absorber mirrors (SESAMs) which passively mode-lock all kind of diode-pumped solid-state lasers without Q-switching instabilities and can be optimized for operation in very different parameter regimes such as laser wavelength, pulse duration, pulse repetition rate and power levels. For the next few years we expect many new exciting developments in the field of diode-pumped solid-state lasers: • For example, new solid-state gain materials, for example Cr2+ :ZnSe which appears to be suitable for the generation of pulses with 20 fs duration or less in a new spectral region around 2.7 µm. • Very high power levels (> 100 W of average power) should become possible with passively mode-locked thin disk lasers. Pulse durations just below 1 ps are feasible and with new materials the regime of 200 fs or even below should become accessible with similarly high powers. • Nonlinear frequency conversion stages (based on second-harmonic generation, sum frequency mixing, or parametric oscillation) will be pumped with high-power mode-locked lasers to generate short and powerful pulses at other wavelengths. The high power makes the nonlinear conversion efficiencies very high with very simple arrangements (Südmeyer, Aus der Au, Paschotta, Keller, Smith, Ross and Hanna [2001]). This will be of interest for application in largescreen RGB display systems, for example. • Simple external pulse compression combined with novel high-average-power solid-state lasers now allows for peak intensities as high as 12 MW with 33fs pulses at the full pulse repetition rate of the laser oscillator (Südmeyer, Brunner, Innerhofer, Paschotta, Furusawa, Baggett, Monro, Richardson and Keller [2003]). This could be focused to a peak intensity of 1014 W/cm2 , a regime where high field laser physics such as high harmonic generation (Ferray, L’Huillier, Li, Lompré, Mainfray and Manus [1988], Lewenstein, Balcou, Ivanov, L’Huillier and Corkum [1994], Salieres, L’Huillier, Antoine and Lewenstein [1999]) and laser plasma generated X-rays (Murnane, Kapteyn, Rosen and Falcone [1991]) are possible at more than 10 MHz pulse repetition rate. This improves signal-to-noise ratios in measurements by 4 orders of magnitude compared to the standard sources at kHz repetition rates. This would be important for low-power applications such as X-ray imaging and microscopy (Schmidt, Bauer, Wiemann, Porath, Scharte, Andreyev, Schönhense and Aeschlimann [2002]), femtosecond EUV and soft-X-ray photoelectron spectroscopy (Bauer, Lei, Read, Tobey, Gland, Murnane and Kapteyn [2001]) and ultrafast X-ray diffraction (Siders, Cavalleri, Sokolowski-Tinten, Toth, Guo, Kammler, Hoegen, Wilson, Linde and Barty [1999], Rousse, Rischel,

1, § 9]

Outlook

97

Fourmaux, Uschmann, Sebban, Grillon, Balcou, Förster, Geindre, Audebert, Gauthier and Hulin [2001]). • Very high pulse repetition rates with > 100 GHz should be possible from passively mode-locked diode-pumped solid-state lasers at different wavelength, even in the wavelength range around 1.5 µm and 1.3 µm for telecom applications. • As an alternative to ion-doped gain media, optically pumped semiconductor lasers with an external cavity and a SESAM for passive mode locking (Hoogland, Dhanjal, Roberts, Tropper, Häring, Paschotta, Morier-Genoud and Keller [2000]) will generate higher output powers and shorter pulses in the regime of multi-GHz pulse repetition rates (Häring, Paschotta, Gini, Morier-Genoud, Melchior, Martin and Keller [2001], Häring, Paschotta, Aschwanden, Gini, Morier-Genoud and Keller [2002]). Thus, all these examples show that the development of ultrafast diode-pumped sources has not come to an end but will continue to deliver superior performances for many established and new applications. In addition, research to produce pulses of even shorter duration is underway. Currently, the most promising path to attosecond pulse generation and attosecond spectroscopy is high-harmonic generation (recent reviews are given in Salieres, L’Huillier, Antoine and Lewenstein [1999] and DiMauro and Agostini [1995]). Nearly single-cycle-excitation optical pulses can produce harmonic frequencies in gases extending even into the water window (2.3–4.4 nm), and a continuum broad enough to support attosecond pulses. Key to this harmonic generation is intense optical pulses of a few cycles, since even slightly longer pulses result in reduced harmonic generation and a lower cut-off energy. It has been speculated early on that pulses from existing sources of high-harmonic generation exhibit attosecond time signature (Corkum [1995], Antoine, L’Huillier and Lewenstein [1996]). First measurements indicate single attosecond pulse duration (Hentschel, Kienberger, Spielmann, Reider, Milosevic, Brabec, Corkum, Heinzmann, Drescher and Krausz [2001]), but also demonstrate that measurement of these short pulses can be even a bigger challenge than the generation of the pulses themselves. We would expect that with attosecond time resolution we open up a new world of physics with as much impact as has been demonstrated in the 70s and 80s with the transition from picosecond to femtosecond time resolution. First time resolved measurements have been done (Drescher, Hentschel, Kienberger, Uiberacker, Yakovlev, Scrinzi, Westerwalbesloh, Kleineberg, Heinzmann and Krausz [2002]). At this point, however, we still have some significant challenges to tackle before we demonstrate standard attosecond pulse generation and attosecond spec-

98

Ultrafast solid-state lasers

[1

troscopy. Solving all of these challenges will make the research in ultrashort pulse generation very exciting and rewarding for many years to come.

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

119

§ 2. Characterization of randomly rough surfaces . . . . . . . . . . . . . .

121

§ 3. Equations for electromagnetic fields and observable quantities . . . .

126

§ 4. Weak localization effects in the multiple scattering of light from randomly rough surfaces. Enhanced backscattering . . . . . . . . . . . .

133

§ 5. Angular intensity correlation functions . . . . . . . . . . . . . . . . .

180

§ 6. Multiple-scattering effects in the scattering of light from complex media bounded by a rough surface . . . . . . . . . . . . . . . . . . . . .

189

§ 7. Near-field effects: localization phenomena for surface waves . . . . .

217

§ 8. Spectral changes induced by multiple scattering . . . . . . . . . . . .

221

§ 9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

229

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

233

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

233

118

§ 1. Introduction One of the first concepts studied in any course in optics or electromagnetic theory is the reflection and transmission of light on a planar interface between two different materials. The importance of this phenomenon is evident from our everyday experience since all material objects are bounded by surfaces. However, any realistic surface is rough to some degree, and most of us are used to making a distinction between the specular reflection caused by a planar surface component and the diffuse reflection, or scattering, caused by the deviation of the surface from planarity. The systematic investigation of the scattering of light from rough surfaces started in the beginning of the 20th century and for the most part of this century was focused on single-scattering phenomena and their theoretical description. Theoretically, the majority of the problems were treated by either a firstorder perturbation theory (small-height expansion) or the Kirchhoff approximation assuming smooth surface roughness with large horizontal scales. The reader interested in the single-scattering of light from rough surfaces is referred to numerous books and review articles addressing this mature subject (Beckmann and Spizzichino [1963], Shmelev [1972], Bass and Fuks [1980], DeSanto and Brown [1986], Ogilvy [1991], Nieto-Vesperinas [1991], Maradudin, Nieto-Vesperinas and Thorsos [1994a, 1994b], Voronovich [1994]). This review is devoted to the topic of multiple scattering of light from randomly rough surfaces. Some foundations of the multiple-scattering theory were generally laid out by earlier workers (see, e.g., Bass and Fuks [1980]) but the mathematical complexities requiring the evaluation of multi-dimensional integrals or the solution of large matrix equations did not allow achieving quantitative results and rendered the investigation of multiple-scattering phenomena impossible. In many instances, the evaluation of light scattering from very rough surfaces (large amplitude and/or large slope) had to be done by unjustifiably pushing the limits of single-scattering theories that allowed some quantitative analysis. The same modeling convenience reasons led to the predominant use of impenetrable surfaces. The situation began to change rapidly in the 1980s when powerful computers became available. 119

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Multiple scattering of light from randomly rough surfaces

[2, § 1

In 1985, it was predicted theoretically (McGurn, Maradudin and Celli [1985]) that the light scattered from a randomly rough surface exhibits a peak in the retroreflection direction. The peak was attributed to a coherent interference multiple-scattering phenomenon, that is similar to the weak localization effect occurring in the electron transport through disordered conductors and, generally, in the propagation of any wave through a disordered volume (Kuga and Ishimaru [1984], Tsang and Ishimaru [1984], van Albada and Lagendijk [1985], Wolf and Maret [1985], Sheng [1995], Barabanenkov, Kravtsov, Ozrin and Saichev [1991]). The experimental confirmation of the enhanced backscattering effect for random surfaces followed shortly thereafter (Méndez and O’Donnell [1987], O’Donnell and Méndez [1987]). In the 1990s, the effect was investigated in more complex systems involving rough surfaces, such as layered media, nonlinear media, materials supporting surface and guided waves, surface systems involving random and deterministic scatterers, etc. It was found that coherent multiple-scattering enhancement can occur in transmission through randomly rough surfaces of penetrable media, and that light scattered diffusely from some surface systems can exhibit satellite peaks that appear in directions other than the retroreflection direction. Coherent signatures were found and analyzed in the angular intensity correlation functions. More recent work done in the second half of 1990s addressed coherent phenomena in the near field and spectral changes in the scattering of polychromatic light induced by multiple scattering. All this effort helped to create adequate theoretical and computational methodologies that are now widely accepted and used by most workers in the field. Progress in the studies of multiple scattering of light from rough surfaces makes it timely to address the key achievements in this review article and stimulate further investigations. Applied science and technology is becoming increasingly interdisciplinary, and many topics addressed in this review have implications in such areas as radar technology, the laser and optics industry, microscopy, control of surface quality in various industries, etc. The structure of this review is as follows. In Section 2, we introduce a statistical formalism for characterizing randomly rough surfaces. In Section 3, we discuss the equations for the electromagnetic field that need to be solved to obtain experimentally observable characteristics of the scattered light. These characteristics are also introduced in Section 3. Section 4 introduces the enhanced backscattering effect, and discusses its origin and the common techniques for its experimental and theoretical investigations. These techniques are also relevant in the study of other multiple-scattering effects discussed in later sections. The coherent multiple-scattering phenomena occurring in the higher-order moments of the intensity of the scattered field are reviewed in Section 5. Section 6 ad-

2, § 2]

Characterization of randomly rough surfaces

121

dresses the studies of multiple-scattering phenomena in rough surface systems, more complex than a simple interface between two linear media. In Section 7, we discuss multiple-scattering phenomena for surface electromagnetic waves and related near-field optics effects. The subject of Section 8 is spectral changes induced by the multiple scattering of polychromatic light from randomly rough surfaces. Finally, in Section 9, we review the up-to-date achievements in the field and discuss some directions for future research. § 2. Characterization of randomly rough surfaces In this section we summarize some of the geometrical and statistical properties of randomly rough surfaces that will be used in the rest of this article. We will restrict our attention to surfaces that are planar in the absence of roughness. Such surfaces can be classified as two-dimensional and one-dimensional. In the former case the departure of the surface from the plane (assumed to be the plane x3 = 0) depends on both of the coordinates x1 and x2 in that plane. In the latter case it depends on only one of the coordinates in that plane, say x1 . Since a one-dimensional random surface can be regarded as a special case of a two-dimensional random surface, we consider the latter type of surface first, and then specialize its properties to the one-dimensional case. The reader interested in the scattering of light from randomly rough surfaces that are not planar in the absence of roughness can find appropriate discussions in the studies by Schiffer [1990], Farias, Vasconcelos, Cesar and Maradudin [1994], and Fuks and Voronovich [1999]. 2.1. Two-dimensional random surfaces The equation defining a two-dimensional surface can be written in the form x3 = ζ (x
),

(2.1)

where the real function ζ (x
) is called the surface profile function. It is a singlevalued function of the two-dimensional position vector in the plane x3 = 0 given by x
= xˆ 1 x1 + xˆ 2 x2 , where xˆ 1 and xˆ 2 are unit vectors along the x1 - and x2 -axes, respectively. The region ζ (x
)min < x3 < ζ (x
)max is called the selvedge region. We will assume that ζ (x
) is differentiable with respect to x1 and x2 as many times as is necessary. In practice this usually means that its first and second partial derivatives exist. The interesting class of continuous, but non-differentiable functions ζ (x
) representing fractal surfaces (Mandelbrot [1977], Rothrock and Thorndike [1980]) is beyond the scope of this review because our central topic

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[2, § 2

– multiple-scattering phenomena – has not yet been studied in enough detail for fractal surfaces. The interested reader is referred to the investigations of the singlescattering phenomena for fractal surfaces by Berry [1979], Jakeman [1986], Jordan, Hollins, Jakeman and Prewitt [1988], Jaggard and Sun [1990], Lin, Lee, Lim and Lee [1995], and Sheppard [1996]. The fact that we are dealing with a randomly rough surface forces us to characterize it by certain statistical properties. Underlying this characterization is the assumption that there is not a single function ζ(x
). Instead there is an ensemble of realizations of this function. Physical properties associated with a randomly rough surface are to be averaged over this ensemble, and it is assumed that this ensemble average does not differ from the spatial average over a single, sufficiently large, surface. 2.1.1. Gaussian random surfaces Most of the existing studies of the scattering of light from a random surface are based on the assumption that the surface profile function ζ(x
) is a zero-mean, strictly stationary, isotropic, Gaussian random process. The characteristic functional of a random process ζ (x
) is defined in general by the formula
   Φ[u] = exp i d2 x
u(x
)ζ (x
) , (2.2) where the angle brackets denote an average over the ensemble of realizations of ζ (x
) and u(x
) is an arbitrary function. Assuming that   (j )  u(x
) = (2.3) λj δ x
− x
, j (j )

where δ(x
) is the Dirac delta-function, and that the {x
} constitute an arbitrary array of points in the plane x3 = 0, we obtain for Φ[u] the expression 
   (j )  λj ζ x
, Φ[u] = exp i (2.4) j

so that multipoint moments can be obtained by differentiation,    (1)  (2)  (k) 1 ∂ k Φ[u]  ζ x
ζ x
· · · ζ x
= k . i ∂λ1 ∂λ2 · · · ∂λk {λj }=0

(2.5)

In the case that ζ (x
) is a zero-mean Gaussian random process, the functional Φ simplifies to (Rytov, Kravtsov and Tatarskii [1989], p. 29)

 

1 d2 x
d2 x
 u(x
) ζ(x
)ζ(x
) u(x
) . Φ[u] = exp − (2.6) 2

2, § 2]

Characterization of randomly rough surfaces

123

When eq. (2.6) is combined with eq. (2.3), and eq. (2.5) is used, it is readily found that multipoint moments are given by  (1)     (α1 )   (α2 )   (αk−1 )   (αk ) 

ζ x
· · · ζ x(k) (2.7) = ζ x
ζ x
· · · ζ x
ζ x

d.p.

 for even k and vanish for odd k, where d.p. denotes the sum over different permutations of pairs from the sequence 1, 2, . . . , k. If the surface profile function is a strictly stationary isotropic Gaussian random process, the two-point moment can be written in the form

  ζ (x
)ζ (x
) = δ 2 W |x
− x
| , (2.8) where

δ 2 = ζ 2 (x
) ,

(2.9)

so that δ is the rms height of the surface. The fact that the surface height autocorrelation function W (|x
− x
|) depends on the coordinates x
and x
only through their difference is a reflection of the assumed stationarity of ζ(x
), while its dependence only on the modulus of x
− x
is a reflection of the assumed isotropy of the random process ζ (x
). The surface height autocorrelation function W (|x
|) possesses several important general properties. It follows from eqs. (2.8) and (2.9) that W (0) = 1. It is also not difficult to show that   −1
W |x
|
1,

(2.10)

(2.11)

by making use of the result [ζ (x
) ± ζ (x
)]2   0. In addition, W (|x
|) tends to zero as |x
| → ∞. This property follows from the fact that on a randomly rough surface the heights of the surface at two widely separated points are uncorrelated. The transverse correlation length of the surface roughness, a, is defined as the distance over which the function W (|x
|) decreases significantly from its value of unity at the origin. For calculational purposes it is convenient, often even necessary, to introduce the Fourier integral representation of the surface profile function,  2 d k
ik
·x
e ζˆ (k
), ζ (x
) = (2.12) (2π)2 where k
is a two-dimensional wave vector, k
= xˆ 1 k1 + xˆ 2 k2 . In the case that ζ (x
) is a zero-mean, stationary, isotropic, Gaussian random process, the Fourier

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Multiple scattering of light from randomly rough surfaces

[2, § 2

coefficient ζˆ (k
) is also a zero-mean, stationary, isotropic, Gaussian random process. Its statistical properties are defined completely by the two-point moment  

  (2.13) ζˆ (k
)ζˆ (k
) = (2π)2 δ k
+ k
δ 2 g |k
| . The function g(|k
|) appearing in eq. (2.13) is called the power spectrum of the surface roughness, and is defined by      g |k
| = d2 x
e−ik
·x
W |x
| . (2.14) The power spectrum g(|k
|) is a non-negative function of |k
|, and is normalized according to   d2 k
 (2.15) g |k
| = 1. 2 (2π) Although several analytic expressions have been assumed for W (|x
|) (see, for example, Ogilvy [1991], pp. 13–14), perhaps the most commonly used representation is the Gaussian expression     W |x
| = exp −x
2 /a 2 . (2.16) The 1/e decay length a entering this expression is the transverse correlation length for this model of surface roughness. The corresponding power spectrum is given by     g |k
| = πa 2 exp −a 2 k
2 /4 . (2.17) 2.1.2. Non-Gaussian random surfaces Although the discussion of the statistical properties of random surfaces up to this point has been based on the assumption that the surface profile function ζ(x
) is a Gaussian random process, many surfaces of practical importance are defined by surface profile functions that are not Gaussian random processes. Well-known examples of such surfaces are a very rough sea (Beckmann [1973]), and terrain with sharp ridges and round valleys (Beckmann [1973]). Nevertheless, the scattering of light from non-Gaussian surfaces, especially multiple scattering effects from such surfaces, has received less attention than the scattering of light from Gaussian surfaces (Kivelson and Moszkowski [1963], Beckmann [1973], Berry [1973], Brown [1982], Jakeman and Hoenders [1982], Henyey [1983], Kim, Méndez and O’Donnell [1987], Wu, Chen and Fung [1988a, 1988b], Henyey, Creamer, Dysthe, Schult and Wright [1988], Michel, Maradudin and Méndez [1989], Creamer, Henyey, Schult and Wright [1989], Jakeman [1991a, 1991b], Tatarskii and Voronovich [1994], Tatarskii [1995a], Tatarskii and Tatarskii [1996]).

2, § 2]

Characterization of randomly rough surfaces

125

One of the chief difficulties in theoretical studies of non-Gaussian random surfaces is that there are not many random functions ζ(x
) for which the characteristic functional Φ[u] is known. In recent work Tatarskii [1995a] has shown how to generate a random function ζ (x
) whose characteristic functional has a non-Gaussian form, and has used the result to calculate an average of the form ζ (x
) exp[iαζ (x
)], which arises in the small-slope approximation of the theory of rough surface scattering (Tatarskii and Tatarskii [1994]). Subsequently, Tatarskii and Tatarskii [1996] generalized the latter result and showed how this approach can be used to generate a two-dimensional non-Gaussian random surface for use in computer simulation studies of rough surface scattering. To our knowledge, such computer simulation calculations based on this model have been carried out yet. 2.2. One-dimensional random surfaces Not all random surfaces of interest in theoretical and experimental studies of rough surface scattering are two-dimensional. In recent years methods for fabricating highly one-dimensional random surfaces in the laboratory have been developed (Knotts and O’Donnell [1990], Méndez, Ponce, Ruiz-Cortés and Gu [1991]). This has made possible detailed comparisons between experimental results obtained for well-characterized one-dimensional random surfaces and results of theoretical calculations for the same surfaces, which are computationally much easier to perform than they are for two-dimensional random surfaces (Knotts, Michel and O’Donnell [1993], West and O’Donnell [1995], Maradudin, McGurn and Méndez [1995]). The height of a one-dimensional random surface above the plane x3 = 0 is given by the equation x3 = ζ (x1 ). The geometrical and statistical properties of the surface profile function ζ (x1 ) follow directly from those of the surface profile function ζ (x
) of a two-dimensional random surface on suppressing all references k2 in the Fourier to x2 in the coordinate representation, and to the wavenumber  representation. In the latter representation all integrals d2 k
f (k
)/(2π)2 must ∞ be replaced by −∞ dk1 f (k1 )/(2π). On the assumption that ζ (x1 ) is a zero-mean stationary Gaussian random process, two forms of the surface height autocorrelation function W (|x1 |) will be considered in this article. These are the Gaussian form,     W |x1 | = exp −x12 /a 2 , (2.18) and the West–O’Donnell [1995] form       W |x1 | = sin(kmax x1 ) − sin(kmin x1 ) / (kmax − kmin )x1 ,

(2.19)

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Multiple scattering of light from randomly rough surfaces

where kmax > kmin . The corresponding power spectra are   √   g |k| = πa exp −a 2 k 2 /4

[2, § 3

(2.20)

and   g |k| =

    π θ |k| − kmin θ kmax − |k| , kmax − kmin

(2.21)

where θ (x) is the Heaviside unit step function. The manner in which kmin and kmax in eqs. (2.19) and (2.21) are defined will be discussed in Section 4.

§ 3. Equations for electromagnetic fields and observable quantities 3.1. Physical quantities studied in rough surface scattering problems In this section, we discuss scattering geometries commonly studied in the literature, and present basic equations and definitions for random fields and their statistical moments. The majority of existing theoretical calculations, studying multiple scattering of light from rough surfaces, has been done within the model of one-dimensional roughness. Such a model reduces the scattering problem to the solution of scalar, one-dimensional equations. As compared to the general twodimensional case, such a simplification reduces both computational cost and effort needed to understand the physics behind the mathematical formalism. Apart from the change of polarization in the scattering process, all multiple-scattering phenomena that exist in the two-dimensional case are present in the one-dimensional case. Thus, we will follow the trend of the existing literature and center our presentation on the one-dimensional case. The studies of scattering from twodimensional rough surfaces will be referenced in appropriate sections, and mathematical formalisms for the two-dimensional case will be introduced when necessary. Figure 1 illustrates two geometries that are most commonly considered in rough surface scattering investigations. A beam of light of frequency ω is incident from the vacuum side (x3 > ζ (x1 )) onto the randomly rough surface of a homogeneous dielectric medium (fig. 1a) or a film structure (fig. 1b). In linear problems, scattering materials are characterized by complex, frequency-dependent dielectric functions ε(ω) and εd (ω), with the case of a perfectly conducting substrate given by the limit ε(ω) → −∞. Assuming that the surface is one-dimensionally rough and that the plane of incidence is the x1 x3 -plane, one can decouple the scattering problems for p-polarized light, characterized by the magnetic field vector H(x1 , x3 ; t) = (0, H2 (x1 , x3 |ω), 0) e−iωt , and for s-polarized light, character-

2, § 3]

Equations for electromagnetic fields and observable quantities

127

Fig. 1. Scattering geometries commonly considered in rough surface scattering investigations.

ized by the electric field vector E(x1 , x3 ; t) = (0, E2 (x1 , x3 |ω), 0) e−iωt . Following the first theoretical study of coherent phenomena in the multiple scattering of light from rough surfaces (McGurn, Maradudin and Celli [1985]), we discuss the mathematical formulation of the rough surface scattering problem in the geometry shown in fig. 1a. The solution of Maxwell’s equations in the region x3 > ζ(x1 )max that represents the sum of an incident plane wave and a scattered field can be written in the form  ∞ dq > inc R(q|k) eiqx1+α0 (q,ω)x3 , U (x1 , x3 |ω) = U (x1 , x3 |ω) + (3.1) −∞ 2π where U > (x1 , x3 |ω) = H2> (x1 , x3 |ω) for p-polarized light and U > (x1 , x3 |ω) = E2> (x1 , x3 |ω) for s-polarized light, U inc (x1 , x3 |ω) = eikx1 −iα0 (k,ω)x3 ,  1/2 , |q| < ω/c, α0 (q, ω) = (ω/c)2 − q 2  1/2 = i q 2 − (ω/c)2 ,

|q| > ω/c,

(3.2) (3.3a) (3.3b)

and R(q|k) determines the amplitude of the wave scattered from the state with wavenumber k = (ω/c) sin θ0 into the state with wavenumber −∞ < q < ∞. The states with |q| < ω/c represent propagating waves that are characterized by the scattering angle θs (see fig. 1) with q = (ω/c) sin θs . In many cases, it is useful to introduce the scattering S-matrix (Brown, Celli, Coopersmith and Haller [1983], Arsenieva and Feng [1993], Sánchez-Gil, Maradudin and Méndez [1995]) 1/2  R(q|k), S(q|k) = α0 (q, ω)/α0 (k, ω)

(3.4)

which is useful in the analysis of such properties of the scattering theory as unitarity and reciprocity.

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Multiple scattering of light from randomly rough surfaces

[2, § 3

The scattered intensity I (q|k) is defined in terms of R(q|k) or S(q|k) as (Malyshkin, McGurn, Leskova, Maradudin and Nieto-Vesperinas [1997a, 1997b]) I (q|k) =

2 2 1 ω  1 ω α0 (q, ω)  R(q|k) = S(q|k) , L1 c α0 (k, ω) L1 c

(3.5)

where L1 denotes the length of the x1 -axis covered by the random surface. Another widely used characteristic of the angular distribution of the intensity of scattered light is the so-called differential reflection coefficient (DRC), ∂R/∂θs , which is defined such that (∂R/∂θs ) dθs is the fraction of the total time-averaged incident flux that is scattered into the angular interval (θs , θs + dθs ). In scattering from a random surface it is the average of the differential reflection coefficient over the ensemble of realizations of the surface profile function ∂R/∂θs  that must be calculated, where here and in all that follows the angle brackets denote this ensemble average. Since specular scattering is not of interest in theoretical studies of enhanced backscattering, even at normal incidence, what is usually calculated is the contribution to the mean DRC from the light that has been scattered diffusely. This is given by 
2 

2  1 ω cos2 θs   ∂R R(q|k) −  R(q|k)  , = (3.6) ∂θs diff L1 2πc cos θ0 where the contribution from the mean, specularly reflected field is subtracted from the total intensity. In addition to the first-order statistical moment (average) of the intensity of the scattered light, it is often useful to study its higher-order moments. The secondorder moment is known as the angular intensity correlation function and is defined by





C(q, k|q  , k  ) = I (q|k)I (q  |k  ) − I (q|k) I (q  |k  ) . (3.7) In some cases (Simonsen, Maradudin and Leskova [1999]), it is convenient to use the normalized angular intensity correlation function, which is equal to C(q, k|q  , k  ) divided by I (q|k)I (q  |k  ). Some authors also define and analyze the properties of unnormalized or normalized amplitude correlation functions, with I (q|k) in eq. (3.7) being replaced by the scattering amplitude R(q|k). Thus, both the DRC and correlation functions are expressed in terms of the scattering amplitude R(q|k). The calculation of R(q|k) is the central problem of rough surface scattering theory. We next outline two alternative methods which are most commonly used to obtain the equations for R(q|k) and note that these equations had been derived long before it became possible to solve them in the multiple-scattering regime.

2, § 3]

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129

3.2. Rayleigh method The Rayleigh method, which we discuss first, is based on the so-called Rayleigh hypothesis which states that the expressions for the fields in the regions x3 > ζ (x1 )max and x3 < ζ (x1 )min can be continued on to the surface itself (Lord Rayleigh [1896, 1907]). Thus, the expression (3.1) and an analogous expression for the field transmitted into the medium can be used in satisfying the boundary conditions at x3 = ζ (x1 ). Toigo, Marvin, Celli and Hill [1977] employed this hypothesis together with Green’s integral theorem and the extinction theorem to derive a single integral equation for the scattering amplitude R(q|k) in the case of p polarization:  ∞ dq (+) M (p|q)R(q|k) = −M (−)(p|k), (3.8) −∞ 2π where M (±) (p|q) =

 [pq ± α(p, ω)α0 (q, ω)]σ  I α(p, ω) ∓ α0 (q, ω)|p − q , (3.9) α(p, ω) ∓ α0 (q, ω)

∞ I (γ |Q) = −∞ dx1 e−iQx1 e−iγ ζ(x1 ) , and α(q, ω) = [ε(ω)(ω/c)2 − q 2 ]1/2 with Re α(q, ω) > 0, Im α(q, ω) > 0. The exponent σ is unity for p-polarization and zero for s-polarization (the latter case was considered by Brown, Celli, Coopersmith and Haller [1983]). Equation (3.8) is called the reduced Rayleigh equation and it is exact within the limits of validity of the Rayleigh hypothesis. The limits of validity of the Rayleigh hypothesis have been explored by several researchers, primarily in the context of electromagnetic and acoustic scattering from deterministic surfaces. Petit and Cadilhac [1966], Millar [1971], Hill and Celli [1978], and Maystre [1984] established that the Rayleigh method is rigorous for sinusoidal surfaces with height to period ratios smaller than 0.07. Since the √ rms height of the sinusoidal surface is 2 times smaller than its amplitude, the range of validity of the Rayleigh method for randomly rough surfaces can be estimated as δ/D < 0.1,

(3.10)

where δ is the rms height and D is the mean distance between consecutive peaks and valleys of the√surface profile ζ (x1 ) (for the Gaussian correlation function (2.20), D = πa/ 6). However, even the early studies we have just mentioned suggest that the practical utility of the Rayleigh method can be extended well beyond the inequality (3.10). Several recent investigations (Soto-Crespo, NietoVesperinas and Friberg [1990], Madrazo and Maradudin [1997]) support this suggestion by comparing the predictions of the Rayleigh method with the exact nu-

130

Multiple scattering of light from randomly rough surfaces

[2, § 3

merical solution and show that the Rayleigh equations can provide accurate results for surfaces with δ/D as large as 0.47. Thus, the reduced Rayleigh equation gives a convenient mathematical formulation of the scattering problem for smooth surfaces. As we will see in Section 4.2, it provides a good starting point for both analytical (perturbative) and numerical calculations. One of the particular advantages of the reduced Rayleigh equation is that its structure (3.8) is generic to many geometries, and not only to the case of the scattering of light from a single, one-dimensional rough interface. In the general case of electromagnetic scattering from a two-dimensional rough surface x3 = ζ (x
), the electric vector is expressed in terms of the 2 × 2 matrix of scattering amplitudes, whose elements Rij (q
|k
) describe scattering of a j -polarized wave with wavenumber k
into an i-polarized wave with wavenumber q
(j = p, s). The reduced Rayleigh equation (3.8) in this case becomes a (±) matrix integral equation. The matrix elements Mij (p
|q
) which appear in this equation were derived by Brown, Celli, Haller and Marvin [1984]. A 2 × 2 matrix reduced Rayleigh equation for the amplitudes of the p- and s-polarized components of the transmitted field have been obtained by Greffet [1988] for the case in which light incident in one dielectric medium enters a second dielectric medium through a two-dimensional random interface. Furthermore, eq. (3.8) was shown to describe the scattering of light from a one-dimensional rough surface of a film in the geometry shown in fig. 1b for both p (Sánchez-Gil, Maradudin, Lu, Freilikher, Pustilnik and Yurkevich [1996]) and s (Sánchez-Gil, Maradudin, Lu, Freilikher, Pustilnik and Yurkevich [1994]) polarizations, when the film is free-standing or deposited on a perfectly conducting substrate. A reduced Rayleigh equation has been derived for the amplitude of an s-polarized electromagnetic field transmitted through a dielectric film whose illuminated face is a one-dimensional rough surface (Leskova, Maradudin, Méndez and Simonsen [2000]). The film is deposited on a planar surface of a substrate of a different dielectric material. A reduced Rayleigh equation for the amplitude of an s-polarized electromagnetic field transmitted through a dielectric film whose back surface is a one-dimensional rough surface while its illuminated surface is planar and which is sandwiched between two different dielectric media has been derived by Maradudin, Méndez, Leskova and Simonsen [2001]. In the case that the latter structure is illuminated by a p-polarized electromagnetic field, the reduced Rayleigh equation for the amplitude of the scattered field has been obtained by Leskova, Maradudin, Shchegrov and Méndez [1998]. It has also been proved to be possible to derive reduced Rayleigh equations for scattering of electromagnetic waves from a dielectric film whose illuminated surface is a two-dimensional random surface while its back surface is planar, and for scattering from a dielectric film whose illu-

2, § 3]

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131

minated surface is planar but whose back surface is a two-dimensional random surface (Soubret, Berginc and Bourrely [2001a]). In the former case the dielectric film was deposited on a perfectly conducting or a metallic substrate. In the latter case it was deposited on a metallic substrate. A set of reduced Rayleigh equations for the scattering of electromagnetic waves from a dielectric film, both of whose surfaces are two-dimensional randomly rough surfaces, has been obtained by Soubret, Berginc and Bourrely [2001b] in the case where the film is deposited on a metallic substrate. The derivation of reduced Rayleigh equations for the scattering of light from, and its transmission through, other multilayered planar structures with interfaces perturbed by one- or two-dimensional roughness appears to be a feasible task if the procedures developed for simpler structures are followed. 3.3. Surface integral equations for electromagnetic fields The mathematical formulation of the scattering problem provided by the Rayleigh method cannot be used for rough surfaces with large slopes. In this case, one has to consider the exact equations for the electromagnetic fields. We present here the exact formulation based on the use of Green’s second integral theorem (Huygens’ principle) in the half-spaces x3 > ζ (x1 ) and x3 < ζ (x1 ) (again, we focus on the geometry of fig. 1 in the one-dimensional roughness case). The scattering amplitude R(q|k) introduced in eq. (3.1) is expressed in terms of the surface values of the total field U(x1 ) ≡ U > (x1 , ζ(x1 )|ω) and its normal derivative V(x1 ) = (∂/∂N )U > (x1 , x3 |ω)|x3 =ζ(x1 ) through (Maradudin, Michel, McGurn and Méndez [1990])  ∞ i R(q|k) = dx1 e−iqx1 −iα0 (q,ω)ζ(x1 ) 2α0 (q, ω) −∞    × i qζ  (x1 ) − α0 (q, ω) U(x1 ) − V(x1 ) . (3.11) The so-called source functions U(x1 ) and V(x1 ) obey a pair of coupled integral equations  ∞   inc dx1 H0 (x1 |x1 )U(x1 ) − L0 (x1 |x1 )V(x1 ) , (3.12a) U(x1 ) = U (x1 ) +  0=

−∞

∞ −∞

  dx1 Hε (x1 |x1 )U(x1 ) − νLε (x1 |x1 )V(x1 ) ,

where Hε (x1 |x1 ) = (i/4)(∂/∂N  )H(1) 0 (1) 

Lε (x1 |x1 ) = (i/4)H0

(3.12b)

  ε(ω) (ω/c)ξ x  =ζ(x  ) ,

  ε(ω) (ω/c)ξ x  =ζ(x  ) , 3

1

3

(3.13a)

1

(3.13b)

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[2, § 3

∂/∂N = −ζ  (x1 )(∂/∂x1 ) + (∂/∂x3 ), H0 (z) is the Hankel function of the first 1/2  , and η is a positive infinitesikind, ξ = (x1 − x1 )2 + (ζ (x1 ) − x3 + η)2 mal. The kernels H0 (x1 |x1 ) and L0 (x1 |x1 ) are obtained by setting ε(ω) = 1 in Hε (x1 |x1 ) and Lε (x1 |x1 ), respectively. Equations (3.11)–(3.13) are exact equations for both p and s polarizations, with ν in eq. (3.12b) equal to ε(ω) for ppolarization and 1 for s-polarization. The inhomogeneous term in eq. (3.12a) is given by U inc (x1 ) ≡ U inc (x1 , ζ(x1 )|ω). Since a numerical solution of eqs. (3.12) is especially difficult due to the infinite limits of integration, it is often useful to replace a plane incident wave (3.2) by a beam of finite width:  π/2 2 2 wω − w ω (θ−θ0 )2 −i ω (x1 sin θ−x3 cos θ) inc dθ e 4c2 e c , (3.14) U (x1 , x3 |ω) = √ 2 πc −π/2 (1)

where w is the half-width of the beam. The half-width of the intercept of the beam with the plane x3 = 0 is g = w/ cos θ0 . The contribution from the diffusely scattered light to the mean DRC has to be modified as compared to eq. (3.6) 
2 cos2 θs ω |R b (q|k)|2 − |R b (q|k)|2 ∂R , = (3.15) ∂θs diff (2π)3/2 wc pinc (θ0 ) wω π due to the change in incident flux. Here pinc (θ0 ) = 12 [erf( √ ( 2 − θ0 )) + 2c

erf( √wω ( π2 + θ0 ))] (Leskova, Maradudin and Shchegrov [1997a]), and R b (q|k) 2c is still given by eq. (3.11) – the superscript b is used to distinguish the scattering amplitude due to the incident beam (3.14) from the scattering amplitude R(q|k) due to plane wave illumination. A detailed derivation of eqs. (3.11)–(3.13) is given by Maradudin, Michel, McGurn and Méndez [1990]. Similar analysis can be implemented for a layered system shown in fig. 1b (Lu, Maradudin and Michel [1991]). In the onedimensional roughness case, each interface adds two source functions (the field and its normal derivative), therefore it also adds two integral equations. When one of the interfaces is perfectly conducting, either the field or its normal derivative vanish identically at that interface, thereby reducing the number of equations by one. Furthermore, for a planar perfectly conducting interface (as in fig. 1b) the method of images allows eliminating the integral equations at that interface altogether (Shchegrov and Maradudin [1999]). The scattering of light from a two-dimensional rough surface x3 = ζ(x
) has to be considered in its full vectorial formulation. This increases the number of unknown source functions and integral equations (Kong [1990], Tran and Maradudin [1994]). Due to its complexity scattering from two-dimensional rough surfaces is

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often studied in the cases of scattering of a scalar plane wave from Dirichlet (Tran and Maradudin [1992, 1993], Macaskill and Kachoyan [1993]) or Neumann (Tran and Maradudin [1993]) surfaces. § 4. Weak localization effects in the multiple scattering of light from randomly rough surfaces. Enhanced backscattering 4.1. The nature of enhanced backscattering effect We start the review of coherent multiple-scattering effects with a qualitative description of the most widely known of these phenomena – enhanced backscattering. The incorporation of multiple scattering into the theories of rough surface scattering began in the 1980s. In 1985, McGurn, Maradudin and Celli [1985] studied the scattering of p-polarized (transverse magnetic) light from a onedimensional randomly rough metal surface, and calculated the mean intensity of the diffusely scattered light by evaluating an infinite sum in powers of the surface profile function. The calculated angular dependence (3.6) of the mean intensity of the light that has been scattered diffusely revealed a well-defined peak in the retroreflection direction. The origin of this novel phenomenon of enhanced backscattering was linked to the coherent interaction of the multiply-scattered, p-polarized, surface electromagnetic waves (surface plasmon polaritons), supported by the vacuum–metal interface, with their reciprocal partners, in which the light and the surface plasmon polaritons interact with the roughness at the same points on the surface, but in the reverse order. This mechanism is illustrated in fig. 2. The partial waves, resulting from such reciprocal scattering events, have the same amplitude and phase if the wave vectors of the incident and scattered waves

Fig. 2. Illustration of the surface plasmon-polariton mechanism for the enhanced backscattering of light. The double-scattered reciprocal waves shown in the retroreflection direction interfere constructively since they have the same phase and the same amplitude. The excitation of intermediate surface waves (surface plasmon polaritons) is essential for backscattering enhancement.

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Fig. 3. The intensity of light scattered in plane from a considerably rough two-dimensional random surface measured as a function of the polar scattering angle. The incident light is s polarized, its wavelength is λ = 633 nm, and the polar angle of incidence is θ0 = −20◦ . (0) s → s scattering data; (+) s → p scattering data (O’Donnell and Méndez [1987]).

are oppositely directed, i.e. these partial waves interfere constructively. For scattering into directions other than the retroreflection direction, the different partial waves become dephased, so that only their intensities add. Thus, the intensity of scattering into the retroreflection direction is a factor of two larger than the intensity of scattering into those other directions, because of the interference terms that contribute to the intensity in the former case. The contribution of single-scattering processes usually significantly reduces this factor of two enhancement, since it is not subject to the coherent backscattering mechanism illustrated in fig. 2. The treatment by McGurn, Maradudin and Celli [1985] relied on the assumption of weakly rough surfaces (with typical amplitudes small compared to the wavelength) that support surface plasmon polaritons. However, the first experimental observation of the enhanced backscattering effect (Méndez and O’Donnell [1987], O’Donnell and Méndez [1987]) was done for surfaces with large roughness amplitudes. These experimental results (fig. 3) stimulated the development of non-perturbative, computational techniques for calculating the multiple scattering of light from very rough surfaces (see Section 4.2.2). This experimental and theoretical work led to the conclusion that the formation of enhanced backscattering from very rough surfaces does not require surface waves and can be interpreted as shown in fig. 4. The large amplitudes and slopes of the surface cause the multiple scattering of light. The intermediate waves traveling from one point of scattering to another are not necessarily surface waves.

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Fig. 4. A double-scattering sequence and its reciprocal partner that contribute to the enhanced backscattering of light incident normally on a very rough random surface. D is the distance between the first and last point on the surface struck by the light, and a is the average distance between consecutive maxima and minima on the surface (a can be approximately identified with the transverse correlation length for the Gaussian power spectrum (2.16)).

Fig. 5. The same geometry as in fig. 4, but showing the partial wave scattered into a direction described by the scattering angle θs .

Figures 2 and 4 illustrate the two main mechanisms for the enhanced backscattering of light from a randomly rough surface. The height of the enhanced backscattering peak on the background of the usually dominant single-scattering contribution is determined by the efficiency of one of these multiple-scattering mechanisms. The angular width δθ of the enhanced backscattering peak can be estimated by considering the phase difference φ between two wave paths shown in fig. 5 (where we assume the geometry of fig. 4). For normal incidence, φ = (2π/λ)θs D, where D is the distance between the first and the last points of the scattering sequence. The angular width θ of the main interference maximum centered at θs = 0 is given by twice the angular position θs = λ/2D of the first interference minimum, i.e. θ = λ/D.

(4.1)

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Multiple scattering of light from randomly rough surfaces

[2, § 4

For the enhanced backscattering mechanisms depicted in figs. 4 and 5, the value of D can be estimated by the mean distance between consecutive hills and valleys on the surface, which is approximately equal to the roughness correlation length a (Maradudin and Michel [1990], Simeonov, McGurn and Maradudin [1997]). For the enhanced backscattering induced by surface waves (the case of fig. 2), D can be approximated by the mean free path l of a surface wave on a random surface. As we will see in the following sections, these estimates agree well with both theory and experiment. To complete our qualitative introduction to the enhanced backscattering effect, we stress that the averaging of the intensity of the scattered light over the ensemble of realizations of the surface profile function is essential for the observation of the effect. The angular dependence of the intensity of light scattered diffusely from any finite area of a rough surface represents a noisy sequence of bright and dark spots – a speckle pattern. In contrast to most peaks that disappear after taking the average over many realizations of the surface profile, the peak in the retroreflection direction remains. The nature of the enhanced backscattering from random surfaces allows relating this effect to a broader class of multiple-scattering phenomena called weak localization. Weak localization was originally studied in the context of electronic properties of disordered conductors (Abrahams, Anderson, Licciardello and Ramakrishnan [1979]). Electrons, which represent quantum wave packets, undergo multiple scattering from randomly distributed impurities. The coherent interference of partial waves due to reciprocal electron trajectories leads to the enhanced probability for an electron to return to its origin. Such weak localization of electrons results in the decrease of the electrical conductivity of a disordered sample. During the 1980s it was realized that weak localization is a general property of all waves in random media (a good discussion on the subject is presented in Sheng [1995]). In particular, the first experimental reports of the enhanced backscattering of light in media with volume disorder appeared in 1984–85 (Kuga and Ishimaru [1984], van Albada and Lagendijk [1985], Wolf and Maret [1985]), and were accompanied by theoretical interpretations describing coherent interference between reciprocal scattered waves (Tsang and Ishimaru [1984, 1985]) and its connection to weak localization (van Albada and Lagendijk [1985], Wolf and Maret [1985]). These investigations triggered the studies of multiple-scattering phenomena in rough surface scattering. Despite the weak localization origin of the enhanced backscattering effect from both random volumes and random surfaces, there is an essential difference. In volumes, a well-defined enhanced backscattering peak is formed after several hundreds or thousands of scattering events. In contrast, for surfaces, double-scattering processes are not

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only sufficient, but usually dominate the contribution of all multiple-scattering processes. The qualitative picture of the enhanced backscattering from randomly rough surfaces, presented in this section, is based on extensive theoretical and experimental work that followed the first efforts of the mid-1980s. In Section 4.2, devoted to theoretical techniques employed in the study of multiple scattering of light from randomly rough surfaces, we will gain deeper insight into the enhanced backscattering effect under different experimental conditions (discussed in Section 4.3). In particular, we will analyze the surface plasmon polariton mechanism for enhanced backscattering in Section 4.2.1 which reviews perturbative techniques for weakly rough surfaces. The other mechanism for enhanced backscattering (fig. 4) will be discussed in the context of numerical simulations used to study scattering from very rough surfaces. 4.2. Theoretical methods employed in the study of multiple-scattering phenomena We next review theoretical methods used in the study of enhanced backscattering of light from a randomly rough surface. The discussion will center on the geometry of a one-dimensional randomly rough surface of a homogeneous medium. However, the methods we will discuss are generic and are also used in investigating other multiple-scattering phenomena in systems with one-dimensional or two-dimensional rough surfaces (see Sections 5–8) with varying complexity of the computational effort required. 4.2.1. Weakly rough surfaces: perturbative techniques 4.2.1.1. Small-amplitude perturbation theory. The weak roughness limit has been at the focus of rough surface scattering theories since the beginning of the 20th century. In this limit, the amplitude of the scattered light is expanded in powers of the surface profile function ζ . The first-order term of this expansion describes single-scattering processes, and from the pioneering work of Mandel’shtam [1913] to the mid-1980s this approximation dominated theoretical investigations of rough surface scattering (Rice [1951], Brekhovskikh [1952], Bass and Fuks [1980]). Since the mid-1980s, several papers have considered the small-amplitude expansion of the scattering amplitude beyond the single-scattering approximation (see, e.g., Jackson, Winebrenner and Ishimaru [1988], Soto-Crespo, Nieto-Vesperinas and Friberg [1990], Maradudin and Méndez [1993], Shchegrov [1998b], O’Donnell, West and Méndez [1998], O’Donnell [2001]). Here we out-

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[2, § 4

line the method developed by Shchegrov [1998b] that allows calculating the scattering amplitude and intensity recursively to any order in the surface profile function. We start from the reduced Rayleigh equation (3.8) and expand the functions M ± (p|q) in powers of the surface profile function, M (±) (p|q) = where

∞ 

ˆ (n) (p − q), µ(±) n (p|q)ζ

n=0  ∞ ζˆ (n) (Q) = −∞ dx1 e−iQx1 ζ n (x1 )

(4.2)

and

 n

µ(±) n (p|q) = (−i) pq ± α(p, ω)α0 (q, ω)

σ 

α(p, ω) ∓ α0 (q, ω)

n−1

,

(4.3) with the same convention for σ as in eq. (3.9). Then, we seek the scattering amplitude in the form of an integral series expansion in powers of ζ , R(q|k) = 2πδ(q − k)R (0) (k) + χ (1) (q|k)ζˆ (q − k)  ∞  ∞  1 ∞ dp1 dpn−1 (n) + ··· χ (q|p1 , p2 , . . . , pn−1 |k) n! −∞ 2π −∞ 2π n=2

× ζˆ (q − p1 )ζˆ (p1 − p2 ) · · · ζˆ (pn−1 − k),

(4.4)

(−) (+) where R (0) (k) = −µ0 (p|q)/µ0 (p|q)

is the amplitude Fresnel reflection coefficient for a planar surface. Substituting eqs. (4.3)–(4.4) into eq. (3.8) and equating the terms that have the same powers of ζ , we find that the functions χ (n) can be calculated recursively, χ (n) (q|p1 , p2 , . . . , pn |k) n−1    n (n−m) = ξ (n) (q|k) − (q|pn−m )χ (m) (pn−m |pn−m+1 , . . . , pn−1 |k), η m m=1 (4.5) (−)

(+)

(+)

where ξ (n) (q|k) = −[µn (q|k) + µn (q|k)R0(k)]/µ0 (q|q), η(n) (q|k) = n (+) µ(+) n (q|k)/µ0 (q|q), and m is the binomial coefficient. It follows from eq. (3.6) that ∂R/∂θs diff is given by  
2   ω cos2 θs 2  (1) ∂R = δ χ (q|k) g |q − k| ∂θs diff 2πc cos θ0   ∞  ∞ ∞  dp1 dpn−1 (2n) 2n ··· Λ (q|p1 , . . . , pn−1 |k) . δ + −∞ 2π −∞ 2π n=2

(4.6)

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Fig. 6. A plot of ∂R/∂θs diff as a function of θs for p-polarized light of wavelength λ = 457.9 nm incident normally on a one-dimensional random silver surface (ε(ω) = −7.5 + i0.24) characterized by the Gaussian power spectrum (2.20) with δ = 5 nm and a = 100 nm. The calculations are based on eq. (4.6) (solid curve) and on eq. (4.37) (dashed curve). The inset shows ∂R/∂θs diff in the vicinity of the retroreflection direction (Maradudin and Méndez [1993]).

The analytical calculation of the functions Λ(2n) is done directly from eqs. (3.6) and (4.4) using the averaging procedure described in Section 2. This task is rather straightforward but increasingly tedious for higher-order terms. The singlescattering term, of O(δ 2 ) is written explicitly in eq. (4.6), and the next, multiplescattering term O(δ 4 ) can be found in the works by Leskova, Maradudin and Shchegrov [1997b] and Shchegrov [1998b]. The contributions to ∂R/∂θs diff from a selected group of terms of O(δ 6 ) has been obtained by O’Donnell, West and Méndez [1998]. In something of a tour-de-force, O’Donnell [2001] has calculated all the terms of O(δ 6 ) and of O(δ 8 ). In the scattering of p-polarized light from a weakly rough one-dimensional random metal surface that supports surface plasmon polaritons, the enhanced backscattering peak first arises in the contribution to ∂R/∂θs diff of the fourth order in δ, as shown in fig. 6 for the case that the surface roughness is characterized by a Gaussian power spectrum (2.20). The peak arises from the term in Λ(4) (q|p1|k) in the integrand in eq. (4.6) that contains the product χ (2) (q|p1 |k)χ (2)∗(q|q + k − p1 |k), and is associated with the coherent interfer-

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Multiple scattering of light from randomly rough surfaces

[2, § 4

Fig. 7. A plot of ∂R/∂θs diff as a function of θs when p-polarized light of wavelength λ = 612 nm is incident normally on a one-dimensional random gold surface (ε(ω) = −9.00 + i1.29) characterized by the West–O’Donnell power spectrum (2.21) with δ = 10.9 nm, kmin = 8.4815 × 10−3 nm−1 and kmax = 13.260 × 10−3 nm−1 . The calculation of ∂R/∂θs diff was carried out on the basis of eq. (4.6) where terms through 0(δ 4 ) were retained (after Maradudin, McGurn and Méndez [1995]).

ence of a doubly-scattered surface plasmon polariton and its reciprocal partner (fig. 2). Enhanced backscattering of light from a one-dimensional random metal surface characterized by a Gaussian power spectrum has yet to be observed experimentally. The visibility of the enhanced backscattering peak is especially high for the West–O’Donnell surfaces, described in Section 2. The power spectrum of these surfaces (eq. (2.21)) forces the incident wave to be scattered only into states with wavenumbers kmin < q < kmax (for normal incidence). By choosing kmin and kmax to lie close to the left and to the right of the wavenumber ksp (ω) of a surface plasmon polariton of frequency ω on a planar surface, we force the system into the multiple-scattering regime. Figure 7 illustrates the angular dependence of the intensity of diffusely scattered p-polarized light calculated to fourth order in δ. The single-scattering contribution ∼ δ 2 appears as two lobes at large scattering angles, while the central part is completely dominated by multiple-scattering processes. Since the single-scattering contribution is absent in the retroreflection direction, the height of the enhanced backscattering peak in this case is twice that of the background. In experiments in which p-polarized light was scattered from a weakly rough one-dimensional random silver surface characterized by a West–O’Donnell power spectrum that is significant at twice the surface plasmon polariton wave number, 2ksp(ω), O’Donnell, West and Méndez [1998] observed an enhanced backscattering peak at large angles of incidence (60◦ –76◦ ) (fig. 8).

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Fig. 8. Measurements at three angles of incidence of the mean diffuse intensity Ip (θs ) in the scattering of p-polarized light of wavelength 1152 nm (ε(ω) = −61.0 + i6.2) from a one-dimensional random silver surface characterized by the West–O’Donnell power spectrum (2.21) with kmin = 9.93 × 10−3 nm−1 , kmax = 1.49 × 10−2 nm−1 , and δ = 11.1 nm (after O’Donnell, West and Méndez [1998]).

This peak is not predicted by the small-amplitude perturbation theory when the expansion of ∂R/∂θs diff in powers of δ 2 is terminated at the fourth-order term (Maradudin and Méndez [1993]). However, the calculation of the contribution to ∂R/∂θs diff from a selected set of terms of sixth order in δ by these authors predicts this peak. In his determination of ∂R/∂θs diff through terms of O(δ 8 ), O’Donnell [2001] observed an enhanced specular peak in the contribution of O(δ 8 ) when the weakly rough one-dimensional random surface was characterized by a West–O’Donnell power spectrum (2.21) and a sufficiently large value of δ was employed (fig. 9). This feature is not present in the contributions to ∂R/∂θs diff of lower orders in δ. This peak has not been observed experimentally up to the present time. These results illustrate the point that interesting new effects can be obtained when small-amplitude perturbation theory is extended to higher and higher orders. The small-amplitude perturbation expansion applied to the scattering of s-polarized light in the same geometries that exhibit the enhanced backscattering effect for p-polarized light show no enhanced backscattering. This is because s-polarized surface plasmon polaritons do not exist on a vacuum–metal interface (see, e.g., Maradudin [1982]). Thus, the small-amplitude perturbation theory provides a simple and fast approach to investigating multiple-scattering phenomena. It allows decoupling and analyzing contributions from single, double, and higher order scatter-

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[2, § 4

(8)

Fig. 9. The calculated enhanced specular peak in the contribution Ip (θs |θ0 ) of eighth order in δ to the mean diffuse intensity in the scattering of p-polarized light of wavelength 457.9 nm (estimated ε(ω) = −9.5 + i0.24) from a random metal surface characterized by the West–O’Donnell power spectrum (2.21) with kmin = 1.70(ω/c), kmax = 2.25(ω/c), and δ/λ = 3 × 10−3 . The angle of incidence is θ0 = 52◦ (after O’Donnell [2001]).

ing processes to the scattered field and intensity – and such an advantage is especially important since in most practical cases the contribution from double-scattering processes greatly exceeds the contribution from other multiplescattering processes (Maradudin and Méndez [1993]). A rather subtle issue concerning the small amplitude perturbation theory is its limit of validity. The first and obvious requirement is the smallness of the rms roughness amplitude δ compared to the wavelength λ of the incident light: δ/λ 1. However, the derivation of the small-amplitude perturbation expansion is typically based on the Rayleigh equations (see Eqs. (4.2)–(4.6)). Therefore, this expansion is often believed to be valid only as long as the Rayleigh hypothesis is valid, i.e. the rms slope of the surface should also be small (Ogilvy [1991]). However, recent investigations (Jackson, Winebrenner and Ishimaru [1988], Tatarskii [1995b]) suggest that the small-amplitude perturbation series derived from the Rayleigh equations and from the exact surface integral equations based on Green’s theorem, coincide. Further, Tatarskii [1995b] argued that the only criterion for the validity of the small-amplitude series is the convergence of this series, which condition does not necessarily require the validity of the Rayleigh hypothesis. 4.2.1.2. Self-energy (many-body) perturbation theory. A somewhat deeper insight into weak localization processes on a weakly rough metal surface and into the nature of the perturbation parameter is provided by another perturbative ap-

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proach developed by McGurn, Maradudin and Celli [1985] where the first theoretical prediction of the enhanced backscattering from randomly rough surfaces was made. In this method, in the case of p-polarized incident light, one seeks the scattering amplitude in the form R(q|k) = 2πδ(q − k)R (0) (k) − 2iG(0)(q)T (q|k)G(0)(k)α0 (k),

(4.7)

where the functions α(k) and α0 (k) are the same as the functions α0 (k, ω) and α(k, ω) entering eq. (3.9) with the argument ω suppressed for brevity. The function G(0) (k) = iε(ω)/[ε(ω)α0 (k) + α(k)] is a Green’s function for surface plasmon polaritons at a planar surface, while the transition matrix T (q|k) is postulated to satisfy  ∞ dq V (q|p)G(0)(p)T (p|k). T (q|k) = V (q|k) + (4.8) 2π −∞ Equations (3.8), (4.7), and (4.8) define the scattering potential V (q|k) which is the solution of the equation  ∞  V (q|k) dp  (+) M (q|p) − M (−) (q|p) 2iα0 (p) −∞ 2π  (+)     = M (q|k) ε(ω)α0 (k) − α(k) − M (−) (q|k) ε(ω)α0 (k) + α(k) ×

1 , 2ε(ω)α0 (k)

(4.9)

where M (±) (q|p) has been defined by eqs. (3.9). To first order in the surface profile function the solution of eq. (4.9) is V (1) (q|k) = i

 ε(ω) − 1  ε(ω)qk − α(q)α(k) ζˆ (q − k). 2 ε (ω)

(4.10)

The use of the approximation to the scattering potential given by eq. (4.10) is called the small-roughness approximation (Toigo, Marvin, Celli and Hill [1977]). Equation (4.8) brings the scattering problem to the form known in the standard wave scattering theory (Sheng [1995]). We next introduce Green’s function for surface plasmon polaritons on the randomly rough metal surface as the solution of  ∞ dp (0) (0) V (q|p)G(p|k) G(q|k) = 2πδ(q − k)G (k) + G (q) −∞ 2π = 2πδ(q − k)G(0) (k) + G(0) (q)T (q|k)G(0)(k).

(4.11)

Recalling eq. (4.7), we find that R(q|k) = −2πδ(q − k) − 2iG(q|k)α0(k).

(4.12)

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[2, § 4

An important role in the theory being described here is played by the averaged Green’s function G(q|k). Due to the stationarity of the surface profile function G(q|k) is diagonal in q and k,

G(q|k) = 2πδ(q − k)G(k), (4.13) where G(k) is related to the unperturbed Green’s function G(0) (k) by G(k) =

1 G(0) (k)−1

− M(k)

.

(4.14)

The function M(k) in this expression is a self-energy that can be calculated as an expansion in powers of ζ (x1 ) as a solution of the equations

M(q|k) = 2πδ(q − k)M(k), (4.15a)  ∞ 

 dp M(q|k) = V (q|k) + M(q|p)G(0)(p) V (p|k) − V (p|k) . (4.15b) −∞ 2π Just as G(0) (k) has simple poles at the wave numbers ±ksp of the surface plasmon polaritons of frequency ω propagating along a planar vacuum–metal interface, G(k) has simple poles at the wavenumbers of the surface plasmon polaritons of frequency ω propagating along the randomly rough vacuum–metal interface. This result can be exploited to simplify calculations in which G(k) appears in the integrand of some integral, by the introduction of the pole approximation to G(k), G(k) ∼ =

C C − , k − ksp − i k + ksp + i

(4.16)

where C=

|ε1 (ω)|3/2 , ε12 (ω) − 1

(4.17)

ε1 (ω) = Re ε(ω), ε2 (ω) = Im ε(ω), and  can be written as  = ε + sp ,

(4.18)

with ε =

ε2 ksp , 2|ε1(ω)|(|ε1 | − 1)

sp = C Im M(ksp ).

(4.19a) (4.19b)

The function ε is the decay rate of the surface plasmon polariton as it propagates along the surface due to ohmic losses, i.e. due to the imaginary part of the dielectric constant, while sp is the decay rate of the surface plasmon polariton due to its roughness-induced conversion into volume electromagnetic waves

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in the vacuum and into other surface plasmon polaritons. In the small-roughness approximation sp is given by  ∞

dp (1) 1 sp = C Im (4.20) V (k|p)G(p)V (1) (p|k) . L1 −∞ 2π In writing eq. (4.16) we have neglected the small renormalization of ksp arising from the real part of M(k), but have taken into account that sp can be comparable to ε . The use of eq. (4.12) in eq. (3.6) yields
 2 

2    1 2 ω3 ∂R = cos2 θs cos θ0 G(q|k) −  G(q|k)  . (4.21) 3 ∂θs diff L1 π c The two-particle Green’s function |G(q|k)|2 is given by the Bethe–Salpeter equation

G(q|k)G∗ (q|k) 2 2





 = G(q|k) G∗ (q|k) + G(q) τ (q, q|k, k) G(k) , (4.22) where τ (q, q|k, k) is the reducible vertex function. The general reducible vertex function τ (q, q  |k, k  ) is related to the irreducible vertex function Γ (q, q  |k, k  ) by



τ (q, q  |k, k  ) = Γ (q, q  |k, k  )  ∞



dp + Γ (q, q  |p, p ) G(p)G∗ (p ) τ (p, p |k, k  ) . 2π −∞ (4.23) We approximate the irreducible vertex function by the sum of the contributions from all maximally-crossed diagrams, which describe the phase-coherent multiple-scattering processes that give rise to enhanced backscattering. It is given in the small roughness approximation by

Γ (q, q  |k, k  )     K(q  + k)   2 2 φ(k) , = 2πδ(q − k − q + k ) W g |q − k| + W φ(q) 1 − K(q  + k) (4.24) where



∞

dp 2 φ (p)G(p)G∗ (Q − p), 2π −∞ √ 1/2   φ(p) = π a exp −a 2 p2 /4 , K(q) = W

2

(4.25) (4.26)

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 1 − ε(ω) ω2 W =δ . (4.27) ε(ω) c2 In obtaining the result given by eq. (4.24) several approximations were made in addition to the small-roughness approximation. The first is that the contribution to the scattering potential V (q|k) that is linear in the surface profile function ζ(x1 ), eq. (4.10), has been approximated by 1 − ε(ω) ω2 (4.28) ζˆ (q − k), ε(ω) c2 which is valid for small q and k. The second is that the imaginary part of the dielectric function ε(ω) has been neglected. The third is that the power spectrum of the surface roughness g(|q − k|) has been approximated by Antsygina, Freilikher, Gredeskul, Pastur and Slusarev [1991] as   √   g |q − k| = π a exp −a 2 (q − k)2 /4  φ(q)φ(k). (4.29) V (1) (q|k) 

This is a good approximation when g(|q − k|) is convolved with a function f (k) that is more sharply peaked than g(|k|) is. These approximations introduce small quantitative errors into the result for Γ (q, q  |k, k  ), but preserve its qualitative features while enabling an analytic result to be obtained. The result given by eq. (4.24) has to be substituted into eq. (4.23), which is then solved by iteration. However, in each of the integral terms of the iterative solution only the contribution associated with the first term on the right-hand side of eq. (4.24) is kept, and all contributions that contain the second term are omitted. The sum of the resulting integral terms is given by K(q − q  ) φ(k). (4.30) 1 − K(q − q  ) This is just the sum of the contributions associated with all the ladder diagrams. It equals the contribution of the second term on the right-hand side of eq. (4.24) when q = −k. Therefore, it cannot be neglected in comparison with the latter contribution. When this result is combined with the one given by eq. (4.24) we obtain as our approximation to the reducible vertex function

τ (q, q  |k, k  ) 2πδ(q − k − q  + k  )W 2 φ(q)

= 2πδ(q − k − q  + k  ) 

   K(q − q  ) K(q  + k) + φ(k) . × W 2 g |q − k| + W 2 φ(q) 1 − K(q  + k) 1 − K(q − q  ) (4.31) The content of the result given by eq. (4.31) is made clearer if we evaluate the function K(Q) with the aid of the pole approximation (4.16) for the Green’s func-

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tion G(k). This approximation yields the result that G(p)G∗ (Q − p) = −

2i 2i C2 C2 + 2 2 Q − 2i (p − ksp ) +  Q + 2i (p + ksp)2 + 2

2πC 2 2πiC 2 δ(p − ksp) + δ(p + ksp ), Q − 2i Q + 2i in the limit of small Q. With this result K(Q) becomes −

(4.32)

4sp (4.33) Q2 + 42 since, with the approximations made in the present calculation we find from eq. (4.20) that K(Q) =

sp = C 2 W 2 φ 2 (ksp ).

(4.34)

With the use of eq. (4.33) the reducible vertex function becomes

τ (q, q  |k, k  ) = 2πδ(q − k − q  + k  )     4sp 4sp × W 2 g |q − k| + W 2 φ(q) + (q  + k)2 + 4ε (q − q  )2 + 4ε

× φ(k) . (4.35) When we set q  = q and k = k  in eq. (4.35) and combine the result with eqs. (4.21) and (4.22), we obtain finally for ∂R/∂θs diff
   2 ω3 ∂R 2 G(q)2 = cos θ cos θ s 0 ∂θs diff π c3 

   sp 4sp 2 2 φ(k) × W g |q − k| + W φ(q) + ε (q + k)2 + 4ε  2 × G(k) . (4.36) The first term on the right-hand side of eq. (4.36) is the single-scattering contribution, the second is the contribution from the ladder diagrams, and the third is the contribution from the maximally-crossed diagrams. The latter two terms are equal for scattering into the retroreflection direction, i.e. when q = −k or θs = −θ0 . The prediction of enhanced backscattering is clearly seen through the presence of the denominator (q + k)2 + 4ε in the last term on the right-hand side of eq. (4.36), which is peaked at q = −k. This term, which is the contribution of the maximally crossed diagrams, expresses the consequence of the coherent interference of multiply-scattered surface plasmon polaritons with their reciprocal partners.

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We have presented the derivation of eq. (4.36) in some detail because the approach used and the approximations made will enable explicit results for angular intensity correlation functions to be obtained in Section 5. However, if the approximations made in obtaining the result given in eq. (4.36) are not made, in particular if the approximations given by eqs. (4.28) and (4.29) are not made, the contribution to the mean differential reflection coefficient from the diffuse component of the scattered light is still given by a very similar expression (McGurn, Maradudin and Celli [1985]), namely 
∂R ∂θs diff  3 2  ω =4 cos2 θs cos θ0 G(q) c

A(q|k) A(−k|k) C 2 2 C2 , + × K(q|k) + 2 1 − (sp /)2 (q + k)2 + 42 1 − (sp /)2 (4.37) where K(q|k) = − − while A(q|k) is a smoothly varying function of q and k. An explicit expression for it is given by McGurn, Maradudin and Celli [1985]. Again, the first term on the right-hand side of eq. (4.37) is the single-scattering contribution, the second is the contribution from the ladder diagrams, and the third is the contribution from the maximally-crossed diagrams. The latter two terms are equal for scattering into the retroreflection direction q = −k. An important difference of the described approach, sometimes referred to as the self-energy perturbation theory (Sánchez-Gil, Maradudin and Méndez [1995]) from the small-amplitude perturbation theory is that the expansion of the scattering amplitude is made in powers of the scattering potential V (q|k) and not just in powers of the surface profile. The scattering potential is calculated to ensure the unitary and reciprocal character of the scattering formalism (Brown, Celli, Coopersmith and Haller [1983]) – these properties are not manifestly present in the small-amplitude expansion, although that theory is unitary and reciprocal. Further, a careful analysis shows that the infinite perturbation series summed to obtain eq. (4.31) has the form of perturbation series in powers of a small parameter sp /. Thus, the perturbative expansion assumes that the ohmic losses exceed the roughness-induced scattering losses for a surface plasmon polariton. When the condition sp / 1 is satisfied, the results expressed by eq. (4.6) through terms of fourth order in δ and eq. (4.37) practically overlap each other, as can be seen in fig. 6 (Maradudin and Méndez [1993]), indicating that the fourth-order δ 2 g(|q

k|)|[(ε(ω) − 1)/ε2 (ω)][ε(ω)qk

α(q)α(k)]|2 ,

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perturbation theory indeed captures the essential behavior predicted in the infinite order perturbation theory. The small differences can be attributed to two reasons. The first reason is the truncation of the scattering potential V (q|k) in eq. (4.10) at the linear term in ζ . The second reason is that the dispersion and decay of a surface plasmon polariton due to roughness effects are not renormalized in the small amplitude perturbation theory. This explains the observed difference in the widths of the enhanced backscattering peaks predicted by the two perturbation theories – as we have seen in Section 4.1, the width is determined by the mean free path of a surface plasmon polariton. It is worth pointing out that the result for ∂R/∂θs diff given by eq. (4.37) is able to predict an enhanced backscattering peak for large angles of incidence, which small-amplitude perturbation theory through terms of fourth order in δ is unable to do. The results of the two perturbation theories, eqs. (4.6) and (4.37), are also in good agreement with the experimental results of West and O’Donnell [1995] (fig. 10), which are the only experimental results to show the enhanced

Fig. 10. A plot of ∂R/∂θs diff as a function of θs for three angles of incidence when p-polarized light of wavelength λ = 612 nm is incident on a one-dimensional random gold surface (ε(ω) = −9.00 + i1.29) characterized by the West–O’Donnell power spectrum (2.24) with δ = 10.9 nm, kmin = 8.4815 × 10−3 nm−1 , and kmax = 13.200 × 10−3 nm−1 . (0) Experimental data of West and O’Donnell [1995]; (−) results based on eq. (4.6); () results based on eq. (4.37) (Maradudin, McGurn and Méndez [1995]).

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backscattering effect in scattering from weakly rough metal surfaces, where the mechanism responsible for the effect is the multiple scattering of the surface plasmon polaritons excited by the incident light. The limit sp /  1, which is not described by the presented theories, requires a fully self-consistent treatment of the scattering problem and is described by Celli, Maradudin, Marvin and McGurn [1985] for the one-dimensional case and by McGurn and Maradudin [1987] in the two-dimensional case. The enhanced backscattering effect was predicted for random surfaces of both dimensionalities. Other theoretical studies of the surface plasmon polariton-assisted enhanced backscattering from a weakly rough metal surfaces were done by Hanato, Ogura and Wang [1997] for the one-dimensional case, and McGurn and Maradudin [1996], Hanato, Ogura and Wang [1997], and Johnson [1999] for the two-dimensional case, with results showing agreement with the studies cited above. To conclude discussion of the enhanced backscattering effect from weakly rough surfaces, we point out that this effect is not specific only to metallic surfaces. In a theoretical work by Leskova, Maradudin and Shchegrov [1997b] it was shown that the effect can be observed for weakly rough surfaces of dipole-active media which support surface phonon polaritons or surface exciton polaritons. 4.2.2. Very rough surfaces: scattering simulations 4.2.2.1. Simulations based on the surface integral equations. The first experimental observations of the enhanced backscattering of light from random surfaces (Méndez and O’Donnell [1987], O’Donnell and Méndez [1987]) were carried out for very rough metal surfaces, for which the perturbation theories described in Section 4.2.1 are invalid. The theoretical approach that has become most commonly used in dealing with such surfaces is based on a numerical solution of the surface integral equations (3.12a), (3.12b), which are exact for surfaces with arbitrary geometrical properties. In this approach, the infinite range of integration in the integral terms of eqs. (3.12a), (3.12b) is replaced by a finite one (−L1 /2, L1 /2), since computers cannot deal with infinitely long surfaces. Then, a realization of a segment of a random surface profile x3 = ζ (x1 ) that covers the interval (−L1 /2, L1 /2) is generated numerically. In most cases, the surface is divided into N intervals of equal length x = L1 /N . Two algorithms for generating random realizations of x3 = ζ (x1 ) on a discrete mesh given the values of the rms roughness amplitude δ and the form of the correlation function can be found in the papers by Maradudin, Michel, McGurn and Méndez [1990] and by Freilikher, Kanzieper and Maradudin [1997] (the appendix of the latter article describes an effective

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but, to our knowledge, unpublished method suggested by V.I. Tatarskii). After the random surface is generated, the integrals in eqs. (3.12a), (3.12b) are approximated by finite sums of N terms each. Thus, the integral equations are replaced (n) with the matrix equations for the values of the field Un = U(x1 ) and its normal (n) derivative Vn = V(x1 ) at the midpoint   x1(n) = −L + n − 12 x,

n = 1, . . . , N,

(4.38)

of each interval. To minimize the edge effects caused by the truncation of the integrals, the incident field is usually chosen to represent a beam of a finite width (eq. (3.14)). The resulting equations have the form Um = 2Uminc −

N  

 (0) Un − L(0) Hmn mn Vn ,

(4.39a)

n=1 N   (ε)  Hmn Un − νL(ε) Um = − mn Vn ,

(4.39b)

n=1

where the factor of two before the incident field term Uminc = U inc (x1(m) ) appears due to the contribution from the singular integration point x1 = x1 in eq. (3.12a). Expressions for the matrices H(0,ε) and L(0,ε) are obtained from eq. (3.13) and can be found in the paper by Maradudin, Michel, McGurn and Méndez [1990]. The system (4.39) of 2N equations is then solved numerically, and the source functions U and V obtained are substituted into the integral expression (3.11) for the scattering amplitude, which is also converted into a finite sum. The procedure must be repeated for many realizations of the surface profile function so that the average characteristics of the intensity of diffusely scattered light can be obtained. Due to the statistical nature of this computational method it is often referred to as scattering Monte Carlo simulations, while the general approach of discretizing the scattering system to convert the integral equations into matrix equations is known as the method of moments. This numerical simulation approach was first implemented for perfectly conducting surfaces by Nieto-Vesperinas and Soto-Crespo [1987]. Within the model of a perfect conductor, one has to solve only N coupled equations given by eq. (4.39a), since one of the source functions vanishes identically (due to the Dirichlet boundary condition U ≡ 0 for s-polarization or the Neumann boundary condition ∂U/∂n ≡ 0 for p-polarization). The enhanced backscattering effect was found for both s and p polarizations for surfaces with the Gaussian correlation function and typical values of the roughness rms amplitude δ and correlation

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length a of the order of the wavelength λ. As expected from the qualitative argument given in Section 4.1, the angular width of the enhanced backscattering peak turned out to be approximately equal to λ/a. The early work cited was followed by numerous subsequent investigations, including those done for one-dimensional surfaces of penetrable media (Tran and Celli [1988], Maradudin, Méndez and Michel [1989], and many others) and a few investigations of scattering from two-dimensional rough surfaces (Tran and Maradudin [1992, 1993]). These studies revealed several points that are essential in optimizing the speed and reliability of the numerical simulation approach as well as some criteria for the existence of pronounced coherent multiple-scattering phenomena. We summarize the most important of these findings below. The first issue concerns the optimal choice of the simulation parameters characterizing the surface (L1 , N , and x = L1 /N ) and the incident beam (half-width of the illuminated region g) for a given frequency ω (or wavelength λ = 2πc/ω), and the number of realizations Np of the surface profile. The discretization interval x must be small compared to λ to properly model wave propagation and scattering and small compared to the correlation length a to properly model the details of the surface profile. On the other hand, the length L1 needs to be large compared to a and to w (it is usually assumed that w  λ). The achievement of the two obvious goals of minimizing x and maximizing L1 is limited by the finite number of points N = L1 /x that can be handled by a computer. Most investigations show that for comparable a and λ simulations of reasonable quality require x
0.2 min(a, λ),

L1  30λ,

g ∼ L1 /5.

(4.40)

These numbers, of course, have to be considered as only empirical estimates, based on the authors’ and other researchers’ experience with scattering simulations. The “marginally suitable” parameters x = 0.2λ and L1 = 30λ yield N = 150, thus requiring the solution of 2N = 300 coupled algebraic equations. This task can now be easily handled by any Pentium-class PC without special memory requirements. Although experiments usually deal with much longer surfaces and illuminated areas whose linear dimensions are larger than a few tens of a wavelength, scattering simulations appear to provide reasonably good modeling of experimental results when the statistical properties of a random surface are accurately known, and are introduced into a computer simulation of the scattering of light from that surface (see fig. 11 adapted from the paper by Knotts, Michel and O’Donnell [1993]). The second issue concerns the testing of the reliability of simulations that has to be done even for an intelligent choice of input parameters. Usually, it is wise to

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Fig. 11. The elements sij of the Stokes matrix and the +45◦ -polarized intensity, when light of wavelength λ = 3.392 µm is incident on a one-dimensional, randomly rough, gold surface with Gaussian power spectrum (2.21) characterized by δ = 1.73 µm and a = 3.43 µm. The angle of incidence is θ0 = 10◦ . Experimental results (solid curves) are compared with numerical results obtained by the use of an impedance boundary condition applied to a surface profile obtained directly by contact profilometry (Knotts, Michel and O’Donnell [1993]).

vary x, L1 , or N to check whether the result is stable against these variations. If not, a further decrease of x and/or increase of L1 is recommended. It is also useful to check the energy conservation requirement R + T = 1 (or R = 1 for perfectly conducting surfaces) for the total intensity reflection (R) and transmission (T ) coefficients. This criterion becomes especially important for surfaces supporting surface waves, which can propagate a distance of a few wavelengths even on rather rough surfaces. If the dissipative loss is expected to be low, then a bad energy conservation test means that the chosen surface is not long enough to ensure the total conversion of excited surface waves into the reflected/transmitted light. The typical energy conservation tests reported in the thoroughly tested numerical simulations in lossless geometries exceed 97%. While the two tests mentioned should be reasonably well satisfied for every realization of the surface profile, one must also obtain the convergence of the average intensity with an increase in the

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Fig. 12. A plot of ∂R/∂θs diff as a function of θs when light of wavelength λ = 632.8 nm is incident normally on a one-dimensional random surface of BaSO4 (nc = 1.628 + i0.0003) characterized by the Gaussian power spectrum (2.20) with δ = 1.2 µm and a = 2 µm. (a) p-polarized incident light (Maradudin, Méndez and Michel [1990]); (b) s-polarized incident light (Maradudin, Michel, McGurn and Méndez [1990]).

number Np of independent surface realizations. As usual in statistical sampling problems, the fluctuations around the average value are expected to decrease as  1/ Np . There has also been much evidence that large rms height, large rms slope surfaces that display a pronounced enhanced backscattering effect induced by the mechanism illustrated in fig. 4 have to be strongly reflective to ensure the efficacy of that mechanism. Such surfaces include, e.g., metallic and semiconductor surfaces in the frequency ranges where Re ε(ω) is negative, and perfectly conducting (model) surfaces. Highly reflective properties can also be expected from dielectrics with positive Re ε(ω) when Re ε(ω) is not very close to unity (high dielectric contrast). Figure 12 provides an illustration that the same one-dimensional randomly rough dielectric surface (ε = 2.25) can display the enhanced backscattering effect for s-polarized light (strongly reflected) and show no sign of the enhanced backscattering for p-polarized light (weakly reflected). The focus of current efforts in scattering simulations seems to be on improving the numerical algorithms to solve matrix equations for the source functions – such improvement is critically needed in dealing with two-dimensional rough surfaces. The abundance of proposed numerical schemes makes it impossible to describe them in detail within this article. In addition, none of these schemes appears to have proved itself as the best choice for all scattering configurations. We present only a brief outline of some of these algorithms. 4.2.2.2. Direct solution of matrix equation. The most straightforward approach, which was used, e.g., in the work of Maradudin, Michel, McGurn and Méndez [1990] cited, consists of the direct solution of the system by, e.g., the LU-

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factorization technique (Press, Teukolsky, Vetterling and Flannery [1992]). The speed of this approach scales as O(Nu3 ) with the number of unknowns Nu (equal to 2N for eq. (4.39a)) and requires a storage capacity of O(Nu2 ). While this exact approach can be safely chosen for Nu smaller than a few thousand (handled by ordinary computers), its utility rapidly decreases for the problems requiring large Nu , e.g., in the case of moderately rough metallic surfaces supporting surface plasmon polaritons that can propagate for considerable distances away from the illuminated area. Large values of Nu are also required in scattering problems with several interfaces (Lu, Maradudin and Michel [1991]). Finally, the necessary number of unknown Nu is especially large for the scattering from a twodimensional rough surface – for a vector field one has Nu = 4N 2 where N is the number of discretization points in the x1 - and x2 -directions, while 4 is the number of independent components of the vector field and its normal derivative (Ong, Celli and Maradudin [1993], Maradudin [1995]). Even a rather small value of N = 100 yields N = 40, 000 which exceeds both storage and speed limitations of best available computers by orders of magnitude. 4.2.2.3. Iterative techniques. A possible approach to circumvent this problem is to use an iterative scheme. For example, in a Neumann–Liouville iterative scheme one takes (2Uminc , 0) for (Um , Vm ) as an initial iteration and continues to carry out iterations without storing kernel matrices until convergence is reached – the implementation of this procedure in the one-dimensional roughness case is presented by Maradudin, Michel, McGurn and Méndez [1990]. The storage and speed of this approach scale as O(Nu2 ) and O(Nu ), respectively. The Neumann–Liouville scheme was used to simulate the enhanced backscattering of scalar waves from Neumann and Dirichlet surfaces with Nu = 2N 2 (Tran and Maradudin [1992, 1993]) and of vector electromagnetic waves (light) from metallic surfaces (Tran and Maradudin [1994]). An attractive feature of the Neumann–Liouville series is that the nth term describes an n-fold scattering process in which the field interacts with the surface n times. Strictly speaking, this identification is valid only in the high frequency or short wavelength limit (Liszka and McCoy [1982]). However, it remains qualitatively valid at lower frequencies as well. Simulations clearly show that the first-order (single-scattering) is normally dominant, but the enhanced backscattering peak appears only if the double-scattering processes are included (fig. 13). Several algorithms have been proposed to improve the convergence of iterations. For example, the method of ordered multiple interactions (MOMI), developed for one-dimensional perfectly conducting surfaces (Kapp and Brown [1996]) and later applied to one-dimensional rough surfaces of penetrable me-

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Fig. 13. Plot of the mean differential reflection coefficient and the contribution from the individual scattering processes. The computer simulations were performed for a two-dimensional rough Dirichlet surface with a Gaussian power spectrum characterized by δ = λ and a = 2λ, illuminated by a normally incident scalar beam. The n-fold scattering processes that are not labeled, decrease monotonically (Tran and Maradudin [1993]).

dia (Tran and Elson [1998a]), modifies the iteration series so that each term contains a large number of scattering events (predominantly scattering events that follow the same direction as the original scattering event). The resulting scheme still shows O(Nu2 ) and O(Nu ) storage and speed scaling, but its convergence is significantly better than that of the Neumann–Liouville iterations. Further improvements can be achieved by breaking up the matrices in the matrix equation into “weak” and “strong” parts (Tsang, Chan, Pak, Sangani, Ishimaru and Phu [1994], Tsang, Chan and Pak [1994]). Such breaking can be applied in both standard and MOMI-reconditioned matrix equations (Tran and Elson [1998b]). Improvements can also be achieved by combining the effective matrix-vector multiplication techniques and conjugate-gradient solvers of the matrix equations. Discussion on such improvements can be found in the papers by Chew and his collaborators developing the so-called FMM or fast multipole method (Michielssen, Boag and Chew [1996], Wagner, Song and Chew [1997]). In the cases considered, the FMM approaches were shown to improve the speed performance from O(Nu2 ) to O(Nu log Nu ), but the applicability of these methods (or necessity of their modification) to coherent multiple-scattering phenomena in rough surface scattering remains to be proven. Currently, the development of these efficient methods continues but none of them appears to have been accepted as a general and most efficient tool for all rough surface scattering simulations. A big part of the problem is that most of that work is done within the perfect conductor model. The matrices of equations for penetrable media are expected to have features (e.g., due to surface waves) that are poorly handled by iterative schemes or perfect-conductor-oriented matrix

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Fig. 14. A plot of the contribution to the mean differential reflection coefficient from the diffuse component of the scattered light when a beam of p-polarized light of wavelength λ = 457.9 nm is incident normally on a two-dimensional randomly rough silver surface (ε(ω) = −7.5 + i0.24) characterized by the Gaussian power spectrum (2.17), with δ = 1.0λ and a = 2.0λ. The solid curves give the contribution from in-plane scattering (φs = 0◦ ), and the dashed curves give the contribution from out-of-plane scattering (φs = 90◦ ) (Tran and Maradudin [1994]).

reconditioning. This is why the iterative series often rapidly diverges for surfaces that support surface waves (Tran [1998]). Another apparent problem with these approaches is difficulties with their extension to the two-dimensional case (e.g., MOMI) or lack of sufficient evidence that they can describe regimes when the enhanced backscattering peak is best visible (FMM). Despite several preliminary reports of the numerically simulated enhanced backscattering effect (Tran and Maradudin [1992, 1993, 1994], Macaskill and Kachoyan [1993], Tran, Celli and Maradudin [1994]) from two-dimensional rough surfaces, the numerical solution of the scattering from a two-dimensional randomly rough surface in the multiple-scattering regime remains the most serious challenge in the field. Current efforts remain centered on the perfect conductor and/or scalar models which reduce the number of equations. However, the effect that involves surface wave excitation and polarization change, observed in the very first experiments on enhanced backscattering, require the solution of the full vector problem for a penetrable medium. A powerful assistance in solving such problems is given by parallel computing, which is traditionally useful in Monte Carlo algorithms. For example, Tran and Maradudin [1994] were able to study a 128 × 128 points mesh modeling a metallic surface that exhibits enhanced backscattering (fig. 14) in this way. 4.2.2.4. Impedance approximation. A powerful technique to reduce the number of unknowns in the surface integral (or matrix) equations is provided by the method of impedance boundary conditions. In the general formulation of this method for a one-dimensional surface (Maradudin [1993]) the starting point is a

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linear relation between the values of the field and its unnormalized normal derivative on the surface x3 = ζ (x1 )  ∞ V(x1 ) = (4.41) dx1 K(x1 |x1 )U(x1 ) −∞

(the source functions U and V are introduced in Section 3.3). The non-local impedance K(x1 |x1 ) is expanded in powers of the optical skin depth d(ω), which is defined by  −1/2 d(ω) = (c/ω) −ε(ω) (4.42) . In most practical cases the local impedance relation  −σ (0) V(x1 ) = ε(ω) K (x1 )U(x1 )

(4.43)

(as before, σ = 1 for p-polarization and σ = 0 for s-polarization) was shown to yield results that agree with those obtained with a more accurate non-local relation (4.41) or with exact integral equations (3.12a) (Maradudin and Méndez [1996]). The accuracy of the impedance approximation was shown to improve for materials with small |d(ω)|. Such materials include most metals in the infrared and visible frequency ranges and dielectrics with Re ε(ω) considerably larger than unity (surprisingly, the values of Re ε(ω) > 2 can often be sufficient to make eq. (4.43) a good approximation). The local surface impedance K (0) (x1 ) in eq. (4.43) is given by

 3 φ(x1 ) [ζ  (x1 )]2 ζ  (x1 ) (0) 2 +O d K (x1 ) = 1 + d(ω) − d (ω) , (4.44) d(ω) 2φ(x1) 8φ 4 (x1 )  where φ(x1 ) = 1 + [ζ  (x1 )]2 . The term of O(1/d) was obtained by Rosich and Wait [1977], the term of O(1) was obtained by Depine [1987, 1988], the term of O(d) was calculated by Garcia-Molina, Maradudin and Leskova [1990], and the term of O(d 2 ) (not shown in eq. (4.44)) was derived by Maradudin [1993] and by Marvin and Celli [1994]. Since ζ2 (x1 ) is essentially the local curvature of the surface1 , it is therefore expected that the first few terms in the expansion (4.44) will provide a good approximation to K (0) (x1 ) when the skin depth is small compared to [ζ  (x1 )]−1 , or to its rms value in the case of a random surface. Several recent papers discuss the improvements and generalizations of the local relation (4.44) – e.g., Maradudin and Méndez [1996] discuss the role and the magnitude of non-local corrections, while Mendoza-Suarez and Méndez [1997] generalize eq. (4.44) to the case of re-entrant surfaces. 1 Strictly speaking, the local curvature of the surface is ζ  (x )/φ(x )3 . 1 1

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159

The use of eq. (4.41) or eq. (4.43) in eqs. (3.12) has the consequence that only the first of these two equations has to be solved, thereby dramatically reducing computational cost. This advantage becomes especially valuable for twodimensional rough surfaces, for which the vector analog of the impedance conditions (4.41) or (4.43) was derived by Ong, Celli and Maradudin [1993], Ong, Celli and Marvin [1994] and Maradudin [1995]. 4.2.2.5. Simulations based on the reduced Rayleigh equations. The scattering simulation approach we discussed in this subsection is based on the integral equations for the electromagnetic field in the coordinate space. Similar Monte Carlo simulations can be done in momentum (wavenumber) space for the reduced Rayleigh equation (3.8). Such simulations were carried out, e.g., by Michel [1994], and showed that the Rayleigh method can be used for fairly rough surfaces that do not allow a perturbative treatment. Madrazo and Maradudin [1997] and Simonsen and Maradudin [1999] performed such simulations for film systems and demonstrated a good overlap of their results with the predictions of the real-space simulations based on the exact surface integral equations. 4.2.3. Other approaches In the previous two subsections, we have described perturbation techniques for weakly rough surfaces and numerical simulation approaches that are usually used for large-amplitude, large-slope surfaces. These two approaches are the most commonly used since most multiple-scattering problems can be treated with one or the other of them. However, much work has been done since the mid-1980s in other directions. We next review these other approaches (without presenting their mathematical details) and separate them into two categories. The first category includes methods that appear to differ from the Rayleigh method (the starting point of perturbative expansions) or the exact surface integral equations (the starting point of numerical simulations) only in formalism but not in their range of validity or generality. The stochastic functional approach (Ogura, Kawanishi, Takahashi and Wang [1995], Ogura and Takahashi [1995], Ogura and Wang [1995], and Kawanishi, Ogura and Wang [1997]), utilizes the Wiener–Itô expansion of the random fields in the weak roughness limit. The stochastic functional approach generally yields results consistent with those obtained by the perturbative solution of the reduced Rayleigh equation and capture the coherent enhanced backscattering if the expansion is done beyond first order in the surface profile function. The phase perturbation theory, first suggested by Maradudin and Shen [1980], utilizes the reduced Rayleigh equation to develop an expression for the scat-

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[2, § 4

tering S-matrix that contains a small-amplitude expansion series in the exponent:  ∞  ∞  (−i)n   Ψ (n) (q|x1|k) , (4.45) dx1 exp −i(q − k)x1 exp S(q|k) = n! −∞ n=1

where Ψ (n)

is O(ζ n )

and the relation of S(q|k) to the scattering amplitude R(q|k) is given by eq. (3.4). Sánchez-Gil, Maradudin and Méndez [1995] obtained the expansion (4.45) in a form that manifestly preserves the reciprocity of the scattering matrix: S(q|k) = S(−k| − q). They also investigated the ranges of applicability of the phase perturbation theory and the two perturbation theories described in Section 4.2.1 by comparing their results with the predictions of numerical simulations for a one-dimensional weakly rough surface of a dielectric with ε = 2.25. It was found that in both s- and p-polarizations the phase perturbation theory produced the best agreement with the simulation data for each angle of incidence considered, while the small amplitude perturbation theory was the least accurate of the three. This accuracy of the phase perturbation theory holds even for large angles of incidence for which both small-amplitude and self-energy perturbation theories do not work very well. The two-dimensional version of the phase perturbation theory was developed within a scalar model for the Dirichlet boundary condition, also in a reciprocal formulation (Fitzgerald and Maradudin [1994]). However, the multiple-scattering formalism provided by the phase perturbation theory has yet to prove its utility in the regime of coherent multiple-scattering phenomena, such as enhanced backscattering. The full-wave approach (Bahar [1987a, 1987b]) obtains approximate expressions for the scattered fields not by iterating the surface integral equations (3.12a) but by switching to local surface coordinates and obtaining approximate integral equations for the scattered field. The key double-scattering term that accounts for enhanced backscattering is written as a 6-dimensional integral even for a onedimensional surface (Bahar and El-Shenawee [1995]), which can be evaluated only when some simplifying assumptions are made (e.g., large surface rms curvatures compared to the wavelength). Unfortunately, the range of applicability for the full-wave approach appears to be unclear. The early theoretical work by Bahar and Fitzwater [1989] where enhanced backscattering was predicted, essentially relied on the approximation of mutual uncorrelation of heights and slopes. This approximation was shown to be invalid (Collin [1992]), even causing the artificial appearance of a retroreflection peak in the predominantly single-scattering regime. Although this contradiction was removed in more recent work (Bahar and El-Shenawee [1995]), the range of applicability of the full-wave method requires further examination.

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The second category includes approaches that are specifically tailored to the scattering geometry. The most famous of them is the Kirchhoff approximation, which has been used for decades (see, e.g., Beckmann and Spizzichino [1963], Bass and Fuks [1980]) to investigate single-scattering of waves from smooth surfaces with large-scale roughness by assuming that the scattered field is formed due to the specular reflection of light from the plane that is tangent to the surface at each point. After the discovery of enhanced backscattering from rough surfaces, the Kirchhoff approximation was extended to the second order, where the light is scattered twice from the surface, each time specularly from the plane tangent to the surface at the points struck by the light, with shadowing incorporated into the result (Ishimaru and Chen [1990], Chen and Ishimaru [1990]). The angular distribution of the intensity of the scattered light, obtained within the second-order Kirchhoff approximation displayed enhanced backscattering for one-dimensional perfectly conducting surfaces and was in agreement with the results of numerical simulations. The theory was later extended to the case of penetrable media (Ishimaru and Chen [1991]). In this context, we mention the paper by Li and Fung [1991] in which the authors reformulated the surface field integral equations for the scattering of light from a two-dimensional rough penetrable surface in such a way that the inhomogeneous terms represented the Kirchhoff approximation to the source functions. The iterative solutions of the reformulated equations were expected to converge rapidly in application to scattering from surfaces for which the Kirchhoff approximation is a good approximation to the exact result. However, no numerical results for the differential reflection coefficient were obtained on the basis of the reformulated equations. Subsequently, Sentenac and Maradudin [1992], reformulated the pair of surface integral equations (3.12) for scattering from a one-dimensional rough penetrable surface in such a way that the Neumann–Liouville series solution of the resulting pair of equations started with the Kirchhoff approximation to each of the source functions. Numerical results for the mean differential reflection coefficient obtained by an iterative solution of the reformulated equations were found to be in good qualitative and quantitative agreement with the results of a direct numerical solution of the original untransformed pair of equations. The same problem was studied by Stoddart [1992], by a different approach that also led to a pair of integral equations for the source functions that had the Kirchhoff approximation to these functions as the inhomogeneous terms. The convergence properties of the iterative solution of the resulting pair of integral equations were examined, but no numerical results were obtained from the transformed equations.

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[2, § 4

The recently developed small-contrast perturbation theory (Leskova, Maradudin and Novikov [2000]) is built on the expansion of the scattering amplitude in powers of the small dielectric contrast η = ε0 − ε,

(4.46)

where ε0 is the dielectric constant of the medium of incidence and ε is the dielectric constant of the scattering medium. This approach can be applied, for example, to the scattering of X-rays from a randomly rough metal surface (Leskova and Maradudin [1997, 1999]), where ε0 = 1 and ε = ε(ω), the dielectric function of the metal. The term of nth order in the expansion of the scattered field in powers of η contains contributions of all orders in the surface profile function. This new expansion yields the enhanced backscattering peak when carried out through terms of fourth order in η(ω). There are also several novel theories that have been developed recently but were not shown to describe coherent multiple scattering phenomena such as enhanced backscattering. This should not be considered as a negative feature of these theories since they proved themselves very effective within their range of validity. These theories include, e.g., small-slope perturbation theory, the operator expansion formalism, and the coherent potential mean-field theory. The small-slope perturbation theory (Voronovich [1985, 1994, 1996]) is akin to the Kirchhoff series. This theory, developed for the scattering of scalar waves from perfectly conducting, smooth surfaces, represents the scattered field as an expansion in powers of small surface slopes. Further investigations including numerical testing against other methods’ predictions (Thorsos and Broschat [1995], Broschat and Thorsos [1997]), showed a robust performance of this approach, including the often problematic case of grazing angles of incidence. The operator expansion formalism is another approach that has been proposed to describe scattering from surfaces with small slopes, first for perfectly conducting surfaces (Milder [1996]) and then extended to surfaces of penetrable media (Smith [1996]). The expansion leads to efficient numerical schemes through the use of fast Fourier transforms. The small slopes assumed in both smallslope and operator expansions realistically model many physical situations such as the scattering of electromagnetic waves from the ocean surface. The coherent potential mean-field theory has been developed for both one-dimensional (Sentenac and Greffet [1998]) and two-dimensional dielectric surfaces (CalvoPerez, Sentenac and Greffet [1999]). In the version of this approach applicable to scattering from one-dimensional surfaces the dielectric constant of the system, ε(x1 , x3 ) = ε> θ (x3 − ζ (x1 )) + ε< θ (ζ (x1 ) − x3 ), where ε> is the dielectric constant of the region x3 > ζ (x1 ) while ε< is the dielectric constant of the region

2, § 4]

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163

x3 < ζ (x1 ), is rewritten in the form ε(x1 , x3 ) = ε(x3 ) + [ε(x1, x3 ) − ε(x3 )], where ε(x3 ) is the average of ε(x1 , x3 ) with respect to x1 . The solution of the scattering problem is then obtained by regarding the fluctuation term [ε(x1, x3 ) − ε(x3 )] as a perturbation on the system described by ε(x3 ). An analogous procedure is followed in the studying of the scattering from a two-dimensional random interface between two dielectric media. This approach does not rely on the smallness of the roughness amplitudes or slopes and was shown to perform well for dielectric constants smaller than 5, but has yet to be extended to the multiple-scattering regime. The approaches reviewed in this subsection probably do not account for the full scope of ongoing theoretical research but certainly demonstrate the high level of activity and creativity of researchers in the very lively rough surface scattering field. 4.2.4. The polarization dependence of scattering from a one-dimensional rough surface In the preceding discussion of the scattering of light from a one-dimensional random surface, where it was assumed that the plane of incidence is normal to the generators of the surface, the plane of scattering coincides with the plane of incidence. We will call scattering in this geometry in-plane scattering. The incident electric field was linearly polarized either parallel or perpendicular to the grooves and ridges of the surface, i.e. it was s- or p-polarized, respectively. In this case the polarization of the scattered field was the same as that of the incident field. The diffuse scattering is therefore described by the two scattering cross sections ∂Rs /∂θs diff and ∂Rp /∂θs diff . However, it was shown by O’Donnell and Knotts [1991] and by Michel, Knotts and O’Donnell [1992] that the complete determination of the incoherent scattering properties (viz. those described by the second-order moments of the field) of a one-dimensional random surface, when the plane of incidence is normal to the generators of the latter, requires not just two but four scattering cross sections. Two of them are effectively ∂Rs /∂θs diff and ∂Rp /∂θs diff , but the other two must be determined by the use of an incident field that is a linear combination of p- and s-polarized fields. To obtain these scattering cross sections we begin by writing the electric vector of a beam of light incident from the vacuum side (x3 > ζ(x1 )) on a onedimensional surface defined by x3 = ζ (x1 ), with the plane of incidence the x1 x3 plane,

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E> (x1 , x3 |ω)inc  π/2     ωw dθ exp − ω2 w2 /4c2 (θ − θ0 )2 = √ 2 π c −π/2   × Ep eˆ p (θ )inc + Es eˆ s (θ )inc   × exp i(ω/c)(x1 sin θ − x3 cos θ ) ,

[2, § 4

|θ0 | < π/2, (4.47)

where eˆ s (θ )inc = xˆ 2 ,

(4.48a)

eˆ p (θ )inc = −ˆx1 cos θ − xˆ 3 sin θ.

(4.48b)

The field given by eq. (4.47) is an exact solution of Maxwell’s equations in vacuum that in the limit (ωw/2c)  1 describes a Gaussian beam propagating in the direction defined by xˆ 1 sin θ0 − xˆ 3 cos θ0 . Its width at the origin of coordinates is 2w. The scattered electric field in the far field is then given by  π/2   i dθs Ep rp (θs )ˆep (θs )sc + Es rs (θs )ˆes (θs )sc E(x1 , x3 |ω)sc = 4π −π/2   × exp i(ω/c)(x1 sin θs + x3 cos θs ) , (4.49) where eˆ s (θs )sc = xˆ 2 ,

(4.50a)

eˆ p (θs )sc = xˆ 1 cos θs − xˆ 3 sin θs

(4.50b)

and the scattering amplitudes rp (θs ) and rs (θs ) are given by  ∞    rp (θs ) = dx1 exp −i(ω/c) x1 sin θs + ζ(x1 ) cos θs −∞

 rs (θs ) =

∞ −∞

    × i(ω/c) ζ  (x1 ) sin θs − cos θs H (x1 |ω) − L(x1 |ω) , (4.51a)    dx1 exp −i(ω/c) x1 sin θs + ζ(x1 ) cos θs

    × i(ω/c) ζ  (x1 ) sin θs − cos θs E(x1 |ω) − F (x1 |ω) . (4.51b) > The source functions H (x1 |ω) and E(x1 |ω) are given by U (x1 , ζ (x1 )), while the source functions L(x1 |ω) and F (x1 |ω) are given by (∂/∂N)U > (x1 , x3 |ω)|x3 =ζ(x1 ) ,

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165

where we recall that U > (x1 , x3 |ω) = H2> (x1 , x3 |ω)(E2> (x1 , x3 |ω)) for p- (s-)polarized light. It is desirable to define the polarizations of the incident and scattered fields in terms of directly measurable quantities. With this in mind it is necessary to calculate the power present in the projection of the incident and scattered fields on polarization states that are linear combinations of the p- and s-polarized states defined by the unit vectors (4.48) and (4.50), respectively. A useful description of polarized light in terms of these powers is through the Stokes vectors (Hecht [1970]), and Stokes matrices (Ishimaru [1978], van de Hulst [1981]). The following pairs of orthonormal vectors are needed to compute the Stokes vectors (Jackson [1975]): 1 eˆ + = √ (ˆep + eˆ s ), 2

1 eˆ − = √ (ˆep − eˆ s ) 2

(4.52)

1 eˆ R = √ (ˆep − iˆes ), 2

1 eˆ L = √ (ˆep + iˆes ), 2

(4.53)

and

where eˆ α with α = p, s, +, −, R, or L must be replaced by eˆ α (θ0 )inc in the case of the incident field, and by eˆ α (θs )sc in the case of the scattered field. The pairs of unit vectors (4.52) and (4.53) define the linearly polarized states at +45◦ , as well as right (R) and left (L) circularly polarized states. If (α, β) denotes any of the pairs (p, s), (+, −), or (R, L), the projection of a field E(x1 , x3 |ω) on eˆ γ (γ = α or β) is Eγ (x1 , x3 |ω) = [ˆe∗γ · E(x1 , x3 |ω)]ˆeγ , and we have that E(x1 , x3 |ω) = Eα (x1 , x3 |ω) + Eβ (x1 , x3 |ω). The elements of the Stokes vector V = (I, Q, U, V ) may be expressed as (Born and Wolf [1980]): I = Pp + Ps = P+ + P− = PR + PL ,

(4.54a)

Q = Pp − Ps ,

(4.54b)

U = P+ − P− ,

(4.54c)

V = PR − PL .

(4.54d)

In the case of the incident field Pα is the total power incident on the surface that is in the projection Eα (x1 , x3 |ω)inc of the incident field on eˆ α . To obtain this total incident power the magnitude of the 3-component of the time-averaged Poynting vector of Eα (x1 , x3 |ω)inc is integrated over the region of the x1 x2 -plane defined by −L1 /2 < x1 < L1 /2, −L2 /2 < x2 < L2 /2, where L1 and L2 are large lengths.

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The Stokes vector of the incident field Vinc thus becomes     |Ep |2 + |Es |2 Iinc  |Ep |2 − |Es |2   Qinc     Vinc =   Uinc  = Pinc  2 Re(Ep E ∗ )  , s Vinc 2 Im(Ep Es∗ )

[2, § 4

(4.55)

where Pinc is given by Pinc ∼ = L2 cw/[8(2π)1/2] when ωw/2c  1. The powers Pα required for the calculation of the Stokes vector of the scattered field given by eq. (4.49) are the differential powers per unit scattering angle Pα (θs )sc obtained from the projection of the plane wave scattered at an angle θs (the integrand on the right-hand side of eq. (4.49)) on the polarization state eˆ α (θs )sc . The function Pα (θs )sc is calculated by integrating the 3-component of the timeaveraged Poynting vector of this projection over the region −L1 /2 < x1 < L1 /2, −L2 /2 < x2 < L2 /2 of a plane in the far field that is parallel to the plane x3 = 0. The Stokes vector v(θs )sc calculated from the scattered field (4.49) with the use of eqs. (4.54) for a single realization of the rough surface is then   |rp (θs )Ep |2 + |rs (θs )Es |2 2  |rp (θs )Ep |2 − |rs (θs )Es |2  c .  v(θs )sc = L2 (4.56) 2 64π ω  2 Re[rp (θs )Ep r ∗ (θs )E ∗ ]  s

s

2 Im[rp (θs )Ep rs∗ (θs )Es∗ ]

A matrix ↔ s (θs ) that relates the Stokes vector of the incident field given by eq. (4.55) to the Stokes vector of the scattered field given by eq. (4.56) through the equation v(θs )sc = ↔ s (θs )Vinc

(4.57)

can be obtained by using the fact that the amplitudes Ep and Es are arbitrary complex amplitudes. This matrix is found to have the form   0 0 s11 (θs ) s12 (θs )  s12 (θs ) s11 (θs ) 0 0  ↔ , s (θs ) =  (4.58)  0 0 s33 (θs ) s34 (θs )  0 0 −s34 (θs ) s33 (θs ) where

2  2   s11 (θs ) = C rp (θs ) + rs (θs ) , 2  2   s12 (θs ) = C rp (θs ) − rs (θs ) ,    s33 (θs ) = C 2 Re rp (θs )rs∗ (θs ) ,    s34 (θs ) = C −2 Im rp (θs )rs∗ (θs ) ,

(4.59a) (4.59b) (4.59c) (4.59d)

2, § 4]

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167

with C=

L2 c2 . 32(2π)2ωPinc

(4.60)

To each realization of the surface profile function ζ(x1 ) corresponds a realizas (θs ). In the scattering of light from a tion of the vector v(θs )sc and of the matrix ↔ randomly rough surface it is the average of the vector v(θs )sc over the ensemble of realizations of the surface profile function,

V(θs )sc = v(θs )sc , (4.61) that is of interest. This equation defines the Stokes vector of the scattered light V(θs )sc . The average of the matrix ↔ s (θs ) over the ensemble of realizations of the surface profile function,

↔ S(θs ) = ↔ (4.62) s (θs ) , is the matrix that enables the Stokes vector of the scattered light to be calculated for any given polarization state of the incident light, ↔

V(θs )sc = S(θs )Vinc . ↔

(4.63)

The matrix S(θs ) is called the Stokes matrix of the randomly rough surface. Just as the mean differential reflection coefficients contain contributions from the specular and diffuse components of the scattered field, so do the elements of the Stokes matrix. Thus, if we replace the scattering amplitudes rp (θs ) and rs (θs ) in eqs. (4.59) by their ensemble averages rp (θs ) and rs (θs ), respectively, we obtain the contribution to the Stokes matrix from the specular component of the ↔ scattered field, S(θs )spec . Similarly, if we replace the scattering amplitudes rp (θs ) and rs (θs ) in eqs. (4.59) by their fluctuating parts rp (θs ) − rp (θs ) and rs (θs ) − rs (θs ), respectively, and then average the resulting matrix elements over the ensemble of realizations of the surface profile function, we obtain the contribution ↔ to the Stokes matrix from the diffuse component of the scattered field, S(θs )diff . It ↔ ↔ ↔ should be noted that S(θs ) = S(θs )spec + S(θs )diff . The elements of the Stokes matrix (4.58) provide a complete description of the polarization of the light scattered from a one-dimensional randomly rough surface when the plane of incidence is normal to the generators of the surface. We note for future reference that the elements of the Stokes matrix for the in-plane scattering of light from a one-dimensional perfectly conducting random surface are obtained by setting ε(ω) = −∞ everywhere in eqs. (4.59). Knotts, Michel and O’Donnell [1993] carried out a comparison of the results of a computer simulation calculation of the elements of the Stokes matrix for the

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[2, § 4

in-plane scattering of light from a one-dimensional randomly rough gold surface with experimental results for these elements obtained in the same work. The frequency of the incident light was in the infrared region of the optical spectrum, so that the impedance boundary condition (4.43)–(4.44) was used in the solution of the integral equations of scattering theory, together with the actual surface profile, obtained by contact profilometry. The agreement between theory and experiment proved to be very good, both qualitatively and quantitatively (fig. 11). From results such as those presented in fig. 11 it can be said that the computational techniques developed for the study of the in-plane scattering of light from one-dimensional random surfaces are impressively accurate. The most detailed information about the polarization properties of light scattered from a one-dimensional random surface, however, is obtained from a study in which the plane of incidence is not perpendicular to the generators of the surface. In this scattering geometry the diffuse portion of the scattered radiation appears on the surface of a cone whose axis is parallel to the generators of the surface, rather than in a plane, because of the translational invariance of the scattering surface parallel to its generators. For this reason the scattering of light in this geometry is called conical scattering. In the earliest experimental studies of the conical scattering of light from a onedimensional random surface (Luna and Méndez [1995], Luna [1996]), and in the earliest theoretical studies of such scattering (Depine [1991, 1993], Li, Chang and Tsang [1994]), the dependence of the elements of the corresponding Stokes matrix on the angles of incidence and scattering was not obtained. The first theoretical determination of the elements of the Stokes matrix for the scattering of light from a one-dimensional random metal surface in a conical configuration, with multiplescattering processes of all orders taken into account was carried out by Novikov and Maradudin [1997]. This work was soon followed by analogous calculations for conical scattering from a perfectly conducting surface by Luna, Acosta-Ortiz and Zou [1998] and Novikov and Maradudin [1999], although in the former paper only single-scattering processes were taken into account. An interesting result of the theoretical study of the conical scattering of light from a one-dimensional perfectly conducting random surface (Novikov and Maradudin [1999]) is that the Stokes matrix in this case has exactly the same form as the Stokes matrix for in-plane scattering from the same surface, and the elements of the Stokes matrix in the former case can be obtained from the elements of the Stokes matrix in the latter case by replacing the wavelength λ of the incident field in the latter elements everywhere by an effective wavelength given by λeff = λ/ cos φ0 .

(4.64)

2, § 4]

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169

The angle φ0 in this expression, called the conical angle, is the angle between the wave vector k, of the incident field and its projection on the x2 x3 plane. This result had been obtained earlier by Maystre [1984] in a study of conical scattering from a perfectly conducting classical grating. In the case of the conical scattering of light from a one-dimensional penetrable random surface, e.g., from a metal surface, all elements of the Stokes matrix are nonzero. The expressions for these elements are rather cumbersome, and the reader is referred to the paper by Novikov and Maradudin [1997] where they are presented in an explicit form. The first experimental results for several elements of the Stokes matrix in the conical scattering of light from a one-dimensional random metal surface were obtained by Luna [2002] and were in fairly good qualitative and quantitative agreement with theoretical results of Novikov and Maradudin [1997]. 4.3. Experimental techniques used in the study of multiple scattering effects, including the enhanced backscattering effect When discrepancies are found in the comparison of scattering theories with experimental data taken with commonly found surfaces, it is difficult to decide if they are due to some approximation made in the electromagnetic treatment of the problem, or to an incorrect choice of the statistical model. Much progress towards understanding the accuracy of various theoretical techniques and observing new phenomena has been made by the use of specially fabricated surfaces. In this subsection we first review the most relevant aspects of that work. We also examine some important issues concerning the relation between practical surface scattering measurements and theoretical work, and describe a typical experimental arrangement for measurements in the optical and infrared regions of the spectrum. Finally, we discuss some experimental results involving multiplescattering effects that are difficult to explore theoretically. 4.3.1. Fabrication of surfaces A simple assumption for theoretical work is that the scattering object consists of a single interface separating two homogeneous media, and that the curve defining the interface constitutes a realization of a stationary Gaussian random process with a Gaussian correlation function (see Section 2.1.1). Surfaces obtained, for example, by processes such as grinding, polishing, and sandblasting, depart from this idealized kind of random process and are difficult to model and characterize; they may present roughness over many scales, be non-stationary, and have illdefined statistics.

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Multiple scattering of light from randomly rough surfaces

[2, § 4

For detailed comparisons between theory and experiment, surfaces with welldefined statistics are required, and methods of fabricating such surfaces have been devised. Moreover, multiple-scattering effects can be enhanced by the use of specially designed surfaces. We restrict our review to the fabrication of random surfaces on photoresist. Typically, the process starts by coating glass substrates with a thin layer of photoresist, which is a light-sensitive material in the blue and ultraviolet regions of the spectrum. The plates are then exposed to some specified light pattern, either sequentially or at once. The procedure followed to prepare and develop the plates varies depending on the kind of photoresist and the nature of the surface to be fabricated. Particular procedures have been described by several authors (e.g., Gray [1978], O’Donnell and Méndez [1987]). With the application of thick metal coatings by vacuum evaporation the sample can behave, in reflection, as the interface between two semi-infinite media. Other measures need to be taken to achieve a similar situation with dielectrics (Chaikina, Hernández-Walls and Méndez [2000], Chaikina, Negrete-Regagnon, Ruiz-Cortés and Méndez [2002]). The minimum standard deviation that can be produced in a controlled way is determined, mainly, by the quality of the substrate and the conditions of the clean room employed in the coating process. A fundamental requirement is that the natural roughness of the substrate and coating must be much smaller than the desired roughness. The maximum standard deviation depends on the thickness of the photoresist coating, and it is common to have samples that contain two or even three layers of photoresist. The most important and widely used method for the fabrication of Gaussiancorrelated Gaussian surfaces was devised by Gray [1978]. Surfaces fabricated with Gray’s method, or some variation of it, have been used extensively in studies involving enhanced backscattering and detailed comparisons with theory (e.g., Méndez and O’Donnell [1987], O’Donnell and Méndez [1987], Sant, Dainty and Kim [1989], Kim, Dainty, Friberg and Sant [1990], O’Donnell and Knotts [1991, 1992], Michel, Knotts and O’Donnell [1992], Knotts, Michel and O’Donnell [1993], Knotts and O’Donnell [1993, 1994], Luna, Méndez, Lu and Gu [1995], Luna and Méndez [1995], Méndez, Navarrete and Luna [1995], Chaikina, Navarrete, Méndez, Martínez and Maradudin [1998], Navarrete, Chaikina, Méndez and Leskova [2002], Chaikina, Hernández-Walls and Méndez [2000], Chaikina, Negrete-Regagnon, Ruiz-Cortés and Méndez [2002]). They have also been employed in studies of non-Gaussian speckle (Levine and Dainty [1982]), scattering by gamma-distributed surfaces (Kim, Méndez and O’Donnell [1987])

2, § 4]

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171

and, more recently, in studies of second harmonic generation by random surfaces (O’Donnell, Torre and West [1996, 1997], O’Donnell and Torre [1997]). In this method, the central idea is to convert the random spatial fluctuations of a speckle pattern into a surface profile, by exposing the photoresist-coated plates to speckle patterns with the appropriate statistical properties. Consider the exposure of the plate to the intensity of a fully-developed speckle pattern, I (x
), produced by the scattering of coherent light from a diffuser. The probability density function associated with the intensity of the pattern is a negative exponential function (Goodman [1984], p. 16), and the intensity covariance function may be written in the form (Goodman [1984], p. 38) 2

 I (x
)I (x
) = C(x
− x
) , (4.65) where I (x
) represents the fluctuations about the mean intensity, and    ∞ q · |x
− x
|   P (q
)2 exp i ω
dq
. C(x
− x
) = c z −∞

(4.66)

Here P (q
) represents the pupil function on the plane of the diffuser that produces the speckle, and z is the distance from the diffuser to the photoresist plate. We note that if the diffuser is illuminated with a Gaussian beam (Gaussian pupil function), C(x
− x
) has also a Gaussian form. Assuming a linear response of the photoresist, the resulting profile is proportional to the exposure. That is ζ(x
) = αI (x
)t,

(4.67)

where ζ(x
) denotes the fluctuations about the mean surface, α is a proportionality constant characterizing the material’s response to the exposing energy, and t is the exposure time. If the linearity assumption is satisfied, the normalized correlation function of the fabricated surface will also be Gaussian. To obtain an approximate Gaussian probability density function, the plate is exposed to several independent speckle patterns of equal mean intensity. The resulting intensity, I, is gamma-distributed (Goodman [1984], p. 23),   I N−1 I , exp − PI (I) = (4.68) (N − 1)!I N I  where N represents the number of exposures, I  = Ij  = I/N , and j = 1, 2, 3, . . . , N . As the number of exposures grows, the distribution of I approaches a Gaussian form, but the contrast of the total exposure decreases, reducing the roughness parameter δ of the fabricated surface. It is considered that a good compromise is obtained with between eight and ten exposures.

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[2, § 4

Fig. 15. Electron micrographs of (a) a surface fabricated in photoresist, and (b) a ground glass surface. The areas shown are, approximately, 30 × 30 µm (O’Donnell and Méndez [1987]).

Surfaces with correlation lengths between a = 1 µm and 50 µm, and standard deviations of between δ = 0.1 µm and 2 µm, have been fabricated in this way. The difference between this kind of surface and others, such as ground glass surfaces is illustrated in fig. 15. The fabrication of what have been called quasi one-dimensional surfaces involves the use of nonisotropic speckle patterns (Sant, Dainty and Kim [1989], Kim, Dainty, Friberg and Sant [1990]). An improvement is possible by scanning the plate while exposing it in the direction of the elongation of the speckles (O’Donnell and Knotts [1991], Luna, Méndez, Lu and Gu [1995]). Through mechanical profilometry it has been verified that Gaussian-correlated surfaces with a Gaussian height distribution can be fabricated with this technique. However, detailed studies of the statistical properties of some surfaces has shown that, even when the height distribution is Gaussian, the higher order statistics of the profile may be incompatible with the assumption of a Gaussian random process (Knotts, Michel and O’Donnell [1993]). The other important technique for the fabrication of rough one-dimensional surfaces was introduced by West and O’Donnell [1995]. The surfaces fabricated with this technique are designed to be weakly rough and produce multiplescattering effects mediated by the excitation of surface waves. The surfaces have Gaussian statistics and a power spectrum g(k) that has constant height between the wavenumbers kmin and kmax , with little spectral power elsewhere (see Section 2.2). Mathematically speaking, the fabrication method is based on a means of generating Gaussian random processes discussed by Rice [1951]. The photoresist plate is exposed, sequentially, to sinusoidal intensity patterns produced by the interference of two intersecting light beams. Assuming a linear response of the

2, § 4]

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173

material, the resulting developed profile ζ (x1 ) is ζ (x1 ) = α

N 

  In 1 + Mn cos(kn x1 − φn ) tn ,

(4.69)

n=1

where N represents the number of exposures, In is the intensity of the nth pattern, Mn is the fringe modulation, kn = nk is a wave number that depends on the angle of intersection of the beams, φn is a phase that determines the position of the fringe pattern, tn is the exposure time, and α is a proportionality constant characterizing the response of the material. The phases φn are chosen as independent random variables, uniformly distributed on the range (0, 2π). Then, ζ (x1 ) =

N 

cn cos(kn x1 − φ),

(4.70)

n=1

where cn = αIn Mn tn . Taking an amplitude cn specified by  1/2 cn = 2g(kn )k ,

(4.71)

it may be shown that ζ (x1 ) has the required roughness spectrum g(k) and, as N increases and k decreases, the statistics converge to the Gaussian form. The method may thus be regarded as a means of constructing a realization of ζ (x1 ) from its randomly phased Fourier components, with appropriate amplitude weighting to produce g(k). Surfaces with standard deviations of heights of as little as δ = 10 nm have been successfully fabricated with this technique. 4.3.2. Scattering measurements We concentrate the discussion on the experimental estimation of the mean differential reflection coefficient. At first sight, this appears to be a simple task, but scattering measurements are full of subtleties (see, e.g., Stover [1995], Chapter 6). There are also some issues concerning the relation between theory and experiment that are worth discussing. The first problem encountered is that the mean differential reflection coefficient involves the estimation of an ensemble average, and averaging over many statistically equivalent experimental samples is not a practical proposition in most cases. Instead, to estimate the mean DRC, the averaging is performed by integration over a finite angular region with the detector. In the following lines, we try to justify this procedure taking, as an example, the scattering by one-dimensional stationary surfaces. For plane wave illumination, the DRC may be written in the form (see Section 3.3)

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Multiple scattering of light from randomly rough surfaces



 ∂R(q) |r(q|k)|2 , = ∂θs F (k)L1

[2, § 4

(4.72)

where F (k) = 8πα0 (k, ω), and  L/2 r(q|k) = F (q, x1) exp{−ıqx1} dx1.

(4.73)

Here, we have defined    F (q, x1 ) = ıqζ (x1 ) − ıα0 (q, ω) U(x1 ) − V(x1 )   × exp −ıα0 (q, ω)ζ (x1 ) .

(4.74)

−L/2

As a function of x1 , the function F (q, x1) is a stationary random process that can be interpreted as the near-field complex amplitude reflected by the surface. The far-field scattering amplitude may be viewed as the Fourier transform of this quantity (the variation of F (q, x1 ) with q is slow), and the statistical estimation of the DRC amounts to the estimation of the spectral density function of the random process F (q, x1). The sample estimator of the DRC can be written in the form    ∞ 1 ∂R(q) C(q, η) exp{ıqη} dη, = (4.75) ∂θs F (k) −∞ where we have defined      ∞ η+ξ 1 ξ rect F (q, ξ )F ∗ (q, η + ξ ) dξ, (4.76) C(q, η) = rect L1 −∞ L1 L1 with rect(ξ ) representing the rectangle function (Goodman [1996], p. 13). The function C (q, η) is also a random function of η, with a slow dependence on q. In the limit of large L1 , it becomes the sample correlation function of the random process F (q, ξ ). Since the expectation value of the sample estimator yields the mean DRC, the estimator is not biased. However, the sample DRC does not converge to any function as the length of the sample is increased. It is known that the variance of the intensity fluctuations does not decrease as the length of the sample tends to infinity. In other words, we know that while the speckle size depends on the length of the surface, the speckle contrast is insensitive to L1 . Spectral estimators may be improved by smoothing the sample estimator (Jenkins and Watts [1968], Section 6.3). Convolution with an aperture function A(q) results in a smoothed estimator with reduced variance. We write    ∞ ∂R(q) dq  ∂R(q  ) A(q − q  ) = , (4.77) ∂θs ∂θs −∞ 2π

2, § 4]

Weak localization effects in the multiple scattering of light

where the aperture function has the property  ∞ A(q) dq = 2π. −∞

175

(4.78)

Making use of the fact that C(q, η) is a slowly varying function of q, we can also write the smoothed estimator in the form  ∞ ∂R(q) 1 = A(η)C(q, η) exp{ıqη} dη, (4.79) ∂θs F (k) −∞ where A(η) is the inverse Fourier transform of A(q). Expressions (4.77) and (4.79) represent standard forms for the estimation of spectral quantities (Jenkins and Watts [1968], p. 243). The expectation value of the smoothed estimator is   ∞ 

∂R(q) dq  ∂R(q  ) A(q − q  ) = . (4.80) ∂θs ∂θs −∞ 2π We can see that, depending on the angular size and shape of the aperture function, the mean smoothed DRC can be a distorted version of the mean DRC. So, if the surface is statistically stationary (at least in the wide sense), the ensemble averaged DRC can be estimated through an angular average. To reduce the statistical fluctuations in the measurements it is necessary to average over an angle that is larger than the speckle size, but smaller than the characteristic features in the angular distribution of the mean intensity. The fulfillment of these conditions determines the minimum size of the sample (the speckle size is inversely proportional to L1 ). This is usually not a problem for scattering measurements in the visible, but can be an important consideration when working with one-dimensional surfaces in the infrared. Although there are instruments that can measure the full hemispherical distribution function, normally, angular light scattering measurements are only performed in the plane of incidence. The instruments employed for these measurements are called in-plane bidirectional scatterometers. A typical arrangement is shown schematically in fig. 16. In the figure, an expanded laser beam is sent through a periscope with a series of mirrors from which it exits horizontally toward the sample. Since the use of lock-in detection techniques is desirable, the illumination beam is usually chopped. For reflective measurements the last mirror of the periscope obstructs the detector in the vicinity of the backscattering direction and, in order to reduce this problem, the beam is focused on this last mirror; the obstruction can then be made narrower, permitting measurements close to the backscattering direction. With this arrangement, which is convenient for backscattering measurements with surfaces that do not present an appreciable coherent

176

Multiple scattering of light from randomly rough surfaces

[2, § 4

Fig. 16. Schematic diagram of a typical experimental setup for scattering measurements. The sample can be rotated about the vertical axis to set the angle of incidence. The detector arm rotates around the same axis and records the scattered intensity.

component, the surface is illuminated by a slightly divergent beam. For weakly rough surfaces that produce a strong coherent component, converging beam or collimated beam illumination can be more convenient. So, the choice of the form of the illumination beam depends on the nature of the surface and the kind of scattering measurement desired. The detection system usually consists of an aperture, a lens, and the detector. The lens is used to collect the scattered light over the region defined by the aperture and, at the same time, to form an image of the illuminated region of the surface. To ensure that the instrument does not distort the angular distribution of the intensity, it must be verified that the detector views the entire illuminated area of the surface, irrespective of the angle of detection. Aperture effects, represented by eq. (4.80), must also be minimized. Accurate scattering measurements are particularly difficult with weakly scattering surfaces and at large angles of incidence and scattering. When the scattering signals are weak in comparison with the strong coherent component, spurious reflections can easily overcome them, interfering with the signal and saturating the detector. The instrument signature (Stover [1995], Chapter 6) depends on the amount of stray light produced by the various elements of the scatterometer (mirrors, lenses, polarization elements, and apertures) and the aberrations of the system. 4.3.3. Observations of enhanced backscattering and other multiple scattering effects The first experimental works on enhanced backscattering with specially fabricated surfaces were reported by Méndez and O’Donnell [1987] (see also O’Donnell and Méndez [1987]). The motivation for that work was a critical evaluation of the

2, § 4]

Weak localization effects in the multiple scattering of light

177

Fig. 17. Photographs of the far-field scattering pattern in normal incidence from a surface similar to the one used for the results of fig. 3. The illumination wavelength is λ = 0.633 µm and the incident polarization is vertical. A and B correspond to orthogonally polarized components (O’Donnell and Méndez [1987]).

available theories. Instead, what was found were results that, quantitatively, are still difficult to reproduce theoretically. Figure 3 shows the angular distribution of the mean intensity for a two-dimensional surface with approximately Gaussian statistics, a Gaussian correlation function, and estimated parameters a = 1.4 µm and δ ≈ 1.0 µm. The parameters represent only estimates because a surface with such large slopes cannot be characterized by mechanical profilometry. There are two noteworthy features in the scattering distributions shown in fig. 3. The first one is the presence of an enhanced backscattering peak; the nature of this effect has already been discussed in Section 4.1. The second important feature is the presence of a relatively large cross-polarized component. Although these results were surprising at the time, their multiple scattering nature was quickly recognized.

178

Multiple scattering of light from randomly rough surfaces

[2, § 4

Fig. 18. Optical micrographs (NA = 0.8, 50×) of an area of a surface illuminated with linearly polarized light and viewed with a parallel (left) and perpendicular (right) analyzer. The areas where changes in the polarization are observed exhibit patterns that are consistent with the explanation of fig. 19 (O’Donnell and Méndez [1987]).

The far-field hemispherical scattering pattern produced by a surface with similar characteristics is shown in fig. 17. The photographs correspond to the co- and cross-polarized scattering patterns. Although the surface is isotropic, the scattering patterns are not. The symmetry is broken by the polarization vector of the incident field. The four-fold symmetry observed in the scattering patterns is also visible in the polarization components of the near-field intensity scattered by the surface, as shown in fig. 18. It is clear that important changes in the polarization take place in the valleys of the surface. These polarization changes may be understood with the help of fig. 19. Consider the incidence of vertically-polarized light upon a valley of a twodimensional perfectly conducting surface. Assume, for simplicity, that the surface can be considered locally flat. The light following the double scattering paths A and B will not lead to any changes in the polarization. However, for the diagonal cross paths C and D, the light will emerge polarized perpendicularly to the initial linear polarization state. The diagonal paths cause a rotation of the polarization vector by 90◦ , explaining the four-fold symmetry observed in the reflected near-field intensities of fig. 18. Measurements of the full hemispherical intensity distribution function of the scattered light from fabricated isotropic metallic surfaces that produce multiple scattering have been reported by Knotts and O’Donnell [1993]. Similar measurements for dielectric surfaces have been reported by Chaikina, Negrete-Regagnon, Ruiz-Cortés and Méndez [2002]. The results provide quantitative data for patterns like the ones shown in fig. 18. We end this section by discussing the problems that have prevented the observation of some interesting multiple-scattering effects.

2, § 4]

Weak localization effects in the multiple scattering of light

179

Fig. 19. Multiple scattering of polarized light from a valley in a two-dimensional perfectly conducting surface. Light scattered along the paths A and B preserve the incident polarization, but the light scattered along the paths C and D is cross-polarized (O’Donnell and Méndez [1987]).

The first theoretical prediction of enhanced backscattering from randomly rough surfaces appeared in 1985 (McGurn, Maradudin and Celli [1985]). The calculations, based on an infinite order perturbation theory, involved a silver Gaussian surface with Gaussian correlation function and parameters a = 100 nm and δ = 5 nm, illuminated with light of wavelength λ = 457.9 nm. Later calculations based on other perturbation theories (Maradudin and Méndez [1993]) and Monte Carlo methods (Michel [1994]) have confirmed the correctness of the prediction. Although the existence of the effect, and the mechanism responsible for its production have been confirmed quite convincingly by the experimental work of West and O’Donnell [1995], it has not been possible to observe enhanced backscattering with Gaussian-correlated weakly rough surfaces, and to obtain quantitative agreement with the theory. The main reason for this surprising fact is that the fabrication of surfaces with such characteristics is a difficult task. These difficulties are compounded by the usual problems associated with scattering measurements with weakly scattering surfaces. The satellite peaks, predicted to occur in the angular distribution of the scattered light by thin films with a random surface (see Section 6.1) have also proved

180

Multiple scattering of light from randomly rough surfaces

[2, § 5

difficult to observe. Here, it is not only the surface characteristics that must be controlled very precisely, but also the thickness of the film.

§ 5. Angular intensity correlation functions The mean differential reflection and transmission coefficients considered in the preceding sections of this review involve the first and second moments of the field scattered from or transmitted through a randomly rough surface. Recently, however, attention has begun to be directed toward the investigation of the conditions under which the amplitudes or intensities of light scattered or transmitted into the far field for different angles of incidence and scattering will be correlated, and the form the resulting correlation functions take. The study of these correlation functions requires the study of not only second moments of the scattered or transmitted fields, but of higher moments as well. Much of the work on multiple-scattering effects on angular correlation functions in rough surface scattering has been carried out for weakly rough random metal surfaces. Thus, Arsenieva and Feng [1993] studied perturbatively the angular correlation function of the scattering amplitude in the cross-polarized scattering of light from a two-dimensional randomly rough metal surface. The squared modulus of this correlation function gives the analog of the C (1) angular intensity correlation function studied earlier in the context of scattering from volume disordered systems (Feng, Kane, Lee and Stone [1988]). By keeping only the contribution from ladder diagrams they obtained what is now called the memory effect peak in the envelope of the C (1) angular intensity correlation function, and related this peak to the enhanced backscattering peak in the angular dependence of the intensity of the light scattered diffusely from the same surface. The same angular amplitude correlation function was also studied perturbatively by Freilikher and Yurkevich [1993]. By keeping only the contribution from maximally-crossed diagrams they obtained what is now called the reciprocal memory effect peak in the envelope of the C (1) angular intensity correlation function, and related it to the enhanced backscattering peak. The analogs of the C (2) and C (3) angular intensity correlation functions discussed earlier in the context of the scattering from volume disordered systems (Feng, Kane, Lee and Stone [1988]) were investigated theoretically in the scattering of p-polarized light from one-dimensional random metal surfaces by Malyshkin, McGurn, Leskova, Maradudin and Nieto-Vesperinas [1997a, 1997b] by small-amplitude perturbation theory. The surface roughness was assumed to be sufficiently weak that these correlations were caused primarily by the multiple

2, § 5]

Angular intensity correlation functions

181

scattering of the surface plasmon polaritons supported by the surface and excited by the incident field through the roughness of the surface. The work of Malyshkin, McGurn, Leskova, Maradudin and Nieto-Vesperinas [1997a, 1997b] also revealed that in rough surface scattering the angular intensity correlation function possesses a contribution that they called the C (10) correlation that is of the same order of magnitude as the C (1) correlation function. This correlation function was overlooked in the earlier studies of Arsenieva and Feng [1993] and of Freilikher and Yurkevich [1993], because the angular amplitude correlation function they investigated was equivalent to the use of the factorization approximation (Shapiro [1986]) in calculating the angular intensity correlation function. This approximation, which was not used in the work of Malyshkin, McGurn, Leskova, Maradudin and Nieto-Vesperinas [1997a, 1997b], is based on the assumption that the amplitude of the field scattered diffusely possesses a circular complex Gaussian joint probability density function (Goodman [1975, 1985]). The fact that it fails to predict the C (10) correlation function shows that the latter assumption is not universally valid. An indication that interesting correlation effects can occur when the factorization approximation is not used had been shown earlier by Nieto-Vesperinas and Sánchez-Gil [1992a, 1992b, 1993] in predicting what they called enhanced long-range correlation functions. In their work Malyshkin, McGurn, Leskova, Maradudin and Nieto-Vesperinas [1997a, 1997b] also showed that there is an additional class of correlations, intermediate between the C (1) and C (2) correlations in the order in the surface profile function in which they first appear. This class was therefore named the C (1.5) correlations. These correlations also arise when the factorization approximation is not made. They are intimately connected with the roughness-induced excitation of forward- and backward-propagating surface plasmon polaritons by the incident light, and introduce peaks into the angular intensity correlation function. Theoretical and experimental studies of angular intensity correlation functions represent a rather new sub-field within the larger field of multiple-scattering effects in the scattering of light from randomly rough surfaces. In this section we present an introduction to this field in the context of the scattering of light from one-dimensional, weakly rough, random surfaces. A study of angular intensity correlation functions in the scattering of light from two-dimensional, weakly rough, random metal surfaces that goes beyond the memory and reciprocal memory effects has been carried out by Malyshkin, McGurn, Leskova, Maradudin and Nieto-Vesperinas [1997b]. The reader interested in this topic is referred to this work.

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Multiple scattering of light from randomly rough surfaces

[2, § 5

Thus, the general angular intensity correlation function C(q, k|q , k  ) we consider here is defined by





C(q, k|q  , k  ) = I (q|k)I (q  |k  ) − I (q|k) I (q  |k  ) , (5.1) where the angle brackets denote an average over the ensemble of realizations of the surface profile function. The intensity I (q|k) entering this expression is defined in terms of the scattering matrix S(q|k) for the scattering of light of frequency ω from a one-dimensional random surface by (Arsenieva and Feng [1993])   2 1 ω  S(q|k) , I (q|k) = (5.2) L1 c where L1 is the length of the x1 -axis covered by the random surface, and the wave numbers k and q are related to the angles of incidence and scattering θ0 and θs , measured counterclockwise and clockwise from the normal to the mean scattering surface, respectively, by k = (ω/c) sin θ0 and q = (ω/c) sin θs . From eqs. (5.1) and (5.2) we can already obtain useful qualitative information about the correlation function C(q, k|q  , k  ). From the fact that the correlation of I (q|k) with itself should be stronger than the correlation of I (q|k) with I (q  |k  ) when q  = q and k  = k, a peak in C(q, k|q  , k  ) is expected when q  = q and k  = k. This peak has come to be called the memory effect peak. At the same time, since S(q|k) is reciprocal, S(q|k) = S(−k| − q), a peak in C(q, k|q  , k  ) is also expected when q  = −k and k  = −q. This peak is called the reciprocal memory effect peak. In terms of the scattering matrix S(q|k) the correlation function C(q, k|q  , k  ) takes the form  

1 ω 2  C(q, k|q  , k  ) = 2 S(q|k)S ∗ (q|k)S(q  |k  )S ∗ (q  |k  ) L1 c



 − S(q|k)S ∗ (q|k) S(q  |k  )S ∗ (q  |k  ) . (5.3) As it stands, the expression for C(q, k|q , k  ) given by eq. (5.3) contains purely specular contributions, i.e. terms proportional to 2πδ(q − k) and/or 2πδ(q  − k  ). If we note that, due to the stationarity of the surface profile function, S(q|k) is diagonal in q and k, S(q|k) = 2πδ(q − k)S(k), we can eliminate these uninteresting specular contributions by rewriting C(q, k|q  , k  ) in terms of the diffuse part of the scattering matrix δS(q|k) = S(q|k) − S(q|k). In addition, if we use the relations between averages of the products of random functions and the corresponding cumulant averages (Kubo [1962]), we can finally write the contribution

2, § 5]

Angular intensity correlation functions

183

to C(q, k|q  , k  ) that is free from specular contributions as C(q, k|q  , k  )  

2 

2 1 ω 2  = 2 δS(q|k)δS ∗ (q  |k  )  +  δS(q|k)δS(q  |k  )  L1 c

 + δS(q|k)δS ∗ (q|k)δS(q  |k  )δS ∗ (q  |k  ) c ,

(5.4)

where · · ·c denotes the cumulant average. The result given by eq. (5.4) is convenient for several reasons. Due to the stationarity of the surface profile function, the average δS(q|k)δS ∗ (q  |k  ) is proportional to 2πδ(q − k − q  + k  ). It therefore gives rise to the contribution to C(q, k|q , k  ) called C (1) (q, k|q , k  ), which contains the memory and reciprocal memory effects. Similarly, the average δS(q|k)δS(q  |k  ) is proportional to 2πδ(q − k + q  − k  ), and contributes the correlation function C (10) (q, k|q , k  ) to C(q, k|q  , k  ). The third term on the right-hand side of eq. (5.4), δS(q|k)δS ∗ (q|k)δS(q  |k  )δS ∗ (q  |k  )c , is proportional to 2πδ(0), due to the stationarity of the surface profile function, and gives rise to the long-range and infinite range contributions to C(q, k|q , k  ) given by the sum C (1.5) (q, k|q  , k  ) + C (2) (q, k|q  , k  ) + C (3) (q, k|q  , k  ). Thus, the contributions to C(q, k|q  , k  ) that have been named C (1) (q, k|q  , k  ) and C (10) (q, k|q , k  ) are explicitly separated out in eq. (5.4). In addition, from eq. (5.4) we can easily estimate the relative magnitudes of the different contributions to C(q, k|q  , k  ). Since 2πδ(0) = L1 in onedimension, when the arguments of the delta-functions vanish the C (1) (q, k|q k  ) and C (10) (q, k|q k  ) correlation functions are independent of the length of the surface L1 , because they are proportional to [2πδ(0)]2 . At the same time the remaining term in eq. (5.4), that yields the sum C (1.5) + C (2) + C (3) , is inversely proportional to the length of the surface, since it is proportional to only the first power of 2πδ(0). Therefore, in the limit of a long surface or large illumination area the long-range and infinite-range correlation functions are small compared to the short-range correlation functions, and vanish in the limit of an infinitely long surface. Consequently, although they contain interesting multiple-scattering effects (Malyshkin, McGurn, Leskova, Maradudin and NietoVesperinas [1997a, 1997b]), they are weak, and therefore will not be considered further here. The preceding results are consistent with the assumptions and conclusions encountered in conventional speckle theory (Goodman [1975, 1985]). Thus, when the surface profile function is assumed to be a stationary random process, and the random surface is assumed to be infinitely long, the scattering matrix S(q|k)

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Multiple scattering of light from randomly rough surfaces

[2, § 5

becomes the sum of a very large number of independent contributions from different points on the surface. On invoking the central limit theorem it is found that S(q|k) obeys complex Gaussian statistics. In this case eq. (5.4) becomes rigorously (Stoffregen [1979])  

2 

2  1 ω 2    δS(q|k)δS ∗ (q  |k  )  +  δS(q|k)δS(q  |k  )  C(q, k|q , k ) = 2 L1 c (5.5a) ≡ C (1) (q, k|q , k  ) + C (10) (q, k|q , k  ),

(5.5b)

because all cumulant averages of products of more than two Gaussian random processes vanish. The third term on the right-hand side of eq. (5.4) therefore gives the correction to the prediction of the central limit theorem due to the finite length of the random surface. If it is further assumed, as is done in speckle theory, where the disorder is presumed to be strong, that δS(q|k) obeys circular complex Gaussian statistics (Goodman [1975, 1985]), then δS(q|k)δS(q  |k  ) vanishes, and the expression for C(q, k|q , k  ) simplifies to  

2 1 ω 2  C(q, k|q  , k  ) = 2 δS(q|k)δS ∗ (q  |k  )  L1 c ≡ C (1) (q, k|q , k  ).

(5.6)

This result is often called the factorization approximation to C(q, k|q  , k  ) (Shapiro [1986]). The degree of surface roughness needed for δS(q|k) to change from a complex Gaussian random process to a circular complex Gaussian random process, and hence for C (10) (q, k|q , k  ) to vanish, is discussed in Section 5.2.

5.1. Memory and reciprocal memory effects: theory and experiment We have seen from eq. (5.6) that the C (1) (q, k|q  , k  ) correlation function is given by C (1) (q, k|q  , k  ) =

2 1 ω2  δS(q|k)δS ∗ (q  |k  )  . 2 2 L1 c

From eq. (4.12) and the relation S(q|k) = (α0 (q)/α0 (k))1/2 R(q|k), we find that δS(q|k) can be written in terms of the Green’s function G(q|k) as 

 1/2 1/2 δS(q|k) = −2iα0 (q) G(q|k) − G(q|k) α0 (k). (5.7)

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185

It follows that the amplitude correlation function δS(q|k)δS ∗ (q  |k  ) takes the form

δS(q|k)δS ∗ (q  |k  ) = 4α0 (q)α0 (k)α0 (q  )α0 (k  )





  × G(q|k)G∗ (q  |k  ) − G(q|k) G∗ (q  |k  ) . 1/2

1/2

1/2

1/2

(5.8)

G(q|k)G∗ (q  |k  )

is solved by the use of the If the Bethe–Salpeter equation for same approximations used in obtaining eq. (4.35), we obtain the result that C (1) (q, k|q  , k  ) 16 ω2 α0 (q)α0 (k)α0 (q  ) L1 c2  2   2           2   2      × α0 (k ) G(q) G(q ) G(k) G(k )  W 2 g |q − k|

= 2πδ(q − k − q  + k  )

 + W 2 φ(q)



2  4sp 4sp  φ(k) +   2  2 (q + k) + 4ε (q − q ) + 4ε

≡ 2πδ(q − k − q  + k  )

1 (1) C (q, k|q , q  − q + k), L1 0

(5.9)

which defines the envelope function C0(1) (q, k|q  , q  − q + k). The term 4sp/[(q  + k)2 + 4ε ] in this result, which arises from the sum of the contributions from the maximally-crossed diagrams, contributes a peak centered at q  = −k, or θs = −θ0 , to the envelope function. This is the reciprocal memory effect peak. The term 4sp/[(q − q  )2 + 4ε ], which arises from the sum of the contributions from the ladder diagrams, contributes a peak centered at q  = q, or θs = θs , to the envelope function. This is the memory effect peak. Both of these peaks have now been observed experimentally in the scattering of ppolarized light from weakly rough, one-dimensional, random gold surfaces (West and O’Donnell [1999]) (fig. 20). From a comparison of the results given by eqs. (4.36) and (5.9), we see that an experimental determination of C (1) (q, k|q  , k  ) in the vicinity of either the memory effect peak or the reciprocal memory effect peak yields essentially the same information as a measurement of ∂R/∂θs diff in the vicinity of the enhanced backscattering peak, but without the experimental difficulties associated with placing a detector at the position of the source. It is this feature of the correlation function C (1) that emerged from the theoretical studies of it by Arsenieva and Feng [1993] and by Freilikher and Yurkevich [1993].

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Multiple scattering of light from randomly rough surfaces

[2, § 5

Fig. 20. Experimental results for the correlation functions C(1) (solid curve) and C(10) (dotted curve) as functions of θ1 for θ0 = 6.3◦ and θs = 8.6◦ when p-polarized light of wavelength λ = 612.7 nm is incident on a one-dimensional randomly rough gold surface (ε(ω) = −9.00 + i1.29). The surface roughness is characterized by the West–O’Donnell power spectrum (2.24) with δ = 115.5 nm, kmin = 8.56 × 10−3 nm−1 , and kmax = 13.3 × 10−3 nm−1 . The peak in C(1) labeled ME is the memory effect peak; the peak labeled RME is the reciprocal memory effect peak (West and O’Donnell [1999]).

We conclude this discussion of the C (1) (q, k|q , k  ) correlation function by noting that the property of a speckle pattern that is characterized by the presence of the factor 2πδ(q − k − q + k  ) in C (1) (q, k|q  , k  ) is that if we change the angle of incidence in such a way that k goes into k  = k + k, the entire speckle pattern shifts in such a way that any feature initially at q moves to q  = q + k. This is why the C (1) correlation function was originally named the memory effect. In terms of the angles of incidence and scattering we have that if θ0 is changed to θ0 = θ0 + θ0 , any feature in the speckle pattern originally at θs is shifted to θs = θs + θs , where θs = (cos θ0 / cos θs )θ0 , to first order in θ0 . 5.2. The correlation function C (10)(q, k|q  , k  ): theory and experiment We see from eqs. (5.5) and (5.6) that the C (10)(q, k|q  , k  ) correlation function is given by C (10) (q, k|q , k  ) =

2 1 ω2  δS(q|k)δS(q  |k  )  . 2 2 L1 c

(5.10)

With the use of eq. (5.7) the amplitude correlation function δS(q|k)δS(q  |k  ) becomes

1/2 1/2 1/2 1/2 S(q|k)δS(q  |k  ) = −4α0 (q)α0 (k)α0 (q  )α0 (k  ) 





 × G(q|k)G(q |k  ) − G(q|k) G(q  |k  ) . (5.11)

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187

If the same approximations are made in solving the Bethe–Salpeter equation for G(q|k)G(q  |k  ) as were made in obtaining G(q|k)G∗ (q  |k  ), we obtain for the correlation function C (10) (q, k|q , k  ): C (10) (q, k|q , k  ) 16 ω2 α0 (q)α0 (k)α0 (q  )α0 (k  ) L1 c2  2  2   2  2    × G(q) G(k) G(q  ) G(k  )  W 2 g |q − k|

= 2πδ(q − k + q  − k  )





2  L(q + q  ) L(k − q  ) + + W φ(q) φ(k)    1 − L(k − q ) 1 − L(q + q ) 2

≡ 2πδ(q − k + q  − k  ) where L(Q) = W 2



∞ −∞

1 (10) C (q, k|q  , q  + q − k), L1 0

dp φ(p)G(p)G(Q − p)φ(p), 2π

(5.12)

(5.13)

which defines the envelope function C0 (q, k|q , q  + q − k). The function L(Q) displays no resonant behavior for values of Q within the range of variation of Q = k − q  and Q = q + q  . Hence the envelope function C (10) (q, k|q , q  + q − k) is a structureless function of q  (θs ) for fixed values of q and k, while k  is determined from the condition k  = q  + q − k(sin θ0 = (10) sin θs + sin θs − sin θ0 ). The envelope function C0 (q, k|q  , q  + q − k) has been determined experimentally recently in the scattering of p-polarized light from weakly rough, one-dimensional gold surfaces (West and O’Donnell [1999]) (fig. 20), and is a structureless function of its arguments. The property of a speckle pattern that is characterized by the presence of the factor 2πδ(q − k + q  − k  ) in C (10)(q, k|q  , k  ) is that if we change the angle of incidence in such a way that k goes into k  = k + k, a feature at q = k − q will be shifted to q  = k  + q, i.e. to a point as much to one side of the new specular direction as the original point was on the other side of the original specular direction. For one and the same incident beam (k  = k), the C (10) correlation function therefore reflects the symmetry of the speckle pattern with respect to the specular direction (in wavenumber space). We have noted in Section 5 that in conventional speckle theory, where the surface roughness is assumed to be strong (Goodman [1975, 1985]), δS(q|k) is found to obey circular complex Gaussian statistics, in which case the correlation function C (10)(q, k|q  , k  ) must vanish. In this section, however, we have shown that (10)

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Multiple scattering of light from randomly rough surfaces

[2, § 5

for weakly rough random surfaces C (10) (q, k|q , k  ) is nonzero, which implies that for such surfaces δS(q|k) must obey only complex Gaussian statistics. How δS(q|k) transforms from a complex Gaussian random process into a circular complex Gaussian random process with increasing surface roughness has been analyzed by Leskova, Simonsen and Maradudin [2002], on the basis of the assumption that the surface profile function ζ (x1 ), and hence δS(q|k), is a stationary random process. The simplest version of phase perturbation theory (Sánchez-Gil, Maradudin and Méndez [1995]) for the scattering of a scalar plane wave from a one-dimensional random surface on which the Dirichlet boundary condition is satisfied was used to calculate δS(q|k). It was found that in the limit as δ/λ tends to zero the joint probability density of the {δS(q|k)} has the complex Gaussian form for a fixed value of the transverse correlation length a. However, as δ/λ increases, the joint probability density of the {δS(q|k)} approaches the circular complex Gaussian form. This can occur for values of δ/λ as small as about δ/λ = 0.6 for a fixed value of a.

5.3. Correlations in film systems Our discussion of angular intensity correlation functions up to now has been limited to those arising in the scattering of light from a one-dimensional random surface bounding a semi-infinite metal. Such surfaces can support at most a single surface electromagnetic mode, and the peaks occurring in the C (1) correlation function are associated with the excitation of this mode. If we turn to a structure that supports two or more surface or guided modes, such as a dielectric film on a reflecting substrate or a free-standing or supported film, it may be expected that, just as the contribution to the mean differential reflection coefficient from the diffuse component of the scattered field exhibits a richer structure – satellite peaks – than is the case for a structure supporting a single surface wave, so will the angular intensity correlation function. The possibility that this should be the case was first noted by Freilikher, Pustilnik and Yurkevich [1994], in the context of the angular intensity correlation function C (1) . In four subsequent theoretical studies the angular intensity correlations of light reflected from a randomly rough dielectric film on a perfectly conducting substrate (Sánchez-Gil [1997], McGurn and Maradudin [1997, 1998a]), or reflected from and transmitted through a randomly rough freestanding metal film (McGurn and Maradudin [1997, 1998b]), were calculated. The resulting correlation functions indeed have a richer structure than the correlation functions calculated for the light scattered from a semi-infinite metal with a randomly rough surface. In particular, the C (1) correlation function acquires additional peaks whose angular positions depend on differences of the wave numbers

2, § 6]

Multiple-scattering effects in the scattering

189

of the guided and surface waves. That is, there are now additional memory effect peaks, and additional reciprocal memory effect peaks. Such peaks, of course, have no counterparts in the scattering of light from a semi-infinite metal. Additional peaks also occur in the C (1.5) correlation function, but these are due simply to the multiplicity of guided and surface waves in the film structures, and their origin is the same as that of the peaks in C (1.5) occurring in the scattering from the surface of a semi-infinite metal.

§ 6. Multiple-scattering effects in the scattering of light from complex media bounded by a rough surface 6.1. Coherent effects associated with the interference of nonreciprocal optical paths Although enhanced backscattering is the best known coherent effect caused by the interference of multiply-scattered optical paths, it is not the only one. Like enhanced backscattering some of the other coherent effects arise from the interference of reciprocal optical paths. However, interesting effects also arise from the interference of nonreciprocal optical paths. In this section we survey these other coherent effects. The simplest and the earliest studied of the coherence effects arising from the interference of nonreciprocal optical paths was discovered in an investigation of the scattering of p-polarized light incident from vacuum on a one-dimensional random metal or n-type semiconductor surface in the presence of a static magnetic field oriented parallel to the generators of the surface (i.e. parallel to the x2 -axis), and hence normal to the plane of incidence, the x1 x3 -plane. The scattering in this case is co-polarized, and the plane of scattering coincides with the plane of incidence. In this geometry the p-polarized surface magnetoplasmon polariton at the frequency ω of the incident light supported by a planar interface between the vacuum and the metal or semiconductor becomes nonreciprocal. The wavenumber k+ (ω) of the surface magnetoplasmon polariton propagating in the +x1 -direction is different from that of the surface magnetoplasmon polariton propagating in the −x1 -direction, k− (ω). Both of these wavenumbers depend on the strength of the applied magnetic field. At each nonzero value of this field k+ (ω) is smaller than k− (ω) for all frequencies ω smaller than a lower critical frequency that depends on the field strength. At frequencies above this critical value only k+ (ω) exists: there is no backward propagating surface magnetoplasmon polariton. Then, at frequencies above a higher critical frequency k+ (ω) also no longer exists. Within the

190

Multiple scattering of light from randomly rough surfaces

[2, § 6

Fig. 21. Double-scattering sequences occurring in the scattering of p-polarized light from a semi-infinite metal or n-type semiconductor, bounded by a one-dimensional random surface, that supports a nonreciprocal surface plasmon polariton when an external dc magnetic field is directed into the plane of the figure.

frequency range where both k+ (ω) and k− (ω) exist, the difference k+ (ω) − k− (ω) increases in magnitude with increasing magnetic field strength. These nonreciprocal properties of the surface magnetoplasmon polariton give rise to an interesting multiple-scattering effect when p-polarized light is incident on the random surface of a semi-infinite metal or n-type semiconductor. Thus, let us consider the system depicted in fig. 21, and consider a doublescattering process in which the incident light of frequency ω excites a forward propagating surface magnetoplasmon polariton of wavenumber k+ (ω) through the breakdown of the infinitesimal translational invariance of the system caused by the roughness. The surface wave thus excited propagates along the surface, is scattered once more by the roughness, and is converted back into a volume wave in the vacuum that propagates away from the surface (fig. 21). All such doublescattering processes are assumed to be uncorrelated due to the random nature of the surface. However, to any given sequence there corresponds a partner, in which the light and surface wave are scattered from the same points, but in the reverse order. This means that the magnitude of the wavenumber of the backwardpropagating surface wave excited is k− (ω). The phase difference between the path (ABCD)+ and the path (A CBD )− , where the subscript denotes the surface wave excited by the incident light, is   φ+− = rBC · (kin + ksc ) − |rBC | k+ (ω) − k− (ω) , (6.1) where kin and ksc are the wave vectors of the incident and scattered light, respectively, while rBC is the vector joining the points B and C. We see from eq. (6.1)

2, § 6]

Multiple-scattering effects in the scattering

191

that in the absence of the magnetic field, when k+ (ω) = k− (ω), constructive interference (φ+− = 0) occurs when ksc = −kin . This condition describes scattering into the retroreflection direction. The complex amplitudes of the reciprocal paths (ABCD)+ and (A CBD )− in this case are equal. They must be added, and the modulus of the sum squared, in calculating their contribution to the intensity of the scattered field. For scattering into directions other than the retroreflection direction the two waves have a nonzero phase difference, and very rapidly become incoherent, so that only their intensities add. Consequently, the intensity of scattering into the retroreflection direction is a factor of two larger than it is for scattering into other directions, due to the contribution from the cross terms in the squared-modulus. Of course, the contribution from the single-scattering processes has to be subtracted in obtaining this factor of two enhancement, since it is not subject to coherent backscattering. This increased intensity of scattering into a narrow angular interval about the retroreflection direction is called enhanced backscattering. However, in the presence of the magnetic field, when k+ (ω) is unequal to k− (ω), we see from eq. (6.1) that constructive interference occurs for scattering into a direction for which ksc = −kin . This gives rise to a peak in the angular distribution of the intensity of the diffuse component of the scattered light at the scattering angle corresponding to this value of ksc . Unlike the enhanced backscattering peak in zero applied field, however, the height of this peak has no simple relation to the height of the background at its position, when the contribution of the single-scattering processes is subtracted off, because the amplitudes of the two interfering waves are not equal in this case. If we note that the points B and C lie on the scattering surface, so that the vector rBC is essentially parallel to the mean scattering plane, we obtain from eq. (6.1) that the scattering angle at which this peak occurs is related to the angle of incidence θ0 by sin θs = − sin θ0 +

 c k+ (ω) − k− (ω) . ω

(6.2)

Although the result given by eq. (6.2) was obtained on the basis of a doublescattering process, it remains unchanged when multiple-scattering processes of all orders are taken into account. Thus, since k+ (ω) is smaller than k− (ω), the coherent interference of the nonreciprocal optical paths shifts the position of the peak away from the retroreflection direction in the direction of larger scattering angles, and the magnitude of the shift is larger the larger is the strength of the applied magnetic field. These predictions have been verified by the results of perturbative (McGurn, Maradudin and Wallis [1991]) and computer simulation (Lu, Maradudin and Wallis [1991]) calculations of the scattering of p-polarized light

192

Multiple scattering of light from randomly rough surfaces

[2, § 6

Fig. 22. The contribution to the mean differential reflection coefficient from the diffuse component of the scattered light when a p-polarized beam of light of wavelength λ = 43.2 µm (ω/ωp = 0.68) is incident on a one-dimensional random surface of n-GaAs, characterized by the Gaussian power spectrum (2.23) with δ = 3.14 µm and a = 15.14 µm. The angle of incidence is θ0 = 5◦ . An external dc magnetic field H is applied parallel to the grooves and ridges of the surface. (a) ωc /ωp = 0 (H = 0); (b) ωc /ωp = 0.2 (H = 2.46 × 104 G); (c) ωc /ωp = 0.4 (H = 3.28 × 10−4 ); (d) ωc /ωp = 0.5 (H = 4.1 × 104 G), where ωc is the cyclotron frequency and ωp is the plasma frequency of the conduction electrons. The position of the enhanced backscattering peak is at θs = −5◦ , −6.8◦ , −7.2◦ , and −7.5◦ , respectively (Lu, Maradudin and Wallis [1991]).

from the structure depicted in fig. 21. In the results of these calculations, especially in those of the latter calculations depicted in fig. 22, it was found that the angle at which the peak in the contribution to the mean differential reflection coefficient occurred departed slightly from the value given by eq. (6.2) when the values of k+ (ω) and k− (ω) for a planar surface were used. This is due presumably to the fact that in the computer simulation calculations the roughness-induced corrections to the values of k+ (ω) and k− (ω) are taken into account. The coherent effect caused by the interference of nonreciprocal optical paths that we have just considered is present in the scattering of p-polarized light from a random surface that supports only a single surface electromagnetic wave, albeit a nonreciprocal one. If we now turn our attention to the scattering of light

2, § 6]

Multiple-scattering effects in the scattering

193

Fig. 23. Double-scattering sequences occurring in the scattering of electromagnetic waves from, and their transmission through, a bounded structure whose illuminated surface is a one-dimensional random surface, that supports several surface or guided waves and their reciprocal partners.

from, or its transmission through, a structure that supports two or more surface or guided waves, new features appear in the angular distribution of the intensity of the incoherent component of the reflected or transmitted light not present in the results described so far, that are also caused by the coherent interference of multiply-scattered nonreciprocal optical paths, even though the surface or guided waves themselves are now reciprocal. Thus, let us consider a free-standing dielectric or metallic film, characterized by an isotropic, complex, dielectric function ε(ω), in the region −d < x3 < ζ(x1 ), with vacuum in the regions x3 > ζ(x1 ) and x3 < −d (fig. 23). This structure is illuminated from the region x3 > ζ (x1 ) by a plane wave of p- or s-polarized light, whose plane of incidence is the x1 x3 -plane. The scattered and transmitted light in this geometry is co-polarized, and the plane of scattering and of transmission is also the x1 x3 -plane. We assume that this structure supports N ( 2) guided or surface electromagnetic waves at the frequency ω of the incident light (N = 2 in the case of a metal film illuminated by p-polarized light), whose wave numbers will be denoted by k1 (ω), k2 (ω), . . . kN (ω). Each of these guided or surface waves is assumed to be reciprocal. The existence of two or more guided or surface electromagnetic waves can add peaks to the angular distribution of the intensity of the light scattered dif-

194

Multiple scattering of light from randomly rough surfaces

[2, § 6

fusely from, or transmitted diffusely through, the film. To see how they arise, we examine double-scattering processes contributing to the scattered and transmitted fields. A typical double-scattering path ABCD and its partner A CBD (fig. 23) are each N -fold degenerate in the sense that along the segment BC there are N “channels” with different phase factors. The phase difference between the path (ABCD)m , and the path (A CBD )n , where the subscript gives the number of the waveguide mode excited by the incident light, is   φnm = rBC · (kin + ksc ) + |rBC | kn (ω) − km (ω) . (6.3) We see from this expression that constructive interference (φnm = 0) can occur when n = m and ksc = −kin , i.e. due to the coherent interference of doubly-scattered reciprocal paths. Such scattering processes give rise to enhanced backscattering. However, we see from eq. (6.3) that constructive interference can also occur for scattering into other directions, ksc = −kin , for some n = m. The satisfaction of these conditions gives rise to satellite peaks in the angular distribution of the intensity of the light scattered diffusely. These are due to the coherent interference of doubly scattered nonreciprocal paths. The scattering angles at which these satellite peaks occur are related to the angle of incidence by   sin θs = − sin θ0 ± (c/ω) kn (ω) − km (ω) , n = m. (6.4) In transmission, the phase difference between the double-scattering path (ABCE)m and its partner (A CBE )n (fig. 23) is   φnm = rBC · (kin + k∗sc ) + |rBC | kn (ω) − km (ω) , (6.5) where the vector k∗sc is obtained from the vector ksc by reversing the sign of the 3-component of the latter. From eq. (6.5) it follows that constructive interference (φnm = 0) can occur when n = m and k∗sc = −kin . This is a new coherent effect arising from the interference of doubly-scattered reciprocal paths. The condition k∗sc = −kin translates into a peak in the angular distribution of the intensity of light transmitted diffusely through the film in the antispecular direction, i.e. in the direction opposite to that for specular reflection from the film. This peak is called enhanced transmission (McGurn and Maradudin [1989]). At the same time, we see from eq. (6.5) that constructive interference can also occur for transmission into other directions k∗sc = −kin for some n = m. The satisfaction of these conditions gives rise to satellite peaks in the angular distribution of the light transmitted diffusely. These peaks are due to the coherent interference of non-reciprocal optical paths. These satellite peaks occur at angles of transmission θt that are related

2, § 6]

Multiple-scattering effects in the scattering

195

to the angle of incidence by   sin θt = − sin θ0 ± (c/ω) kn (ω) − km (ω) ,

n = m,

(6.6)

(Freilikher, Pustilnik, Yurkevich and Maradudin [1994], Sánchez-Gil, Maradudin, Lu and Freilikher [1995]). Although the preceding discussion has been based on the double-scattering processes depicted in fig. 23, the conclusions reached, in particular eqs. (6.4) and (6.6), remain valid when all higher-order multiple-scattering processes are taken into account. Of the N(N − 1) satellite peaks predicted to occur at the angles of scattering and transmission given by eqs. (6.4) and (6.6), respectively, not all may be observed, however. This will be the case when the magnitude of the right-hand side of eq. (6.4) or eq. (6.6) is larger than unity. Furthermore, among the real satellite peaks that should appear when the magnitude of the right-hand sides of eqs. (6.4) and (6.6) is smaller than unity, not all of them might be intense enough to be observable. We also note that the occurrence of satellite peaks in general appears to be limited to the case of the scattering of p- or s-polarized light from a one-dimensional random surface of a structure that supports two or more surface or guided waves. In the case of the scattering of light from an isotropic two-dimensional surface of such a structure the ensemble averaging of the scattered intensity restores isotropy in the mean scattering plane, and thereby eliminates the occurrence of special scattering angles at which satellite peaks could occur (Kawanishi, Ogura and Wang [1997]). It should be noted, however, that in their study of the scattering of light from the same structure as the one studied by Kawanishi, Ogura and Wang [1997], although by a different approach, namely small-amplitude perturbation theory, Soubret, Berginc and Bourrely [2001a] predicted satellite peaks in addition to the enhanced backscattering peak in the angular dependence of the intensity of the scattered light. However, as the authors point out, the contribution to the mean differential reflection coefficient of third order in δ was larger than the first-order contribution, which casts some doubt on the validity of the small-amplitude perturbation theory for this structure. The first theoretical calculation to predict the existence of satellite peaks was devoted to the scattering of s-polarized light from a dielectric film deposited on the planar surface of a perfect conductor when the illuminated surface of the film was a one-dimensional random surface (Freilikher, Pustilnik and Yurkevich [1994]). This structure supported two guided waves, and two satellite peaks were observed in the angular dependence of the intensity of the light scattered diffusely, in addition to an enhanced backscattering peak between them. This was

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Multiple scattering of light from randomly rough surfaces

[2, § 6

soon followed by a calculation of the scattering of s-polarized light from, and its transmission through, a free-standing dielectric film bounded by parallel planar surfaces, in the case that the dielectric constant of the film was a random function of the coordinates parallel to the plane of incidence (Freilikher, Pustilnik, Yurkevich and Maradudin [1994]). In this case satellite peaks were present in the angular dependence of the intensity of the light scattered diffusely from, and transmitted diffusely through, the film, in addition to enhanced backscattering and enhanced transmission peaks. Subsequently theoretical calculations of the scattering of light from a dielectric film whose interface with vacuum is a one-dimensional random surface, but which is deposited on the planar surface of a perfect conductor, whose results display satellite peaks, have been carried out by several authors (SánchezGil, Maradudin, Lu, Freilikher, Pustilnik and Yurkevich [1994], Sánchez-Gil, Maradudin, Lu, Freilikher, Pustilnik and Yurkevich [1996], Wang, Ogura and Takahashi [1995], Madrazo and Maradudin [1997]). The scattering of light from a dielectric film whose interface with vacuum is planar, but which is deposited on the one-dimensional randomly rough surface of a perfect conductor, was investigated by Simonsen and Maradudin [1999]. In contrast with the earlier calculations, in which the roughness of the random surface was characterized by a Gaussian power spectrum (2.20), in the work of Madrazo and Maradudin [1997] and of Simonsen and Maradudin [1999], a West–O’Donnell power spectrum (2.21) was used for this purpose as well. It was found that the use of the latter power spectrum increased the height and sharpness of the satellite peaks in comparison with those obtained when a Gaussian power spectrum, that yielded the same rms height and rms slope as the West–O’Donnell power spectrum, was used. This was particularly the case in the scattering of p-polarized light, where the satellite peaks are not very well defined when a Gaussian power spectrum is used. The scattering of p-polarized electromagnetic waves from, and their transmission through, a thin, free-standing, metal film, whose illuminated surface was a one-dimensional randomly rough surface, while its back surface was planar was studied by Sánchez-Gil, Maradudin, Lu and Freilikher [1994, 1995], and by Lu and Maradudin [1997]. In the results of these calculations satellite peaks were present both in scattering and transmission, in addition to enhanced backscattering and enhanced transmission peaks. The calculations of Lu and Maradudin [1997] also showed that the use of a West–O’Donnell power spectrum (2.21) to characterize the surface roughness that is nonzero in a range of wavenumbers that includes the wavenumbers of the two surface plasmon polaritons supported by the film, enhances the satellite peaks both in scattering and transmission with respect to those that occur when a Gaussian power spectrum is assumed. Similar results

2, § 6]

Multiple-scattering effects in the scattering

197

were obtained in the case that the metal film was not free-standing but instead was deposited on a dielectric film of finite thickness (Chaikina, García-Guerrero, Gu, Leskova, Maradudin, Méndez and Shchegrov [2000]). The enhanced transmission peak has been observed experimentally (Celli, Tran, Maradudin, Lu, Michel and Gu [1991], Gu, Maradudin, Méndez, Ponce and Ruiz-Cortés [1991], Gu, Dummer, Maradudin, McGurn and Méndez [1991], den Outer [1995]). In fig. 24 we present a plot of the relative transmission of

Fig. 24. An experimental plot of the relative transmission of p-polarized light of wavelength λ = 0.6328 µm through a vacuum-deposited silver film deposited on a BK-7 glass plate (nd = 1.51) of finite thickness, whose bottom surface is coated with a very thin layer of MgF2 to reduce reflections from it. The angle of incidence is θ0 = 20◦ . The illuminated surface of the silver film is a two-dimensional random surface characterized by the Gaussian power spectrum (2.17) with δ = 118.2 ± 10 Å and a = 1185.0 Å. The plane of transmission coincides with the plane of incidence, and it is the p-polarized component of the transmitted light that is measured (after Gu, Maradudin, Méndez, Ponce and Ruiz-Cortés [1991]).

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p-polarized light through a vacuum-deposited silver film deposited on a BK-7 glass plate of finite thickness, whose bottom surface is coated with a very thin layer of MgF2 to reduce reflections from this surface (Gu, Maradudin, Méndez, Ponce and Ruiz-Cortés [1991]). The plane of transmission coincides with the plane of incidence, and it is the p-polarized component of the transmitted light that is measured. The angle of incidence is θ0 = 20◦ . A sharp peak in the transmittance is observed at an angle of transmission θt = −20◦ . This is enhanced transmission. The fact that satellite peaks were not observed in this measurement is due to the two-dimensional nature of the roughness of the illuminated surface of the silver film, which has the effect of washing out the satellite peaks. The only observation of satellite peaks in the scattering of light from a random surface was carried out for a structure that differs from the one depicted in fig. 23 (Méndez, Chaikina and Escamilla [1999]). It utilized the double passage of light through a two-dimensional random phase screen placed in front of a mirror. A piece of birefringent material (calcite) in the form of a beam-displacing prism, was placed in the space between the random phase screen and the mirror (fig. 25). The incident light was in a +45◦ linear polarization state, i.e. it consisted of a linear superposition of p- and s-polarized components with equal amplitudes. The beam-displacing prism was oriented in such a way as to produce a lateral displacement of the p-polarized component of the incident light. This displacement removed the mean rotational symmetry of the random phase screen that washed out the satellite peaks in the experiments of Celli, Tran, Maradudin, Lu, Michel and Gu [1991], Gu, Maradudin, Méndez, Ponce and Ruiz-Cortés [1991], Gu, Dummer, Maradudin, McGurn and Méndez [1991], and den Outer [1995].

Fig. 25. A schematic diagram showing the scattering geometry employed in the observation of satellite peaks through the double passage of light through a random phase screen and a beam-displacing birefringent crystal (Méndez, Chaikina and Escamilla [1999]).

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The birefringence of the calcite prism led to propagation paths that were different for waves polarized along two orthogonal directions, and the inclusion of a linear polarizer between the prism and mirror mixed these two components of polarization, allowing the interference of waves that had picked up the same random phase (traversing the random phase screen) but that had traveled along different deterministic, optical paths, when the transmission axis of the polarizer made an angle of 45◦ with the x-axis. The mean intensity of the light scattered from this structure into the far field has been calculated on the basis of the thin random phase screen model (Escamilla, Méndez and Hotz [1993]), in which a delta-correlated random phase screen in contact with a Gaussian aperture was assumed. The result has the form (Escamilla, Méndez and Chaikina [2001])

I (q|k)     w2 w2 2 2 = 2 + exp − (k + q) + exp − (k + q ± p) 4 4         ω 2 w2 p2 w2 p 2 + 2 cos . exp − + exp − k+q ± c 4D 4 4 4 2 (6.7) In this expression w is the half-width of the Gaussian aperture in the plane of the random phase screen, D is the distance from the random phase screen to the mirror,  is the lateral shift of the p-polarized component of the light transmitted through the random phase screen, p = (ω/c)(/D), and k = (ω/c) sin θ0 , q = (ω/c) sin θs , where θ0 and θs are the angles of incidence and the scattering, respectively. The + sign in the exponentials of the second and third terms applies to the p-polarized component of the scattered light; the “−” sign applies to the s-polarized component. The sign of q in the last three terms on the right-hand side of eq. (6.7) is opposite to that in the corresponding expression given by Méndez, Chaikina and Escamilla [1999], because here we define the scattering angle θs in the conventional manner, as in fig. 23. The first term on the right-hand side of eq. (6.7) represents a constant background; the second produces a peak in the retroreflection direction q = −k; the third gives rise to two satellite peaks at q = −k ± p; the last term produces satellite features at q = −k ± (p/2), that can be either peaks or dips, depending on the sign of the cosine factor, i.e. on the distance D of the mirror from the random phase screen. No peak in the specular direction q = k is present in the expression (6.7) for I (q|k): the “roughness” of the random phase screen was assumed to be sufficiently strong that the specular component of the scattered light was strongly

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suppressed, and only the diffuse component contributed to the mean scattered intensity in the far field. All of these features of I (q|k) are observed in the measurements of this mean scattered intensity carried out by Méndez, Chaikina and Escamilla [1999] (fig. 26), and their relative strengths are well described by eq. (6.7).

6.2. Scattering from the random surface of an amplifying medium The metallic and dielectric random surfaces considered in this review up to now have all bounded passive (absorbing) media, i.e. media characterized by dielectric constants whose imaginary parts are positive. The question naturally arises: what effects will be displayed by the angular dependence of the intensity of the light reflected specularly or diffusely from the random surface of an active (amplifying) medium, i.e. a medium characterized by a dielectric constant whose imaginary part is negative? This question attracts attention because the transmission of light through, and its reflection from, volume disordered amplifying media, have been the subjects of several recent theoretical studies (Zyuzin [1994a, 1994b, 1995], Pradhan and Kumar [1994], Zhang [1995], Feng and Zhang [1996], John and Pang [1996], Beenakker, Paasschens and Brouwer [1996], Paasschens, Misirpashaev and Beenakker [1996], Freilikher, Pustilnik and Yurkevich [1997]), as well as of experimental investigations (Genack and Drake [1994], Lawandy, Balachandran, Gomes and Sauvain [1994], Sha, Liu and Alfano [1994], Wiersma, van Albada and Lagendijk [1995], Cao, Zhao, Ho, Seelig, Wang and Chang [1999]). A review of such investigations has been given by Wiersma and Lagendijk [1997]. The interest in the interplay between phase-coherent multiple scattering and stimulated emission that these studies represent is due in large measure to the possibility of using such systems as random lasers (Kempe, Berger and Genack [1997]), i.e. lasers in which the feedback is provided by multiple scattering due to disorder rather than by confinement by mirrors. In these investigations it was predicted (Zyuzin [1994a, 1994b], Wiersma, van Albada and Lagendijk [1995]), and was verified experimentally (Wiersma, van Albada and Lagendijk [1995], that the enhanced backscattering cone narrows as a result of stimulated emission below the lasing threshold; the observation of random laser action in a semiconductor powder has been reported (Cao, Zhao, Ho, Seelig, Wang and Chang [1999]); it was shown that in one-dimensional random systems amplification suppresses transmittance in just the same way as absorption does (Paasschens, Misirpashaev and Beenakker [1996]); and that all moments of the transmittance of

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Fig. 26. (a) Experimental results for the p-polarized component of the mean scattered intensity showing two satellite peaks in addition to the enhanced backscattering peak; (b) experimental results for the p-polarized component of the mean scattered intensity showing a satellite peak and a satellite dip, in addition to the enhanced backscattering peak (after Méndez, Chaikina and Escamilla [1999]).

a one-dimensional random amplifying medium are infinite (Freilikher, Pustilnik and Yurkevich [1997]).

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In the first theoretical study of the scattering of light from the random surface of an amplifying medium (Tutov, Maradudin, Leskova, Mayer and Sánchez-Gil [1999]), the system studied consisted of a dielectric film of mean thickness d deposited on a perfectly conducting substrate. The film–substrate interface was planar, while the vacuum–film interface was a one-dimensional random interface. characterized by the Gaussian power spectrum (2.20). The film was characterized by an isotropic, frequency-independent, complex dielectric constant ε = ε1 + iε2 , whose real part ε1 was assumed to be positive. The imaginary part ε2 was assumed to be positive or negative. In the former case the film was a passive (absorbing) medium; in the latter case it was an active (amplifying) medium. Taking ε2 to be negative is the simplest way to model stimulated emission in this system. The film was illuminated from the vacuum by an s-polarized plane wave, whose plane of incidence was normal to the generators of the random surface, and the reflectivity and the diffuse scattering of the light were calculated by the methods of manybody perturbation theory. It was found that the reflectivity of an amplifying film with a randomly rough surface depends strongly on whether or not the film supports a leaky guided wave, i.e. a guided wave whose wavenumber has a real part that lies in the radiative region (−ω/c, ω/c), at the frequency of the incident light, in addition to true guided waves, i.e. guided waves whose wavenumbers have a real part in the nonradiative region. If the scattering structure supports a leaky wave, the reflectivity of the rough film is smaller than that of the corresponding film with a planar vacuumdielectric interface, and can be smaller than unity for some range of angles of incidence. If it does not, the reflectivity of the rough film is larger than that of the corresponding film with a planar vacuum-dielectric interface. In contrast, the reflectivity of an amplifying film with a planar vacuum-dielectric interface is greater than unity, irrespective of whether or not the film supports a leaky guided wave at the frequency of the incident light, in addition to the true guided waves at that frequency that it supports. The contribution to the mean differential reflection coefficient from the diffuse component of the scattered light displayed an enhanced backscattering peak and satellite peaks (the latter if the scattering structure supports two or more guided waves). It also can depend significantly on whether the scattering system does or does not support a leaky guided wave at the frequency of the incident light. If no leaky guided wave is present in addition to true guided waves, the overall scattered intensity and the height of the enhanced backscattering peak decreases monotonically with increasing |ε2 |, while the width of the enhanced backscattering peak increases monotonically with increasing ε2 . In contrast, for an amplifying film the overall scattered intensity and the height of the enhanced backscattering peak

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initially increase with increasing |ε2 |, and then decrease, while the width of the enhanced backscattering peak initially decreases with increasing |ε2 |, and then increases. However, whether a leaky wave is present or not, the contribution to the mean differential reflection coefficient from the diffuse component of the scattered light for an amplifying film with a given value of |ε2 | is larger than that for an absorbing film with the same magnitude of ε2 but of opposite sign. In a subsequent study of the scattering of s-polarized light from the same system (Simonsen, Leskova and Maradudin [2001]), the random roughness of the vacuum-dielectric surface was characterized by the West–O’Donnell power spectrum (2.21), and the corresponding reduced Rayleigh equation was solved rigorously by a numerical approach for a large number of realizations of the surface profile function. It was found that the enhanced backscattering peak and the satellite peak supported by this system became narrower and taller as the amplification of the medium is increased, although the quality of the numerical results were such that this conclusion for the widths of the satellite peaks has only a qualitative nature. More interesting, it was found that the positions of the satellite peaks shifted to smaller scattering angles as |ε2 | was increased for negative values of ε2 , and an explanation for this shift was suggested. The position of the enhanced backscattering peak did not change with a change of ε2 . An experimental study was carried out of the enhanced backscattering of light from the randomly rough surface of a laser dye-doped polymer (Gu and Peng [2000]). The sample was a slice of a pyrromethene-doped polymer coupled with a rough gold layer with a large rms slope. When the sample was illuminated by an s-polarized He–Ne laser and was pumped by a cw argon-ion laser, amplified backscattering was observed. The amplitude of the enhanced backscattering peak increased sharply, and its width narrowed, for a sample with a small negative value for the imaginary part of its dielectric constant. 6.3. Scattering from a nonlinear medium bounded by a rough surface This section is devoted to the study of multiple-scattering effects in the diffuse component of the second-harmonic light generated in the interaction of light with a randomly rough metal surface. The problem constitutes one manifestation of a subject that is of interest in many branches of physics; namely, the interaction of randomness and nonlinearity. Experimental and theoretical studies of second harmonic generation in the specular reflection from planar metal surfaces go back more than three decades, to the first experimental observation of the effect (Brown, Parks and Sleeper [1965], Brown and Parks [1966]) and its first theoretical descriptions (Jha [1965, 1966],

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Bloembergen, Chang and Lee [1966]). In these early works, it was already established that the surface played an important role in the generation of the second harmonic signal. Significant contributions to the understanding of the problem were made by Bloembergen, Chang, Jha and Lee [1968] and Rudnick and Stern [1971]. The second harmonic specular reflection from flat surfaces has remained a subject of studies throughout the years (see, e.g., the review by Sipe and Stegeman [1982]) and has become a useful nondestructive probe that is sensitive to surface structure at the monolayer level (see, e.g., Lim, Downer, Ekerdt, Arzate, Mendoza, Gavrilenko and Wu [2000]). Second harmonic generation by metal surfaces involving the attenuated total reflection geometries of Otto [1968, 1970] and Kretschmann [1971] have also been the subject of several studies. Surface polaritons can be excited efficiently in these kind of geometries and the strong fields created in the vicinity of the interface lead to increased second harmonic signals (Simon, Mitchell and Watson [1974, 1975], Sipe, So, Fukui and Stegeman [1980]). In the last two decades, considerable attention has been given to the generation of second harmonic light by non-planar surfaces. An initial motivation for this was the possibility of obtaining an enhancement of the second harmonic intensity through the excitation of surface plasmon-polaritons with a periodic profile (Farias and Maradudin [1984], Coutaz, Neviere, Pic and Reinisch [1985], Simon, Huang, Quail and Chen [1988]). Nevertheless, the case of random surfaces has also received some attention. The angular scattering distribution of the second harmonic light was studied in the lowest-order perturbation theory by Deck and Grygier [1984]. More recently, due to the growing interest in the broader area of interference effects occurring in the multiple scattering of electromagnetic waves from random media, there has been an increase in the scientific activity in the area. It had been hoped that nonlinear optical interactions in random media would give rise to new features due to the interference of multiply-scattered electromagnetic waves at the generated frequencies. However, attempts to observe such effects in the second harmonic light generated by powdered nonlinear media produced negative results (Loo, Lee, Takiguchi and Alfano [1989]), apparently due to a phase mismatch between the scattered fundamental and harmonic fields. Nonetheless, rough surfaces have proven to be more interesting in this respect. In the first theoretical study of multiple-scattering effects in the second harmonic light produced at weakly rough random metal surfaces McGurn, Leskova and Agranovich [1991] predicted, on the basis of a perturbative calculation, that a narrow enhancement peak should occur not only in the retroreflection direction but also in the direction normal to the mean surface. The peak predicted at the

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surface normal was attributed to interference effects in the multiple scattering of surface plasmon polaritons of the fundamental frequency, excited by the incident light through the surface roughness, while the peak in the retroreflection direction was attributed to interference effects in the multiple scattering of surface plasmon polaritons of the harmonic frequency. This work stimulated several experimental studies of second harmonic generation in the scattering of light from random metal surfaces (Wang and Simon [1991], Simon, Wang, Zhou and Chan [1992], Aktsipetrov, Golovkina, Kapusta, Leskova and Novikova [1992], Wang and Simon [1993], Kuang and Simon [1995], Bozhevolnyi and Pedersen [1997]). In these experiments, however, the scattering system was not simply the interface between free space and a semi-infinite metal; in order to excite surface polaritons and, thus, increase the second harmonic signal the authors adopted the attenuated total reflection geometry. Experimental studies of multiple scattering effects in the second harmonic generation of light scattered from a clean one-dimensional random vacuum–metal interface have been carried out by O’Donnell and his colleagues in a series of papers (O’Donnell, Torre and West [1996, 1997], O’Donnell and Torre [1997]). These experiments are unusual in many ways. They were carried out with characterized, specially fabricated one-dimensional silver surfaces, and the scattering data were absolutely normalized, permitting quantitative comparisons with theoretical results. It was found that, for both weakly (O’Donnell, Torre and West [1996]) and strongly (O’Donnell and Torre [1997]) rough silver surfaces, a dip is present in the retroreflection direction in the angular dependence of the intensity of the second harmonic light, rather than the peak that occurs in the fundamental scattering. No peak in the direction normal to the mean surface was observed in these experiments. Many of the features of second harmonic scattering found in these papers are reproduced by the rigorous numerical simulations of second harmonic generation from random surfaces carried out by Leyva-Lucero, Méndez, Leskova, Maradudin and Lu [1996], and by Leyva-Lucero, Méndez, Leskova and Maradudin [1999]. There remain, however, some differences that may be attributed to uncertainties in the phenomenological nonlinear surface susceptibilities. In this section we first outline the theoretical approach employed by LeyvaLucero, Méndez, Leskova, Maradudin and Lu [1996] and Leyva-Lucero, Méndez, Leskova and Maradudin [1999] for the study of multiple-scattering effects in the second harmonic generation of light in reflection from randomly rough metal surfaces. Typical results for the nonlinear scattering problem from three kinds of surfaces are also presented, together with a discussion of the main features arising from multiple scattering.

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6.3.1. Theoretical approach One promising theoretical approach to the study of multiple-scattering effects in the second harmonic light generated at randomly rough surfaces is a Monte Carlo technique (Leyva-Lucero, Méndez, Leskova, Maradudin and Lu [1996], LeyvaLucero, Méndez, Leskova and Maradudin [1999]). This method is capable of dealing with both weakly and strongly rough surfaces. Only one-dimensional random surfaces have been treated, because it is only for such surfaces that multiplescattering effects can be calculated readily at the present time. For brevity, we consider here only the case of incident light of p polarization, which produces the strongest harmonic field. To solve the scattering equations at the harmonic frequency, the Maxwell equations satisfied by the electromagnetic fields in regions I (x3 > ζ(x1 )) and II (x3 < ζ (x1 )) have to be supplemented by boundary conditions at the rough metal interface x3 = ζ (x1 ). At the fundamental frequency ω these boundary conditions express the continuity of the tangential components of the magnetic and electric fields across the interface. Considering now the harmonic fields, we first note that homogeneous and isotropic metals possess inversion symmetry. Therefore, the dipolar contribution to the bulk nonlinear polarization vanishes. The presence of a surface breaks the inversion symmetry. Both the electromagnetic fields and the material parameters vary rapidly at the surface, and their gradients give rise to the optical nonlinearity of the metal surface. The normal component of the electric field is discontinuous at the surface and its normal derivative behaves as a delta function. Thus, the surface term is normally more important than the bulk one in the second harmonic generation from centrosymmetric media. In what follows we will neglect the small contribution to the nonlinearity coming from the bulk of the metal and the possible anisotropy of the material constants. So, in contrast with the situation for the fundamental fields, the tangential components of the harmonic fields change abruptly across the interface and the generation of the second harmonic field can be accounted for through the boundary conditions. The nonlinear boundary conditions for the harmonic fields are obtained by integrating Maxwell’s equations for them across the interface layer, and then passing to the limit of a vanishing layer thickness. In carrying out this calculation it is convenient to introduce a local coordinate system (x, y, z) with its origin at each point of the surface x3 = ζ (x1 ) that is defined by the unit vectors {ˆx, yˆ , zˆ }, where xˆ = (1, 0, ζ  (x1 ))/φ(x1 ), yˆ = xˆ 2 , zˆ = (−ζ  (x1 ), 0, 1)/φ(x1), and φ(x1 ) = [1 + (ζ  (x1 ))2 ]1/2 . The unit vectors xˆ and zˆ are, respectively, tangent and normal to the interface in the plane perpendicular to its generators. On integrating the tangential component of the equation ∇ × H = −(2iω/c)D, namely, ∂Ht /∂z = (2iω/c)ˆz × Dt + ∇t Hz (with ∇t = [ˆx(∂/∂x) + yˆ (∂/∂y)]) across the

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vacuum–metal interface, and with the use of the relation D = ε(2ω, z)E + 4πPNL , we obtain the first nonlinear boundary condition,   2iω (II) H(I) (6.8) zˆ × 4πPst (x|2ω) , t (x|2ω) − Ht (x|2ω) = c where we have defined the tangential component of the nonlinear surface polarization as  η   dz PNL Pst (x|2ω) = lim (6.9) t (x, z|2ω) , η→0 −η

PNL t (x, z|2ω) represents the tangential component of the

and where nonlinear polarization vector. The subindex t denotes quantities that are tangential to the surface and we shall also use a subindex z to denote quantities that are normal to the surface. The components of the second-order surface susceptibility tensor χijs k are the constants of proportionality relating the amplitude of the components of the nonlinear surface polarization to the fundamental field amplitudes at the surface, according to  (I) (I) χijs k Ej (x|ω)Ek (x|ω). Pis (x|2ω) = (6.10) jk

χijs k

has 27 components, but for the surface of a centrosymmetric In general, medium only three distinct components are necessary to characterize the response s . The tangential component of of the system. These are χtst z = χtszt , χzts t , and χzzz the nonlinear surface polarization can be written as (I)

Pst (x|2ω) = χtst z Et (x|ω)Ez(I) (x|ω),

(6.11)

where we are using the convention that the permutation of the fields should yield no additional contribution to Pst (x|2ω) (Shen [1984], p. 38). The boundary condition expressed by eq. (6.8) can then be written as 2iω s (I) χ zˆ × E(I) (6.12) t (x|ω)Ez (x|ω). c ttz Similarly, on integrating the tangential component of the equation ∇ × E = (2iω/c)H, namely ∂Et /∂z = −(2iω/c)ˆz × Ht + ∇t Ez , across the vacuum–metal interface, we obtain the second nonlinear boundary condition,   (I) (II) Et (x|2ω) − Et (x|2ω) = −∇t 4πPzs (x|2ω) , (6.13) (II) H(I) t (x|2ω) − Ht (x|2ω) = 4π

where we have defined the normal component of the surface nonlinear polarization as   η  NL Pz (x, z|2ω) s Pz (x|2ω) = lim (6.14) dz , η→0 −η ε(2ω, z)

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with ε(2ω, z > η) = 1 in vacuum, and ε(2ω, z < −η) = ε(2ω) in the metal. The use of eq. (6.10) and the symmetry properties of the medium enable us to write  (I) 2 (I) (I) s Ez (x, |ω) + χzts t Et (x|ω) · Et (x|ω), Pzs (x|2ω) = χzzz (6.15) from which we express the second boundary condition as (II) E(I) t (x|2ω) − Et (x|2ω)  s  (I) 2  (I) (I) Ez (x, |ω) + χzts t Et (x|ω) · Et (x|ω) . = −4π∇t χzzz

(6.16)

The solution of the linear problem, together with the nonlinear surface susceptibilities involved in the boundary conditions (6.12) and (6.16), provide the required information for the calculation of the second harmonic field. The susceptibilities could, in principle, be determined experimentally, but we are not aware of any works in this respect. Alternatively, their values can be estimated with microscopic models of the material, through the nonlinear polarization. To deal with the problem of second harmonic reflection from metallic surfaces, several models or the nonlinear polarization have been proposed in the literature (Bloembergen, Chang, Jha and Lee [1968], Rudnick and Stern [1971], Sipe, So, Fukui and Stegeman [1980], Sipe and Stegeman [1982], Agranovich and Darmanyan [1982], Maystre, Neviere and Reinisch [1986], Corvi and Schaich [1986], Maytorena, Mochán and Mendoza [1995]), leading in turn to different nonlinear constants and to a variety of notations. The parameters µ of Agranovich and Darmanyan [1982] are related to the surface nonlinear susceptibilities of other authors (e.g., Mendoza and Mochán [1996]), and to the dimensionless frequency dependent parameters a(ω) and b(ω) of Rudnick and Stern [1971]. One finds that   1 µ1 ε(ω) − 1 2 s , =− a(ω) = χzzz (6.17a) 2 64π ne e ε(ω) 4π χzts t =

µ2 , 4π

χtst z = −

(ε(ω) − 1)2 µ3 1 b(ω) = , 2 32π ne e ε(ω) 4π

(6.17b) (6.17c)

where e is the magnitude of the electron charge and ne is the bulk electron density. In the calculations presented in this review, we use nonlinear constants calculated on the basis of the free-electron model (Bloembergen, Chang, Jha and Lee [1968], Sipe and Stegeman [1982], Shen [1984], p. 10, Maystre, Neviere and Reinisch [1986], Neviere, Vincent, Maystre, Reinisch and Coutaz [1988]), mainly

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because it leads to simple algebraic expressions. In this model, the nonlinear polarization takes the form PNL (2ω) = γ ∇(E · E) + β(∇ · E)E,

(6.18)

where β = e/(8πmω2 ), γ = e3 ne /(8m2 ω4 ), and m is the electron’s mass. With this form of PNL (2ω), from eqs. (6.9), (6.11), (6.14), and (6.15), and using the fact that in the free electron model, the dielectric function is given by 4πe2ne mω2 and, therefore, that γ = β(1 − ε(ω, z))/4, we find that    ε(ω) 2 (ε(ω) − 1)(ε(ω) − 3) 2 s − ln χzzz =− β , 3 2ε2 (ω) 3 ε(2ω) ε(ω, z) = 1 −

χzts t = 0,   ε(ω) − 1 χtst z = β . ε(ω)

(6.19)

(6.20a) (6.20b) (6.20c)

Apart from the sign, these nonlinear susceptibilities coincide with those obtained by Mendoza and Mochán [1996] by a different approach. The solution of the linear problem, and the knowledge of the nonlinear constants permit the calculation of the discontinuities of the harmonic fields across the boundary, thus determining the strength of the nonlinear source. From the boundary conditions (6.12) and (6.16) it can be readily shown that, in general, the second harmonic field contains both, s- and p-polarized components. In our cylindrical geometry, pure s- or p-polarized incident fields generate only p-polarized light at the harmonic frequency. So, neglecting anisotropies of the material and contributions from the bulk, an s-polarized second harmonic field can only be generated by a mixture of s- and p-polarized fundamental fields. The generation of the harmonic field is known to be more efficient in the case of a p-polarized incident field and, for brevity, we limit this review to that case. The fields entering the boundary conditions (6.12) and (6.16) are written in the local coordinate system (x, y, z). On returning to the laboratory coordinate system (x1 , x2 , x3 ) we note that when the incident electromagnetic field is p-polarized, the nonlinear sources on the right-hand sides of eqs. (6.12) and (6.16) are nonzero only for p-polarized fields of frequency 2ω, so that only p-polarized second harmonic light is generated in reflection from a one-dimensional random surface. In this case it is convenient to work with the single nonzero component of the magnetic field in the system. The nonlinear boundary conditions (6.12)

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and (6.16) then take the forms H (I) (x1 |2ω) − H (II) (x1 |2ω) = 4π

2ic χtst z (I) d (I) H (x1 |ω), L (x1 |ω) ω φ 2 (x1 ) dx1 (6.21a)

1 L(II) (x1 |2ω) ε(2ω) 2     (I) 2 2ic d 1 d (I) s s L = 4π H (x |ω) + χ (x |ω) , χ 1 1 zt t ω dx1 φ 2 (x1 ) zzz dx1 (6.21b) where we have introduced the source functions L(I) (x1 |2ω) −

H (I,II) (x1 |2ω) = H2(I,II) (x1 , x3 |2ω)|x3 =ζ(x1 ) , L(I,II) (x1 |2ω) =

(6.22a)

∂ H (I,II) (x1 , x3 |2ω)|x3=ζ(x1 ) , ∂N 2

(6.22b)

and where ∂ ∂ ∂ = −ζ  (x1 ) + ∂N ∂x1 ∂x3

(6.22c)

is a derivative along the normal to the interface. As in the case of the fundamental fields, application of Green’s theorem to the two semi-infinite media that define the surface yields two integral equations (Leyva-Lucero, Méndez, Leskova and Maradudin [1999]),    ∞ ∂ 1 (2ω)   (I)  dx1 G (x , x |x , x ) H (I) (x1 |2ω) + 1 3 1 3 H (x1 |2ω) 4π −∞ ∂N  0

(2ω) , − G0 (x1 , x3 |x1 , x3 )L(I) (x1 |2ω) x3 =ζ(x1 )+η x  =ζ(x  ) 3 1

0=

1 4π



∞ −∞

dx1

(6.23a)



 ∂ (2ω)   (II)  G (x , x |x , x ) (x1 |2ω) 1 3 1 3 H ∂N  ε

  (II)  (x , x |x , x )L (x |2ω) − G(2ω) 1 3 ε 1 3 1

x3 =ζ(x1 )+η x  =ζ(x  ) 3 1

,

(6.23b)

  where G(2ω) 0,ε (x1 , x3 |x1 , x3 ) are the Green’s functions for free-space and the metal at 2ω, respectively. Note that at the frequency 2ω there is no incident field and the required coupling between these two equations is provided by the nonlinear boundary conditions (6.21a) and (6.21b). The equations are solved numerically in

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the same way as the corresponding equations at the fundamental frequency (see Section 4.2.2), and their solution allows the calculation of the scattered field at the harmonic frequency. It is convenient to normalize the scattered power in such a way that the results are independent of the incident power and the illuminated area. We therefore define the normalized second harmonic scattered power as (2ω) = Psc

Psc(2ω) S, 2 Pinc

(6.24)

where S is the illuminated surface area. It is worth mentioning that other authors use here the cross-section of the illuminating beam. If we choose a plane wave of amplitude H0 for the incident field, the normalized diffuse component of the intensity of the generated harmonic light may be written as

|r(θs |2ω|2  − |r(θs |2ω)|2 , I (θs |2ω) diff = 2ωL cos θ0 |H0 |4

(6.25)

where L is the length of the surface, r(θs , 2ω)    ∞  (I) 2ω   (I) dx1 i = ζ (x1 ) sin θs − cos θs H (x1 |2ω) − L (x1 |2ω) c −∞

 2ω  x1 sin θs + ζ(x1 ) cos θs , × exp −i (6.26) c and θs is the scattering angle. 6.3.2. Discussion To illustrate the consequences of some of the physical processes that can take place on the surface, we present scattering distributions at the harmonic frequency for three silver surfaces whose profiles represent samples of Gaussian random processes. The surfaces have different power spectra and their statistical parameters were chosen to illustrate three kinds of multiple-scattering mechanisms that are involved in the generation of the second harmonic scattering distribution. They are: (i) Multiple-scattering of fundamental and harmonic waves in the valleys of a deep surface (see, e.g., the diagrams in fig. 27(a) and (b)), (ii) multiple-scattering mechanisms in a shallow surface involving the excitation of fundamental surface polaritons (see, e.g., the diagrams in fig. 27(c) and (d)), and

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Fig. 27. Illustration of some of the scattering processes that give rise to the second harmonic radiation.

(iii) multiple-scattering mechanisms in a shallow surface involving the direct excitation of second harmonic surface polaritons (see, e.g., the diagram in fig. 27(e)). First, we consider a surface with a Gaussian height autocorrelation function. The statistical parameters that characterize the random profile are chosen as a = 3.4 µm and δ = 1.81 µm, where a represents the correlation length (e−1 value of the normalized height correlation function) and δ is the standard deviation of heights. This allows a comparison between the numerical calculations and the experimental results of O’Donnell and Torre [1997]. This is a rough (δ is comparable to λ), high-sloped surface. The linear scattering calculations at the fundamental wavelength, λ = 1.06 µm produce a curve whose most noteworthy feature is the presence of a well-defined enhanced backscattering peak. For such rough surfaces, the backscattering enhancement is due to the multiple (mainly double) scattering of waves within the valleys of the sur-

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Fig. 28. The diffuse component of the mean normalized second harmonic intensity as a function of the scattering angle θs for the scattering of p-polarized light from a random silver surface characterized by the Gaussian power spectrum (2.17) with the roughness parameters a = 3.4 µm, and δ = 1.81 µm, and dielectric constants ε(ω) = −56.25 + i0.60 and ε(2ω) = −11.56 + 0.37. The curve represents the numerical simulation results averaged over Np = 2000 realizations of the surface. The angle of incidence is θ0 = 6◦ and the fundamental wavelength is λ = 1.064 µm. The vertical line indicates the backscattering direction.

face (Méndez and O’Donnell [1987], O’Donnell and Méndez [1987], Maradudin, Michel, McGurn and Méndez [1990], Maradudin, Méndez and Michel [1990]). In the far field, the waves that follow multiple-scattering paths are coherent in the vicinity of the backscattering direction, where they interfere constructively. Computer simulation results for the mean normalized second harmonic scattering intensity are shown in fig. 28. The scattering curve has some similarities with the linear scattering curve but, significantly, instead of a peak there is now a dip in the backscattering direction. On the other hand, there is reasonable qualitative agreement between these results and the experimental results of O’Donnell and Torre [1997]. Quantitatively, there is a discrepancy of about two orders of magnitude (the theoretical curves are higher). Given the relative simplicity of the assumed nonlinear polarization, and the fact that there are no fitting parameters, this is understandable. It is now known (O’Donnell and Torre [1997], Leyva-Lucero, Méndez, Leskova and Maradudin [1999]) that the minimum in the backscattering direction in the angular distribution of the mean second harmonic intensity is a consequence of multiple scattering (and nonlinear mixing of the light) within the valleys of the surface (see the diagrams in Figs. 27(a) and (b)). As in the case of linear optics, it is natural to expect that some of these processes are coherent, and that the backscattering

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effects are due to the interference between multiply scattered waves. However, in contrast with the linear case, at the second harmonic frequency the interference is destructive. That interference effects are indeed a plausible explanation for the dip may be recognized from the fact that the amplitude reflection coefficient for second harmonic generation is an odd function of the local angle of incidence on the surface (O’Donnell and Torre [1997], Leyva-Lucero, Méndez, Leskova and Maradudin [1999]). In the linear problem the interference between some pairs of paths is constructive because the paths are reciprocal in the backscattering direction. However, when nonlinear phenomena are involved reciprocity does not necessarily apply. Moreover, in this case, there seems to be some degree of antireciprocity when source and detector are interchanged. We now consider a surface that is designed to efficiently excite surface polaritons at the fundamental frequency. It is a surface with a West–O’Donnell power spectrum (West and O’Donnell [1995]) centered at the surface polariton wavenumber ksp(ω) = Re{(ω/c)[ε(ω)/(ε(ω) + 1)]1/2} with θmax = 15◦ and δ = 28.3 nm. The manner in which the power spectrum and θmax are defined is discussed in Section 2.2. Computer simulation results for the mean normalized second harmonic scattering intensity for this surface are shown in fig. 29. Due to the design of the surface, for angles of incidence smaller than 15◦ surface plasmon polaritons are launched both to the left and to the right. This creates

Fig. 29. The diffuse component of the mean normalized second harmonic intensity as a function of the scattering angle θs for the scattering of p-polarized light from a random silver surface with a West–O’Donnell power spectrum centered at ksp (ω) with θmax = 15◦ and δ = 28.3 nm. The dielectric constants are ε(ω) = −56.25 + i0.60 and ε(2ω) = −11.56 + 0.37. The numerical simulation results represent the average of Np = 2000 realizations of the surface. The angle of incidence is θ0 = 8◦ and the fundamental wavelength is λ = 1.064 µm. The vertical lines indicate the normal and backscattering directions.

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a relatively intense field on the surface. For this surface, the main mechanisms for the generation of the second-harmonic field are the nonlinear mixing of a polariton and the incident field (fig. 27(d)), and the nonlinear mixing of two polaritons at the fundamental frequency, traveling in the same (fig. 27(c)) or opposite directions. The parallel components of the wavevectors involved in the excitation and deexcitation of the fundamental and harmonic surface polaritons satisfy well-known momentum conservation laws. Using these rules, it is not difficult to determine where the second-harmonic scattered light should appear. For instance, the second harmonic scattered light arising from the nonlinear mixing of the incident field and a polariton satisfies the condition (Deck and Grygier [1984]) kinc (ω) ± ksp (ω) = ksc (2ω),

(6.27)

where kinc (ω) = (ω/c) sin θ0 and ksc (ω) = (ω/c) sin θs . So, for θ0 = 8◦ , we expect a relatively large second harmonic intensity along θs = −26◦ and 35◦ . Bands of light in those positions are evident in fig. 29. The mechanism that produces these bands is illustrated in fig. 27(d). Two fundamental polaritons traveling in the same direction produce, through nonlinear mixing, a second-harmonic polariton traveling in the same direction (see fig. 27(c)). This 2ω-polariton can be converted into volume waves due to interaction with the roughness. This mechanism produces the two broad distributions around ±(30◦–40◦ ). For a flat surface, two fundamental polaritons traveling in opposite directions produce no second-harmonic far-field radiation. However, it has been argued (McGurn, Leskova and Agranovich [1991]) that for rough surfaces they could produce features along the normal direction to the surface. In this respect, we point out that in the second-harmonic distribution of fig. 29 there are no peaks in the backscattering and normal directions, but there are hints of dips in those directions. The results of fig. 29 can be compared directly with the experimental data of fig. 8 of the paper by O’Donnell, Torre and West [1997]. The main features are reproduced. The power levels are nearly the same (the theoretical curve is higher by less than a factor of two) and there are only subtle differences between the distributions. The third kind of surface we consider is a surface with a West–O’Donnell power spectrum centered at ksp (2ω) with θmax = 12.2 ◦ and δ = 11.1 nm. Again, these correspond to the parameters of one of the surfaces employed by O’Donnell, Torre and West [1996]. Computer simulation results for the mean normalized second harmonic scattering intensity for this surface, illuminated at normal incidence, are shown in fig. 30. The spatial frequencies for this surface are too high to provide, for near normal incidence, a coupling mechanism into fundamental surface polaritons. However, due to the finite length of the surface, the edges can serve

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Multiple scattering of light from randomly rough surfaces

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Fig. 30. The diffuse component of the mean normalized second harmonic intensity as a function of the scattering angle θs for the scattering of p-polarized light from a random silver surface with a West–O’Donnell power spectrum centered at ksp (2ω) with θmax = 12.2◦ and δ = 11.1 nm. The dielectric constants are ε(ω) = −56.25 + i0.60 and ε(2ω) = −11.56 + 0.37. The numerical simulation results represent the average of results obtained for Np = 2000 realizations of the surface. The angle of incidence is θ0 = 0◦ and the fundamental wavelength is λ = 1.064 µm. The vertical lines indicate the normal and backscattering directions.

as coupling points. Thus, the narrow peaks around 30◦ are due to the nonlinear mixing of the incident field with the polaritons excited at the ends of the surface. The surface is designed in such a way that it permits, through the nonlinearities, the direct excitation of second harmonic polaritons on the surface (see fig. 27(e)), traveling both to the left and to the right. These polaritons can couple out of the surface through interaction with the roughness, producing the broad central region around |θs |
θmax . The most striking result in this figure is the presence of a minimum in the backscattering direction. In linear scattering, the backscattering peak is due to the constructive interference between the scattering contributions arising from the right-traveling and left-traveling polaritons. In the nonlinear case, and by similarity with the physical mechanism responsible for the backscattering dip of fig. 28, it is reasonable to assume that there is also a phase difference between these two contributions, producing thus the destructive interference. The features observed in fig. 30 are in agreement with those present in the experimental data of O’Donnell, Torre and West [1996]. It should be mentioned, however, that the power contained in the theoretical distribution is roughly a factor of six higher than that of the experimental data. Moreover, in the numerical work, the backscattering dip can become a peak for larger angles of incidence and the amount of scattered power increases rapidly with the angle of incidence. On the

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other hand, no peaks were observed in the experiments, and the power level is fairly constant for small angles of incidence. We mention that, in all cases, the experimental data can be fitted very well by the simulation if we choose suitable values for the nonlinear susceptibilities. The quantitative differences between the experimental and theoretical sets of results, and the fact that the disagreement depends strongly on the geometry and the nature of the scattering mechanisms, highlights the need for more realistic models of the nonlinear polarization and for reliable experimental data in a variety of situations.

§ 7. Near-field effects: localization phenomena for surface waves In Sections 4 and 5 we discussed how surface waves (called surface plasmon polaritons on metallic surfaces) stimulate coherent phenomena in the multiple scattering of light from rough surfaces. In the case of light scattering, surface plasmon polaritons are excited as intermediate waves that can undergo multiple scattering. One might also ask what happens if the incident wave is a surface plasmon polariton, instead of a beam of light. The enhanced backscattering of light suggests that a similar, weak localization effect should be present for surface plasmon polaritons themselves. Furthermore, one might expect to find conditions under which a strong (Anderson [1958]) localization of surface plasmon polaritons occurs. Anderson localization is an interference phenomenon, first predicted for electrons (quantum waves), that occurs when the wave becomes trapped in a random potential, and the transmission coefficient through a disordered system decreases exponentially T ∝ e−L/ lloc with the length L of the system, where lloc is the localization length (Sheng [1995]). The possibility of localization phenomena for surface plasmon polaritons on a rough surface was first explored by Arya, Su and Birman [1985] with the aid of the self-energy perturbation theory in the context of strong localization. However, for some time the issue had seemed non-practical since the intensity of surface plasmon polaritons is localized within a few wavelengths from the surface and, therefore, very difficult to measure directly. Very recently, such measurements have become possible with the development of NSOM – near-field scanning optical microscopy (see, e.g., a review by Greffet and Carminati [1997]). Since the intensity of a surface plasmon polariton is concentrated within a few wavelengths from the surface, this excitation can be considered as a powerful probe of surface properties. Furthermore, it turns out that the study of the localization of surface plasmon polaritons by surface roughness opens up opportunities unavailable in typical localization experiments.

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The unusual property of a surface plasmon polariton is that it is a twodimensional wavelike excitation whose field and intensity extend in three dimensions. The third (x3 ) dimension gives an opportunity to monitor wave localization by a random potential in situ. This represents a striking contrast with the conventional localization problem where a classical wave or an electron propagating through a disordered slab can be studied experimentally only through analyzing the transmission coefficient (conductivity) but whose intensity cannot be accessed along its propagation path. Extensive experimental near-field investigations of the localization of surface plasmon polaritons on randomly rough surfaces of gold were performed by Bozhevolnyi and his co-workers (Bozhevolnyi, Smolyaninov and Zayats [1994], Bozhevolnyi, Vohnsen, Smolyaninov and Zayats [1995], Bozhevolnyi [1996]). The remarkable feature of these experiments is the possibility of probing both the surface topography and the near-field intensity. By exciting a surface plasmon polariton on a rough metal surface, Bozhevolnyi and his co-workers explored different regimes of strong and weak scattering. In the regime of strong scattering, the near-field image consists of bright spots that indicate the spatial localization of electromagnetic intensity, associated by the authors with Anderson localization (fig. 31). In the regime of weaker scattering, the images suggest the existence of preferential backscattering that could be attributed to weak localization, but this evidence appears to be inconclusive since the backscattering enhancement in this case could also be attributed to the single-scattering component and thus could have no relation to localization phenomena (Bozhevolnyi [1996]). Less direct experimental evidence of the multiple-scattering regime for surface plasmon polaritons on rough Ag and Au substrates was found by Wang, Feldstein and Scherer [1996], who studied the ultrafast excitation and decay of surface waves and fitted their dynamics data by the diffusion model. Their conclusion regarding the presence of a localization regime seems to require further proof. Overall, the existing experimental data undoubtedly show the presence of interferent multiplescattering phenomena for surface plasmon polaritons on random surfaces, but unambiguous conclusions regarding localization effects cannot be reached yet since alternative interpretations of experimental data are not ruled out. Theoretical investigations of localization effects for surface plasmon polaritons have been developing rather actively in recent years, with a primary focus on a strong localization. By analyzing the poles of the scattering matrix in the complex frequency plane, Maystre and Saillard [1994] found that a surface plasmon polariton becomes trapped within certain regions on the surface at frequencies corresponding to those poles.

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Fig. 31. Grayscale near-field 4 × 4 µm2 optical images due to a surface plasmon polariton excited on a randomly rough gold surface (Bozhevolnyi, Vohnsen, Smolyaninov and Zayats [1995]).

Several theoretical papers studied the strong localization of surface waves on a rough surface that is periodic on average. The motivation for the periodic geometry is twofold. First, it was argued (John [1987]) that the Ioffe–Regel criterion kls < 1 for strong localization, where k is the wave number and ls is the scattering mean free path, is easier to satisfy for frequencies near the edges of gaps that open up in the spectra of surface plasmon polaritons due to the periodicity of the surface. The criterion, which means that the wave can no longer be traveling since its mean free path becomes shorter than its wavelength, is easier to satisfy because the wave number k is replaced by the “crystal momentum” kcryst which vanishes at the Brillouin zone boundary. Second, the attenuation of a surface wave near the band gap edge due to its conversion into other surface and volume waves is suppressed. Thus, the suppression of its traveling nature cannot be attributed to leakage, i.e. to the transformation of the surface plasmon polariton into volume electromagnetic waves, rather than to localization. Using this periodic-on-average geometry, Saillard [1993] carried out numerical experiments that showed the possibility of localization of the surface wave field in a portion of the rough surface. The conclusion was further supported by Pincemin and Greffet [1996] who pre-

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sented a detailed discussion of the strong localization phenomenon at frequencies near the band gap edge. The same geometry was studied by Sánchez-Gil and Maradudin [1997] by analytical and numerical calculations. They considered one of the central points in the theory of strong localization for surface plasmon polaritons – the competition between leakage and localization. A number of cases considered in that work show that there are regimes when localization predominates over leakage. All of these calculations were done within the one-dimensional roughness model. The only two-dimensional calculation appears to have been done in the work by Shchegrov [1998a], who applied the self-energy perturbation theory based on the two-dimensional version of the reduced Rayleigh equation in the weak roughness limit. The work demonstrated weak localization phenomena for surface waves that are manifested in an enhanced backscattering peak (fig. 32). The active investigation of the localization of surface plasmon polaritons by surface roughness continues and the practical possibility of strong localization still seems to remain open – the proof will require at least matching the conditions

Fig. 32. Polar plot showing the intensity of scattered surface polaritons due to an incident plane-wave surface polariton propagating on a rough surface of Ga0.18 Al0.82 As with δ = 0.1 µm and a = 8.0 µm. The vacuum wavelength λ0 = 2π c/ω = 25.6 µm. The dashed line shows the second-order contribution in δ only, while the solid line gives the sum of the second-order and fourth-order contributions in δ calculated with the self-energy perturbation theory (Shchegrov [1998a]).

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of a real experiment (two-dimensional roughness, surfaces that are non-periodic on average) and theoretical calculations. The localization effects for surface waves do not exhaust the list of novel near-field topics in multiple scattering from randomly rough surfaces that have appeared in recent years. Other topics include the spatial reconstruction of a surface profile from near-field intensity data (Madrazo and Nieto-Vesperinas [1997a], Pascual, Zierau, Leskova and Maradudin [1998]), the influence of objects placed in the near-field (subwavelength) zone of the rough surface on the far-field intensity pattern (Shchegrov and Maradudin [1997], Madrazo and NietoVesperinas [1997b]), the detection of such objects (Zhang, Tsang and Kuga [1997], Shchegrov and Maradudin [1997], Zhang, Tsang and Pak [1998], Gu [1998]), etc. We believe that the most important breakthroughs in these areas are yet to come, especially due to the remarkable development of experimental techniques and tools, and leave these topics out of the limited space of this review, referring the reader to the papers cited above and the review article by Greffet and Carminati [1997].

§ 8. Spectral changes induced by multiple scattering An underlying assumption in optical spectroscopy is that the spectrum of light is invariant upon propagation. This assumption remained unchallenged until a few years ago when E. Wolf, in a series of papers (Wolf [1986, 1987a, 1987b, 1987c]), predicted that the far-field spectrum of the light intensity emitted by a stationary source could appear to be red-shifted or blue-shifted with respect to that of the source. These spectral changes, caused by the correlation properties of the source fluctuations, are now known as the Wolf effect. Because of the close analogy between radiation and scattering, this prediction stimulated investigations of the spectral changes of the light scattered by volume disordered systems (Wolf, Foley and Gori [1989], Wolf and Foley [1989], Foley and Wolf [1989], Wolf [1989], James and Wolf [1990]; see also the review by Wolf and James [1996]). There are many ways in which the spectrum of the light may change when it interacts with a stationary linear system. Absorption, propagation through dispersive media, and wavelength-dependent single-scattering properties (like Rayleigh scattering) can cause such changes. More interesting, however, are cases in which this wavelength dependence is due to a collective property of a random system, such as the multiple-scattering processes that give rise to the enhanced backscattering phenomenon. This section is devoted to a brief review of this topic. The first treatment of spectral changes of light scattered by volume disordered media in which multiple-scattering effects were taken into account was given by

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[2, § 8

Lagendijk [1990a, 1990b], who pointed out that the enhanced backscattering of light from a strongly scattering random medium, which is due to the coherent interference between multiply-scattered optical paths and their reciprocal partners, can be regarded as being due to the re-emission of light from an extended source that possesses just the type of source correlation needed to produce a red shift of the spectrum of the scattered light. To illustrate matters, let us consider a polychromatic field that is incident on a scattering system. We write the statistically stationary optical field as a superposition of linearly polarized, monochromatic plane waves  ∞ dω inc U (x1 , x3 |ω) exp(−iωt), U inc (x1 , x3 |t) = (8.1) −∞ 2π where U inc (x1 , x3 |ω) represents a member of an ensemble of monochromatic fields. We further assume that each member of the ensemble can be written as   U inc (x1 , x3 |ω) = F (ω) exp ikx1 − iα0 (k, ω)x3 , (8.2) where F (ω) represents a zero-mean complex random process with the property ∗

F (ω)F (ω ) F = 2πδ(ω − ω )S0 (ω). (8.3) In eq. (8.3) S0 (ω) is the spectral density of the incident field, and the average is taken over an ensemble of monochromatic wavefields {U inc (x1 , x3 |ω) exp(−iωt)}. The field scattered by the surface may be represented by another ensemble, U sc (x1 , x3 |ω), of space-frequency realizations which, in terms of the scattering amplitude that results from illuminating the surface with a unit amplitude plane wave, R(q|k|ω), can be written as  ∞   dq sc R(q|k|ω) exp iqx1 + iα0 (q, ω)x3 . U (x1 , x3 |ω) = F (ω) (8.4) −∞ 2π Let



W sc (x1 , x3 , x1 , x3 , ω) = U sc∗ (x1 , x3 , ω)U sc (x1 , x3 , ω) F

(8.5)

be the cross-spectral density (Mandel and Wolf [1995], p. 62) of the reflected field emerging from the surface, where the outer brackets denote an average over an ensemble of realizations of the surface. The spectral density can be found by setting x1 = x1 and x3 = x3 . Using eq. (8.4), the spectrum of the scattered light can be written as  ∞  dq ∞ dq  sc R(q|q  |ω) S (x1 , x3 , ω) = S0 (ω) −∞ 2π −∞ 2π     × exp −i(q − q  )x1 − i α0∗ (q, ω) − α0 (q  , ω) x3 , (8.6)

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where

R(q|q  |ω) = R ∗ (q|k|ω)R(q |k|ω) ,

(8.7)

may be recognized as the angular amplitude correlation function of the scattered field. This quantity is related to the speckle size and depends on the size of the illuminated portion of the surface. For a long (L  λ) or infinite surface we can use the approximation 2

 R(q|q  |ω) = 2πδ(q − q  ) R(q  |k|ω) . (8.8) Then, the spectral density of the far-field scattered light S sc (ω) can be written as  ∞ dq sc S (q, ω), S sc (ω) = (8.9) −∞ 2π where the angle-resolved far field spectral density is given by 2

 S sc (q, ω) = S0 (ω) R(q|k|ω) .

(8.10)

This equation expresses, in abbreviated form, the influence of a static scattering system on the spectrum of the far-field scattered light, providing also a simple interpretation of the effect. The spectral changes are due to the different weights introduced by the process of scattering and propagation on the spectral components provided by the source. Notice that the properties of the scatterer enter only in a multiplicative way [even in eq. (8.6)]. From this perspective, if the spectral changes are viewed as shifts, these are restricted in magnitude to values that are of the order of or smaller than the effective widths of the spectral lines. In other words, the shift of a spectral line cannot be larger than its width. It is also clear that the spectrum of a monochromatic wave would remain invariant. In order to obtain changes in the spectrum of light scattered from a random medium that are large enough to be measured, it is necessary for the intensity of the monochromatic light scattered from the medium to possess features that depend strongly on the frequency of the incident light. The width of the enhanced backscattering peak is wavelength-dependent, but the magnitude of the red shifts calculated by Lagendijk for scattering angles close to the backscattering direction are small, e.g., δω/ω0 ≈ 10−5 , where ω0 is the central frequency of the incident light, for a linewidth ω/ω0 ≈ 10−2 . These spectral shifts have not been observed in optical experiments up to the present time. In more recent studies of spectral changes of the light scattered from random media, attention has been directed at structures that possess features that depend strongly on the wavelength. The scattering systems that give rise to satellite peaks, reviewed in Section 6.1, possess this property.

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We discuss first the case of a rough waveguide that supports more than one mode, illuminated from the outside (Leskova, Maradudin, Shchegrov and Méndez [1997]). The system consists of a ZnS film of mean thickness d = 475 nm, deposited on the planar surface of a perfect conductor. The dielectric constant of the film was assumed to be ε = 5.6644 + i0.005 and frequency-independent in the range of interest. The vacuum-ZnS interface is a one-dimensional randomly rough surface. At normal incidence and the monochromatic wavelength λ = 632.8 nm, this structure supports four guided waves that give rise to eight satellite peaks, at θs = ±10.84 ◦, ±25.34◦, ±38.02 ◦, ±44.27 ◦ in the angular dependence of the intensity of the scattered light (Sánchez-Gil, Maradudin, Lu, Freilikher, Pustilnik and Yurkevich [1994, 1996]) in addition to an enhanced backscattering peak at θs = 0◦ . The scattering angles at which these peaks occur depend strongly on the frequency of the incident light. In the work of Leskova, Maradudin, Shchegrov and Méndez [1997], the incident spectrum was of Gaussian form with central frequency ω0 and a 1/e halfwidth ω. It was assumed that ω/ω0 = 0.05 and that the central wavelength λ0 = 2πc/ω0 = 632.8 nm. The spectrum of the scattered light was then found to be given by a somewhat distorted Gaussian centered at a frequency ωm (θs ), where θs was the angle of scattering. The relative frequency shift δω/ω0 (with δω = ωm (θs ) − ω0 ) was calculated as a function of θs . This quantity is plotted in fig. 33 as a function of θs , for s-polarized light incident from vacuum on the rough film. The magnitude of the relative shift in the position of the maximum of the spectrum of the scattered light as a function of the scattering angle θs , |δω|/ω0 , could reach values as large as 0.012 for θs in the vicinity of the angle at which a satellite peak occurs. For narrower bandwidths, corresponding to laser sources, the shifts were smaller, e.g., at θs = 0 ◦ , δω/ω0 = −4.4 × 10−4 for ω/ω0 = 5 × 10−3 . Even so, the Wolf shifts were still about two orders of magnitude larger than those predicted for disordered volume scattering for the same value of ω/ω0 . Mainly due to the difficulties in obtaining or fabricating structures with the desired parameters, at present, these effects have not been observed with this kind of system. Another system that gives rise to satellite peaks is a modified version of the double passage scattering configuration discussed in Section 6.1 (see also Méndez, Chaikina and Escamilla [1999], and Escamilla, Méndez and Chaikina [2001]). The arrangement employs a piece of birefringent material (calcite) in the space between the random phase screen and the mirror. The birefringence of the calcite prism leads to propagation paths that are different for waves polarized along two orthogonal directions, and the inclusion of a linear polarizer mixes these two

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Fig. 33. The apparent shift (ωm (θs ) − ω0 )/ω0 as a function of the scattering angle θs , when p-polarized light of central wavelength λ0 = 632.8 nm is incident normally on a rough ZnS film of mean thickness d = 475 nm deposited on a perfectly conducting substrate (Leskova, Maradudin, Shchegrov and Méndez [1997]).

components of the polarization allowing the interference between waves that have followed nonreciprocal paths. The mean intensity reaching the far field is given by eq. (6.7). The term multiplied by the cosine produces satellite features that can be either peaks or dips. The positions of these satellite features are fairly insensitive to the wavelength, but their amplitudes have a strong dependence on it. In the direction of this feature [sin θs = − sin θ0 ∓ (/2D)] we have that    ω 2 sc S (ω) ∝ S0 (ω) 1 + cos (8.11) , c 4D where  represents the lateral shift introduced by the calcite prism and D is the distance from the phase screen to the diffuser. Calculations corresponding to illumination of the described system with a superluminescent diode are shown in fig. 34 for the case of normal incidence and three different distances D between the random phase screen and the mirror: D = 23.10 cm, D = 22.57 cm, and D = 22.10 cm. The spectrum is approximately Gaussian with a central wavelength λ0 = 0.850 µm and spectral width ω/ω0 = 1.88 × 10−2 . The lateral shift introduced by the prism is  = 2.7 mm. The normalized spectrum of the incident light is denoted by the curves with circles in the figure. The spectrum of the scattered light in the direction in which the

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[2, § 8

Fig. 34. Normalized spectral densities of the incident and scattered light for three different distances D between the random phase screen and the mirror for the case of normal incidence. The incident spectrum corresponds to the curve with the white circles, and the normalized spectrum in the direction θs = sin−1 (/2D) corresponds to the curve with the black squares (Chaikina, Méndez and Escamilla [2001]).

satellite peaks and dips occur is denoted by the curves with black squares. From the figure it is clear that in this direction the normalized spectrum can undergo important modifications. It can appear to be shifted towards lower or higher frequencies, and situations in which the spectral line appears to split can also be found. In contrast, we point out that in the backscattering direction, and in most other directions, the spectrum of the scattered light practically coincides with that of the incident light. For the spectral width of the illumination employed, the magnitude of the apparent shift can be larger than δω/ω0 = 0.01, making the experimental observation of the effect a practical proposition. Recently, spectral changes have been observed with this configuration (Chaikina, Méndez and Escamilla [2001]).

2, § 8]

Spectral changes induced by multiple scattering

227

In view of the fact that only sources with certain spectral coherence can give rise to radiated light whose spectrum is invariant as it propagates away from the source (Wolf [1986]), it was expected that the spectrum of scattered light measured at any distance from the scattering system should differ from that of the source. In particular, the spectrum of light scattered from a randomly rough surface, measured in the near-field region, i.e. at a sub-wavelength distance from the surface, should be an interesting object of study, because it includes contributions from the evanescent, non-radiative components of the scattered field, which never reach the far-field region, as well as from the radiative components, which do. In addition, the near-field amplitude contains contributions from both the incident and the scattered fields. The total field above the surface can be written as the sum of the incident and scattered fields. We then write U > (x1 , x3 |ω) = U inc (x1 , x3 |ω) + U sc (x1 , x3 |ω),

(8.12)

which leads to a spectrum of the form S > (x1 , x3 , ω) = S inc (x1 , x3 , ω) + S sc (x1 , x3 , ω)

  + 2 Re U inc (x1 , x3 |ω)U sc∗ (x1 , x3 , ω) F .

(8.13)

The behavior of the spectrum of the near-field scattered light (i.e. the term S sc (x1 , x3 , ω)) has been studied by Shchegrov and Maradudin [1999] for a scattering system that consists of a rough BaSO4 film deposited on the planar surface of a perfect conductor. This structure supports two guided waves, which in the far field give rise to satellite peaks at θs = ±25◦ , in addition to an enhanced backscattering peak at θs = 0 ◦ . In fig. 35 we reproduce their results for the evolution of the relative spectral shift on propagation of the scattered light from the near to the far field. The normally incident light is s-polarized and its central wavelength is λ0 = 2πc/ω0 = 632.8 nm. In this figure x is the radial distance from the illuminated spot on the random surface to the point of observation. This figure shows the relative position of the spectral maximum ωm (θs ) as a function of the angle of observation θs , at different radii x. It is seen that for any value of x the spectrum is red-shifted, but the angular dependence changes considerably with increasing x. No coherent phenomena (enhanced backscattering, satellite peaks) are reflected in the dependence of (ωm − ω0 )/ω0 on θs ; they form only in the far field. In fig. 35(f), which corresponds to x = 50λ0 , strong shifts associated with the satellite peaks are seen in the vicinity of θs = ±25◦ , and features associated with single-scattering processes are present near θs = ±55◦ . For the parameters used in these calculations, even in the near field, relative Wolf shifts as large as 0.014 are obtained, and they become as large as 0.05 in the far field.

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Multiple scattering of light from randomly rough surfaces

[2, § 8

Fig. 35. Evolution of the relative spectral changes (ωm (θs ) − ω0 )/ω0 on propagation from the near to the far field. (a) x = 0.1λ0 ; (b) x = 0.3λ0 ; (c) x = 3λ0 ; (d) x = 6λ0 ; (e) x = 15λ0 ; (f) x = 50λ0 (Shchegrov and Maradudin [1999]).

On the other hand, it seems that the interference term of eq. (8.13) has not been given much consideration. In regions that are a few wavelengths away from the surface it can cause important modifications of the incident spectrum. To illustrate this, let us consider the reflection from a flat perfectly conducting surface in s polarization. We have that Rs (q|k|ω) = −2πδ(q − k), which leads to the following expression for the total spectrum: 

 ω S > (x3 , ω) = 2S0 (ω) 1 − cos 2 x3 cos θ0 . c

(8.14)

(8.15)

Thus, in the neighborhood of the surface the spectrum is a function of x3 and, in general, can be quite different from the spectrum of the incident light. This is simply due to the interference between the incident and scattered fields that produces the well known Wiener fringes (Born and Wolf [1999], pp. 311–313). The situation is similar to that studied by Wolf [1987b] in which he considered the apparent red and blue shifts produced by two correlated sources.

2, § 9]

Conclusions

229

Spectral changes in the polychromatic light scattered from a disordered birefringent medium have also been investigated theoretically within the first Born approximation (Ponomarenko and Shchegrov [1999]). The authors find changes in the spectrum that are due to the combined action of disorder and anisotropy. From the results reviewed here we conclude that observable changes in the spectrum can occur at scattering angles in the vicinity of multiple-scattering features in the angular dependence of the scattered intensity. The changes are larger for systems whose angular scattering pattern contains features that depend strongly on the wavelength of the illumination. These spectral effects can also be large in the near field of the random surface. § 9. Conclusions In concluding this review we would like to indicate several directions in which studies of multiple-scattering effects in rough surface scattering could go in the next several years. The scattering of light from two-dimensional random metallic and dielectric surfaces poses challenging computational problems for theorists. The development of techniques for solving this problem accurately and rapidly would represent a significant advance in this field. The original formally rigorous approaches to the scattering of a scalar wave from a two-dimensional Dirichlet or Neumann surface (Tran and Maradudin [1992, 1993], Macaskill and Kachoyan [1993]), and to the scattering of electromagnetic waves from two-dimensional perfectly conducting and metallic surfaces (Tran, Celli and Maradudin [1994], Tran and Maradudin [1994]) are time consuming. Recent work on this problem has proceeded in two directions. The first is the exact solution of the integral equations of scattering theory by numerical methods that are faster than a straightforward application of the method of moments followed by an iterative solution of the resulting matrix equation, an O(N 2 ) approach, where N is the number of unknowns to be determined. For example, Wagner, Song and Chew [1997] have developed a fast multipole fast Fourier transform method to calculate the scattering of an electromagnetic wave from a small height, two-dimensional, randomly rough, perfectly conducting surface that is an O(N log N) method. For surfaces with larger roughness they have shown that another O(N log N) approach, the multilevel fast multipole algorithm is more efficient. These approaches have yet to be applied to scattering from metallic and dielectric surfaces. The other direction that has been taken is the approximate solution of the exact integral equations. In the sparsematrix canonical grid method of Tsang, Chan and Pak [1993, 1994] the matrix elements connecting two close points on the surface are treated exactly, while

230

Multiple scattering of light from randomly rough surfaces

[2, § 9

those connecting two distant points are treated approximately, in an iterative solution of the matrix equations obtained by the method of moments. Initially applied to the scattering of a scalar wave from a two-dimensional random surface (Tsang, Chan and Pak [1993, 1994]), it has subsequently been applied to the study of electromagnetic wave scattering from a two-dimensional, randomly rough, perfectly conducting surface (Pak, Tsang, Chan and Johnson [1995], Johnson, Tsang, Shin, Pak, Chan, Ishimaru and Kuga [1996]), and more recently to the scattering of an electromagnetic wave from a two-dimensional, randomly rough, dielectric surface (Pak, Tsang and Johnson [1997]). This approach has been elaborated and made faster by Johnson and his colleagues, resulting in O(N) methods in some cases, see, e.g., Torrungrueng, Chou and Johnson [2000] where scattering from a twodimensional, randomly rough, perfectly conducting surface is calculated. Despite these advances, the calculation of the scattering of electromagnetic waves from two-dimensional random surfaces, especially those bounding penetrable media, remains a computationally intensive problem in need of further improvements. A good recent survey of numerical simulation methods used in rough surface scattering, including scattering from two-dimensional random surfaces, can be found in the review by Warnick and Chew [2001]. The problem of grazing angle scattering of electromagnetic waves from oneand two-dimensional random surfaces still awaits a satisfactory solution. This is an important problem, especially but not exclusively, in the context of the remote sensing of the ocean’s surface. The difficulty with the standard computer simulation approach to grazing angle scattering is that if the width of the incident beam is 2w, its intercept with the mean scattering plane is 2g = 2w/ cos θ0 , where θ0 is the angle of incidence, and as a result grows significantly as θ0 approaches 90 ◦ . Thus, for example, if θ0 is 85◦ , 2g = 11.47(2w). Since the length L of the x1 axis covered by the random surface in simulations for one-dimensional surfaces is of the order of 5g, we find that L = 28.68(2w). The value of w used is typically 10λ (and it is much larger in experiments), so that the value of L becomes 574λ. If a sampling interval x = λ/10 is used, 5740 abscissas must be used in converting the integral equations for the source functions into matrix equations. For a penetrable surface a pair of equations must be solved, and this leads to systems of 104 equations in the same number of unknowns. This gives rise to significant computational demands at the present time if rigorous solutions are desired. Several approximate theoretical and computational schemes have been developed in recent years to meet these demands. These include several of the approaches mentioned earlier in this review, such as small-amplitude perturbation theory, the small-slope approximation, the second-order Kirchhoff approximation with shadowing added, the use of an impedance boundary condition, the use of an

2, § 9]

Conclusions

231

approximate Green’s function for a system consisting of vacuum separated from the scattering medium by a rough surface, and the use of Monte Carlo simulations based on an iterative method of moments in which the matrices arising in the solution of the integral equations of scattering theory are decomposed into near- and far-field interactions. However, it seems fair to say that not all of the features observed in experimental data have been explained by the results of these schemes, especially in the case of two-dimensional random surfaces. An accurate picture of where the state of the art in this field currently resides can be found in the special issue of the IEEE Transactions on Antennas and Propagation devoted to low-grazing-angle backscatter from rough surfaces (Brown [1998]). Another area that needs to be developed further is that of the scattering of electromagnetic waves from the random surface of an inhomogeneous medium. An example of a system to which such a theory would be applicable is a polycrystalline sample of a metal such as beryllium. The optical properties of a beryllium crystallite are anisotropic, and the orientations of the crystallites vary randomly across a sample. In the earliest efforts to deal with such systems (Elson [1984, 1988, 1997], Elson, Bennett and Stover [1993]), single-scattering approximations were used in which the effects of the surface roughness and of the dielectric inhomogeneities contributed independently to the scattered field, and to the intensity of the scattered field. A numerical simulation approach was applied by Pak, Tsang, Li and Chan [1993] to a study of the scattering of light from an inhomogeneous dielectric medium bounded by a one-dimensional random surface. In these calculations multiple scattering was taken into account. The scattering of a scalar wave from a two-dimensional random surface bounding an inhomogeneous dielectric medium was studied by the techniques of many-body perturbation theory and the use of an impedance boundary condition by Mudaliar and Lee [1995]; however, no results for the mean differential reflection coefficient were presented in this work. More recently Mudaliar has studied perturbatively the specular (Mudaliar [1999]) and diffuse (Mudaliar [2000]) scattering of a scalar wave from a layer of a random medium bounded by a random surface. However, these treatments are purely formal, and no numerical calculations are based on them. A rigorous theory of the scattering of electromagnetic waves from a two-dimensional random surface bounding an inhomogeneous medium that takes into account the vector nature of the scattering problem and can yield numerical results is still lacking. An important special case of an inhomogeneous random medium bounded by a random surface is one in which the dielectric inhomogeneity is due to the presence of a strong reflector, for example a metal plate or cylinder in an otherwise homogeneous (or even inhomogeneous) substrate. The ability to detect such strong

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Multiple scattering of light from randomly rough surfaces

[2, § 9

reflectors by ground penetrating radar, especially if one could also characterize their shape and dielectric properties, would have many practical applications. Much of the work done to date in the area of rough surface scattering has been devoted to the direct problem, namely given the statistical properties characterizing a randomly rough surface, to calculate the angular and polarization dependence of the intensity of the scattered field, or higher moments of the intensity. The inverse problem, namely given scattering data, to reconstruct the surface profile, or even just such statistical properties of it as its power spectrum, or in some cases just its rms height, has been less intensively studied until recently, and deserves to be more widely pursued. We have not discussed inverse problems associated with the reconstruction of surface profiles in this article because, for the most part, they are based on single-scattering theories, both in the far field (Wombell and De Santo [1991a, 1991b], Quartel and Sheppard [1996a, 1996b]) and in the near field (Greffet and Carminati [1997], García and Nieto-Vesperinas [1993]), and thus fall outside the scope of this article. A method for reconstructing the surface profile function from far-field amplitude data, that can tolerate some multiple scattering has been presented by Macías, Méndez and Ruiz-Cortés [2000, 2002]. Surface profile reconstructions based on evolutionary strategies, using farfield intensity information and a multiple-scattering theory, have been reported by Macías, Olague and Méndez [2002, 2003]. An approach to the reconstruction of the power spectrum of a one-dimensional random surface that is based on a multiple-scattering theory has been presented by Malyshkin, Simeonov, McGurn and Maradudin [1997], but it has yet to be applied to the reconstruction of the power spectrum of a two-dimensional random surface. A version of the inverse problem that seems to us to deserve more study is the design of random surfaces with specified scattering properties. There are many practical situations in which it would be desirable to have optical diffusers whose light scattering properties could be controlled. In recent work it has been shown how to design one-dimensional random surfaces that act as band-limited uniform diffusers, that is surfaces that scatter light uniformly within a specified range of scattering angles, and produce no scattering outside this range (Leskova, Maradudin, Novikov, Shchegrov and Méndez [1998], Méndez, Martinez-Niconoff, Maradudin and Leskova [1998], Méndez, García-Guerrero, Leskova, Maradudin, Muñoz-López and Simonsen [2002a, 2002b]), and such surfaces have now been fabricated (Chaikina, García-Guerrero, Gu, Leskova, Maradudin, Méndez and Shchegrov [2000]). It has also been shown how to design one-dimensional random surfaces that act as band-limited uniform diffusers in transmission (Leskova, Maradudin, Méndez and Simonsen [2000], Maradudin, Méndez, Leskova and Simonsen [2001]), surfaces that suppress single-scattering

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processes (Maradudin, Simonsen, Leskova and Méndez [1999], Méndez, GarcíaGuerrero, Leskova, Maradudin, Muñoz-López and Simonsen [2002b]), and surfaces that act as one-dimensional Lambertian diffusers (Maradudin, Simonsen, Leskova and Méndez [2001], Méndez, García-Guerrero, Leskova, Maradudin, Muñoz-López and Simonsen [2002b]). Two-dimensional random surfaces have been designed and fabricated that scatter light uniformly within a rectangular region of scattering angles and produce no scattering outside this range (Méndez, García-Guerrero, Escamilla, Maradudin, Leskova and Shchegrov [2001]), that act as band-limited uniform diffusers within a circular region of scattering angles (Maradudin, Méndez and Leskova [2002a, 2002b]), and that act as twodimensional Lambertian diffusers (Maradudin, Méndez and Leskova [2002a]). The design of such one- and two-dimensional surfaces has been based on approximations such as the Kirchhoff approximation and phase perturbation theory. What is needed is an approach that can be applied to the design of two-dimensional random surfaces that possess specified scattering properties within more general regions of scattering angles.

Acknowledgements A.V. Shchegrov would like to express his gratitude to Professor E. Wolf and his research group for their hospitality during the author’s stay at the University of Rochester where a large part of this review was written. The research of A.A. Maradudin and E.R. Méndez was supported in part by Army Research Office grant DAAD 19-02-1-0256. They are also grateful to T.A. Leskova for discussions concerning various aspects of this review. E.R. Méndez is also grateful to K.A. O’Donnell, C.I. Valencia, and E.I. Chaikina for their critical reading of some sections of the manuscript.

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E. Wolf, Progress in Optics 46 © 2004 Elsevier B.V. All rights reserved

Chapter 3

Laser-diode interferometry by

Yukihiro Ishii Department of Electronic System Engineering, University of Industrial Technology, Sagamihara, 4-1-1, Hashimotodai, Sagamihara, 229-1196, Japan

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(03)46003-4 243

Contents

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

245

§ 2. Laser-diode operation . . . . . . . . . . . . . . . . . . . . . . . . . .

246

§ 3. Modulation methods in laser-diode interferometers . . . . . . . . . .

250

§ 4. Laser-diode phase-shifting interferometers . . . . . . . . . . . . . .

251

§ 5. Sinusoidal phase-modulating interferometry . . . . . . . . . . . . .

283

§ 6. Feedback interferometry . . . . . . . . . . . . . . . . . . . . . . . .

285

§ 7. Heterodyne interferometry . . . . . . . . . . . . . . . . . . . . . . .

286

§ 8. Optical coherence function synthesized by tunable LD . . . . . . . .

299

§ 9. Holographic interferometer and phase-conjugate interferometer by tunable LD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

301

§ 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

302

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

303

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

303

244

§ 1. Introduction Optical interferometers give extremely accurate measurements and have been used as investigative metrological techniques for almost a hundred years. The great progress in heterodyne and phase-shifting interferometers in the last three decades can be attributed to the development of digital computers and solid state detector sensors as well as to accurate electronic phase-measuring techniques. The sensitivity and speed of phase measurements have increased remarkably and measurement accuracy is of the order of 1/100 wavelength (Creath [1988], Schwider [1990]). Many reviews on this field have been written, for example, by Greivenkamp and Bruning [1992], and Malacara, Servin and Malacara [1998]. Phase modulators are needed for generating the modulated interferograms in the interferometer, such as a translating mirror, a tilting plane-parallel plate, a rotating half-wave plate, a moving grating, an acousto-optic device, and an electrooptic device (Wyant and Shagam [1978]). A piezoelectric transducer (PZT) phase shifter is commonly employed for phase modulation: a piezoelectrically driven mirror is the major translation-type phase modulator although it has the disadvantages of hysteresis, nonlinearities and mechanical vibration. Many attempts have been made to minimize the errors caused by PZT disadvantages since repeated samples can be taken at different phases in a detection system (Bruning, Herriott, Gallagher, Rosenfeld, White and Brangaccio [1974]). Furthermore information from the interferometric test can be used to calibrate the nonlinear motion of the PZT if interferometric data have repeatability. An interesting and potentially significant recent advance is the direct modulation of the output wavelength of laser diodes (LDs) to produce stepwise or continuous phase shifts in optical interferometry (Ishii [1991, 1999]). Laser-diode interferometry is based on the fact that the phase difference in an interferometer is proportional to the product of the optical path difference and the wavelength or frequency change. Laser diodes are characterized by compact size, high efficiency, single-mode operation and frequency tunability: changes in laser current produce frequency modulation by altering the refractive index of the active region and consequently the optical path length of the laser cavity (Kobayashi, Yamamoto, Ito and Kimura [1982]). Thus the LD can be considered to be a simple current245

246

Laser-diode interferometry

[3, § 2

controlled oscillator in the frequency range of a few gigahertz. The capability of direct frequency modulation was applied quite early to a phase-compensation scheme in an interferometric fiber-optic sensor (Dandridge and Tveten [1982]) with a phase-nulling technique (Hotate, Sagehashi and Niwa [1983]) in a fiber interferometer. This article presents a phase-measuring technique in an unbalanced interferometer using direct frequency modulation of LD sources by changing currents. The wavelength is stepwise, sinusoidally or rampwise changed to introduce a timevarying phase difference between the two beams of the interferometer. The operation of the LD and the phase-extracted method from measured interferograms are demonstrated. In particular, a two-wavelength LD interferometer has been elaborately constructed to extend the range of interferometric phase measurement by using a long wavelength synthesized by two wavelengths. Features of the recent development in LD interferometers are described on a feedback interferometer to calibrate the phase shifts and to lock the interferometer on a phase-shift condition by controlling the bias and modulation currents of the LD. The system can eliminate the phase shift error between the target phase shift and the real phase shift. The changes in laser-diode power violate the assumption of constant intensity in the phase-extraction algorithm. An electrically normalizing technique with a photodiode or a phase-extraction algorithm insensitive to LD-power changes has been applied to alleviate this problem. The LD phase-shifting interferometer has also been applied to distance measurements with a wide-tunable laser diode. The frequency-chirped LD interferometer can be employed as an optical radar such as in optical frequency-domain multiplex imaging. Selective extraction of multiplex images can be performed by electronic tuning of multiple interference beat signals. Multiple interference fringes corresponding to image holograms in a heterodyne interferometer can be processed to reconstruct the amplitude and phase of objects. Various measured phase maps together with amplitude maps are shown. Laser-diode interferometry has been applied to holographic interferometry and phase-conjugate interferometry with LD wavelength tuning, keeping interferometric capabilities.

§ 2. Laser-diode operation Many electronics manufacturers have produced index-guided LDs. A selection of LDs suitable for interferometric experiments can be made following the requirement for longitudinal single-mode lasing, yielding a long coherence length. Double-heterostructure LDs with different structures are suitable as light sources

3, § 2]

Laser-diode operation

247

for interferometric use. Laser diodes consisting of a multiple stack of quantum wells (Saleh and Teich [1991]) have been commercially available for optical disk applications, and are suited to interferometric measurement owing to their singlemode operation with low threshold current. Use of a stabilized LD operating in a single mode permits operation of an unbalanced interferometer (see Section 3) at widely differing path lengths between the two arms. An increase in LD current over a threshold current yields a single-mode emission spectrum.

2.1. Single-mode laser diodes The single-frequency feasibilities can be checked by using a high-finesse scanning Fabry–Perot interferometer (Newport Supercavity) with a free spectral range of 6 GHz. The typical power spectrum is shown in fig. 1 for an AlGaAs LD light source (Sharp LT024) with an operating wavelength of λ = 792 nm at a 70-mA current. The linewidth (FWHM) is ∼ 2 MHz measured by a Supercavity (Newport SR-140). The corresponding coherence length is ∼ 10 m, which is adequate to perform interferometric experiments with a path length of several centimeters between reference and object paths. To minimize the effects of temperature on the mode fluctuations, we mounted the LD on a copper plate whose temperature was regulated to 20 ± 0.03 ◦ C by an automatically driven temperature-controlled

Fig. 1. Spectral profile of an LD (λ = 672 nm) measured by a high-finesse Fabry–Perot interferometer.

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Laser-diode interferometry

[3, § 2

(ATC) assembly attached to a Peltier element. An ATC system consists of a temperature transducer attached to the LD which senses the temperature of the LD. The temperature difference from the preset value is transformed into a pulse-width modulated signal whose average voltage corresponds to the laser temperature fluctuations; this signal is fed back to the drive current of a Peltier element. This ATC system (Ishii [1991]) enables the temperature control to operate at a high repetition rate.

2.2. Wavelength tunability in laser diodes The laser wavelength can be modulated directly by injection current because the longitudinal mode is modulated by changing the refractive index and the cavity length. The typical wavelength tunability is indicated as follows. The emitting wavelength of an AlGaAs LD (Hitachi HL7801) is measured by an optical spectrum analyzer attached to a Czerny–Turner grating monochromator with a resolution of 0.1 nm (Anritsu MV02). Figure 2 shows the measured wavelength as a function of the injection current. The operating wavelength is λ = 788 nm at 55mA current. Wavelength tunability is produced by changing the injection current from 46.1 mA to 49.9 mA over the linear range with no appearance of a mode hop.

Fig. 2. Wavelength tunability of the LD as a function of the injection current at a temperature of 20 ◦ C.

3, § 2]

Laser-diode operation

249

Fig. 3. Transistor control of the current injected into an LD to provide stepwise or continuous frequency modulation.

As the current increases, the wavelength becomes longer, the so-called red shift. The current-tuning ratio α is calculated as α = 0.006 nm/mA (β = 3 GHz/mA) from the slope in the linear range of fig. 2. The current-tuning ratios α and β are related by β = cα/λ2 , where c and λ denote the light velocity and the optical wavelength of the light source, respectively. The linear frequency change of this type of LD by changing the current is restricted to ∼ 30 GHz by mode hops. Control of wavelength tunability is made easier by using a combination of an operational amplifier and a transistor. A circuit as shown in fig. 3 is commonly used to drive the LD. The optical frequency ω can be changed by varying current i of the LD. The circuit includes transistor control of the current injected into the LD to provide analog modulation in stepwise or continuous frequency of the emitted light. The performance in the circuit involves both power and wavelength variations by changing LD current. Fluctuations in the intensity of the emitted light may be stabilized by the use of optical feedback in which the emitted light is monitored and used to control the injected current. It needs that an optical feedback of the circuit is off if the frequency modulation is applied to the LD interferometer. Laser diodes with wide tuning ranges of frequency are available for resolvable measurement in depth. A grating-tuned external-cavity laser-diode has a frequency change of ∼ 5 THz and is used as a light source in wavelength-shift speckle interferometry (Barnes, Eiju and Matsuda [1996], Tiziani, Franze and Haible [1997]). A distributed Bragg reflector LD (DBR-LD) is used to perform selective image extractions with a synthesized coherence function for the large frequency change of ∼ 400 GHz (He, Mukohzaka and Hotate [1997]).

250

Laser-diode interferometry

[3, § 3

§ 3. Modulation methods in laser-diode interferometers A laser-diode Twyman–Green interferometer for testing surface profiles, with an unbalanced optical path length l, is shown in fig. 4. The modulation current in the laser diode is supplied by a waveform generator (function generator) with an arbitrary waveform. The laser diode converts the chirp current waveform into a light waveform where the emitting wavelength and power are proportional to the driving current. The laser light is sent into an interferometer equipped with a polarizing beam splitter (PBS) and quarter-wave (λ/4) plates. This configuration is used to reduce feedback from the coupling optics to the LD. The light returning from the object mirror and those returning from the reference mirror interfere through an analyzer. The intensity distribution of the interference pattern in the photodetector is
  I (x, y) = IM (x, y) 1 + γ (x, y) cos φ(x, y) ,

(3.1)

where IM is the bias intensity, γ is the visibility and φ is the tested phase. Phase φ is given by φ(x, y) = k · l where l is the optical path difference (OPD) between the two arms of the interferometer, and k is the wave number, k = 2π/λ. When the wave number is changed from k to k + k by linearly varying the LD current by i, the intensity I (x, y) in eq. (3.1) by shifting the wavelength of LD

Fig. 4. A laser-diode interferometer with an unbalanced optical path length l.

3, § 4]

Laser-diode phase-shifting interferometers

251

becomes   I (x, y) = IM (x, y) 1 + γ (x, y) cos(φ + φ)   = IM (x, y) 1 + γ (x, y) cos(k · l + k · l) .

(3.2)

The modulated phase φ (= k · l) in eq. (3.2) is produced by the wavelength diversity λ due to the current change i of the LD with a change in the wave number of k = −2πλ/λ2 . In the phase modulation φ concerned, k is changed to k = −2π/(λ2 /λ) = −2π/Λ of the synthetic wavelength Λ that is able to measure the phase in an extended range (Ishii and Onodera [1991]). For an optical path difference l(x, y) introduced by a phase object inserted into one arm of the interferometer in fig. 4, Takeda and Ha-se [1995] have demonstrated how well a virtual wave front l · k(x, y) is generated as compared to a real wave front l(x, y) · k. It is important that the wavelength of the LD is synchronized, by an amount of wave number k(t), with the raster scan of the image sensor detecting an interferogram in a scan-synchronized wavelength-shift interferometer. The temporal variation of wave number k(t) in eq. (3.2), caused by a linear current change of the LD, can be transformed into a spatial variation of wave number k(x, y). The interpretation of the virtual wave front generation is applied to phase-locked wavelength-shifting interferometry (see eq. (5.3) in Section 5). The final goal in laser-diode interferometry is to obtain the unknown phase φ by varying the modulated phase φ with the LD current change following various demodulation techniques. This method is based on the fact that the phase difference in an “unbalanced” interferometer is proportional to the product of the OPD and the LD temporal frequency; wavelength tunability produces a stepping phase, a sinusoidal phase or a ramping phase as shown by the three categories of modulation methods in fig. 5.

§ 4. Laser-diode phase-shifting interferometers Phase-shifting interferometry (PSI) was invented by Bruning, Herriott, Gallagher, Rosenfeld, White and Brangaccio [1974]. One technique steps known phases between intensity measurements, whereas another integrates the intensity while the phase is being shifted. This is referred to as the integrating-bucket technique (Wyant [1975]).

252 Laser-diode interferometry [3, § 4

Fig. 5. Modulation methods in a laser-diode interferometer.

3, § 4]

Laser-diode phase-shifting interferometers

253

4.1. Single-wavelength phase-shifting interferometry The dependence of double-heterostructure AlGaAs LD wavelength on injection current and temperature was first investigated by means of a Michelson interferometer by Yonemura [1985]. An area that shows great promise for simplifying the PSI performance is to obtain phase shifts through direct modulation of the output wavelength of the LD by its current (Ishii, Chen and Murata [1987], Tatsuno and Tsunoda [1987]). Temperature modulation of the laser diode wavelength has been applied to an LD phase-shifting speckle interferometer to avoid the large alteration in bias fringe intensity caused by the current change in LD (Kato, Yamaguchi and Ping [1993]). 4.1.1. Phase-stepping techniques In phase-shifting interferometry (see the left panel of fig. 5), the PSI technique can demodulate the phase φ from the measurement of N -stepped interferograms Ij (j = 1, . . . , N) induced by known phase shifts of N -stepped modulated phase (k · l0 )j . In this case, the wavelength changes by λ (= αi) if the current of the LD varies linearly with time from i to i + i. The unbalanced OPD l in fig. 4 is composed of the constant uniform path length l0 and the unknown path length of the test object w(x, y), l(x, y) = l0 + 2w(x, y).

(4.1)

By using eq. (4.1), the modulated phase φ + φ in eq. (3.2) is expressed by   2πλl0 2πλ2w 2π φ + φ ∼ (4.2) (l0 + 2w) − + . = λ λ2 λ2 As the current and its resultant wavelength change through a range of one wavelength in four steps as shown in the left panel of fig. 5, the j th phase shift δj [= −(k · l0 )j ], using eq. (4.2) under the condition of l0  w, is given by 2πl0 αi 2πl0 βi = (j − 1) 2 c λ π (j = 1, . . . , 4), = (j − 1) (4.3) 2 where i is the minimum current increment to give one phase step. The j th sampled intensity Ij has the same form as eq. (3.2) when δj is written instead of φ, i.e. 
 Ij (x, y) = IM (x, y) 1 + γ (x, y) cos φ(x, y) − δj . (4.4) δj = (j − 1)

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Laser-diode interferometry

[3, § 4

The data required for PSI are a series of interferograms recorded with different phase shifts introduced by current changes of the LD. We move to the lower left panel of fig. 5 to get the actual phase φ(x, y) = k · l in eq. (3.2) with the fourstepped measurements Ij∗ taken at the four known phase shifts δj in eq. (4.3),  the least-squares criterion, 4j =1 (Ij∗ − Ij )2 , is minimum. The phase φ including the test object is extracted by solving the matrix equation (Morgan [1982], Greivenkamp [1984]), giving B(δj )A(x, y) = C(x, y, δj ),

(4.5)

where B is the 3 × 3 matrix containing only the known phase shift δj and C is the 3 × 1 matrix consisting of terms related to the known intensity Ij∗ and phase shift δj . These matrices are     A1 IM A =  A2  =  IM γ cos φ  , A3 IM γ sin φ   N N cos δj sin δj N j =1 j =1 N N   B= N cos δj cos2 δj j =1 sin δj cos δj  , and jN=1 N j =1  N 2 j =1 sin δj j =1 sin δj cos δj j =1 sin δj  N  ∗ j =1 Ij    I ∗ cos δj  . C= N (4.6) jN=1 j∗ j =1 Ij sin δj The matrix A in eq. (4.5) can be solved by computing the matrix inversion and multiplication B−1 C with at least three intensity readings. The object profile w(x, y) can be extracted by using a conventional four-step technique, giving    ∗ ∗ λ λ −1 A3 (x, y) −1 I4 − I2 w(x, y) = (4.7) tan = tan , 4π A2 (x, y) 4π I1∗ − I3∗ where the phase shifts δj are arranged as 0, π/2, π , and 3π/2 for the case of N = 4 within one period of the fringes as the current changes equally in four steps. Among many phase-extraction algorithms (Surrel [1996]), the algorithm in eq. (4.7) is symmetric, simple and fast. The phase shift δj in eq. (4.3) is exactly dependent on the test surface profile w(x, y) and is not the same over the whole test surface. The phases φ and φ  are assumed to be measured from the uniform OPD l0 and the OPD l in eq. (4.1) that includes the test surface roughness, respectively. Let us show the phase error, φ  − φ, and the fact that its value could be negligible in a common LD interferometric setup. The phase φ  is written as φ  (x, y) = φ(x, y) + (∂φ/∂w)d(w) =

3, § 4]

Laser-diode phase-shifting interferometers

255

φ(x, y) − π[3 cos(2φ) + 1]w/(2l0 ). In the derivation of this equation it has been assumed to be that the test surface profile w is extracted by the four-step algorithm in eq. (4.7). From | cos 2φ|
1, the maximum phase error is reduced to |φ  − φ|max = 2πw/ l0 . If the best phase-measurement accuracy in phaseshifting interferometry could be gained, such as 2π/100 (Greivenkamp and Bruning [1992]), the phase error is followed by the condition |φ  − φ|max = 2π/100. This inequality is reduced to l0  100w. An LD interferometer commonly has an unbalanced OPD of several tens of millimeters in the experiment, so that this condition is fully satisfied. The phase shift δj in eq. (4.3) allows one to produce a constant phase shift independent of the test surface profile over the entire aperture. In addition, the long optical path difference of tens of millimeters can alleviate the problem of mode hopping (fig. 2), when the changes in injection current required for a 2π phase shift are small. For the measurement, the arrangement in fig. 4 uses an LD (HL 7801) with an operating wavelength of 788 nm to test a convex lens inserted into the object arm. A flat mirror at the focus of the lens is employed to ensure that asymmetric aberrations such as coma are canceled out, leaving only the spherical aberration and astigmatism. The interferogram (fig. 6, top) was taken at marginal focus with spherical aberrations for a small lens of f/No. = 2.8. By setting δ5 = 2π in eq. (4.3), the OPD l0 and the current increment i are found to be 26.6 mm and 1.25 mA, respectively, for α = 0.0046 nm/mA. The phase-extraction algorithm in eq. (4.7) can be performed by a computer with 8-bit digital data and 128 × 128 sample points, and the measured phase map is displayed on an x − y plotter. The phase w is shown in fig. 7a, exhibiting 2π phase discontinuities that result from the inverse tangent in the phase-retrieval equation. By correcting the 2π phase gaps, the spherical aberration is depicted in fig. 7b where the wave aberration at the margin of the aperture is ∼ 2λ. The four-step algorithm in eq. (4.7) has been performed by LD current modulation, thus avoiding the need for active phase modulation within a fiber-optic interferometer (Hand, Carolan, Barton and Jones [1993]). Direct injection-current modulation of an LD has been demonstrated for phase-stepping illumination in a dynamic speckle shearing interferometer (Huang, Ford and Tatam [1996]). Phasestepping illumination by LD has been implemented in an electronic speckle pattern interferometer for vibration analysis with no additional phase modulator (Olszak and Patorski [1997]) and for supplying laser-diode parameters within a laser pulse (Hinsch, Joost, Meinlschmidt and Schlüter [1997]). Speckled interferograms from a rough steel plate have been generated with a small change of the LD wavelength to determine the in-plane rotation angle of the object (Nassim, Joannes and Cornet [1999]). In this case, the wrapped phase has been calcu-

256

Laser-diode interferometry

[3, § 4

Fig. 6. Interference pattern and one-dimensional straight fringes used for calibration in phase shifts.

lated from five π/2 phase-shifted images with a five-frame algorithm (Schwider, Burow, Elssner, Grzanna, Spolaczyk and Merkel [1983], Hariharan, Oreb and Eiju [1987]). 4.1.2. Integrated-bucket techniques The PSI technique can be applied to a four-bucket-integrated method for measuring the phase φ. The advantage of the integrated-bucket technique is that the LD current can be modulated at a constant frequency rather than stepwise, and hence the coherent noise of the LD can be decreased by integrating the interfer-

3, § 4]

Laser-diode phase-shifting interferometers

257

Fig. 7. (a) Three-dimensional (3D) plot of discontinuous phase distribution. (b) 3D plot of the spherical aberration made by correcting the 2π phase gaps in (a).

ence signal and by means of the phase measurement algorithm that provides a frame-to-frame subtraction as depicted in eq. (4.7). In the four-bucket-integrated technique, the interference signal of each set of data is integrated over the time in which the injection current linearly varies through a π/2 change in phase. If the integration for the interference intensity in eq. (4.4) is carried out after the manner of fig. 8a and the phase shift with the fundamental frequency νs of the signal, δj = −2πνs t = −ωs t, varies continuously, the four detected signals have the form

(4.8)

258

Laser-diode interferometry

(a)

[3, § 4

(b)

(c) Fig. 8. Integrated-bucket techniques with a laser diode. (a) Interference intensity integrated by phase-shift changes. (b) Triangular currents in synchronism with field pulses. (c) 3D plot of a step thin film.

 I1 (x, y) =

−π/4

 I2 (x, y) =  I3 (x, y) =

  IM (x, y) 1 + γ (x, y) cos(φ + 2πνs t) d(ωs t),

π/4


3π/4


  IM (x, y) 1 + γ (x, y) cos(φ + 2πνs t) d(ωs t),

π/4 5π/4


  IM (x, y) 1 + γ (x, y) cos(φ + 2πνs t) d(ωs t),

(4.9)

3π/4

 I4 (x, y) =

7π/4


  IM (x, y) 1 + γ (x, y) cos(φ + 2πνs t) d(ωs t).

5π/4

Substitution of the integrated interferograms in eq. (4.9) into eq. (4.7) and calculation can yield the phase distribution φ (Tatsuno and Tsunoda [1987]). For measurement, a photodetector such as the CCD in fig. 4 is employed to integrate the time-varying intensity signal (Chen, Ishii and Murata [1988]). The integration times of field signals in the CCD are overlapped, and the charge cor-

3, § 4]

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responding to these fields is read out in turn. For the integrated-bucket technique, the modulation currents in the LD must be synchronized with either the even or the odd field pulses in the CCD driving circuit. Figure 8b shows an oscillograph demonstrating for the even field pulse positioning the cursor dashed lines with a 1/30-s interval which is equal to the integrating time. The trigger pulse produced by the even field pulse is fed to a function generator which generates a triangular frequency-modulated current in fig. 8b driven to the LD. The integration procedure of each bucket is repeated six times between even field pulses so that the phase ωs t in eq. (4.8) is shifted between −π/4 and π/4 six times for the first frame, and then between π/4 and 3π/4 six times for the second frame, and so on. The currents change successively from i to i + i for the first frame, and from i + i to i + 2i for the second frame, and so on. The many times integration affects the decrease in the coherent noise of the interference signals. An experimental result is shown in fig. 8c in which the phase step of ∼ λ/6 (λ = 788 nm) on an evaporated thin film has been measured. 4.1.3. High-speed phase-shifting interferometers High-speed phase measurements are needed for applications such as dynamic measurements of surface deformation in optical testing or refractive variations in living tissues. Onodera and Ishii [1999b] have developed an LD phase-shifting interferometer (fig. 9) that can capture high-speed phase-shifted interferograms at a frame rate of 1/30 s on a video tape with high storage capacity. An iron heater is inserted into the object arm in fig. 9, causing air turbulence. The four-step current variation is synchronized with the field pulses of a CCD camera using a timing controller in the same manner as in Section 4.1.2. Phase-shifted interferograms at the frame rate are measured with the CCD camera which is operated in field accumulation mode. The field accumulation mode which requires a one-frame period for integration is selected to record the phase-shifted interferograms at the frame rate. The video signal from the CCD camera is sent to a video recorder and is recorded on tape for a few hours. The video frame number is simultaneously recorded on the audio track of the video tape. A frame memory triggered by the freeze pulse converts for successive phase-shifted interferograms of I1 to I4 that are substituted into eq. (4.7). The dynamic phase φ can be calculated with a computer. Figure 10 shows an experimental result for 16 frames corresponding to a time interval of (1/30 s) × 16. Four frames at each line demonstrate the steadystate interference patterns. A π/2 phase shift between the interference fringes can be seen by comparing the lines. A two-wavelength surface contouring system based on electronic speckle pattern interferometry has been constructed (Zou, Pedrini and Tiziani [1996]) in

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Fig. 9. Experimental setup of a high-speed frame-rate phase-shifting interferometer with an LD.

which the interferograms at two wavelengths were measured in both fields of a single video frame by synchronizing the CCD camera with the current modulation of a laser diode. Two states of the test object corresponding to each wavelength of the laser can be recorded in both fields of a single video frame. This is achieved by synchronizing the interline-transfer CCD camera in a noninterlaced mode with the current modulation of the laser diode. Video-rate fringes have been stabilized by a feedback interferometer (see Section 6) immune to vibration (Liu, Yamaguchi, Kato and Nakajima [1996a]). The interferometer uses the closed-loop control of the LD current to compensate for fringe shift. A video image of the locked fringe is real-time analyzed to deliver an unwrapped phase from three-step algorithm.

4.2. Phase-shift calibration The measurement accuracy in PSI needs the known phase shifts to be exact. One way to calibrate the preset phase shifts (Ishii, Chen and Murata [1987]) is as follows. Four sets of one-dimensional intensity profiles across one tilt fringe (straight fringe) over the range marked by the vertical lines in fig. 6 are taken as I1 , . . . , I4 from a two-dimensional tested intensity profile. Four least-squareserror criteria are used to fit these values to four cosine functions of tilt fringe

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Fig. 10. Sixteen successive frame-rate captured interference patterns of an iron object.

in order to get accurate phase shifts. A tilt interference fringe Ij is described as Ij (x, y, δj ) = Aj (x, y) + Bj (x, y) cos(2πx/dj − δj ). Four unknowns, that is, the bias intensity Aj , the modulation Bj , the period of tilt fringe dj , and the relative phase δj at the fringe center, are allowed to vary during the iterative damped-least-squares routine at every extracted one-dimensional intensity. For the procedure to converge to the correct solution, the algorithm must have goodquality initial guesses. Good initial guesses can be made by estimating from a one-dimensional intensity profile so that the convergence is rapid with a few repetitions. The resultant relative phase shifts of interest to the first phase step δ1 = 0 are δ2 = π/2 + 0.097π , δ3 = π − 0.069π , and δ4 = 3π/2 − 0.096π . For calibrating the bias intensity and the modulation of each interference fringe, subtraction of the intensity averaged over whole samples from the intensity profile provides the constant part of bias intensity, and the intensity data divided by the average

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modulation among three appropriate maxima and minima produce the constant part of modulation. After the calibration, the phase of the wave front is calculated according to the matrix equation (4.5) with Gaussian elimination algorithms by using the calibrated phase shifts δj in the formula. In this case, the computation involves the off-diagonal terms of B in eq. (4.5) since the calibrated phase steps are unevenly spaced. As a comparison of measured phases and by measuring φ 10 times, the rms phase fluctuation is reduced from ∼ 0.23 radian for an uncalibrated case to ∼ 0.17 radian for a calibrated case. The technique for finding the phase shifts with the least-squares estimates has been applied to straight fringes in a PZT phase-shifting interferometer (Lai and Yatagai [1991]). Unlike phase-shift calibration but using the least-squares method, Okada, Sakuta, Ose and Tsujiuchi [1990a, 1990b] have elaborately constructed an LD phase-shifting interferometer with which surface shapes and refractive index of a glass plate can be measured separately based on the leastsquares fitting in various phase-shifted fringes introduced by the wavelength change of LD. Next let us show how to calibrate the preset phase shifts by using a feedback interferometer to lock the phase shifts to 2π phases with the frequency tuning in LD (Onodera and Ishii [1992], Ishii and Onodera [1993]). Precalibration of the phase shift was performed by using the signal IM γ cos φ which is the difference between IM + IM γ cos φ and IM − IM γ cos φ in eq. (3.1) detected by two photodiodes, and was employed by a feedback system. These signals can be generated from the combined optics of a λ/2 plate and another PBS (not shown in fig. 4) through an interference signal noted in eq. (3.2). The bipolar fringe signals biased to zero can be obtained from the difference between both interference signals. The phase shift introduced by the bipolar triangular modulation current can be matched to the exact phase ranging from −π to π during a half period of a triangular modulation current as illustrated by fig. 11. Following the demonstration in fig. 11, the three stages of the feedback operation are summarized: (a) The bipolar fringe signal detected on a PD (not shown in fig. 3) that is used as the feedback signal. (b) The adjustment of LD bias current maximizes the error signal marked by a shaded area that we produced by integrating the fringe signal over the halfperiod of the triangular current. (c) The adjustment of the range of LD modulation current sets the error signal at zero, and the feedback system is then forced to lock the phase shift to the 2π change in phase. The laser-diode frequency is controlled via its drive current during the rising period of a triangular waveform (15-Hz) in fig. 12a to produce the interference signal synchronized with the modulated current. First, the bias current of LD is controlled to lock the interferometer on a preset π -phase point of the interference signal to the midpoint of the triangular

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Fig. 11. Timing chart for the calibration of phase shifts: (a) the feedback loop is opened, (b) the error signal becomes maximum when the bias current is adjusted, and (c) the phase shift is matched to the phase ranging from −π to π by the feedback system.

current in the top panel of fig. 11. Second, the modulation current is controlled to synchronize the interference signal (fringe signal) with a preset 2π phase in the middle panel of fig. 11 by equalizing the integration of signal IM γ cos φ over the rising period of the triangular wave to zero. An oscilloscope trace of the integration is shown in fig. 12c. After that, the continuous interference signal depicted in fig. 12b can be produced using the feedback system with correct precalibration. A rising excursion of the triangular current in fig. 12a is divided by four, and this current is driven to a transistor circuit in fig. 3 and can generate a calibrated π/2 phase shifting. If the phase shift δj has some inaccuracies, i.e. δj = δj (1 + ε) from eq. (4.3), the measured phase φ  differing from the true phase φ is written as φ  (x, y) = φ(x, y) + (∂φ/∂ε)ε, where ε is a small quantity. The phase error is due to wavelength instabilities or the absence of calibration in phase shifts. Following the same mathematical procedure as in Section 4.1, the systematic phase error due to the phase shift error becomes φ  − φ ∼ = ε sin 2φ. This phase error varies sinusoidally with half the period of the fringes (Schwider, Burow, Elssner, Grzanna, Spolaczyk and Merkel [1983], Hariharan, Oreb and Eiju [1987]). Hariharan [1989b] has hinted that accurate measurements require the laser wavelength to be stabilized during each measurement. To do this, the laser wavelength is locked to a resonance of the reference cavity. This procedure has the advantage that if the length of the reference cavity is four times the separation of the mirrors in the interferometer, phase steps of π/2 can be introduced by locking the laser wavelength to successive resonances of the reference cavity. The relative

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Fig. 12. (a) A triangular modulation current, (b) interference signal in synchronism with the triangular current and (c) integration signal used for feedback control.

phase difference of π/2 has been calibrated by monitoring the phase-shift signal, which is obtained by integrating the fringe intensity over a local window (Kato, Yamaguchi and Ping [1993]). In this case, the wavelength tunability of LD is accomplished by operating a temperature controller attached to the LD package.

4.3. Laser-diode Fizeau interferometry Phase-shifting Fizeau interferometers are now employed widely as commercially available testing instruments of surface roughness (Greivenkamp and Bruning [1992]). Developments have included a phase-measuring Fizeau or Mirau interferometric microscope with a PZT developed by Biegman and Smythe [1988]. To test spherical surfaces in the Fizeau interferometer, a spherical reference surface is needed such that the required phase shifts translated by the PZT are not the same over the whole surface. The phase shifts introduced by the wavelength changes of the LD are the same in both axial and off-axial rays at the optical path differences between test spherical surfaces and a reference surface (Chen and Murata [1988]). A laser-diode phase-shifting Fizeau interferometer has been employed to increase the range of surface finishes with a long wavelength of 1.55 µm (de Groot [1994]). Wavelength tuning in LD is an effective way of performing the phase-shifting technique to test large optics with no need of a PZT stage (Fairman, Ward, Oreb, Farrant, Gilliand, Freund, Leistner, Seckold and Walsh [1999]).

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Fig. 13. A Fizeau interference microscope with a wavelength-shifted LD and the measured surface of a surface-relief grating.

An LD Fizeau interference microscope (Ishii, Chen, Onodera and Nakamura [2003]) (fig. 13) has been demonstrated which is based on the four-step PSI of eq. (4.7); this can be implemented by modulating the optical path difference between the plane-parallel Fizeau plates with change in LD current. The light source is an InGaAlP laser diode (Toshiba TOLD 9211) with an operating wavelength of λ = 671.8 nm at a current of 43 mA. The light emitted from the LD is collimated by a 2× objective with a numerical aperture (NA) of 0.1. The light reflected from a polarizing beam splitter (PBS) is led to the Fizeau interferometer through a 5× microscope objective (Mitutoyo FS) of NA = 0.12 with a long working distance of 32 mm. This long working distance makes it easier to incorporate the interferometer into a microscope body. Part of the collimated beam is reflected from the reference dielectric nonabsorbing mirror of 25%. The rest of the beam passes through the reference plane-parallel plate with an intensity transmittance of 75% and illuminates a relatively low-quality test surface. A PBS together with a quarter-wave (λ/4) plate can reduce backreflections from the interferometric optics into the LD. The light returning from the test surface and that returning from the reference surface combine and interfere. The interference fringe localized at the test surface is imaged onto a CCD through a tube lens. The lateral

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resolution in the microscope is limited only by the optics and CCD array pixel size; it is typically ∼ 3 µm. The inset at the right in fig. 13 shows the surface profile of a surface-relief grating measured with the Fizeau interferometer. The result involves the surface roughness of a test grating surface with its figure tolerance of ∼ λ/20. A step height of ∼ 0.073 µm and a grating pitch of ∼ 50 µm can be achieved. The experiment is performed using the four-step method of eq. (4.7) with the averaging technique in two-run measurements as will be shown in Section 4.3.1. 4.3.1. Reduction of phase error due to multiple-beam interference Common phase-extraction algorithms are based on the assumption of a sinusoidal interference fringe produced by two-beam interference. The visibility of the interference fringes falls off rapidly as the reflectance of the test surface increases. The intensity distribution in the interference fringes will no longer be sinusoidal because of multiple reflective beams, resulting in significant systematic errors. Hariharan [1987] has stated that the phase error due to multiply reflected beams has a period of 90◦ corresponding to one-fourth of the fringe spacing. The phase error is eliminated to a first-order approximation when the four-step algorithm in eq. (4.7) is applied to the Fizeau interferometer. If Fizeau fringes are formed in reflection between two identical lossless dielectric surfaces and all the multiply reflected beams are taken into account, the j th intensity Ij with the phase shift δj is given by Ij = 1 −

(1 − r12 )(1 − r22 ) 1 + r12 r22

− 2r1 r2 cos(φ − δj )

=1−

b , 1 − e cos(φ − δj )

(4.10)

where r1 and r2 are the amplitude reflection coefficients in the reference and the front surface of the test plate in fig. 13, respectively. The constants b and e in eq. (4.10) depend on the amplitude reflection coefficients of sandwiched mirrors; b = (1 − r12 )(1 − r22 )/(1 + r12 r22 ) and e = 2r1 r2 /(1 + r12 r22 ). The values of b and e are positive and smaller than unity. The phase error, ε = φ  − φ, due to the multiple reflection between the Fizeau plane parallel plates is demonstrated as follows. The intensity differences (I4 −I2 ) = (2be sin φ)/(e2 sin2 φ −1) and (I1 − I3 ) = (2be cos φ)/(e2 cos2 φ − 1) are substituted into the four-step algorithm in eq. (4.7), and a term tan φ  , including a phase φ  with an error, is established as tan φ  = tan φ − tan φ(e2 cos2 2φ)/(1 − e2 sin2 φ). The phase error sin(φ  − φ) under φ  ≈ φ is written as sin(φ  − φ) ∼ =−

e2 sin 4φ

e4 2 e2 ∼ sin 4φ − sin φ sin 4φ. − = 4 4 4(1 − e2 sin2 φ)

(4.11)

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The systematic phase error ε due to the multiple reflection in eq. (4.11) becomes 2

e ε = φ − φ ∼ = − sin 4φ, 4

(4.12)

by using a binomial approximation under the condition that the low reflectance of the sandwiched mirrors has been assumed to be r1 < 1, r2 < 1, and then e < 1. The phase error ε in eq. (4.12) has a 90◦ periodicity corresponding to one-fourth of the fringe spacing in the test phase φ. In view of the periodicity of the systematic phase error, the method to eliminate this error involves introducing an additional phase shift of half a period and recording a second set of phase data. The average of the two sets of phase data is free from the systematic phase error (Schwider, Burow, Elssner, Grzanna, Spolaczyk and Merkel [1983]). The averaging technique is applied to reduce the phase error due to multiply reflected beams (Ishii, Chen, Onodera and Nakamura [2003]). Among the first measurements, four intensity patterns, I1 , I2 , I3 , and I4 , are measured sequentially with a stepwise increase of current in an LD to equally give a π/2 phase shift. Figure 14a shows the experimental phase error as a function of the phase difference φ in the tilted test mirror surface obtained by subtraction of the tilt component from the measured phase result by using the LD Fizeau interferometer in fig. 13. Since the phase error ε in eq. (4.12) has a spatial frequency four times that of the fringe pattern of the test phase, the error can be cancelled by averaging these two sets of phase data with the offset phase of π/4. In a second measurement, and changing the dc-bias current to give a δ1 phase angle in first measurement plus 45◦ , four sequential intensity patterns are acquired and the measured phase in the same tilted surface as the first measurement is obtained by the phase-extraction algorithm of eq. (4.7). The phase error as a function of phase difference is demonstrated in fig. 14b. Comparing fig. 14a with 14b, it is seen that the phase-error curves are mutually inverted and the two runs are in phase opposition to each other with about twenty-nine peaks over seven fringes on the measuring area. This verifies the numerical analysis given by eq. (4.12). Figure 14c demonstrates the feasibility of averaging over both sets of phase data, minimizing the rms error to 0.15 rad as opposed to 0.23 rad in fig. 14a and 0.25 rad in fig. 14b. The averaging result in fig. 14c, over two runs with different bias currents of the LD, exhibits an rms accuracy of ∼ λ/84 that is derived from the rms error divided by two at the test mirror surface itself to reduce the systematic phase error due to multiple reflections. Wavelength tunability in LDs plays an important role in adjusting the offset phase of π/4 in between two-run measurements.

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Fig. 14. Experimental phase error versus object phase difference; two sets of measurements (a) and (b) with phase offset π/4 are averaged in (c).

4.3.2. Fourier description in wavelength-scanning Fizeau interferometry Laser-diode wavelength-scanning interferometry adequately extracts the desired modulated phase from superimposed fringes in flatness testing of transparent

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plates in the Fizeau interferometer. Okada, Sakuta, Ose and Tsujiuchi [1990b] demonstrated the simultaneous measurement of the front surface shape and the refractive index inhomogeneity of a parallel plate in a wavelength-scanning Twyman–Green interferometer (fig. 4). The algorithm was derived by a leastsquares method that can eliminate the two dominant interference noise components generated by the reflected beam from the rear surface of the test plate, but neglects the smaller multiple-reflection noise of order R, where R is the reflectance of the plate, and obtains a peak-to-valley (P-V) value of λ/20. A set of six interferograms obtained by changing the wavelength of an LD (λ = 795 nm) has been used to measure separately both the surface shapes and the refractive index inhomogeneity of a glass plate. The plate is put in the object arm, and six interference fringes are formed: between the mirror-image plane of the reference mirror and the front and rear surfaces of the glass plate, between the two surface reflections, between the mirror-image plane of a reference and the object mirror, and between the object mirror and the front and rear surfaces of the glass plate. Since each of the six superimposed interferograms corresponds to a different optical path difference, each phase shift varies at a different rate, and six algebraic equations for least-squares fitting are obtained to solve thirteen unknowns and to separate the six quantities to be measured. The superimposed interference fringes cause errors that are analyzed (Okada, Sakuta, Ose and Tsujiuchi [1990a]), taking into account the multiple-beam fringes. By using a series of interferograms generated by wavelength shifts of a distributed feedback (DFB) LD (λ = 830 nm), the phase distribution of a plane mirror and the phase shifts to be calibrated can be simultaneously measured for the least-squares estimates with the acquisition of thirty-two fringes (Okada, Sato and Tsujiuchi [1991]). Fizeau interferometry with a wavelength-shifted LD can suppress interference modulations from spurious reflections with a desired phase-shifted fringe by LD (de Groot [2000]). Multiple-beam interference noise in a phase-shifting interferometer can be identified and suppressed with a new algorithm by LD modulation frequencies (Hibino [1999], de Groot [2000], Hibino and Takatsuji [2002], Hibino, Oreb and Fairman [2003]). Multiple-reflection noise can be eliminated efficiently by algorithms in the Fourier domain. An unbalanced Fizeau interferometer between a reference plate and a transparent test plate is shown in fig. 15 with a stepwise wavelength change in the LD source. The LD is modulated by a sawtooth current with period T and amplitude i. The change i of the LD current generates an optical-frequency variation ν, producing the relation ν = βi with the current-tuning ratio β (see Section 2.2 and fig. 5). The temporal carrier frequencies νk are produced by ramping

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Fig. 15. Configuration of the extraction of a desired phase by FMCW techniques in a Fizeau interferometer.

the currents on an unbalanced OPD lk as βilk . (4.13) Tc The carrier frequencies are called beat frequencies following the frequencymodulated continuous-wave (FMCW) technique (Giles, Uttam, Culshaw and Davies [1983]). The beat frequency νk is proportional to the optical path difference of each pair of interfering beams. The open circles in fig. 15 stand for the phase-shifted interference fringes associated with the stepped current changes. The interference intensity observed by the CCD shown in fig. 15 is formed by the multiple-beam interference from the three surfaces with three optical path differences: l1 = 2d, l2 = 2nth and l3 = 2(d + nth ), where d is the air-gap width, and n and th are the refractive index and the thickness of the transparent plate, respectively. The interference intensity Ij (x, y, tj ) as a function of j th time tj in stepwise change is given by   3    Ij (x, y, tj ) = IM (x, y) 1 + (4.14) γk cos 2πνk tj + φk (x, y) , νk =

k=1

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in accordance with the j th phase shift δj , i.e. δj = −2πν1 tj from eq. (4.8), considering the detection of a surface profile of a test plate. Most phase-shifting algorithms can be written as  N    j =1 sj Ij (x, y; tj ) −1 −1 p1 . = tan φ = tan (4.15) N p2 j =1 oj Ij (x, y; tj ) The principle of heterodyne phase measurements as correlations of the intensity Ij (tj ) with two phase-stepping functions pn is given by the following relation,   ∞   ∗ pn = 2 Re (4.16) FT Ij (tj ) Fn (ν) dν (n = 1, 2), 0

where Re stands for the real part, FT[. . .] denotes the Fourier transform of a function, and the asterisk denotes the complex conjugate. The Fourier transforms of the coefficients sj and oj in eq. (4.15) are written as N    F1 (ν) = FT f1 (t) = sj ei2πν(−tj ) , j =1

  F2 (ν) = FT f2 (t) =

N 

(4.17) oj ei2πν(−tj ) ,

j =1

with the two sampling functions f1 (t) =

N 

sj δ(t − tj )

and

j =1

f2 (t) =

N 

oj δ(t − tj ).

j =1

The resulting filter functions Fn (ν) in eq. (4.17) are transfer functions in the frequency domain (Freischlad and Koliopoulos [1990], Larkin and Oreb [1992]) under the conditions F1 (ν) = −F1 (−ν) and F2 (ν) = F2 (−ν). For the extraction of the fundamental frequency ν1 in FT[Ij (tj )], the following conditions (Hibino, Oreb and Fairman [2003]) for zero amplitudes at higher harmonics should be satisfied: F1 (νk ) = F2 (νk ) = 0 [k = 0, 2, . . . , 10] for an equal phase-shift interval of δj = π/6 and a thickness setup of nth = 3d. Another condition for the suppression of a detuning error (phase-shift error) at the fundamental frequency ν1 is given by d[F1 (ν)]/dν = d[iF2 (ν)]/dν at ν = ν1 by using a Taylor expansion in Fn (ν) with the inclusion of an in-quadrature (90◦ out of phase) relation. The

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Fig. 16. Experimental results of the surface shape of the BK7 plate; (a) grey-scale map of the phase, (b) cross-sectional profile of the surface, and (c) Fourier components of sampling functions in a 19-sample algorithm used. (After Hibino, Oreb and Fairman [2003].)

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index dispersion is considered up to the second order of the reflectance of the test BK7 plate. Substitution of both sampling coefficients, sj and oj , and the relation −2πν1 tj = π(j − 10)/6 for j = 1, . . . , 19 (= N) in eq. (4.17) into the abovestated conditions gives nineteen linear algebraic equations. A set of determined coefficients of sj and oj is substituted into eq. (4.17), and the relative amplitude in Fourier components of the sampling functions Fn (ν) is shown in fig. 16c. These functions have equal amplitudes and matched gradients at the fundamental frequency ν1 , which leads to the elimination of the error from the detuning of the sampling frequency. The amplitudes are smaller for other frequencies 3ν1 to 10ν1 . The surface of a BK7 plate (n = 1.5) with a diameter of 230 mm was measured by the wavelength-scanning Fizeau interferometer of fig. 15 with a Littman-type external-cavity laser diode of 18-GHz wide tunability (Hibino, Oreb and Fairman [2003]). Figures 16a and 16b show a grey-scale map of the measured phase and its cross-sectional surface along the horizontal axis, respectively. From the results, it is evident that the multiple-interference noise has been eliminated by the nineteen phase-stepping algorithms. The minimum reflection noise with this algorithm for testing a glass plate is ∼ λ/600.

4.4. Phase-extraction algorithm insensitive to changes in LD power It is a problem that the power of the laser diode source changes during wavelength scanning, because diode gain and cavity gain are different at different wavelengths. Therefore, in LD PSI, measurement accuracy is decreased by intensity modulation of the interferogram induced by laser-power changes. One way to solve this problem is to electrically normalize the intensity of the interferogram with LD output power (Ishii [1991]). The normalization procedure can also be performed numerically. The problem of the intensity modulation of the LD is then solved using an iterative method for finding both the phase distribution φ of the wavefront and the value of the phase shift δj (Okada, Sato and Tsujiuchi [1991]). The components of the interferogram that change with the phase shift are calculated in this method. Then the phase distributions are measured that are insensitive to the laser-power variation associated with the phase shift. Sandoz, Gharbi and Tribillon [1996] have shown that the variation in background intensity due to the emitted power increase is compensated by considering pairs of optical irradiances with a relative phase-shift that is a multiple of 2π . In the normalization process one needs to consider a sufficiently large number of phase-shifted interference fringes. The same method has been applied to four-step LD Fizeau interferometry (see Section 4.3): experimentally a small rms phase error of 0.09 rad is found

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with compensation compared to 0.20 rad without compensation (Ishii, Chen, Onodera and Nakamura [2003]). An amplitude-stabilized LD source has been developed that utilizes a feedback system with a superluminescent diode (SLD) as an external light controller (Takahashi, Yoshino and Ohde [1997]). This constantamplitude controlled LD device with an SLD has been applied to heterodyne interferometry (see Section 7) (Onodera, Ishii, Ohde, Takahashi and Yoshino [1995]). Let us now demonstrate a phase-extraction algorithm that is insensitive to changes in LD power (Ishii and Onodera [1995], Onodera and Ishii [1996a]). Assuming a linear relationship between injection current and output power of the LD, the bias intensity IM associated with the phase-shift change in eq. (4.3) is 0 (x, y) + 2δ I (x, y)/π , where I 0 is the bias inrepresented by IM (x, y) = IM j M tensity without current modulation and therefore with phase shift zero, and I is the intensity change produced by the current variation required to produce a phase-shift interval of π/2. As the current changes by an equal amount in each of the N steps, the j th sampled intensity Ij is written, using eq. (4.4), as 
 0 Ij (x, y) = IM (1 + 2δj ρ/π) 1 + γ (x, y) cos φ(x, y) − δj (j = 1, . . . , N),

(4.18)

0 is the rate of change of the bias intensity. When a tested where ρ = I /IM phase is calculated from a least-squares fit to a function of cosine form as used in conventional PSI (see Section 4.1.1), the best estimate of the phase φ is obtained when the least-squares criterion is minimized. The least-squares criterion 0 , ρ, γ , and φ. The four algebraic equaincludes four independent unknowns; IM tions become nonlinear, and the solution may be difficult to derive. Alternatively, eq. (4.18) can be treated as if it contained six unknowns (N = 6). With a matrix equation similar to that in eq. (4.5), a 6 × 6 matrix in B similar to that in eq. (4.6), consisting of four 3 × 3 submatrices, is constructed at known phase shifts, i.e. see also δj = π(j − 7/2)/2 (j = 1, . . . , 6). Phase φ, tested is found with six measurements Ij∗ , is solved by computing a matrix inversion and multiplying B−1 C by six readings; i.e.   3I1∗ − 5I2∗ + 5I5∗ − 3I6∗ φ(x, y) = tan−1 (4.19) , −I1∗ − 3I2∗ + 4I3∗ + 4I4∗ − 3I5∗ − I6∗

where C is a 6 × 1 matrix containing the known measured intensity Ij∗ and phase shift δj . The Fourier description noted in Section 4.3.2 can be applied to derive the algorithm in eq. (4.19) (Onodera and Ishii [1996a]). The time-varying interference signal Ij (x, y; tj ) obtained from eq. (4.18) through eq. (4.8) is Fourier-transformed, and then substituted into eq. (4.16), giving the phase-stepping functions pn . For

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extracting the fundamental frequency component of FT[Ij (tj )], it is necessary to filter out the various terms in pn except the fundamental frequency component, through Fn (ν) in eq. (4.17). So the following conditions for F1 (ν) and F2 (ν) should be satisfied: F1 (0) = F2 (0) = 0, d{F1 (ν)}/dν = d{F2 (ν)}/dν = 0 at ν = 0, and d{F1 (ν)}/dν = d[F2 (ν)}/dν = 0 at ν = ν1 . The differentiation operation arises from the Fourier transform of the product of time tj and the exponential function exp(i2πν1 tj ), explaining the linear power change with time. In addition, the filter functions F1 (ν1 ) and F2 (ν1 ) should be in quadrature and equal in magnitude: F1 (ν1 ) = iF2 (ν1 ). A set of coefficients of sj in odd symmetry and oj in even symmetry can be solved by using the six linear algebraic equations in eq. (4.17) derived from the above-stated conditions. The determined coefficients sj and oj are equivalent to those in the phase-extraction routine of eq. (4.19). Surrel [1997] demonstrated that the six-step algorithm in eq. (4.19) is also insensitive to miscalibration, and a new algorithm was developed using characteristic polynomials with the same properties as eq. (4.19) but with one intensity sample less, i.e. five steps. The algorithm involves a minimal number of samples without phase-shift error. The five-step algorithm insensitive to power changes in LD was derived using the condition of free phase-shift error, i.e. d{F1 (ν)}/dν = d{iF2 (ν)}/dν at ν = ν1 (see Section 4.3.2) (Afifi, Nassim and Rachafi [2001]). According to an analysis by Phillion [1997] of the algorithm of eq. (4.19), it is sensitive to quadratic detector nonlinearity and insensitive to a linear drift of the sinusoidally varying interference signal. An unknown phase is found by direct integration of its gradient, avoiding the evaluation of the arctangent function (Páez and Strojnik [1997, 1999]). The method is insensitive to spatial nonuniformity of the illuminating intensity, in particular, the power change of a laser-diode source. Experimental verification is performed by measuring a chrome-coated step object in a Twyman–Green interferometer (see fig. 4). The single-frequency light source is an LD (TOLD9211) with an operating wavelength of λ = 672 nm at 43 mA current. With an OPD of l0 = 40 mm and a measured tuning rate of α = 0.01 nm/mA, six phase shifts sampled symmetrically in equal space are produced by increasing the LD current in a range corresponding to a power change of 0.6 mW. Figure 17 shows the measured phase calculated from the phase-extraction routine in eq. (4.19) (top) and the measured phase φ  that includes the phase error (bottom) from a four-step algorithm with intensity readings (I1∗ , . . . , I4∗ ) in a similar algorithm of eq. (4.7), giving  ∗  I1 − I2∗ − I3∗ + I4∗  −1 . φ (x, y) = tan (4.20) −I1∗ − I2∗ + I3∗ + I4∗

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Fig. 17. 3D plots of the measured phase of a step object calculated from (top) the six-step algorithm in eq. (4.19) and (bottom) the four-step algorithm.

The algorithm in eq. (4.20) involves four steps that are arranged symmetrically as the injection current changes equally. The result in the top panel of fig. 17 demonstrates freedom from the systematic error caused by LD power changes. The bottom panel shows that the periodic phase error has the same period as the interference fringe, i.e. 360◦ . This phase error is verified as follows. Substitution of eq. (4.18) into the intensities Ij instead of Ij∗ in eq. (4.20) produces a phase error sin(φ  − φ) under the assumption that φ  ≈ φ: sin(φ − φ) ∼ = 

√ ρ[(2 2/γ ) sin φ − cos 2φ] . √ ρ[2 − (2 2/γ ) sec φ − tan φ] − 2

(4.21)

  Equation √ (4.21) can be simplified to an expression for φ − φ, that is, φ − φ ≈ ρ[−( 2/γ ) sin φ + (1/2) cos 2φ]. The first term is dominant over the second term

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√ as 2/γ  1/2. Thus the phase error varies sinusoidally with the same period as the fringes, i.e. 360◦ , as shown in the bottom panel of fig. 17. Hariharan [1989a] has pointed out that the phase error due to changes in laser output caused by wavelength scanning had been identified at an early stage, and at that stage was within acceptable limits for most workshop measurements with several tens of millimeters in OPDs. The laser-diode phase-stepping technique has been applied to electronic speckle pattern interferometry for measuring mechanical beam displacement utilizing the algorithm in eq. (4.19) (Patorski and Olszak [1997]). 4.5. Two-wavelength phase-shifting interferometry In interferometric measurements, the phase distribution across the interferogram is measured modulo 2π . This problem limits the phase measurement range of PSI using a single wavelength. One solution is to use two shorter visible wavelengths to generate a longer synthetic wavelength, which makes it possible to measure distances much greater than the optical wavelength without introducing 2π ambiguities (Polhemus [1973]). A synthetic wavelength Λ (= λ1 λ2 /|λ1 − λ2 |) is obtained from two wavelengths λ1 and λ2 . Laser diodes are pertinent light sources with their tunabilities in emitting wavelengths for two-wavelength interferometry. They permit one to choose an appropriate synthetic wavelength. Let us demonstrate two-wavelength phase-shifting interferometry with dual wavelength-tunable LDs (Ishii and Onodera [1991], Onodera and Ishii [1994]): they are frequency-modulated by mutually inverted ramped currents on an unbalanced interferometer. In the unbalanced Twyman–Green interferometer shown in fig. 18, the intensity distributions of the interference pattern for the wavelengths λk (k = 1, 2) are Ik (x, y) = IMk (x, y){1 + γk (x, y) cos[φk (x, y)]}, where IMk is the bias intensity, γk is the visibility, and φk is the phase given by φk (x, y) = 2π{2w(x, y) + l0 ]/λk . In the same manner as stated in Section 4.1.1, and as the current changes in discrete steps, the j th phase shift δkj is given by 2δkj = j (2πl0 αk ik )/λ2k when the injection currents of the LDs change independently by amounts ik . The j th sampled intensity aj (x, y) of the two-wavelength interferogram is written as the incoherent sum of both intensities as aj = I1 + I2 , and has a moiré pattern. If γ1 IM1 = γ2 IM2 is taken into consideration, the moiré pattern aj becomes aj (x, y) = IM1 (x, y) + IM2 (x, y) + 2γ1 (x, y)IM1 (x, y)  
 × cos φ1 (x, y) + φ2 (x, y) /2 − δ1j − δ2j  
 × cos φ1 (x, y) − φ2 (x, y) /2 − δ1j + δ2j .

(4.22)

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Fig. 18. A two-wavelength phase-shifting interferometer with dual wavelength-scanning LDs.

If the phase shifts are arranged so that δ1j = −δ2j , eq. (4.22) becomes aj (x, y) = IM1 (x, y) + IM2 (x, y) + 2γ1 (x, y)IM1 (x, y)     × cos Ψ (x, y) cos Φ(x, y) − 2δ1j ,

(4.23)

where Ψ = (φ1 + φ2 )/2 = π(2w + l0 )/Γ with Γ the average wavelength given by Γ = λ1 λ2 /(λ1 + λ2 ), and Φ = (φ1 − φ2 )/2 = π(2w + l0 )/Λ with Λ the synthetic wavelength given by Λ = λ1 λ2 /|λ1 − λ2 |. The first cosine term of eq. (4.23) is independent of modulated phases, and the argument of the second cosine term is a function of the phase shift, which makes this moiré pattern applicable for PSI technique. Thus, the envelope component of the two-wavelength interferogram is scanned but its carrier component remains unchanged. The twowavelength phase-shifting intensity in eq. (4.23) has no measurement sensitivity when cos(Ψ ) = 0 at Ψ = π(2m + 1)/2 (m; integer). Using aj of eq. (4.23), one can calculate phase Φ at a synthetic wavelength Λ using the four-step algorithm by replacing 4πw/λ and Ij∗ in eq. (4.7) with Φ(x, y) and aj , respectively. It is interesting to note that the synthetic phase in eq. (4.23) is produced by the addition of two single-wavelength interferograms so that the phases are equally shifted in opposite directions. The time-varying two-wavelength interference sig-

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nal aj (x, y; tj ) in eq. (4.22) using eq. (4.8) has been analyzed by Onodera and Ishii [1997] with the Fourier description discussed in Section 4.3.2. The synthetic phase is measured modulo π and has a half period compared with the conventional two-wavelength interferometry (Bartolini, Fornetti, FerriDeCollibus, Occhionero and Papetti [1991]): the synthetic phase (φ1 − φ2 ) can be retrieved (modulo 2π ) by electrically processing the beat signal through a squaring device and a low-pass filter. For the measurements with the interferometer shown in fig. 18, two collimated beams emitted from LD1 (λ1 = 672 nm, TOLD 9211) and LD2 (λ2 = 788 nm, HL7801) are incident on two neutral-density filters to equalize the modulation intensities for both LDs. A synthetic wavelength of Λ = 4.6 µm is produced. Two laser sources of the same linear polarization are used to illuminate simultaneously the interferometer arms with an OPD of l0 = 40 mm. Half of the two-wavelength (moiré) patterns accompanied by variations in phase shifts is measured with a CCD camera. The rest of the interference signal is transmitted through two interference filters IF1 and IF2 to two sets of PBSs with λ/2 plates to detect an interference signal for each wavelength. According to the phase-shift calibration described in Section 4.2, the desired phase-shift relation of δ1j = −δ2j is accomplished by adjusting the frequency tunings of the LDs. Each phase shift introduced by the bipolar triangular modulation current can be matched to the exact phase ranging from −π to π during the half-period of a triangular modulation current as illustrated in fig. 19. The current variations in the two LDs coincide with the corresponding interference signals. The interference signals in fig. 19 are each

Fig. 19. Variations in the calibrated currents of two LDs (upper two traces) and the corresponding interference signals (lower two traces). Two LDs are driven by mutually inverted triangular currents.

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Fig. 20. A two-wavelength moiré pattern of the test step object.

normalized by the power monitors of two LDs with two electronic dividers. The modulated phases δ1 and δ2 are equally shifted in opposite directions by changing mutually inverted currents. Figure 20 shows a two-wavelength moiré pattern in accordance with eq. (4.22), captured by a CCD camera upon simultaneous illumination by the two LDs of a step object located on the test object plane in fig. 18. The experimental layout illustrated in fig. 18 has been used to test a diffraction grating with dual LDs with a synthetic wavelength Λ = 4.6 mm. The bias currents of LD1 and LD2 used in the experiment were 43.8 mA and 57.0 mA, respectively, and the total diversities of the modulation currents in the four steps were 1.26 mA and 1.80 mA, respectively. The current increment steps were thus i1 = 0.32 mA and i2 = 0.45 mA. The current changes of both LDs can satisfy the phase-shift condition. The two LD currents vary counterstepwise so that their resultant phases are shifted in opposite directions. The measured 3D phase map is presented in the bottom panel of fig. 21. The measured height of the grating is ∼ 0.46 µm which is in good agreement with the manufacturer’s calibrated result of 0.45 µm. The top panel of fig. 21 shows the measurement made at a single wavelength (λ1 ) with the PSI technique, in which the measured height of 0.13 µm is subtracted half a wavelength from the actual height, causing an ambiguous measurement. The rms repeatability with the electronic calibration shown in fig. 19 is obtained

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Fig. 21. 3D phase maps of (top) single-wavelength (λ1 ) and (bottom) two-wavelength (Λ) measurements, respectively.

as ∼ Λ/78 (Ishii and Onodera [1993]). In addition, a seven-step phase-extraction algorithm in two-wavelength interferometry has been developed that is insensitive to power changes of dual LDs (Onodera and Ishii [1994]). The two-wavelength interference pattern typically shown in fig. 20 has been used to measure the phase at the synthetic wavelength by using the Fouriertransform method. Takeda, Ina and Kobayashi [1982] have developed a sophisticated technique of phase measurement by the Fourier-transform method that performs the filtering process in the spatial frequency domain. A first-order frequency spectrum for a λ1 wavelength and a minus-first-order frequency spectrum for a λ2 wavelength are selected from a two-wavelength interferogram with spatial carrier frequencies and are translated toward the origin in the frequency domain. A phase profile at the synthetic wavelength is obtained by the inverse Fourier transformation of these spectra (Onodera and Ishii [1998b]). The two-color (two-wavelength) interferometer with dual LDs in fig. 22 yields Λ = 15 µm synthesized by λ1 = 785 nm and λ2 = 828 nm (de Groot and Kishner [1991]). Interference filters F1 and F2 are used to separate the two wavelengths λ1 and λ2 and permit a common-path geometry. The phases φ1 and φ2 are measured by the five-step algorithm mentioned in Section 4.1.1. The reference mirror is moved axially to five discrete positions by the PZT, and the photodiode (PD) outputs in fig. 22 are sampled at both wavelengths for each of five positions. The

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Fig. 22. Two-color interferometer implementation using two LDs (LD1, LD2) and common-cavity stabilization of the synthetic wavelength. (After de Groot and Kishner [1991].)

analysis in the two-color measurement is performed to determine the relative synthetic fringe number M from any initial point of reference that is obtained from the differences in single-wavelength fringe numbers m1 and m2 derived from the phases φ1 and φ2 , respectively. The frequency stabilization involves simultaneous control of the two LDs by electronic feedback from a common Fabry–Perot resonator (FPR) as shown in fig. 22. By using this method, de Groot [1991] presented three-color interferometry. A pair of multimode laser diodes generates the synthetic wavelengths of 720 µm and 20 µm that progressively reduce the uncertainty in the distance measurement. Two of three-wavelength LDs have also been stabilized on two consecutive resonances of a common FPR and an electronic beat-frequency calibration between another LD tuned from one to another frequency through the FPR to measure a distance (Dändliker, Hug, Politch and Zimmermann [1995], Zimmermann, Salvadé and Dändliker [1996]). Phase stepping is accomplished by using the wavelength shifts of three LDs in an oblique-incidence interferometer of an OPD with l0 = 70 mm (Franze and Tiziani [1998]). The three wavelengths are λ1 = 690 nm, λ2 = 787 nm and λ3 = 826 nm. All three wavelengths can be simultaneously illuminated into a metallic object. The phase of the rough surface in the metallic object is determined for each of the wavelengths by the four-step method. The phase difference due to λ2 and λ3 is first determined because of the large synthetic wavelength, Λ = 106.3 µm. This is obtained with a combination of λ1 and λ2 , leading to a synthetic wavelength

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of Λ = 26.76 µm. Finally, the accuracy is increased by combining the above two measurements with the phase obtained from one wavelength.

§ 5. Sinusoidal phase-modulating interferometry The phase modulation in sinusoidal phase-modulating interferometry (Sasaki, Takahashi and Suzuki [1990]) as schematized in fig. 5 is very simple. The signal is a sinusoidal function of the phase difference between the object and reference waves of the Twyman–Green interferometer in fig. 4. Direct frequency modulation in LDs is often utilized in this type of interferometer. A sinusoidal current change i with amplitude a and carrier frequency ωc , written as i = a cos ωc t, is driven into an LD driver circuit. The interference signal in eq. (3.2) is now changed to
 I (x, y; t) = IM (x, y) 1 + γ (x, y) cos(φ + z cos ωc t) , (5.1) where the modulation depth z = 2πl0 αa/λ2 is derived from eq. (4.3). Letting the initial phase of the sinusoidal current variation in eq. (5.1) be equal to zero for simplicity, the sinusoidal phase modulating interference intensity in eq. (5.1) is transformed into
  I (t) = IM 1 + γ cos φ J0 (z) − 2J2 (z) cos(2ωc t) + · · ·   − γ sin φ 2J1 (z) cos(ωc t) − 2J3 (z) cos(3ωc t) + · · · , (5.2) where the function Jn (z) is the nth order Bessel function. As the Fourier transformation of the intensity I (t) in eq. (5.2) is defined as F (ω), the value of z can be determined from the relation |F (3ωc )|/|F (ωc )| = |J3 (z)|/|J1 (z)| derived from eq. (5.2). By analyzing the effect of the additive noise contained in the signal I (t), Sasaki and Okazaki [1986] proved that z = 2.63 is the most suitable amplitude. Using this value of z, the phase φ can then be extracted by taking the ratio of the ωc -Fourier spectrum F (ωc ) divided by J1 (z) to the 2ωc -spectrum F (2ωc ) divided by J2 (z). This calculation has been performed by a phase detection circuit that can measure the object displacement in real time (Suzuki, Sasaki, Higuchi and Maruyama [1989]). In addition to the sinusoidal phase modulation by LD and by controlling the vibration mirror with a feedback system, the phase fluctuation can be eliminated (Sasaki, Yoshida and Suzuki [1991]). OPDs up to sub-millimeter range can be measured in terms of the ratio of the (2n + 2)th- and 2nth-order Bessel functions in eq. (5.2) for the sinusoidal wavelength modulation of an LD (den Boef [1987]). Since the phase is modulated by a sinusoidal current with frequency ωc , the time-varying interference signal is detected by a CCD camera

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Fig. 23. Flow diagram for the measurement procedure in a laser-diode feedback interferometer.

instead of a photodiode, and the phase φ can also be measured from interval integrating interference signals using the four-integrated-bucket technique in much the same way as shown in fig. 8a (Sasaki, Okazaki and Sakai [1987]). When the current feedback to a light source LD (Yoshino, Nara, Mnatzakanian, Lee and Strand [1987]) is applied to a sinusoidal phase modulating interferometer as shown in fig. 23, the test object w in eq. (4.1) can be measured by controlling the LD current to lock the modulated phase 2π(l0 + 2w)/(λ + λ) in eq. (4.2) to the fixed phase 2πl0 /λ, i.e. 2πl0 /λ = 2π(l0 + 2w)/(λ + λ). Accordingly, the modulated phase tracks the test surface of an object (Suzuki, Sasaki and Maruyama [1989]). It yields the object profile, l0 λ l0 αi = , (5.3) λ λ from measurements of the control current i of the LD. The current is controlled with a proportional integral controller by using a feedback signal, −2J1 (z) sin φ, in eq. (5.2). A two-wavelength interferometer with dual laser diodes has been constructed to use sinusoidal phase modulation for phase detection and feedback control of currents in dual LDs for eliminating external disturbances (Sasaki, Sasazaki and Suzuki [1991]), and the distance can be measured real-time with a twowavelength sinusoidal phase-modulating interferometer with dual LDs (Suzuki, Kobayashi and Sasaki [2000]). The system is constructed in such a way as to allow the control current i to be fed directly to two different laser diodes. To increase the detection sensitivity for the phase-modulation amplitude, a DBR laser diode with wide tunability is employed in a double sinusoidal phase-modulating interferometer (Suzuki, Suda and Sasaki [2003]). The displacement of an object 2w =

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is obtained by a lock-in amplifier with sinusoidal current modulation in the LD, and the path-length modulation by a PZT reference mirror (Chang, Chien and Lee [1999]). The displacement signal is extracted with a 3ωc component in the sinusoidal interference signal and then by use of an electric gating circuit followed by an electric mixer. § 6. Feedback interferometry Yoshino, Nara, Mnatzakanian, Lee and Strand [1987] have shown feedback interferometry in which the interference signal detected by a photodiode is fed back to the LD current (fig. 23, dashed line) in Twyman–Green and self-coupling (optical feedback) interferometers to measure the mirror displacement. Referring to the feedback interferometer described in Section 5, the object displacement can be measured by controlling the LD current to lock the modulated phase φ in eq. (3.2) to the fixed phase φ so that the displacement can be extracted from the feedback current i following eq. (5.3). This technique, based on phase-nulling detection, is also employed as an optical fiber sensor (Hotate and Jong [1987]). The feedback interferometer does not require a vibration-isolated table. Fringes can be stabilized in spite of environmental disturbances such as mechanical vibrations and air turbulence, and laser-frequency variations (Liu, Yamaguchi, Kato and Nakajima [1996b]). In this scheme, a servo control is coupled to fringe intensity, which causes feedback instability at phases of ±π/2. To overcome this, the fringe phase of ±π in the output of a vector voltmeter (see fig. 24) has been used as a feedback signal with constant gain at any fringe point (Yoshino and Yamaguchi [1998]). The interferometer in fig. 24 is servo-controlled in the phase domain, where the phases are measured by two-frequency optical heterodyning. The demodulated phase is utilized as a feedback signal so that the phase can be measured modulo 2π . The LD beam (λ = 633 nm, HL3212G) is converted by acousto-optic frequency shifters (FS) into a beam consisting of two orthogonally linearly polarized frequency components with a frequency difference of 1 MHz. The interference intensity from a Fizeau interferometer with an OPD of 2d (fig. 24), is put into the vector voltmeter with a reference beat signal of 1 MHz. The demodulated phase is fed back into the LD current, thereby constituting a closed-loop interferometer. An improved stabilization has been established by an improved feedback control system to perform closed-loop phase-shifting interferometry (Yokota, Asaka and Yoshino [2003]). A feedback interferometer (self-mixing interferometer) can be constructed to detect the variation of output power by the change in the Q-value of the LD due to a change in the distance between the laser and an external mirror. In the feed-

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[3, § 7

Fig. 24. Configuration of the feedback system for stabilized laser-diode phase-shifting interferometry. (After Yokota, Asaka and Yoshino [2001].)

back interferometer, the reflected beam from a target mirror is coherently mixed externally into the laser diode with the source light itself as the reference beam. Distance and velocity can be measured by the homodyne technique with selfmixing (de Groot, Gallatin and Macomber [1988], Takahashi, Kakuma and Ohba [1996], Gouaux, Servagent and Bosch [1998]). A feedback phase-locked differential interferometer has been constructed by using the orthogonally polarized light of a frequency-modulated LD (Nakatani [2003]). Instead of the phase-locking scheme of eq. (5.3), phase-shift locking has been achieved by using a sawtooth current change i in the LD and a preset phase shifter  produced by a quarter-wave plate and polarizers (Onodera and Ishii [2003]). According to eq. (4.2), the phase shift (modulated phase) φ induced by current change, −i, is locked to a phase difference . The OPD, l0 , is given by l0 = (/2π) × (λ2 /λ) = (/2π)Λ. When the phase difference  is set to 2π , the OPD, l0 , can be measured real-time up to a range of several centimeters in the synthetic wavelength Λ. § 7. Heterodyne interferometry Heterodyne detection of the interferometric output is an attractive phase-extracted technique, but an additional frequency shifter such as a Bragg cell (Saleh and Teich [1991]) is required. The basic principle of heterodyne phase measurements

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consists of mixing a periodic signal, with two sinusoidal signals 90◦ out of phase, from a local oscillator and applying a low-pass filter. Thus, the real and imaginary parts of the Fourier component of the signal at the frequency of the local oscillator are determined. The phase of the signal is obtained as the arctangent of the ratio of the imaginary part and the real part. 7.1. Single-wavelength heterodyne interferometry Many interferometric methods based on the FMCW technique (see Section 4.3.2) have been developed in which temporal carrier frequencies are produced by ramping the LD currents on an unbalanced interferometer as shown in fig. 25. The detection scheme in the FMCW technique does not require an additional optical frequency shifter (Jackson, Kersey, Corke and Jones [1982]). Nonlinearities in the frequency ramp produce unwanted sidebands in the frequency spectra of the system. The limitations and noise sources have been discussed (Economou, Youngquist and Davies [1986]) by the LD frequency tuning. The fast flyback (retrace) portion of the ramp waveform induces the noise in the phase measurement. The injection current of the LD in the heterodyne interferometer of fig. 25 is shown modulated by a ramp waveform with period of T , frequency ωs (= 2π/T ),

Fig. 25. Heterodyne interferometer with saw-tooth modulation of an LD.

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[3, § 7

and amplitude of i in the Twyman–Green interferometer. The optical frequency changes with the current variation as illustrated in fig. 5. The current modulation results in output-power changes of the LD as noted in Section 4.4. When a linear light–power relationship with current change is assumed, the variation of the output intensity is 1 + ρt/T , where ρ is defined in Section 4.4 by the ratio of the laser-power change to the dc power. The temporal interference signal becomes   0 (1 + ρt/T ) 1 + γ cos(ωb t + φ) , s(t) = IM (7.1) which is similar to eq. (4.18). An interference beat frequency is generated between the two beams, as shown in the solid and dashed lines of fig. 5, giving ωb = 2πνk = ωs βil/c from eq. (4.13). Here the time lag τ is written as τ = l/c. By adjusting the amplitude of the modulation current i, we make the beat frequency equal to the ramp frequency, i.e. ωb = ωs . The beat signal s(t) is fed into a lock-in amplifier to measure the phase φ of eq. (7.1) synchronized to the ramp frequency ωs as a reference signal in fig. 25. The experiment with the LD heterodyne interferometer uses as a light source a GaAlAs LD (Sharp LT015) with an operating wavelength of λ = 825 nm at 61 mA current. The injection current of the LD is modulated by a ramp waveform with frequency ωs = 47 Hz in which the intensity modulation index and the visibility are ρ = 1.0 and γ = 0.5 in eq. (7.1), respectively. Figure 26 shows the experimental results for the phase measured with the feedback loop off (a) and on (b). The feedback loop is used to stabilize the LD intensity in the case of ρ = 0.03. The SLD acts as a Fabry–Perot resonator and can be used as an external light-power controller by changing the gain with the SLD current (see Section 4.4; Takahashi, Yoshino and Ohde [1997]). The SLD output through the LD is electrically fed back to the SLD driving circuit so as to control the gain of the SLD. The vertical axis in fig. 26 is the optical path length introduced by the displacement of the object mirror attached to the PZT. The phase changes from 0 to 2π periodically with the optical path length. The phase error with 360◦ periodicity is illustrated in fig. 26a and is the same periodicity as the interference signal in the lower panel of fig. 17. The rms error in fig. 26a is 0.29 rad in comparison with 0.09 rad in fig. 26b, which demonstrates an improvement in the measurement accuracy with the intensity stabilization of the LD source. The phase measurement was performed under the condition of ωb = ωs . There is commonly a difference between the ramp frequency ωs and the beat frequency ωb which causes a systematic phase error of with a 180◦ periodicity (Onodera and Ishii [1996b]). An electronic calibration method equalizing the beat frequency with the ramp frequency can eliminate this kind of phase error. A half period, π/ωb , of the interference beat signal is detected by taking first and second zerocrossing points on the beat signal. The time sequence of the half-period can be

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Fig. 26. The phases are measured from the heterodyne technique in (a) absence and (b) presence of a feedback loop to the SLD. A phase error with 2π periodicity can be seen in (a).

matched to a half-modulation period, π/ωs , by changing the modulation current i of the LD to which the output of a photodiode detecting the fringe signal is fed back (see fig. 23). The feedback loop makes the beat frequency ωb equal to the ramp frequency ωs . The LD is a GaAlAs LD (HL7801) with an operating wavelength of 780 nm at 51 mA current. The frequency-modulated laser light is coupled into an the interferometer of fig. 25 with an OPD of l0 = 60 mm. The intensity of the interference signal is detected by a photodiode and is fed to a lock-in amplifier through an electric divider to measure the phase φ in eq. (7.1). To eliminate the intensity alteration caused by the current change, the photodiode output of the fringe signal is normalized by an electric divider incorporating a monitor photodiode assembled with the LD. Figure 27 shows the experimental result of the measured phase φ for ωb /ωs = 1.3. The optical path length is altered by the movement of the object mirror attached to the PZT. The electronic calibration circuit was operated over the period from Feedback On to Feedback Off in fig. 27. The measured phase error with 180◦ periodicity due to the frequency discrepancy is removed when the feedback is on, demonstrating improved measurement accuracy.

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Fig. 27. The phase is measured by the LD heterodyne interferometer of fig. 25 during the movement of a PZT mirror.

Heterodyne techniques with a frequency-tunable LD have been applied to various fields of interferometric measurements. Spatiotemporal specklegrams produced by a wavelength-shifted interferometer (Takeda and Yamamoto [1994]) have been analyzed to measure the three-dimensional shape of a diffuse object by using the Fourier-transform method (Takeda, Ina and Kobayashi [1982]). Simultaneously with wavelength scanning, a two-dimensional FMCW technique can be applied to the specklegram. The unwanted intensity variation introduced by the LD current change has been filtered out from the shape measurement. A continuous wavelength change of LD is used to measure the effect of the modal noise in optical fibers (Ohtsubo and Kourogi [1989]). A Doppler beat signal is measured by using the frequency shifting in laser-diode velocimetry (Ohtsubo and Aoshima [1989]). Frequency modulation by LD current changes can play a role in separating the Doppler frequency from the low-frequency drift component. A laser diode with polarization optics, which has a two-frequency beam with mutually orthogonal polarizations (Otani, Tanahashi and Yoshizawa [1996]), is used as a heterodyne source, and LD heterodyne polarimetry (Oka, Takeda and Ohtsuka [1991]) is performed by the FMCW technique. 7.1.1. Distance measurement Beheim and Fritsch [1985] have remotely measured displacements to minimum distances of 8 mm with a resolution of 2 µm with the interferometer in fig. 25

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using the FMCW technique in which the inverse period of the beat signal ωb in eq. (7.1) was found to be proportional to the OPD to high accuracy. Subnanometer change of the vibration amplitude on a mirror is measured by a heterodyne technique with use of a real-time phase demodulator at which the frequency of the LD is ramped (Imai and Kawakita [1990]). In fiber-optic interferometers, an optical-fiber sensor has been constructed (Kersey, Jackson and Corke [1983]) in which interferometer outputs in quadrature are generated by adjusting the current change i in such a way that a π/2 phase happens in each LD frequency switching. Fiber fault location (Kingsley and Davies [1985]) has been checked by using the FMCW technique; the measurement range was limited by the source linewidth since coherent detection is required (Uttam and Culshaw [1985]). An optical step frequency reflectometer has been exploited in which a wide frequency tunability over mode hopping is attained by stepping the temperature of the Nd-YAG laser, and a high-resolution fiber fault locator was implemented (Iizuka, Imai, Freundorfer, James, Wong and Fujii [1990]). The amplitude and phase at the location can be measured by the heterodyne technique with an acousto-optical modulator (AOM) reference frequency. Kubota, Nara and Yoshino [1987] developed a laser-diode interferometer using the triangular current modulation of the LD. The two interference fringes with orthogonal polarizations are 1 + γ cos(ωb t + φ) and 1 + γ sin(ωb t + φ) for ρ = 0 in eq. (7.1), respectively; their phase quadrature outputs can measure the sign of the displacement as well as its magnitude by the fringe-counting method. Triangular (dual-slope) frequency modulation in the LD eliminates the noise generated during the fast flyback portion of the ramp waveform (Chien and Pan [1991]). For ρ = 0 in eq. (7.1), two multiplexed Mach–Zehnder fiber-optic interferometers with two different OPDs generate two sets of two different beat signals such that s1 (ω1 ; φ1 ) and s2 (ω2 ; φ2 ) in the upward slope, and s3 (ω1 ; φ1 ) and s4 (ω2 ; φ2 ) in the downward slope of the triangular waveform. The phase can be measured with the sensitivity of 2φ1 and 2φ2 by comparing the beat signals s1 (ω1 ; φ1 ) and s3 (ω1 ; φ1) [or s2 (ω2 ; φ2 ) and s4 (ω2 ; φ2 )] with demodulation circuits such as phase-locked loops (PLLs). The common-mode noise in the phase measurement can be eliminated. This type of double Mach–Zehnder interferometer with a frequency-tunable short external-cavity LD has been implemented in a monolithic optical circuit to measure distances (Gorecki [1996]). Triangular frequency modulation of the LD has given absolute distance and velocity measurements by using the upward and downward ramps of the interference signal (Chebbour, Gorecki and Tribillon [1994], Chien, Chang and Chang [1995]). Dual-channel interference fringes in phase quadrature in a Michelson interferometer (see fig. 4)

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provide distance and velocity measurements for time-division sinusoidal and triangular modulations of the LD, respectively (Chang, Chien and Chang [1997]). Next to the heterodyne interferometric technique, homodyne demodulation has been demonstrated as a means of directly measuring OPDs with a sinusoidalmodulated LD (Carlisle, Warren and Riris [1996]). An LD single beam with amplitude and FM modulations from 0.1 to 100 MHz is square-law detected by a high-speed photodiode to extract a cosinusoidal signal proportional to the modulation frequency as well as the path length to be measured. Wavelength tuning of an LD has been applied to distance measurements by using the measurement of the phase shift φ in eq. (3.2). Kikuta, Iwata and Nagata [1986, 1987] have first reported the measurement of the absolute distance l by detecting the modulated phase change φ (= k · l) in the interference fringe of eq. (3.2) with the acoustic-optical heterodyne demodulated technique when the wavelength λ and the wavelength change λ of the LD were known in advance. The laser-diode temperature must be stabilized to within 0.02 ◦ C at 17.5 ◦ C to measure a 1.5-mm distance with an accuracy of 1 µm. Another method of measuring φ has been reported (Watanabe and Yamaguchi [2002]). When the wavelength of the LD is shifted sinusoidally, the interference beat signal modulated by the frequency difference between two AOMs is fed into a computer. The computer calculates two interference signals in phase quadrature by Hilbert transform. The time-varying waveform with the sinusoidal modulation frequency is differentiated, and the amplitude of the sinusoidally varying waveform is directly proportional to the OPD to be measured. 7.1.2. Three-dimensional imaging by FMCW technique In a laser-diode interferometer, the wavelength shift of the LD produced by the current ramp introduces temporal carrier frequencies ωb proportional to the path difference described in Section 7.1. By using the generation of a temporal carrier frequency by the LD, Takeda and Kitoh [1992] have made a space-time interferogram for adding spatial carrier frequencies that are orthogonally arranged toward temporary frequency; two different frequencies provide flexibility to the location of the fringe spectra, permitting the efficient use of spatiotemporal bandwidth product and the extraction of multiple-phase objects. The multiple interferograms through a fiber multiplexer with different OPDs by a frequency-ramped LD consist of a superposition of multiple heterodyne signals whose temporal carrier frequencies are proportional to the OPDs and whose phase change is due to the temperature change in the fiber (Yamaguchi and Hamano [1995]). The temperature is sensed at different points by performing the Fourier-transform algorithm

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(Takeda, Ina and Kobayashi [1982]) from multiple interferograms. An interferometer of multiple-layer objects with a free-running multi-mode LD generates the interference fringe in a spectral region through an optical spectrometer (Tan-no, Ichimura, Funaba, Anndo and Odagiri [1994]). The power spectrum as a function of a distinct image distance can be formed by Fourier-transforming this interference fringe, following the Wiener–Khintchine theorem. The method does not need the ramp frequency sweeps, but does need a spectrometer compared with the method of optical frequency domain reflectometry (OFDR). A laser-diode interferometer has been applied to three-dimensional (3D) imaging using the OFDR techniques to reduce the scattering noise from a reflecting glass plate (Yoshimura, Masazumi and Shigematsu [1997]). Let us demonstrate holographic 3D imaging of phase objects using the FMCW technique (Onodera and Ishii [1995a, 1998a]), following the technique of an optical holographic radar (Marron and Schroeder [1993]). A frequency-ramped LD is placed in the heterodyne interferometer shown in fig. 25, in which phase objects Oi are located in the object arm with various OPDs, li , where the subscript i denotes an index from 1 to N and N is the total number of sliced objects. The interference beat frequencies ωbi become ωbi = ωli /(T c) from eq. (4.13); they are proportional to the OPDs li . The electronic tuning makes it possible to extract selectively the interference beat signal assigned by the object on li . The intensity distribution of image holograms consisting of multiple interferograms involving the depth information is given by the sum of the beat signals:   N    s(x, y, t) = IM (x, y) 1 + (7.2) γi (x, y) cos ωbi t + φi (x, y) , i=1

which is the same as eq. (4.14) when kth is replaced with ith OPD. Here φi is the phase distribution of the ith sliced object. When the interference beam on a dissector-camera plane is sampled at a frequency synchronized with the beat frequency, the beat signals with temporal carrier frequencies can be converted into interference fringes with spatial carrier frequencies. This serial-to-parallel conversion produces the spatial carrier frequency given by ξi = ωbi /(fs p), where fs is the sampling frequency of the detector and p is the scan size. The ith interference fringe signal can be selected by tuning the center frequency to ωbi . The tuned interferogram s(x, y, ξi x) corresponding to the image hologram is spatially modulated by the phase distribution of the ith object and has no bias intensity. In order to reconstruct the phase objects separately, the difference, (ωbi+1 − ωbi )/2π , of temporal frequencies between the (i + 1)th and ith beat frequencies must be greater than the bandwidth 2B of the band-pass filter in the tuner: (ωbi+1 − ωbi )/2π  2B. The condition is rewritten as l  li /Q,

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Laser-diode interferometry

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Fig. 28. Three-dimensional imaging of phase objects by the LD heterodyne interferometer of fig. 25.

where l (= li+1 − li ) stands for the range resolution with an OPD between the (i + 1)th and ith sliced objects, and Q is the quality factor of the tuner given by Q = (ωbi+1 − ωbi )/(2π × 2B). The experiments are performed on two sliced phase objects (N = 2) such as the letters G at l1 = 30 mm and B at l2 = 90 mm (fig. 28), whose difference l (= 60 mm) in OPDs is set longer than a resolvable range of l = l2 /Q = 33 mm for Q = 2.7 used for the experiment. The relation of temporal carrier frequencies becomes ωb1 = 3ωb2 . The image hologram shown in the left panel of fig. 28 is focused onto the image dissector camera. The spatial carrier frequencies in the holograms are ξ1 = 3.2π mm−1 and ξ2 = 9.6π mm−1 , which reduce to 1.6 and 4.8 fringes/mm, respectively. The performance in 3D imaging by using the frequency-ramped LD is provided by the Fourier-transform relationship that exists between the spatial-carrier image hologram s(x, y, ξi x) in eq. (7.2) and the 3D reconstruction in amplitude and phase profiles of objects as demonstrated in the right panel of fig. 28. The amplitude distribution of the letter B at position O2 shows the diffraction effect due to defocusing.

7.2. Two-wavelength heterodyne interferometry According to the discussion in Section 4.5 and in order to extend the range of unambiguity, a synthetic wavelength Λ from λ1 and λ2 is used in the heterodyne interferometer. The synthetic phase has been conventionally measured by subtract-

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295

ing two phases obtained by using two lock-in-amplifiers with λ1 - and λ2 -timevarying interference signals (Fercher, Hu and Vry [1985]). Here two-wavelength LD interferometry is demonstrated using the single heterodyne-detection technique (Onodera and Ishii [1995c]). It can be regarded as a time-domain version of the two-wavelength PSI as described in Section 4.5. In the Twyman–Green interferometer shown in fig. 29, two LDs are frequency-modulated by mutually inverted ramp currents with a ramp frequency ωs as shown in the right-hand panel of fig. 5. Each interference beat signal in eq. (7.1) is detected by two photodiodes by separating each signal into its different polarization states. The sum of the interference electric signals, normalized by electric dividers with monitor photodiodes, becomes s(t) = 1 + γ1 (x, y) cos(ωb1 t + φ1 ) + γ2 (x, y) cos(ωb2 t + φ2 ),

(7.3)

from eq. (7.2) for N = 2, and is fed to one lock-in amplifier. The beat frequencies ωbk for wavelength λk are generated by the difference lk /c in propagation time between the reference and object arms as depicted in fig. 29. The condi-

Fig. 29. A two-wavelength LD heterodyne interferometer with one phase meter built into a lock-in amplifier.

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Laser-diode interferometry

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tion ωb1 = −ωb2 = ωs for the beat frequencies implies that the sign of ωb1 is the opposite of that of ωb2 for the mutually inverted ramp modulation of the LD current. This condition assures separate adjustment of the amplitudes of the ramp currents in the two LDs. The combined interference signal s(t) in eq. (7.3) is demodulated with a bandpass filter in synchronism with ωs , and the tested phase is measured using the heterodyne technique. The bandpass-filtered signal is given by sf (t) = πγ1 [exp(iφ1 ) + exp(−iφ2 )] exp(iωs t) = 2πγ1 cos(Ψ ) exp(iΦ) exp(iωs t) for γ1 = γ2 (see eq. (4.23)); this signal gives the measurement in the synthetic phase Φ [= (φ1 − φ2 )/2] with one phase meter. Figure 30 shows the experimental result in which the optical path length is changed from 0 to 2Λ by moving the object mirror at a constant speed with the PZT whose applied voltage VPZT is exhibited. The phase φ1 is measured with a lock-in amplifier using the interference beat signal for a single wavelength (λ1 ). The phase φ1 changes from 0 to 2π periodically with varying optical path length. The phase Φ at a synthetic wavelength Λ = 4.6 µm is illustrated in the lower part of the figure. The phase Φ changes from 0 to π which demonstrates the extension of the measurement range in the object mirror in fig. 29. A two-wavelength interferometer with fractional fringe techniques (Onodera and Ishii [1995b]) has been constructed by using dual frequency-ramped LDs with same-directional ramp currents (see fig. 29). The two phases at the respective wavelengths, φ1 and φ2 , are measured by the heterodyne technique with two lockin amplifiers by synchronizing the interference fringes with the ramped frequency of the LD. An electric circuit has been used to unwrap 2π -phase jumps in the synthetic phase Φ produced by the difference between the fringe orders of φ1 and φ2 . To extend the 3D profile of a step object by using this heterodyne technique, it is necessary to move both PD1 and PD2 (fig. 29) synchronously to cover the fullview measurement of a step object. This is hard to realize due to the precise adjustment required. Two interference fringes are detected by one photodiode attached to an X–Y motorized stage whose output is sent into two lock-in amplifiers. Two interference signals assigned by different frequencies can be filtered out without crosstalk. A 3D step object profile in a 3-mm square area is measured by fractional fringe techniques scanning the stage as shown in fig. 31. The double OPD for the step height is longer than a single wavelength, but the measurement is performed with a synthetic wavelength Λ (= 4.6 µm) by two-wavelength LD interferometry. Dändliker, Thalmann and Prongué [1988] have suggested that the superheterodyne detection scheme in fig. 32 permits high-resolution measurements at arbitrary synthetic wavelength without the need for interferometric stability at the optical wavelengths λ1 and λ2 . The superheterodyne interferometer in fig. 32 consists of two independent heterodyne interferometers working at different frequen-

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Fig. 30. The result for an optical-path-length change longer that a wavelength λ1 . φ1 is the phase at wavelength λ1 . The result for phase Φ demonstrates the PZT movement by two-wavelength heterodyne interferometry.

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Laser-diode interferometry

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Fig. 31. A 3D step object profile is measured by the two-wavelength LD interferometer with fractional fringe techniques scanning the stage.

Fig. 32. Two-wavelength superheterodyne interferometer with dual LDs. (After Dändliker, Thalmann and Prongué [1988].)

cies ν1 and ν2 . The interference signal Ir , having a beat frequency of (f1 − f2 ), is detected by a photodiode. The phase φ at the interference intensity I depends on the synthetic wavelength Λ and can be synchronously examined for displacement L equal to an OPD with the reference frequency (f1 − f2 ). Using two stable LDs emitting different wavelengths, the heterodyne frequency shifts can easily be obtained with two AOMs with f1 and f2 . Margheri, Giunti, Zatti, Manhart and Maurer [1997] have produced the adjacent wavelengths gained from the diffracted and undiffracted beams from a Bragg cell through a single-mode LD for superheterodyne sources. A superheterodyne frequency can be generated by using a reference interferometer with two AOMs. The range reflected from a diffuse rough surface can be measured by using a long synthetic wavelength Λ (= 60 cm) so that a dif-

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Optical coherence function synthesized by tunable LD

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fuser in the target behaves like a mirror. Two-wavelength double heterodyne interferometry corresponding to superheterodyne interferometry is applied for topographic measurements on rough surfaces to improve system stability with a beam deflection arrangement by a grating (Sodnik, Fischer, Ittner and Tiziani [1991]). In the two-wavelength scanning spot interferometer presented by den Boef [1988], two LDs with no modulations generate a set of in-quadrature interference signals at two photodiodes; sin(2πl/Λ) and cos(2πl/Λ) in the Twyman–Green interferometer. In this interferometer, a vibrating mirror is used to introduce the sinusoidal phase modulation. The measurement of a rough aluminum plate is performed with a synthetic wavelength Λ = 50 µm by using λ1,2 ∼ = 780 nm for both LDs. A dual laser-diode heterodyne differential interferometer was developed by Tiziani, Rothe and Maier [1996], who used two interference fringes for two LDs to measure a synthetic phase Φ by point by point heterodyning two laser beams. The two interference beat signals by two AOMs are sent to a lock-in amplifier with the measurement of step heights for a synthetic wavelength Λ = 33.6 µm from λ1 = 810 nm and λ2 = 830 nm. Dual time-multiplexed LDs modulated by sinusoidal currents have been used to measure the displacement driven by a PZT in a fiber-optic interferometer at a synthetic wavelength Λ = 18 µm (Beheim [1986]). The time-multiplexed interference signals, γ cos(ωb t + φ1 ) and γ sin(ωb t + φ2 ), are generated by switching the injection current between two levels in an LD emitting two wavelengths whose difference is 1.2 nm, giving a synthetic wavelength Λ = 280 µm (Williams and Wickramasinghe [1986]). The synthetic phase Φ{= (φ1 − φ2 )/2} is determined by a FM demodulator with a carrier frequency ωb /2π from an AOM. The timemultiplexed beat signal is produced by driving the two LDs with on/off ramp currents and is detected by a single photodiode (Onodera and Ishii [1999a]). The phase at the synthetic wavelength Λ = 4.3 µm can be measured by synchronous detection with one beat signal at the signal input and another beat signal at the reference input of a lock-in amplifier. A two laser-diode mode can vary the spacing of two slits located on an external mirror at a Littman-type external-cavity LD. This type of two-wavelength source has been implemented in a two-wavelength interferometer in the second-order correlation method (Wang, Chuang and Pan [1995]). § 8. Optical coherence function synthesized by tunable LD The possibility of direct frequency modulation by changing the current in the LD has been utilized in the synthesis of an optical coherence function (OCF) (Hotate and Kamatani [1989]). OCF plays an important role in the extraction of

300

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a slice of a three-dimensional transparent object at arbitrary depth (Hotate and Okugawa [1994]). The OCF used for this purpose can be synthesized to have a delta-function-like shape by modulating the laser-diode frequency with the current waveform, and its frequency change consists of multiple pairs of frequencies as shown in fig. 33a. A multi-peak power spectrum with frequency spacing fs is

(a)

(b)

(c) Fig. 33. Synthesis of the coherence function: (a) modulation current wave form of an LD, (b) power spectrum, and (c) delta-function-like coherence function as a coherence length. (After Hotate and Okugawa [1994].)

3, § 9]

Holographic interferometer and phase-conjugate interferometer by tunable LD

301

obtained in the sense of time averaging in fig. 33b. The degree of coherence |γ | in fig. 33c is equal to the coherence length lc = c/(2fs ), assuming that the principle is based on the Wiener–Khintchine theorem in which the coherence function is expressed by a Fourier transform of the spectral power density. The information in the plane of a three-dimensional (3D) object is selectively recorded with a holographic technique by synthesizing the OCF. When the peak of the deltafunction-like coherence length (fig. 33c) is matched to the position to be selected in the 3D object, the image in the desired plane is recorded on a hologram. The image in one plane can be reconstructed through this hologram with the modulation of the laser-diode frequency. An arbitrary coherence function can be synthesized by controlling both amplitude and phase of the power spectrum. The phase in the power spectrum is controlled by a phase modulator inserted into the beam path in the interferometer. The synthesis of an optical coherence function has been applied to fiber-optic distributed sensing systems such as distributed fiber strain sensors through Brillouin scattering (Hotate [2002]).

§ 9. Holographic interferometer and phase-conjugate interferometer by tunable LD Referring to holographic interferometry with efficient use of a frequency-tunable laser diode, real-time phase-measuring holographic interferometry has been demonstrated based on the phase-stepping technique (Ishii [1988]). Additionally, time-average holographic interferometry with sinusoidal phase modulation by a wavelength-shifting LD has been presented for offsetting the argument of the Bessel-function fringes, and then for shifting the bright irradiance to a large vibrating amplitude region (Ishii [2001]). The vibration phase can be extracted from irradiance measurements taken by a detector from three kinds of time-average holographic interference fringes. LD phase-measuring interferometry with a BaTiO3 self-pumped phase-conjugate mirror (SPPCM) has been demonstrated that offers improvements in measurement accuracy through the use of phase-shifting, which can be implemented by modulating the uniform phase of the conjugate wave by varying the laser-diode frequency (Ishii and Uehira [1993]). An SPPCM in an interferometer can cancel out any nonuniform phase changes such as aberrations and turbulence. The same measurement accuracy as with the conventional interferometer has been achieved by use of a BaTiO3 SPPCM with a polarization phase shifter through a frequency-doubled Nd–YAG laser (Krause and Notni [1996]). A phase-conjugate wavefront-matched interferometer with dual BaTiO3 SPPCMs for distance measurement has been presented that can be implemented by varying in a wide range

302

Laser-diode interferometry

[3, § 10

the wavelength of an external-cavity laser diode (Ishii and Takahashi [2002]). Two cat mirrors in an interferometer have been employed to enhance the immunity of the wavefront matching to spatially nonuniform phase distortions. The cat-mirror displacement can be derived from the measurement of the interference beat frequency by the FMCW technique. The system has been characterized over a wide dynamic range from 0.09 mm (wavelength diversity λ = 2.49 nm) to 15.75 mm (wavelength diversity λ = 0.08 nm). The dynamics of an LD with optical feedback from a photorefractive phaseconjugate mirror were experimentally studied by Murakami and Ohtsubo [1999]. An LD with a BaTiO3 photorefractive resonator shows dynamics similar to those with a conventional mirror resonator. Tomographic imaging can be achieved by changing the wavelength-scanning diversity in the LD using a BaTiO3 photorefractive crystal (Sasaki, Yamagishi, Manabe and Suzuki [2000]). The interference pattern between an object and a reference is recorded on a crystal by two-wave mixing. By using the FMCW technique in the sawtooth wave modulation of the LD, the magnitude of the linear phase modulation of the interference pattern (beat signal) caused by wavelength scanning in the LD is proportional to the OPD (object position to be imaged) from eq. (4.13). Moreover, a weak signal beam from the objects can be amplified by two-wave mixing.

§ 10. Conclusions This article reviews the interesting and potentially significant recent advance of laser-diode (LD) interferometry in the direct modulation of the output wavelength of a laser diode to produce a phase shift. The laser diode has many strong points; it is small and lightweight, has high efficiency, and provides the possibility of wavelength scanning by changing its current. LD interferometry is based on the phase difference in an unbalanced interferometer being proportional to the product of the optical path difference and the laser-diode wavelength shift. The laser wavelength can either be stepped to implement the phase-shifting interferometric approach or ramped to perform the heterodyne approach. The wavelength-tunable LD eliminates the need for the mechanical motion of a PZT and the associated high-voltage drive circuit. The Fourier transform of the time-varying intensity of the convolution algorithm renders the phase-extraction routine insensitive to changes in laser-diode power together with the least-squares estimate. A feedback interferometer is used to calibrate the phase shifts to ensure measurement accuracy. In heterodyne interferometry with the FMCW technique, a feedback system can make the beat frequency equal to the modulation frequency without

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introducing the 180◦ periodicity in phase error. Two-wavelength LD interferometry has been developed to extend the phase-measurement range. The phases have to be equally shifted in opposite directions to form a moiré pattern which can be applied to the PSI technique. Two-wavelength LD heterodyne interferometry with one phase meter is a time-domain version of two-wavelength LD PSI. It has been shown experimentally that LDs are promising light sources in single- and twowavelength interferometers. External-cavity laser diodes with wide tunability are now available for a two-wavelength interferometer that controls the interferometric measurement range.

Acknowledgements This article has been achieved with the continuously proficient guidance and encouragement of T. Asakura. The author gratefully acknowledges him. The author thanks R. Onodera for interesting discussions and collaborations during the research. Sincere thanks are due to J. Chen and T. Takahashi for various support in completing the work.

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E. Wolf, Progress in Optics 46 © 2004 Elsevier B.V. All rights reserved

Chapter 4

Optical realizations of quantum teleportation by

Julio Gea-Banacloche Department of Physics, University of Arkansas, 226 Physics Building, Fayetteville, AR 72701, USA

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(03)46004-6 311

Contents

Page § 1. A brief primer on quantum teleportation . . . . . . . . . . . . . . . .

313

§ 2. Optical teleportation of discrete variables . . . . . . . . . . . . . . . .

326

§ 3. Optical teleportation of continuous variables . . . . . . . . . . . . . .

342

§ 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

350

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

350

312

§ 1. A brief primer on quantum teleportation 1.1. Introduction Quantum teleportation, first introduced by Bennett, Brassard, Crépeau, Jozsa, Peres and Wooters [1993], has become one of the most important “primitive” functions in the rapidly-growing field of quantum information theory (see, for instance, Gottesman and Chuang [1999]). It has also been demonstrated experimentally, almost exclusively, so far, in optical systems. There is some very interesting physics behind both the concept and many of the practical implementations, and the purpose of this article is to serve as an introduction to it. The literature on quantum teleportation is already vast, and it would be impossible to do full justice to it, especially to all the many proposed extensions of the original concept. Hopefully, the topics presented here, and the references, will be enough to serve as an adequate starting point for an interested reader.

1.2. Teleportation of a two-state system (qubit) Even for the simplest quantum mechanical system of all, a two-state system (or “qubit”, short for “quantum bit”, in the jargon of quantum information theory), a full specification of its state may require a potentially infinite amount of information. Indeed, such a state may be written generically as |ψ
= α|0
+ β|1
,

(1.1)

where |0
and |1
are arbitrary basis states and α and β are complex numbers (satisfying |α|2 + |β|2 = 1), whose specification to any desired accuracy may require, in principle, any number of (decimal, or binary) digits. Hence, even if one has a known state, and wants to send enough information to allow another party to recreate the state at a remote location, one may need to send, in principle, an infinite number of bits over a classical communication channel. Things are even harder when one is given a system in an arbitrary, unknown state. It is intuitively obvious that there can be no way to obtain all that information from a single measurement of the system’s state. Indeed, it is, in general, 313

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Optical realizations of quantum teleportation

[4, § 1

impossible to ascertain, from a single measurement on a quantum system, what its state was prior to the measurement. Consider, for instance, the case when the system is a single photon, and one wishes to determine its polarization. The states |0
and |1
in (1.1) above could then be orthogonal linear polarization states, and an arbitrary state of polarization would be represented by appropriate values of the numbers α and β. Yet a single measurement, with, for instance, a polarizer aligned along the direction represented by |0
, only discloses the information that α must not have been initially zero, if the photon passes the polarizer; or that β must not have been initially zero, if the photon fails to pass. In other words, the single measurement discloses no more than one bit of information about the original state. Hence, the idea of simply measuring the state of the system and sending that information to a remote party is, generally, not very useful. One would need many copies of the original state, and many different measurements, one on each copy, to gather a sufficiently accurate representation of the coefficients α and β. Yet – in another very interesting twist – quantum mechanics does not allow one to make an exact copy of an unknown state. This celebrated result is known as the “nocloning theorem” (Wootters and Zurek [1982]). From all of the foregoing, it might appear as if the only way to transport the unknown state |ψ
to the remote location were to physically move the system in question to that location. Quantum teleportation, however, provides an alternative, both to this “unknown state” problem and to the “known state” problem mentioned earlier (namely, the need to send an, in principle, infinite number of bits of information to prepare a known state remotely). What teleportation does require, in return, is that both sender and receiver share ahead of time a maximally entangled1 state of two identical particles (in the simplest case, identical, also, to the particle whose state they are trying to send). It is straightforward to show formally how teleportation works for a qubit. In accordance with convention, let us call the sender Alice and the receiver Bob, and assume that initially each one has a particle (numbered 2, for Alice, and 3, for Bob) which are jointly in the state
(−)   1 
Ψ 23 = √ |01
23 − |10
23 2

(1.2)

1 Throughout this paper, a number of standard terms of quantum information theory (such as en-

tanglement, quantum gates, quantum key distribution, etc.) will be used without definition. Readers unfamiliar with these terms may wish to consult some of the standard texts in this field, such as Nielsen and Chuang [2000] and Bouwmeester, Ekert and Zeilinger [2000]. These references, incidentally, also contain good discussions of quantum teleportation.

4, § 1]

A brief primer on quantum teleportation

315

(alternatively, any of the maximally entangled states of the form given below, in eqs. (1.3) and (1.4), could be used for this purpose). In addition to this, Alice has a particle (numbered 1) in a state |ψ
, which may be known or unknown to her, but that she wants to re-create at Bob’s location. It turns out that she can accomplish this simply (in theory!) by just measuring the joint state of particles 1 and 2 in the so-called “Bell operator basis” consisting of the four states
(±)   1 
Ψ (1.3) 12 = √ |01
12 ± |10
12 2 and
(±)   1 
Φ 12 = √ |00
12 ± |11
12 . 2

(1.4)

To see what the possible outcomes of such a measurement can be, note that the (−) joint state |ψ
1 |Ψ23
(with |ψ
1 given by (1.1)) can be rewritten as
(+) 
(−)  1 
(−)    |ψ
1
Ψ23 = 2 −
Ψ12 α|0
+ β|1
3 +
Ψ12 −α|0
+ β|1
3
(−) 
(+)     +
Φ12 α|1
+ β|0
3 +
Φ12 α|1
− β|0
3 . (1.5) This shows that if, for instance, the result of Alice’s measurement is the state (−)
(something which would occur with probability 1/4), then Bob already |Ψ12 has in his hands the state |ψ
, “teleported” to him through the correlation shared by his particle 3 and Alice’s particle 2. For Alice’s other possible measurement outcomes, Bob only needs to apply some simple transformations to his particle to reproduce the state |ψ
. For instance, in the case of a photon, with |0
and |1
(+) orthogonal linear polarization directions, the outcome |Ψ12
would only require Bob to establish a π -radian phase difference between the two components (which (+)
requires, in can be done with a half-wave plate), whereas the outcome |Φ12 addition to this, to exchange the x- and y-directions. In any case, note that the information about the |ψ
state completely disappears from Alice’s side. This feature of the scheme, which is, perhaps, the one most reminiscent of the “teleportation” found in science-fiction stories, is, of course, required by the no-cloning theorem, since otherwise one would have succeeded at cloning the original state2 . All four outcomes occur with equal probability, so, if 2 Residual information may be left on Alice’s side only when the teleportation is “imperfect”, that

is, when it fails to generate a 100% faithful replica of the original state; this is consistent with the fact that quantum mechanics does allow for “imperfect cloning” (see Buzek and Hillery [1996]). Imperfect teleportation is discussed later in this article.

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Optical realizations of quantum teleportation

[4, § 1

the state |ψ
was unknown to Alice to begin with, she cannot even use the result of her measurement to gather any information on the values of α and β. Meanwhile, on his side, Bob does need to know the result of Alice’s measurement in order to know what needs to be done to his particle 3 to re-create the state |ψ
. Without this information (which is equivalent to two classical bits, for the four possible outcomes), his knowledge about the state of particle 3 is represented by a density matrix consisting of all four alternatives present in eq. (1.5), all with equal weights. It is easily seen that such a state is equivalent to the totally mixed (unpolarized) state ρ = 12 |0
0| + 12 |1
1|.

(1.6)

This shows that no superluminal communication takes place in this scheme: Bob does not really know anything about the state of his qubit until he receives (through a conventional, that is, subluminal, channel) the information about the outcomes of Alice’s measurements. Nonetheless, it is remarkable that just those two classical bits are enough to (in principle) recreate exactly a state whose complete specification, as argued above, may require an infinite number of bits. Bennett, Brassard, Crépeau, Jozsa, Peres and Wooters [1993] claim that teleportation naturally splits the information into quantum and classical parts; the quantum part is transmitted “instantaneously” through the entanglement shared by particles 2 and 3, when the joint measurement of 1 and 2 is carried out. That the “classical” part amounts to two bits per qubit can be taken to be related to the phenomenon now called “superdense coding” (Bennett and Wiesner [1992]) and referred to as “four-way coding” by Bennett, Brassard, Crépeau, Jozsa, Peres and Wooters [1993]. Clearly, this seemingly very simple protocol already raises interesting questions about the nature of quantum and classical information, and how they are stored and manipulated in the process. Some articles addressing these issues in various ways are Braunstein [1996], Deutsch and Hayden [1999], Griffiths [2002], Koashi and Imoto [2002]. One feature of the scheme that needs to be stressed is the fact that (unlike in most science-fiction stories) quantum teleportation cannot work unless the receiver and the sender already share a pair of quantum particles, identical to the particle being teleported, in a highly entangled state. This situation may have come about in a number of ways, but in all likelihood it means that, at one time, there was a “quantum channel” (that is to say, some means to send a quantum system while preserving its state, and its possible entanglement to other systems) open between Alice and Bob. That being the case, why would Alice choose to

4, § 1]

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317

use teleportation, instead of just sending the state |ψ
down the quantum channel directly? A possible answer is provided by the assumption of a channel that is not 100% reliable. If there is a probability that the state of the particle sent may be damaged during the transmission, then it would make sense to first send, possibly, a large number of entangled partners down the channel; then use distillation techniques (see Bennett, DiVincenzo, Smolin and Wootters [1996]) to extract from them a small fraction, which would have a large probability of being in a maximally entangled state; and then use one of those pairs as a resource to teleport the state |ψ
(which, unlike the state of the entangled pairs, is probably unique and unknown, and hence not expendable). In this way, teleportation can be used to turn a noisy channel into a virtually noiseless one, at the cost of a reduction in the transmission rate, and provided only that storage of the entangled pairs for the required amount of time be feasible.

1.3. Generalization to an N -state system In their original paper, Bennett, Brassard, Crépeau, Jozsa, Peres and Wooters [1993] did not just consider two-dimensional systems, but also provided a generalization of the teleportation protocol for systems of arbitrary dimension N . The initial entangled state of particles 2 and 3 shared by Alice and Bob should be any member of the set
 1  2πij n/N |ψnm
= √ (1.7) e |j
⊗
(j + m) mod N N j (all indices run from 0 to N − 1), and Alice’s joint measurement on particles 1 and 2 should be one that completely discriminates between all the N 2 states |ψnm
in (1.7). As before, the precise transformation that Bob needs to apply to his particle depends on the outcome of Alice’s measurement, and hence on 2 log2 N bits of classical information which he needs to receive from Alice. As an example, if the initial state of particles 2 and 3 is chosen to be |ψ00
23 and the outcome of Alice’s measurement is nm, then Bob’s needed transformation is 
 e2πikn/N |j
(j + m) mod N
Unm = (1.8) k

as can be easily verified by direct calculation, that is, by applying (1.8), acting on particle 3, to 12 ψnm |ψ
1 |ψ00
23 , where |ψ
is an arbitrary state of particle 1 of the form N−1 j =0 cj |j
1 .

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Bennett, Brassard, Crépeau, Jozsa, Peres and Wooters [1993] did not provide any particular interpretation of this procedure or of the set of states (1.7), and, of course, in general the states |j
appearing there could be any orthogonal basis of the N -dimensional Hilbert space. Nonetheless, with the benefit of hindsight, the following possible realization may be found very enlightening. Suppose that the system in question is a particle in one dimension of space, and the states |j
correspond to wavefunctions localized in a region of width a around √ the point with x = j a (e.g., rectangles of height 1/ a and width a); suppose also that periodic boundary conditions apply, so that |j = N
= |0
. By treating the states |j
as a complete set, in this “discretization” scheme, we are essentially ignoring any spatial details on a scale smaller than a, so there is a natural momentum cutoff of π h/a. Also, the boundary conditions lead to momentum quantization in ¯ units of 2π h¯ /L, where L = Na. Under these conditions, the states in (1.7) can be seen to correspond to a situation in which each particle (say, particles 2 and 3) is completely delocalized over the length L = Na of the whole lattice, but the difference between the two positions is well-defined and has the value x3 − x2 = ma. Moreover, the joint momentum wavefunction can be calculated from (1.7) as φnm (p2 , p3 ) =

g(p2 )g(p3 )  −ip2 j a/h¯ −ip3 (j +m)a/h¯ 2πij n/N e e e , √ 2π h¯ N j

(1.9)

where g(p) is the diffraction function for a rectangular opening, which, like the overall factor of e−ip3 ma/h¯ , is not really relevant for what follows. When the sum (1.9) is evaluated, and the quantization condition for p2 and p3 is taken into account, it is easily seen to vanish unless 2π h¯ 2π h¯ mod . (1.10) L a In short, then, the states (1.7) are, in this discretization scheme, simultaneous eigenstates of x2 − x3 with eigenvalue ma, and of p2 + p3 with eigenvalue 2nπ h¯ /L; in other words, they are a discretized version of the original Einstein– Podolsky–Rosen (EPR) state(s) (Einstein, Podolsky and Rosen [1935]). Note also that the measurement which projects the pair (1, 2) onto one of the states (1.7) is essentially a measurement of the commuting observables x1 − x2 and p1 + p3 , just like the one originally envisioned by Einstein, Podolsky and Rosen. These results will make it very easy to appreciate the generalization to continuous variables to be discussed in the next subsection, but before leaving the subject of multi-(discrete) dimensional quantum teleportation one should mention the systematic study by Stenholm and Bardroff [1998] of the types of entangled p2 + p3 = n

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states and operations useful for that purpose. (Note also that van Enk [1999] has presented a construction of discrete teleportation from continuous teleportation which is, in a way, the inverse of the approach followed here.)

1.4. Continuous-variable teleportation Vaidman [1994] presented the first scheme for continuous variable quantum teleportation: that is, a scheme to reproduce the complete wavefunction of a particle 1 in a particle 3 at a remote location, through measurements made locally on particles 1 and 2, and classical communication. His method can be understood as an exact parallel to that of Bennett, Brassard, Crépeau, Jozsa, Peres and Wooters [1993], for the multidimensional case, only generalized to continuous degrees of freedom as suggested at the end of the previous subsection. Indeed, in Vaidman’s proposal Alice and Bob must share an EPR state of particles 2 and 3, such that, for instance, x2 − x3 = 0,

p2 + p3 = 0

(1.11)

(of course, any value other than 0 would also work). Then Alice makes a joint measurement of x1 − x2 and p1 + p2 on her particles 1 (as before, the one whose state is to be teleported) and 2. Say that the outcomes are a and b, respectively, so that x1 − x2 = a,

p1 + p2 = b.

(1.12)

Vaidman then claims that the original correlation (resp., anticorrelation) (1.11) between particles 2 and 3, together with the correlation (resp., anticorrelation) (1.12) between 1 and 2, leads to a correlation between particles 1 and 3, so that one now has x3 = x1 − a,

p3 = p1 − b

(1.13)

and the probability (amplitude) of each of these values is weighted by ψ(x1 ), the original state of particle 1, so that, at the end of the process, the state of particle 3 is e−ibx3 /h¯ ψ(x3 + a). A moment’s reflection shows that this is, indeed, the original state, only displaced in momentum by −b and in position by −a. (A trivial sign error in Vaidman’s original article has been corrected here.) The reasoning above is intuitively appealing, but one might have some doubts as to its applicability, for, although there is no question that (1.13) follows directly from (1.12) and (1.11), it is also a fact that the measurement on particles 1 and 2 disturbs the total wavefunction in such a way that, once (1.12) holds, (1.11) does

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not anymore. More precisely, since p1 + p2 does not commute with x2 − x3 , one would expect that the first of eqs. (1.11) would not be applicable, if x1 − x2 was measured after p1 + p2 ; conversely, the second of eqs. (1.11) would not apply if p1 + p2 was measured after x1 − x2 . But these concerns are unfounded: the essential point is that x1 − x2 and p1 + p2 do commute with each other, and hence it cannot matter in which order they are measured. Thus, both conclusions expressed by eqs. (1.13) are correct, even though only one can be established at a time. A more formal proof for truly continuous wavefunctions (as opposed to the “discretized” ones considered in the previous subsection) can be given along the lines, for instance, of Section II of Milburn and Braunstein [1999]. A simultaneous eigenstate of the operators xˆ2 − xˆ3 and pˆ2 + pˆ 3 , with eigenvalues X, P , can be written formally as |X, P
23 = e−ipˆ 2 xˆ3 /h¯ |X
2 |P
3 ,

(1.14)

where the states |X
2 and |P
3 are defined as xˆ2 |X
2 = X|X
2 and pˆ 3 |P
3 = P |P
3 . It is easy to verify that (xˆ2 − xˆ3 )|X, P
23 = X|X, P
23 and (pˆ 2 + pˆ3 )|X, P
23 = P |X, P
23 directly from the explicit expression (1.14) and the known results [x, ˆ f (p)] ˆ = ih¯ ∂f/∂ pˆ and [p, ˆ f (x)] ˆ = −ih¯ ∂f/∂ x. ˆ Now consider another particle in state |ψ
1 and a joint measurement of xˆ1 − xˆ2 , yielding a value X , and pˆ1 + pˆ 2 , yielding a value P  . The (unnormalized) state of the whole system after the measurement is given by the projection 12 X



P  |ψ
1 |X, P
23 = 1 X | 2 P  |eipˆ 1 xˆ2 /h¯ e−ipˆ 2 xˆ3 /h¯ |ψ
1 |X
2 |P
3 = 1 X |eipˆ 1 xˆ3 /h¯ e−iP

 xˆ

¯ 3 /h

2 P



|X
2 eipˆ 1 X/h¯ |ψ
1 |P
3

1   e−iP X/h¯ ψ(X + X + xˆ3 )e−iP xˆ3 /h¯ |P
3 . =√ 2π h¯ (1.15) Here, the Baker–Hausdorff formula has been used to commute the two exponentials on the first line, and the fact that eipˆ 1 xˆ3 /h¯ is a position displacement operator when acting on the state |ψ
1 has been used to go from the second to the third line; ψ(x) = 1 x|ψ
1 is the wavefunction of particle 1. Inserting now a complete set of states

1 = |x
33 x| dx (1.16) in (1.15), just in front of the state |P
3 , immediately shows that (apart from normalization and an unimportant phase factor) the wavefunction for particle 3 is

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indeed given by ψ(x + X + X )ei(P −P

 )x/h

¯,

(1.17)

which is the same as the original wavefunction for particle 1, only displaced both in momentum and position. The amount of the displacement needs to be communicated by Alice to Bob, so he can apply a suitable transformation to restore the original wavefunction; this information is contained in the outcome of Alice’s measurement, X and P  (since X and P were presumably initially known to both parties). Note that, since X and P  are, in principle, arbitrary real numbers, it takes a potentially infinite amount of classical communication from Alice to Bob to perform continuous-variable teleportation with arbitrary accuracy. The most interesting aspect of continuous-variable teleportation from an optics point of view is that, as first realized by Braunstein and Kimble [1998], the socalled “squeezed vacuum states” of the electromagnetic field (which can be produced, for instance, in optical parametric oscillators; see, e.g., Chapters 2 and 16 of Scully and Zubairy [1997]) may provide a good approximation to the EPR states such as (1.14), and could, therefore, be used to teleport arbitrary states of the quantized electromagnetic field. This will be discussed in more detail below, in Section 3.

1.5. Imperfect teleportation The previous sections have considered the ideal teleportation of an arbitrary state: if the quantum resource is a maximally entangled state, and the necessary measurements and unitary transformations are perfect, the output state will be an identical copy of the input. In practice, of course, imperfections of one sort or another will be unavoidable. The standard figure of merit to characterize the performance of a real-life teleportation setup is, as almost everywhere else in quantum information, the fidelity of the output state ρout (at Bob’s station, after the whole process is over) to the input state |ψin
(originally at Alice’s station). Here it is assumed that the initial state is pure, but the final state could be mixed, and therefore described by a density matrix. The fidelity in this case is defined as F = ψin |ρout |ψin


(1.18)

and can be interpreted as the probability to measure the state |ψin
in the output ρout (for a pure-state output, it is just the absolute value squared of the overlap, or inner product, between the two states). Of course, if ρout = |ψin
ψin |, the teleportation is perfect and F = 1.

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When the teleportation protocol is not perfect, the fidelity (1.18) is typically found to depend on the initial state; in that case, theoretical treatments typically concentrate on the average of the quantity (1.18) over all possible input states with an appropriate measure. As an example of this procedure, and also to introduce a useful (although perhaps overused) benchmark, consider what happens in the extreme case in which the state shared by Alice and Bob is not entangled at all. In this case, Alice’s measurements can have no effect at Bob’s location, and their only practical resource is the classical channel. The maximum fidelity achievable under these circumstances is sometimes called the “classical fidelity”, and can be calculated as follows for a two-dimensional system (i.e., a qubit). As mentioned in the Introduction, Alice can get at most one bit of information by making a measurement on her system. Suppose its state (unknown to her) is |ψin
= α|0
+ β|1
, with |α|2 + |β|2 = 1. Let her then measure this state in the |0
, |1
basis. With probability |α|2 she will obtain the result |0
, which she can communicate to Bob through the classical channel; likewise, with probability |β|2 she will obtain |1
and communicate this to Bob. If Bob then simply prepares whatever state Alice obtains, the final state can be described by the density operator ρout = |α|2 |0
0| + |β|2|1
1|, and, by eq. (1.18), the fidelity of this to the initial state will be θ θ F = |α|4 + |β|4 = cos4 + sin4 (1.19) 2 2 using a natural parametrization for the qubit’s “Bloch sphere”, with the angle θ ranging between 0 and π . The average fidelity is then obtained by averaging over this angle, with a density 12 sin θ (which corresponds to a uniform distribution of |α|2 between 0 and 1). The result is Fcl = 23 .

(1.20)

To do better than this, on average, Bob and Alice need to share some entanglement. The dependence of the average fidelity on the kind of entanglement shared, however, has turned out to be a nontrivial question, with a rather interesting history. It was first observed by Popescu [1994] that it is possible to find entangled states that do not violate any Bell inequalities, yet they are sufficiently “nonlocal” to allow for an average fidelity greater than Fcl when used for the teleportation of qubits (note that all pure entangled states violate a Bell inequality, but there are entangled mixtures that do not: see Werner [1989]). Popescu naturally asked then whether all entangled states must yield an improvement in teleportation fidelity over the result (1.20), regardless of whether they violate a Bell inequality or not. A partial answer to this question was provided by Horodecki, Horodecki and Horodecki [1996], who showed that (a) all entangled states of qubits that violate

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Bell’s inequalities are “useful” for teleportation, in the sense of leading to an average F > Fcl , and (b) there are, however, entangled states which are “not useful” in this sense. The more complete result presented in Horodecki, Horodecki and Horodecki [1999] relates the usefulness of a state for teleportation to its “singlet fraction” or “fully entangled fraction” f , defined as (Bennett, DiVincenzo, Smolin and Wootters [1996]) f (ρ) = maxψ|ρ|ψ
,

(1.21)

where ρ is the density matrix characterizing the state that Alice and Bob share, and the maximum is taken over all the maximally entangled states |ψ
of the system of two particles (of arbitrary dimension N ; for instance, over the set (1.7)). As shown in Horodecki, Horodecki and Horodecki [1999], the average teleportation fidelity achievable with a state that has a given f is equal to = f N + 1. F (1.22) N +1 For a non-entangled state (equivalent to the case when only a classical channel is available), f = 1/N , yielding the N -dimensional generalization of the formula (1.20): 2 (1.23) N +1 (see Barnum [1998]). Thus, in order to be “useful” for the teleportation of an N -dimensional system, the state shared by Alice and Bob has to have a fully entangled fraction f > 1/N . The existence of states that are entangled, yet have f < 1/N , is actually a surprising result of entanglement theory, as is the fact that the “usefulness” of some of these states for teleportation can, in some cases, actually be increased, from f < 1/N to f > 1/N , through local operations only (Badziag, Horodecki, Horodecki and Horodecki [2000]); the subject will, however, not be discussed further here. It should be noted that the measure used to compute the “classical” fidelity may be (and generally is) chosen differently depending on the set of states one may actually consider transmitting in a given experiment. For instance, for the teleportation of continuous variables, an average over all possible input states would yield an Fcl of zero, as N → ∞ in (1.23), but in practice one typically chooses to average over a much smaller subset of possible input states, resulting in a nonzero Fcl (see Section 3 below). Although Fcl provides a useful reference point, its actual importance as a “threshold” value may have been overemphasized in the literature. It would be wrong to assume, for instance, that any teleportation experiment yielding an Fcl =

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F < Fcl is, in some sense “not quantum”. Even calling “useful” for teleporta > Fcl , as was done above, may be somewhat tion only those states which yield F misleading, as shown by the following considerations. It is clear that, in the “classical-channel only” protocol, when Alice measures the state of her particle in the {|0
, |1
} basis, any prior entanglement that might exist between that particle and another one is lost. Yet, from this it does not follow that any teleportation with fidelity
Fcl must “break” such prior entanglement. It is, in fact, a simple exercise to see that if one of the states with f = 1/2 given in, for instance, Badziag, Horodecki, Horodecki and Horodecki [2000], is used for teleportation, and particle 1 is initially maximally entangled with another particle 1 , then at the end of the teleportation process particle 3 will be, in turn, partially entangled3 with 1 . (This process is an example of partial “entanglement swapping”; see Zukowski, Zeilinger, Horne and Ekert [1993].) In other words, even teleportation with F
Fcl may (partially) preserve nonclassical features of the states being teleported.

1.6. Possible applications and extensions Today, in the quantum information community, teleportation is widely expected to become eventually a preferred method to transport quantum information from one site to another. Once a maximally entangled state has been set up between two parties, it can be used to teleport an arbitrary state (of the appropriate dimensionality), and, if the requisite quantum measurements can be carried out with sufficient accuracy, the transfer can be made essentially noiseless. Setting up the initial entangled state may be a time-consuming process if the quantum channel connecting the two parties is very noisy, but one can always, in principle, distill a maximally entangled state from a sufficiently large number of imperfect trials, provided the fidelity of the transfer does not drop below a certain threshold, which is typically more forgiving for the entanglement purification schemes than for the quantum error-correcting code schemes (Bennett, DiVincenzo, Smolin and Wootters [1996]). Based on this idea, a number of authors have proposed networks involving quantum repeaters (Briegel, Dür, Cirac and Zoller [1998]) which could 3 Formally, this follows from the fact that Alice’s Bell measurement on particles 1 and 2 can be seen equivalently as either setting up the (imperfect) teleportation of the state of 1 to 3, through the partial entanglement shared by 2 and 3, or as setting up the teleportation of the state of 2 to 1 , through the maximal entanglement shared by 1 and 1 . The latter operation would leave 1 and 3 sharing the same kind of entanglement originally shared by 2 and 3; local unitary operations at the locations of 1 or 3 would not change the magnitude of this entanglement.

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provide secure and virtually noiseless transfer of quantum information over long distances (see also, for other quantum network proposals, Cirac, van Enk, Zoller, Kimble and Mabuchi [1998], and references therein). Essential to the feasibility of these schemes, however, is the need for very long-lived quantum memories where the entangled states can be stored while the overall purification process is under way. Another potentially very useful application of quantum teleportation would be in quantum computing. In the course of a typical quantum computation, one may need to carry out quantum logical operations involving distant qubits, yet only a few quantum computer architectures have been proposed which would allow a direct interaction between arbitrary pairs of qubits; most schemes rely, instead, on local interactions, limited to nearest-neighbor pairs. Under these conditions, one way to carry out an operation on a distant pair would be to use a sequence of “swap” operations, in which the state of one qubit is moved along the computer until it is next to the qubit with which it is supposed to interact. This scheme, however, may lead to long delays and an accumulation of errors along the way. An alternative scheme would instead set up, ahead of time, an entangled pair of qubits near the locations of the two that need to interact, and then use teleportation to perform the quantum logical operation (Brennen, Song and Williams [2003]). Note that an entangled resource actually allows one to teleport an operation as well as a state; see, e.g., Gottesman and Chuang [1999], Zhou, Leung and Chuang [2000]. In this “teleportation bus” scheme, suppose that distilling a maximally entangled state requires, on average, a time of the order of Nτ , where τ is the time needed for an elementary quantum logical operation (between nearest-neighbor qubits, for instance). Then one can “look ahead” in the computation to identify the next N operations involving non-neighboring qubits, and start setting up the appropriate entangled resources, concurrently, but with a delay of the order of τ between consecutive starting times. By the time Nτ , the first entangled pair needed will be ready; a time τ afterwards the next pair will be ready to use, and so on. Clearly, overall, the calculation could proceed in this way, after an initial delay of the order of Nτ , at a rate set by τ , even for nonlocal operations. Note, however, that this scheme also requires long-lived memories, of the order of Nτ , to store the intermediate pairs generated during the distillation process. Lastly, one should mention that there have been many extensions of the original teleportation concept, too many to review in detail here. The following list is intended only to provide some representative examples: multiparty teleportation (Karlsson and Bourennane [1998], Dür and Cirac [2000]), telecloning (Murao,

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Jonathan, Plenio and Vedral [1999]), teleportation of operations (Huelga, Vaccaro, Chefles and Plenio [2001]), remote state preparation (Bennett, DiVincenzo, Shor, Smolin, Terhal and Wootters [2001]), and many proposals using different kinds of states as the entangled resource. Of the latter, which are really too numerous to mention, only a few (those having, so far, led to experimental realizations) will be explicitly discussed in the next couple of sections.

§ 2. Optical teleportation of discrete variables 2.1. The difficulties of Bell-state measurements As mentioned in the introduction, the two orthogonal polarization states of a photon provide a natural two-state system. More importantly, optical parametric amplifiers naturally generate polarization-entangled pairs of photons, so producing the entangled resource, of the form (1.2), that Alice and Bob need to share, is, in principle, a straightforward matter. The difficulty arises, however, when considering the Bell-state measurement that Alice needs to perform on her photons 1 and 2. This is supposed to be a projective measurement that can discriminate between the four possible two-photon states |Ψ (±)
and |Φ (±)
in eqs. (1.3) and (1.4), where |0
and |1
represent orthogonal polarization states. Now, in principle, if one could establish an interaction between the two photons strong enough to perform a “CNOT” quantum logical gate with, say, photon 1 as the control and photon 2 as the target, the task would become straightforward. The CNOT operation leaves the state of the target unchanged if the state of the control is |0
and flips it if the state of the control is |1
, so the states (1.3) and (1.4) are transformed as  1  |Ψ (±)
→ √ |0
± |1
|1
, 2  1  |Φ (±)
→ √ |0
± |1
|0
2

(2.1)

and then a simple measurement of the polarization of both photons in the appropriate bases allows one, in principle, to discriminate perfectly all the four states (2.1). Unfortunately, the experimental schemes currently available to make photons interact (typically involving nonlinear optical media) are not yet capable of producing the very large phase shifts needed to achieve something like a CNOT gate, at the single-photon level. When planning to use teleportation in, for instance, a

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quantum computing context, it is only natural to assume that a CNOT gate will be, eventually, available, in one form or another; but, in the meantime, experimenters anxious to demonstrate the principles of quantum teleportation with photons have been forced to resort to other, generally more indirect, methods, to accomplish something resembling the ideal Bell-state measurement assumed in the previous section. An interferometric scheme, which uses only linear optical devices (such as beam-splitters and polarization analyzers), and allows, in principle, to distinguish two of the Bell states from the other two, was first proposed by Weinfurter [1994], and independently by Braunstein and Mann [1995]. The method uses a variation of the celebrated Hong–Ou–Mandel interferometer (Hong, Ou and Mandel [1987]). The “interference” here refers to what happens when two photons arrive simultaneously on both sides of a beam splitter, and there are two different ways for them to exit in a particular configuration; the probability amplitudes corresponding to these two different quantum mechanical “paths” then interfere, either constructively or destructively, depending on their relative phases. Let a and b denote the input modes of√a 50–50 beam splitter, √ and c and d the output modes; assume that a = (c + d)/ 2 and b = (c − d)/ 2, and denote by a  , b  , c , d  the corresponding modes with the orthogonal polarization. If two photons in states |Φ (±)
or |Ψ (±)
are sent into the beam splitter, the incoming and outgoing states are, respectively, 1 1 |Ψ (+)
= √ (ab + a  b)† |vac
= √ (cc − dd  )† |vac
, 2 2 1 1 |Ψ (−)
= √ (ab − a  b)† |vac
= √ (dc − cd  )† |vac
, 2 2 |Φ

(+)

1 1
= √ (ab + a  b )† |vac
= √ (cc − dd + c c − d  d  )† |vac
, 2 2 2

(2.2)

1 1 |Φ (−)
= √ (ab − a  b )† |vac
= √ (cc − dd − c c + d  d  )† |vac
, 2 2 2 where |vac
represents the four-mode vacuum. Thus, in the case in which the incoming photons have the same polarization, and are therefore completely indistinguishable (the two states |Φ (±)
), quantum interference, just as in the original Hong–Ou–Mandel experiment, rules out the possibility of each photon coming out at one port: the final state, as shown by (2.2), is a superposition of both photons leaving through the first channel and both photons leaving through the other channel. The relative phase of the states in this superposition does depend on whether the input state is |Φ (+)
or |Φ (−)
,

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Fig. 1. Scheme to distinguish two of the four Bell states. The diamonds are polarizing beamsplitters, reflecting one polarization and transmitting the orthogonal one.

but it is otherwise unobservable in this particular polarization basis; consequently, the method based on eqs. (2.2) and shown in fig. 1 cannot discriminate between |Φ (+)
or |Φ (−)
. On the other hand, it can discriminate between these states and |Ψ (±)
, as well as among |Ψ (+)
and |Ψ (−)
; as eqs. (2.2) show, when the incoming photons are in the antisymmetric state |Ψ (−)
, the outgoing photons are invariably separated, whereas, if they are in the symmetric state |Ψ (+)
, they both leave the beamsplitter along the same path, but since they have different polarizations this case can be told apart from the case(s) |Φ (±)
. Hence, the arrangement of beamsplitters, polarizing beamsplitters, and detectors shown in fig. 1 can detect the states |Ψ (±)
unambiguously. If the input to the first beamsplitter is the unknown state of particle 1 and the EPR-pair-member particle 2, eq. (1.5) shows that half of the time the outcome of the measurement will be either |Ψ (−)
12 or |Ψ (+)
12 . If the outcome is |Ψ (−)
(experimentally, if one of the (c, c ) and one of the (d, d  ) detectors click), then, by eq. (1.5), the state of particle 1 has in fact been teleported to particle 3. If the outcome is |Ψ (+)
(detectors c and c click, or detectors d and d  click) the state teleported is −α|0
+ β|1
, and Bob needs to introduce a π phase shift between the two polarization components of his photon 3 (with a half-wave plate) to recover the initial state. If, on the other hand, the outcome is none of the above (that is, if only one of the detectors in fig. 1 clicks, as both photons impinge on it), then the teleportation has failed; since we do not know whether particles 1 and 2 were left in state |Φ (+)
or |Φ (−)
, the state of photon 3 is a 50–50 mixture of the third and fourth terms in eq. (1.5), which is to say, |α|2 |1
1| + |β|2 |0
0|. One may ask how general this result is, and if there is no way to improve on it by further manipulations of the photons. The question was addressed in Vaidman and Yoran [1999] and Lütkenhaus, Calsamiglia and Suominen [1999],

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where it was shown that any attempts at Bell-state analysis based only on linear optics and single photon detectors can only discriminate two of the four Bell states unambiguously. Generalizations of these results, to, for instance, include auxiliary modes in the vacuum state (Calsamiglia and Lütkenhaus [2001]), and to higherdimensional systems (Calsamiglia [2001]) have subsequently been given. Partly to address this difficulty, in the context of quantum computation with linear optics, a modified teleportation protocol was introduced in Knill, Laflamme and Milburn [2001], which requires 2n auxiliary photons to teleport the state of a single one, and can be realized with linear optics and photon-number detectors, with a failure probability equal to 1/(n + 1). This scheme will be discussed in detail in Section 2.5, after the experiments based on the standard teleportation protocol (with perhaps slight modifications) have been considered. The impossibility to carry out a full Bell-state measurement with linear optics has led, thus far, to basically three different experimental approaches to discretevariable optical teleportation. One modifies the original protocol so that a single photon plays the part of photons 1 and 2; others simply content themselves with a probabilistic outcome of the sort indicated above, where teleportation only actually happens some of the time; and others use very large numbers of input photons to overcome the inefficiency of nonlinear optical processes at the two-photon level. These three approaches will be discussed at some length in the next three sections.

2.2. Teleportation using entanglement between different degrees of freedom of the same photon In what was arguably the first experiment on quantum teleportation, Boschi and coworkers transferred the polarization state of a photon to another one with which the first one was momentum-entangled (Boschi, Branca, DeMartini, Hardy and Popescu [1998]). In this scheme only two photons, rather than three, are used altogether, with one of them playing a dual role: its polarization provides the state to be teleported, and its momentum entanglement with the other photon provides the entanglement resource needed to make teleportation work. How the scheme works in theory may be best explained by simply describing the experiment. The entangled photon pair is obtained through type-II degenerate down conversion in a nonlinear optical crystal, β-barium borate, pumped by a 200 mW UV cw argon laser with wavelength 351.1 nm. As described by, e.g., Kwiat, Mattle, Weinfurter, Zeilinger, Sergienko and Shih [1995], there are two privileged directions along which the down-converted 702.2 nm photons are nat-

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urally polarization-entangled, in a state that can be written as  1  √ |v
1 |h
2 + |h
1 |v
2 , 2

(2.3)

where h and v denote vertical and horizontal polarizations, respectively. Next, the two photons are made to pass through calcite crystals which send them in different directions according to their polarization. Suppose, for instance, that crystal 1 sends a v polarized photon along direction k1 and an h-polarized photon along direction k1 , and similarly for crystal 2. The resulting state will be  1  √ |k1 , v
|k2 , h
+ |k1 , h
|k2 , v
. 2

(2.4)

Now suppose that the experimental arrangement is such that, if the photon is traveling along direction k1 or k2 , it goes (eventually) to Alice, whereas if it is traveling along k1 or k2 it goes to Bob. Then the state (2.4) can be written as  1  √ |k1
A |k2
B + |k2
A |k1
B |v
A |h
B 2

(2.5)

which shows that Alice’s and Bob’s photons are momentum entangled. We can say that, for Alice’s photon, the labels “k1 ” and “k2 ” play a similar role to the labels “0” and “1” in the entangled state (1.2), and likewise with the labels “k1 ” and “k2 ” for Bob’s photon. The state to be teleported is the polarization state of Alice’s photon. To this end, a “preparer” performs an arbitrary rotation of |v
A , for instance, by inserting appropriate (and identical) optical elements along both paths k1 and k2 . When this is done, the total state becomes    1  √ α|v
+ β|h
A |k1
A |k2
B + |k2
A |k1
B |h
B 2

(2.6)

(−) in close analogy with the state |ψ
1 |Ψ23
considered in Section 1.2 (only now we (+) are using an entangled state that looks more like |Ψ23
, but this does not make any substantial difference). The polarization degree of freedom of Alice’s photon plays the role of particle 1, with the labels “v” and “h” being the equivalent of “0” and “1” in eq. (1.1). The polarization state of Bob’s photon, |h
is, for the time being, a mere spectator. Teleportation can then proceed very much as indicated in Section 1.2. The reason why the Bell state measurement is now straightforward is that the equivalent of the “CNOT gate” considered in Section 2.1 (and which yielded the transformation (2.1)) is trivial when “0” and “1” refer to different degrees of freedom of the

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same photon. That is to say, to rotate the polarization conditioned on the path all one has to do is insert an appropriate optical element in the appropriate path. (±) (1.3) and (1.4) Explicitly, the equivalent of the states |Φ (±)
, |Ψ √
of eqs. (±) (±)
= (|k
|h
±
= (|k
|v
± |k
|h
) / 2, and |Ψ may be written as |Φ 1 2 A 1 √ |k2
|v
)A / 2, and in terms of these states (2.6) can be written as         1 1 2 |Φ+
A β|k1
+ α|k2
B |h
B + 2 |Φ−
A −β|k1
+ α|k2
B |h
B     + 12 |Ψ+
A α|k1
+ β|k2
B |h
B + 12 |Ψ−
A −α|k1
+ β|k2
B |h
B . (2.7) Alice’s measurement in the |Φ (±)
, |Ψ (±)
basis proceeds (experimentally) as fol-

lows. The k2 photon is rotated in polarization another 90◦ , so that |k2
|v
becomes −|k2
|h
, and |k2
|h
becomes |k2
|v
. Then the |Φ (±)
states become associated with vertical polarization, and the |Ψ (±)
states with horizontal polarization only. The parity (+ or − sign) in the superposition can be determined by combining the paths k1 and k2 at an ordinary 50/50 beamsplitter, with a phase shift set so that the superposition corresponding to each sign goes out through one or the other port. To finish the teleportation, it is necessary to transfer the momentum state of Bob’s photon to its polarization. This is accomplished by, first, rotating the polarization in the path k1 by 90◦ , and then combining the two paths k1 and k2 at a polarizing beamsplitter that transmits horizontal and reflects vertical polarization. Then the state (2.7) can be rewritten as     1 1 2 |Φ+
A β|v
+ α|h
B + 2 |Φ−
A −β|v
+ α|h
B     + 12 |Ψ+
A α|v
+ β|h
B + 12 |Ψ−
A −α|v
+ β|h
B (2.8) times a common factor denoting a single direction of propagation for Bob’s photon (the transmitted k2 ), which is no longer important. The final manipulation required to bring Bob’s photon into the form α|v
+ β|h
depends on the result of Alice’s measurement, as in the standard teleportation protocol of Section 1.2 (compare eq. (1.5)). In the actual experiment of Boschi, Branca, DeMartini, Hardy and Popescu [1998], several linear as well as elliptically polarized states were teleported, with good visibility. While this experiment may be said to successfully implement (in a formal way) the teleportation protocol of Bennett et al., the substitution of two degrees of freedom of the same particle for what, in the original idea, were two completely independent particles, has some shortcomings: in particular, there can be no question, in this scheme, of teleporting to Bob the state of a separate particle that somebody

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else has prepared, and handed to Alice. On the other hand, the scheme certainly succeeds at transferring to Bob’s photon an arbitrary state of polarization prepared at Alice’s location, using only their shared entanglement and two bits of classical communication.

2.3. Teleportation with less than a full Bell-state measurement (conditional teleportation) Very shortly after the experiment of Boschi et al., discussed in the previous subsection, was reported, Bouwmeester, Pan, Mattle, Eibl, Weinfurter and Zeilinger [1997] presented the results of an experiment in which an entangled photon pair was used to teleport the state of polarization of a third, independent photon, in accordance with the original protocol, except that (for the reasons discussed in Section 2.1) only one of the Bell states, the antisymmetric state |Ψ (−)
was actually discriminated; teleportation therefore succeeded, on average, only 1/4 of the time. The entangled pair (particles 2 and 3, see fig. 2) was generated through typeII parametric down-conversion in a nonlinear crystal, using UV pump light at 394 nm. The 200 fs pump pulse was then retroreflected through the same crystal, to create a second photon pair. One of these 788 nm photons would play the role of particle 1; the other one (call it particle 4) went to a detector, where its arrival would serve to signal that the photon to be teleported was on its way. (Note, however, that because of the very low efficiency of the parametric down conversion process, the likelihood that each pump pulse would produce the needed four photons was quite low; most often, even if there was a photon 1 in the device, there would not be a 2–3 pair to teleport it with.)

Fig. 2. Scheme of the beam generation for the experiment of Bouwmeester et al. C is the nonlinear crystal where the entangled pairs are generated; B is the beamsplitter where 1 and 2 are combined for a partial Bell-state measurement.

4, § 2]

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The Bell-state measurement is produced by combining the photons 1 and 2 at an ordinary 50/50 beamsplitter and looking for the outcomes in which each photon leaves in a different output port (the dc and d  c outcomes, in fig. 1), which are the signature of the |Ψ (−)
state. To make sure that the simple analysis encapsulated in eqs. (2.2) correctly describes what happens at the beamsplitter, the photons 1 and 2 must be made to arrive “simultaneously”, to within their coherence time; this is ensured by sending them through very narrow (4 nm) bandwidth filters, which results in a coherence time of 520 fs. In the experiment, several polarization states of photon 1 were teleported, including linear and circular states. The visibilities were generally high; for the fourfold coincidence experiments (all four photons are detected; photon 3, after passing through an appropriate analyzer), visibilities of the order of 70% were reported, which are a direct measure of the degree of polarization of the teleported photon. The fourfold coincidence experiments are important, because detection of any fewer photons is actually insufficient, in this experiment, to be sure that teleportation has taken place. The reason is that, just as likely as the generation of the two pairs (1, 4) and (2, 3), is the generation of two (1, 4) pairs or two (2, 3) pairs. All of these possibilities may contribute to coincidences at the Bell-state detectors (for instance, if two “1” photons arrive at the beamsplitter, there is a 50–50 chance that one will be reflected and the other one transmitted); moreover, the case in which two (1, 4) pairs are produced would also give an event at the detector 4, indicating the presence of an input particle 1, yet in this case, even if a coincidence was observed at the Bell-state analyzer, there would actually be no teleported particle (photon 3). In principle, this need to detect photon 3, so as to ascertain that teleportation has actually taken place, could be eliminated by using a more sensitive detector to distinguish between the occurrence of one and two photons of the type 4; hence, this particular problem cannot really be considered a fundamental limitation of this experimental approach. More serious is the fact that, due to the incomplete nature of the Bell-state analysis, the teleportation is probabilistic (or conditional, as it is sometimes called): most of the time, the state of the incoming particle is not teleported at all, and yet it is destroyed in the process. Obviously, this approach could not really be used in a quantum communication network, unless one started out by protecting the states to be used with a sufficient amount of redundancy, and had some means (such as quantum error-correcting codes) to restore encoded states damaged in the transmission. (An important exception to this would be quantum key distribution, where the states sent by Alice are chosen randomly,

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Optical realizations of quantum teleportation

[4, § 2

and it does not matter whether a particular one arrives at Bob’s station or not, provided that the ones that do arrive are transmitted faithfully.) In spite of these shortcomings, the experiment of Bouwmeester et al. is an impressive demonstration of the possibility of carrying out an entangling operation on two independent photons, and, as a result of this, having the state of one of them transferred to a third one, with which it was not originally entangled. This is, arguably, the essence of the teleportation process as it was originally conceived. Very recently, Marcikic, de Riedmatten, Tittel, Zbinden and Gisin [2003] have performed an experiment involving the probabilistic teleportation of photonic qubits over long distances (2 km of optical fiber) and at the telecommunication wavelength of 1.55 µm. A difference with the previous experiments is that the qubits were encoded in “time bins”, instead of in the polarization states of the photons. This is a more resilient encoding for propagation through optical fibers. The idea of time-bin encoding is that the logical |0
and |1
states correspond to having a photon in one or another of two possible, temporally separated, wavepackets. Figure 3 shows how an arbitrary superposition of such states can be created. A single photon goes into a beamsplitter with amplitude reflection coefficient α and transmission coefficient β; this produces the state α|1
r |0
t + β|0
r |1
t . The two alternative paths have unequal lengths, and additional phase shifters can be inserted in one or the other, before they are made to recombine at the “switch” S. This is a device which, in principle, is set to wholly transmit a photon at the time that a photon would arrive if it took the short path, and then, immediately afterwards, it is set to wholly reflect the photon which took the long path. In this way, the photon ends up travelling in a well-defined direction, but in the superposition state |ψ
1 = α|0
+ eiφ β|1
,

(2.9)

Fig. 3. Time-bin encoding. The beamsplitter B generates a superposition of two alternative wavepackets that are then sent along the same direction by the switch S. If S is replaced by an ordinary beamsplitter, the total state also has a component exiting along the direction of the dashed arrow.

4, § 2]

Optical teleportation of discrete variables

335

where |0
represents a wavepacket corresponding to the long path and |1
a wavepacket corresponding to the short one. In practice, in the experiment of Marcikic et al., a passive coupler, with reflectivity and transmittivity of 50%, was used instead of the switch S. This means that in addition to the “long path-reflected” and “short path-transmitted” possibilities √ present in (2.9) (which now have an additional weighting factor of 1/ 2 ), the true state of the√ system includes two more possibilities, “long √ path-transmitted” (with weight α/ 2 ) and “short path-reflected” (with weight β/ 2 ). The “conditional branch” represented by the state (2.9) may be expected to be “really there” only half of the time; its actual presence is ascertained, ultimately, through the detection of two photons at the Bell-state measurement time. A pump pulse, consisting of a large-amplitude coherent state, when passed through a device similar to that in fig. 3, would be split into two mutually coherent, time-delayed wavepackets. If these are then sent through a nonlinear crystal, they can, to lowest order, produce a pair of photons in the time-bin entangled state  1  |Φ
23 = √ |00
+ |11
. (2.10) 2 Here the two possibilities being superposed correspond to the two photons being (simultaneously) created either by the earlier (|11
) or later (|00
) part of the pump pulse. The state |Φ
23 can be used as the entangled resource for teleportation of the state |ψ
1 . To this end, a (partial) Bell-state measurement has to be carried out on the photon in the state (2.9), and one of the photons in the state (2.10), simultaneously. In the experiment of Marcikic et al., the two photons were combined at a 50–50 fiber coupler (which acts as a 50–50 beamsplitter), with the outputs sent to separate detectors. The timing has to be such that photons in corresponding time bins in (2.9) and (2.10) arrive at the beamsplitter simultaneously. When this happens, the analysis of the beamsplitter action is, formally, the same as when the two time bins represented orthogonal polarization states (cf. eqs. (2.2)): the only possibility that leads to detection events at the two separate output channels of the beamsplitter is the singlet state  1  |Ψ (−)
12 = √ |1
1 |0
2 − |0
1 |1
2 2 and when this happens the state of photon 3 automatically becomes eiφ β|0
− α|1
.

(2.11)

(2.12)

To turn this output state (2.12) into the same as the input state (2.9), additional (local) operations are required, which were not carried out in the experiment; rather,

336

Optical realizations of quantum teleportation

[4, § 2

it was simply verified that the state (2.12) appeared at Bob’s station whenever the incomplete Bell measurement was successful. Note that the state (2.11) implies that one of the photons detected took the long path and the other took the short path, so the times at which the detectors click will differ by precisely the time difference between the two time bins. Unlike the experiment of Bouwmeester, Pan, Mattle, Eibl, Weinfurter and Zeilinger [1997], this one used two separate nonlinear crystals to produce the photon to be teleported, on the one hand, and the shared entangled pair, on the other. A pump pulse from a femtosecond laser was split in half at a beamsplitter. One half was used to generate twin photons in a crystal, one to go to Alice, and the other one to be discarded, whereas the other half was passed through an unbalanced interferometer, and, as it exited, split into two different time bins, it was sent through another nonlinear crystal to produce the state (2.10). The part of that state that goes to Bob was then sent through 2 km of optical fiber. Additionally, the photons were passed through interference filters of 10 nm spectral width, to increase their coherence length, before going into the detectors. Although, even under ideal conditions, the teleportation only works, in this scheme, 1/8 of the time (only one of the four Bell states is measured, and the timebin encoded input state (2.9) is only created successfully half the time), Marcikic et al. suggest that this approach could be useful to increase the distance over which qubits could be sent reliably through an optical fiber in quantum cryptographic applications. The idea is that, if a photonic qubit has a good chance of arriving intact after traveling a distance L from Alice to Charlie, then Charlie can teleport its state to Bob, a distance 2L away from him, by using one of a pair of entangled photons emitted by a source located halfway between him and Bob (the other photon of the pair is, of course, sent to Bob). This multiplies by 3 the distance a qubit can travel safely in the fiber, at the expense of a reduction in the overall transmission rate.

2.4. Low-efficiency teleportation with complete Bell-state measurement As explained in Section 2.1, nonlinear interactions between photons are necessary for a complete Bell-state measurement; as these are typically very weak at the two-photon level, a possible approach is to start with a large number of input photons, all in the same polarization state, sent simultaneously into a nonlinear crystal, where they may interact with the one photon of an entangled pair. The probability of the desired interaction taking place, which is extremely low for any individual input photon, may be boosted into a near-certainty by the large numbers

4, § 2]

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of input photons involved. Naturally, since the entangled resource consists of only two photons in any case, Bob receives only one photon. The above approach was used in an experiment by Kim, Kulik and Shih [2001]. Their input photon is one of many (perhaps 1010 ) in a macroscopic laser pulse, of central √ wavelength 800 nm, all of them in a polarization state |ψ
1 = (α|0
+ β|1
)/ 2, in an appropriate polarization basis. In the same basis, the entangled pair of photons 2 and 3, produced through nondegenerate type-I downconversion (−) in a pair √ of crystals by a pump pulse at 400 nm, is in the state |Φ
23 = (|00
− |11
)/ 2; the wavelength of photon 2 is 885 nm and that of photon 3 is 730 nm. The two terms being superposed correspond to the two photons being generated in one or the other of the two crystals; appropriate compensators are used to ensure that the two possibilities are really quantum-mechanically indistinguishable. The Bell-state measurement is carried out by sending the 800-nm laser pulse (photon 1) and the 885-nm photon 2 together through a series of sum-frequency generating crystals, two of type I and two of type II. The first type-I crystal is set so that it will convert an input pair |11
12 into a single 420-nm photon with horizontal (“H ”) polarization, whereas the second one is set to convert a |00
12 pair into a single vertically polarized (“V ”) photon. After photons 1 and 2 pass through these two crystals, and appropriate compensating elements, a dichroic mirror is used to send the 420-nm photon (assuming it was actually produced) down into a polarization beamsplitter set at a 45◦ angle, so as to separate out the |H
+ |V
and |H
− |V
possibilities. Since these correspond to the input pair states |Φ (+)
and |Φ (−)
, respectively, this accomplishes the first half of the Bell-state measurement. In a similar way, the two type-II crystals are set to convert the state |01
12 (resp. |10
12 ) into a single 420-nm photon with horizontal (resp. vertical) polarization. Again, a dichroic mirror is used to separate this 420-nm photon out and send it into a polarization beam-splitter that sends the |H
+ |V
and |H
− |V
states to different detectors. These two possibilities now correspond to the input states |Ψ (+)
and |Ψ (−)
, which completes the Bell-state measurement.

2.5. Near-deterministic teleportation of photonic qubits using only linear optics In a recent, seminal paper by Knill, Laflamme and Milburn [2001] on the possibility of quantum computation with linear optics, a modified teleportation protocol was introduced which is especially suited to work with a particular encoding of a qubit in a photon state, and with linear optical elements and photon-counting detectors. As a first step towards their general scheme, it is helpful to consider the

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following simplest case, in which a single photon provides the entangled resource for probabilistic teleportation (Villas-Bôas, de Almeida and Moussa [1999], Lee and Kim [2000]). Suppose that the qubit state to be teleported is encoded as |ψ
1 = α|0
+ β|1
,

(2.13)

where |0
now refers, literally, to the absence of a photon from a particular propagation and/or polarization mode (denoted as mode 1), and |1
, instead, to the single-photon state of that mode. The question of how a state like (2.13) could be prepared in practice will be considered later. For the moment, however, note that with this kind of encoding a single √ photon entering a beamsplitter and leaving in the state (|0
r |1
t + |1
r |0
t )/ 2 (a 50–50 superposition of the “transmitted” and “reflected” possibilities) provides, by itself, a state of the same form as the standard maximally entangled states used in the conventional teleportation protocol. Only, now, it is the modes (rather than a pair of photons) that are entangled. With these conventions, one can now see how probabilistic teleportation of the state (2.13) of the input mode 1 to one of the beamsplitter’s output modes (say, mode t) is possible. The overall state of the two-photon, three-mode system is  1  √ α|001
1rt + α|010
1rt + β|101
1rt + β|110
1rt . 2

(2.14)

Now let mode 1 and mode r be combined at yet another √ 50–50 beamsplitter. The input state |01
1r is turned into (|01
1 r  − |10
1 r  )/ 2 in terms of the√exit modes 1 and r  , whereas the state |10
1r is turned into (|10
1 r  + |01
1 r  )/ 2. Hence, detection of one photon in mode 1 and no photons in mode r  will project the state (2.14) into −α|0
t + β|1
t , whereas detection of one photon in mode r  and no photons in mode 1 will result in the state α|0
t + β|1
t . If no photons are detected in either output (assuming perfect detectors), this corresponds to the first term in eq. (2.14), which occurs with probability |α|2 /2; if two photons are detected in the same output mode, this corresponds to the last term in eq. (2.14), which occurs with probability |β|2 /2. In either of these last two cases, the teleportation attempt fails, so the total failure probability is (|α|2 + |β|2 )/2 = 1/2. Thus, half the time teleportation with this protocol is successful. Just as in the standard protocol, a phase shift may be necessary to complete the process, but note that no “bit flip” (changing |1
to |0
) is required at the receiver’s end; this is good, because such an operation is nontrivial when |0
is the vacuum state. Thus far, this approach is not any better than teleportation with the standard protocol and an incomplete Bell-state measurement. To see how the success probability can be increased by using more photons, consider the following two-photon,

4, § 2]

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339

four-mode entangled resource:  1  |t2
1234 = √ |0011
1234 + |1001
1234 + |1100
1234 . (2.15) 3 Here the notation of Knill, Laflamme and Milburn [2001] has been adopted. Modes 1 and 2 are on the sender’s side, and modes 3 and 4 are on the receiver’s side. The state to be teleported will be assigned to “mode 0” and written as α|0
0 + β|1
0 . Then the overall initial state, written in a way that separates out the modes 0, 1 and 2, on which a joint measurement is to be made, from the modes 3 and 4, where the teleported state is to appear, is 1  √ α|000
0123|11
34 + α|010
012|01
34 + β|100
012|11
34 3

 + α|011
012 |00
34 + β|110
012|01
34 + β|111
012|00
34 .

(2.16)

A careful comparison of the states (2.14) and (2.16) reveals the important similarities. Consider, for instance, the second and third terms in (2.16). In both of these there is only one photon in the modes 0, 1 and 2. If these modes can be combined, in a generalized beam-splitter fashion, so that the photon can be detected without knowing from which of the three modes it came originally, the state (2.16) will collapse to a coherent superposition of α|01
34 and β|11
34 , possibly with some relative phase; in this case, then, the input state of mode 0 is “teleported” to mode 3, and mode 4 is left in a one-photon state. Similarly, the fourth and fifth terms in (2.16) correspond to having two photons in the modes 0, 1 and 2. Again, if these two photons are detected without revealing which mode they came from originally, the overall state will collapse to a coherent superposition, this time of α|00
34 and β|01
34 , so now the teleported state appears in mode 4. The two terms in (2.16) that do not lead to a coherent superposition are the first one, corresponding to 0 photons in modes 0, 1, 2, and the last one, corresponding to 3 photons in those modes. The first one occurs with probability |α|2 /3, and the second one with probability |β|2 /3, so the total failure probability is 1/3, equal to 1/(n + 1), where n = 2 is the number of auxiliary photons used. The general protocol proposed in Knill, Laflamme and Milburn [2001] makes use of the n-photon entangled state of 2n modes |tn
1...(2n) =

n 

|1
j |0
n−j |0
j |1
n−j ,

(2.17)

j =0

|a
j

where means |a
|a
. . . j times, and the modes, labeled 1, . . . , 2n, are understood to be written in order, left to right, in (2.17). This state is to be com-

340

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bined with an input state of the standard form, α|0
0 + β|1
0 , and then the modes 0, 1, . . . , n are “scrambled”, before the number of photons in them is measured, by the following kind of “generalized multiport beamsplitter” transformation: n  1  e2πijj /(n+1) aj† aj† = √ n + 1 j =0

(2.18)

(here shown as acting on the “input” mode creation operators aj† , 0
j
n, to

produce the output mode creation operators aj† , 0
j
n ). Eq. (2.18) is actually an n + 1 point Fourier transform, and has efficient linear optics implementations (Weihs, Reck, Weinfurter and Zeilinger [1996]). After the transformation (2.18), the number of photons in modes 0 , . . . , n is measured. If this number equals 0 or n, the procedure fails, and the total probability of this happening is 1/(n + 1). Otherwise, if k photons are detected, the input state of mode 0 appears in (i.e., is teleported to) mode n + k, modulo a possible relative phase that can be corrected in a standard way. A question that still needs to be addressed is how an input state like (2.13) is to be created in the first place. The entangled resources of the form (2.17) all have a well-defined number of photons, and could presumably be created, if reliable sources of single photons “on demand” were available, by sending the desired number of photons into an appropriate multiport interferometric device; but a state like α|0
+ β|1
is, if taken literally, one in which one is not really sure whether one has a photon or not. How can such a state be produced, and information be carried by it? The simplest answer is that the coefficients α and β need not be numbers; they could, themselves, be quantum states, such as number states for another mode. Thus, an expression like (2.13) could really expand to, for instance, something like α  |1
h |0
v + β  |0
h |1
v ,

(2.19)

where h and v denote horizontal and vertical polarization modes traveling in the same direction, in which case (2.19) simply represents a general polarization state for a single photon. If the state (2.19) is put through a polarization beamsplitter, one of the exit modes, say, v, could be said to be in a state like (2.13); only, properly speaking, the coefficients α and β would have to be equal to α  |1
h and β  |0
h , respectively. Put another way, the teleportation schemes discussed in this section are wellsuited to teleport “one half” of a two-mode entangled state such as (2.19). Such a state can, of course, be used to encode a “photonic qubit”; to reconstruct the

4, § 2]

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Fig. 4. Teleportation of a polarization state using single-photon entanglement. The initial state to be teleported is prepared by the polarization rotator R. The diamonds are polarizing beamsplitters. The (ordinary) beamsplitter B1 produces the single-photon entangled resource, whereas B2 is used for the partial Bell-state measurement.

full qubit at the end of the process, the teleported state would have to be physically combined with the other mode (of course, this other mode could also be teleported, separately, to the same location). An example of how this could work is shown in fig. 4. The information-carrying photon entering at the far left is given an arbitrary polarization by the rotator R, then sent into a polarizing beamsplitter. One of the modes exiting the beamsplitter is then teleported using a single-photon two-mode entangled resource, such as the one shown at the beginning of this section to be generated by an ordinary beamsplitter (B1 in fig. 4). By combining mode v with mode r at another beamsplitter, B2, and measuring the photon numbers at the output modes v  and r  , the state of mode v (including its prior entanglement with h) can be teleported, with probability 1/2, to the other mode t. Finally, to reconstruct the full initial state, mode h needs to be combined with t at another polarization beamsplitter; suitable delays need to be introduced to ensure that the two paths overlap exactly, of course. In addition, depending on the outcome of the measurements, an extra phase shift may need to be applied to t. A couple of experiments essentially like the one sketched here (only using an ordinary beamsplitter, rather than a polarizing one, to create the input state (2.13)) were recently carried out by Lombardi, Sciarrino, Popescu and De Martini [2002] and Giacomini, Sciarrino, Lombardi and De Martini [2002]. The second of these was noteworthy in that it was the first time that an active manipulation of the received photon, conditioned on the result of the measurement at the sender, was,

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in fact, performed; all other prior experiments in single-photon teleportation had used passive means to simply verify that the teleported state had the expected form. At this point one may wonder what good this whole teleportation scheme is, if Alice has to send part of the initial photon state (the mode h in the figure) directly to Bob, anyway, in order to reconstruct the full initial state. Why not just send Bob directly the input photon, without any further manipulation? Of course, a partial answer is that Alice could, in principle, also teleport that mode’s state to Bob separately. Perhaps a more relevant answer, however, is that in the scheme of Knill, Laflamme and Milburn, teleportation is used, not so much to take a state from one place to another, but, rather, as a tool to apply quantum logical operations to photonic qubits, in the spirit of Gottesman and Chuang [1999]. In that work, it was shown that one could “teleport a qubit through a gate” by preparing an appropriate entangled resource in such a way that, at the end of the teleportation process, the transmitted qubit’s state would be not just |ψ
in , but U |ψ
in , where U is any desired unitary operator, with similar constructions being possible for two-qubit gates. The logic of this procedure is that, under some conditions, it may be easier to prepare and verify the entangled resource, whose desired state is known in principle, than to act directly on the input qubit, whose state is unknown and may not be replaceable if it is damaged in the process. In the scheme of Knill et al., the teleportation is, in fact, used to implement two-photon logical gates, through the preparation of an appropriate entangled resource, which can only be constructed probabilistically; by proceeding in this fashion, one puts off acting on the information-carrying qubit until preparation of the entangled resource is known to have succeeded, at which point the desired quantum logical operation can be applied “almost deterministically”, along lines similar to those indicated above. Clearly, this is already somewhat removed from the initial teleportation concept, and will not be discussed further here, although it is currently a topic of substantial interest. Further constructions for quantum logic with linear optics can be found in Knill, Laflamme and Milburn [2001] and subsequent publications such as Franson, Donegan, Fitch, Jacobs and Pittman [2002], and references therein.

§ 3. Optical teleportation of continuous variables 3.1. Squeezed states as EPR states The possibility of using optical means to demonstrate continuous-variable teleportation stems from the fact that a single mode of the electromagnetic field is

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formally the same as a simple harmonic oscillator. One can then define operators xˆ and p, ˆ formally equivalent to position and momentum, from the standard photon creation and annihilation operators a † and a, as xˆ =

a + a† , 2

(3.1) a − a† pˆ = 2i (see, e.g., Chapter 2 of Scully and Zubairy [1997], where these “quadrature” operators are called X1 and X2 , respectively). These are dimensionless, canonically conjugate, continuous-variable operators, whose commutator is equal to i/2. One could use them to write a wavefunction description of the state of the field mode, just as one would use x and p for an ordinary particle in one dimension. Moreover, when one considers two such modes, there exist states, called “two-mode squeezed states”, with the property that the variances (xˆ1 − xˆ2 )2 =

e−2r , 2

(3.2) e−2r (pˆ 1 + pˆ 2 = , 2 where r is a parameter known as the “squeezing parameter”. Equations (3.2) follow directly, for instance, from eqs. (2.8.2)–(2.8.8) of Scully and Zubairy [1997], for the case θ = π , noting that xˆ1 − xˆ 2 is proportional to b1 for δ = π (in the notation of that book), and pˆ1 + pˆ2 is proportional to b2 for δ = 0. Two-mode squeezed states received a great deal of theoretical and experimental attention in quantum optics about twenty years ago (see the references in Scully and Zubairy [1997]). The result (3.2) implies that they can be used, formally at least, as an approximation to a two-particle EPR state, provided that the squeezing parameter r is sufficiently large, since in this case the dispersion in xˆ1 − xˆ2 and pˆ 1 + pˆ2 goes to zero, meaning that the values of these quantities are simultaneously sharply defined. Mathematically, the two-mode squeezed state results from applying the operator )2

S(ξ ) = eξ

∗a

† † 1 a2 −ξ a1 a2

(3.3)

to the two-mode vacuum state, where ai is the annihilation operator for the ith mode, and ξ = reiθ . Physically, multimode squeezed states of this form (only involving a continuum of modes, typically correlated in pairs around a central frequency) are naturally generated by optical parametric oscillators (see Scully

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and Zubairy [1997], Chapter 16). Such a pair of modes (let us label them, in the following, 2 and 3, for consistency with the notation of earlier sections) can then be used to provide the entangled, EPR-like resource that Alice and Bob must share for successful continuous variable teleportation. The next question is how Alice is to carry out the joint measurement of xˆ1 − xˆ2 and pˆ1 + pˆ 2 , where mode 1 is in an unknown (to her) state, and 2 is her part of the two-mode squeezed state. This turns out to be very simple. First, modes 1 and 2 must be combined in an ordinary 50–50 beamsplitter, yielding output modes c and d with annihilation operators 1 c = √ (a1 + a2 ), 2

(3.4) 1 d = √ (a1 − a2 ) 2 and then the “p”-like component of mode c and the “x”-like component of mode d (with “x” and “p” as in eqs. (3.1)) are measured separately. It is well known in quantum optics that any field “quadrature” of the form (aeiφ + a † e−iφ )/2 can be measured by homodyne detection, using a local oscillator with an appropriate phase (see Scully and Zubairy [1997], Section 4.4.2). As explained in Section 1.4, this measurement automatically collapses the state of Bob’s mode 3 to a replica of the original state of mode 1, only possibly displaced in x and p by amounts that depend on the outcome of Alice’s measurement. To faithfully reproduce the original state of mode 1, all that is necessary is for Bob to undo these displacements, using the information sent to him by Alice. Note that “x” and “p” in eqs. (3.1) are basically the real and imaginary parts of a phasor representing the complex field amplitude a; hence the required displacements can be reduced to manipulations of the phase and amplitude of Bob’s field mode. (In practice, this is actually carried out by having Bob prepare a very intense coherent state with the right amplitude and phase, and combining it with beam 3 on a highly reflecting beam splitter; how this works is examined in more detail in the next subsection.) The above is, in broad outline, the essence of the optical approach to continuous-variable teleportation, first proposed by Braunstein and Kimble [1998] (see also van Loock, Braunstein and Kimble [2000], for an extension of the theoretical treatment to broadband fields), and demonstrated experimentally by Furusawa, Sørensen, Braunstein, Fuchs, Kimble and Polzik [1998] (see also Bowen, Treps, Buchler, Schnabel, Ralph, Bachor, Symul and Lam [2003] and Zhang, Goh, Chou, Lodahl and Kimble [2003] for details of the most recent experiments).

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3.2. Teleportation fidelity for coherent states As with discrete variable teleportation, it is natural to judge the success of a continuous-variable teleportation experiment by the fidelity of the final state of mode 3 to the initial state of mode 1, defined in the standard way (see eq. (1.18)). The first experiment by Furusawa et al. reported a fidelity of 0.58, and the most recent results of Bowen et al. and Zhang et al. have reached fidelities of 0.64 and 0.62, respectively. A natural question to ask is how large a fidelity can be achieved for a given degree of squeezing, since the two-mode squeezed state is not exactly an ideal EPR state for any finite r. The general answer is complicated (see Furusawa, Sørensen, Braunstein, Fuchs, Kimble and Polzik [1998], Braunstein, Fuchs and Kimble [2000], and Braunstein, Fuchs, Kimble and van Loock [2001], for details), as it depends on the kind of state to be teleported as well as on various efficiencies and on the gain (amplification) necessary at Bob’s end, but in the simplest case in which the state to be teleported is a coherent state, neglecting losses, and with Bob’s gain equal to 1, the theoretical result is 1 (3.5) . 1 + e−2r In particular, this predicts that, in the absence of squeezing (r = 0), the maximum achievable fidelity is 1/2, and for infinite squeezing F → 1. Note that when r = 0 the resource shared by Alice and Bob is simply the vacuum state, which is clearly not entangled, and Alice’s measurement of the “p”-like component of mode c and the “x”-like component of mode d may be regarded as an attempt to determine x and p simultaneously for mode 1, by dividing it at a beamsplitter and measuring x on one side and p on the other. Such a simultaneous determination is forbidden, of course, by the uncertainty principle, and the vacuum noise introduced by the beam splitter ensures that Alice’s measurement does not violate this fundamental constraint. More precisely, consider the problem of reproducing, at a remote location, an initially unknown coherent state of the field, sending only classical information. The first thing that needs to be done is to measure the field state – its amplitude and phase, or its “quadratures” x1 and p1 . The problem is that these are not classical quantities, that could in principle be determined with arbitrary precision: they are quantum properties, that have some intrinsic “quantum uncertainty”. In a coherent state, this quantum uncertainty x1 = p1 = 1/2 is the same as the one in the vacuum state, or “vacuum noise”. Thus, the measured values for the quadratures will typically not be the same as the expectation values x1
or p2
. If Bob simply prepares a coherent state centered about Alice’s measured values, F=

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this will typically be “off” – i.e., not overlap exactly with – the original coherent state. When the probabilities of various outcomes are considered, and the possible fields generated by Bob are thus represented, as a mixed state, by an appropriate density operator, this field will have the right averages x1
and p2
, but increased fluctuations about those average values, relative to those of a pure coherent state. In the r = 0 case, that is, in an experiment without EPR beams, Alice simply √ divides her field’s initial amplitude by 2 at her beamsplitter, and attempts√to measure√ x1 on one side and p1 on the other side. In fact, she thus measures x1 / 2 and p1 / 2, with a probable error δxd and δpc of the order of the vacuum noise in both cases; this is because, even though the beamsplitter reduces the original √ √ noise in, say, x1 to δx1 / 2, it also lets in uncorrelated vacuum noise δvac / 2, so 2 /2 remains at the vacuum level (and similarly the total noise power δx12 /2 + δvac for √ the other output port). Bob then needs to multiply Alice’s measurement by 2, so the coherent amplitude √ of the field he creates may be off from that of the original field by as much as 2δxd , and, in addition, the coherent state he creates will have its own uncorrelated vacuum-noise-level fluctuations. In power units, √ what all this adds up to is a field that has ( 2δxd )2 + (δvac )2 = three times the uncertainty of the coherent state that was originally handed to Alice (Braunstein and Kimble [1998]). The way the teleportation setup gets around this difficulty is, basically, by arranging for the fluctuations introduced at the mode-2 port of Alice’s beamsplitter to be subtracted at Bob’s station, by combining the field generated by Bob with the correlated mode 3 at a very asymmetric beamsplitter, t  1. √ √ of transmissivity Say that Bob prepares a field with coherent amplitude ( 2/t)(x1 / 2 + δxd ) (and a similar thing for the other quadrature), and normal vacuum-level noise. At the beamsplitter, most of Bob’s vacuum noise is reflected away and replaced by the noise of mode 3, which is correlated 2, so the√ field’s amplitude √ with that of mode√ √ now becomes something like x1 + 2δxd +δx3 = x1 + 2(δx1 / 2 −δx2 / 2 )+ δx3 ; the noise cancellation between modes 2 and 3 then results in a field with fluctuations of the same size as those of the original coherent state. This view of the teleportation process as a “noise subtraction” process is an interesting one. Note that, at the opposite end of r → ∞ (which would result, according to eq. (3.5), in perfect teleportation, i.e. teleportation with unit fidelity), the noise δx2 and δp2 introduced by mode 2 at Alice’s beamsplitter is effectively infinite (see Furusawa, Sørensen, Braunstein, Fuchs, Kimble and Polzik [1998]), completely masking any information on the original state’s phase or amplitude. This is as it should be, since, for teleportation to be fully successful, no information on the initial state must remain anywhere in classical form (which can be

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copied without any restrictions). Thus, continuous variable teleportation may be said to work by using one of a pair of correlated states to add enough noise to the state to be teleported to make it unrecognizable, sending that completely masked signal as classical information to another location, where it may be used to prepare a seemingly totally random quantum state, and then using the other member of the correlated-state pair, at the remote location, to remove all the added randomness from that state in such a way as to precisely re-create the original one. For the teleportation of a coherent state, the fidelity F = 1/2, corresponding to r = 0 in eq. (3.5) and hence, as argued above, to the generation, at Bob’s station, of a field with three times the fluctuations of the original one, is usually referred to as the “classical teleportation fidelity”, in analogy to its discrete-variable counterpart introduced in Section 1.5. (Note that the restriction to only coherent-state inputs is important; for an unrestricted input, the classical teleportation fidelity (1.23) in an infinite-dimensional Hilbert space is necessarily zero.) This naturally became, at the beginning, “the number to beat” experimentally, even if, as argued in Section 1.5, the importance of such a benchmark is more symbolic that fundamental (let alone practical). With that particular threshold already crossed by the first experiment, however, other (equally symbolic) benchmarks have subsequently been proposed, such as F = 2/3 (see Grosshans and Grangier [2001]), which is the greatest fidelity with which an unknown coherent state could be cloned (Cerf and Iblisdir [2000])4.

3.3. The role of “classical” fields, and the nature of laser fields As described above, the experiments in optical continuous-variable teleportation require a large number of optical frequency fields which must be either all locked to the same reference phase, or at least phase stabilized to the point that their relative phases do not drift uncontrollably over the duration of the experiment. The simplest way to ensure this, and the one, so far, followed in practice, is to use the same laser to generate all the required “classical” fields. These include: the pump field for the parametric oscillator that generates the EPR beams; the 4 The fact that one can do better by cloning than by classical teleportation is not altogether surprising, since these are different protocols addressing different problems. In particular, for optimal cloning no attempt is actually made to learn the identity of the state to be cloned, unlike for classical teleportation. Cloning does not, in fact, solve the teleportation problem: Alice may clone her state if she wishes, but in order to send it to Bob with reasonable fidelity she needs to either have an open quantum channel (in which case she might as well have sent the original in the first place) or prior shared entanglement, in which case she could have used teleportation to begin with.

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local oscillators for Alice’s quadrature measurements; the displaced coherent state generated by Bob; and the local oscillators used at the verifying station, where the teleported state is analyzed. In addition, the same laser is also used to provide the “unknown” input coherent state to be teleported by Alice. With all of this going on, one may worry that the integrity of the teleportation protocol may have been compromised somehow, and that there may be an open quantum channel connecting Alice and Bob more or less directly in the experiment. This is, however, not the case. Information (meaning, here, quantum information) on the nature of Alice’s unknown state could certainly have been sent directly to Bob if one had chosen to, but this is more or less the case in all teleportation experiments to date: it would always have been, physically, much easier to send the original state directly to Bob, but this was simply (and, of course, purposely) not done. There is no path, in this or the other experiments, through which quantum information on the unknown state could have “leaked through” from Alice’s side to Bob’s side. All of the phase locking required amounts, ultimately, to nothing more than clock synchronization, and could, in principle, be done classically if one had clocks at optical frequencies. The alleged “classical” nature of all these laser fields has, however, been questioned by Rudolph and Sanders [2001], who argue (among other things) that if the laser field is not correctly represented, as it is usually done, by a quantum coherent state, then when it is split at a beam splitter the two outputs are not, in general, in a factorizable, but, rather, in an entangled state. This would mean that, in the real experiments, Bob and Alice would, in principle, share much more entanglement than just that provided by the EPR beams, since they do share at least two halves of a laser beam (one for Alice’s quadrature measurements, and one for Bob to prepare his coherent state). More generally, if the laser field is not, at least to a good approximation, a coherent state, then a description of what may actually be going on in a particular instance of the experiment becomes extremely complicated, to the point that Rudolph and Sanders can argue that it is not clear whether the existing experiments do demonstrate teleportation at all. The suggestion that an individual laser’s field is not really in an (approximate) coherent state is based on an argument put forward by Mølmer [1997] (see also Gea-Banacloche [1998a] and Mølmer [1998] for further discussion). Mølmer argued that there is not, in the lasing process, any mechanism that can actually generate coherences between atomic levels, and hence actual radiating dipoles, with definite (albeit unknown) phases; this lack of atomic coherence, in turn, translates into a lack of actual coherence in the field the atoms radiate. Note that the (approximate) result, derivable from general laser theory (see, e.g., Scully and

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Zubairy [1997], Chapter 11) for the steady-state density matrix of a laser field ρss = e

−|α|2

∞  |α|2n n=0

1 |n
n| = n! 2π



2π
√


ne ¯ iφ

√ iφ
ne ¯
dφ

(3.6)

0

(where |α|2 = n¯ is the average number of photons in the laser cavity at steady state) is certainly consistent with the assumption that an individual laser is in a √ iφ coherent state | ne ¯
, with some unknown phase; but it is also, in principle, consistent with other interpretations, such as that the cavity field is instead in a number state, with a probability as given by the first term in (3.6). Rudolph and Sanders argue that, in view of Mølmer’s work, the first interpretation is unlikely, and the second one would certainly lead to shared entanglement between Alice and Bob, if the number state is split among the two using a standard beamsplitter. Interpretational matters aside, though, it is easy to lay to rest the latter claim, from a consideration of the actual equations of motion for the laser field. It was shown by the present author (Gea-Banacloche [1998b]) that, if an individual laser is assumed to be in a number state at the time t = 0, this state is very rapidly destroyed (i.e., turned into an orthogonal mixture), by a combination of cavity losses and atom emission, over a time of the order of 1/nκ, ¯ where κ is the cavity decay rate. Since this time can easily be of the order of femtoseconds for even a milliwatt laser, it is clear that any description of what may be going on at the beamsplitter, based on the assumption of an initial number state, will have to include a time average that is very likely to wipe out all entanglement, as soon as enough terms in the mixture (3.6) are included. (Note that the complete sum (3.6), when incident on a beam splitter, does lead to unentangled outputs, since the coherent states on the right-hand side all split without entanglement.) A converse point made by Rudolph and Sanders was that, if the full density matrix (3.6), rather than a single coherent state with an unknown phase, is used to describe the state of the beam that pumps the parametric oscillator in the experiment of Furusawa, Sørensen, Braunstein, Fuchs, Kimble and Polzik [1998], the output beams do not show any nonlocality or entanglement. This point also has been effectively rebutted by van Enk and Fuchs [2002], who show that, provided Alice is also given some of the laser light used to generate the twin beams (as was indeed the case in the actual experiments), the total field state does contain distillable entanglement, even in this full density matrix formulation. Finally, there can be little doubt that the experiments do show the noise cancellation which was argued in the previous subsection to be a characteristic of continuous-variable teleportation. Regardless of how one chooses to imagine the

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initial field state5 , it is clear that a close approximation to it is produced at the end of the process, by following the teleportation protocol, as argued above.

§ 4. Conclusions The field of quantum information processing is still in its infancy, and all the experimental demonstrations of quantum teleportation discussed here must be considered very preliminary, but they are enough to prove the principle, and, in so doing, illustrate a remarkable property of quantum systems: the possibility to encode a quantum state in two separate channels, a quantum one and a classical one, and to eventually combine the information carried by both, so as to fully reconstruct the original state. This is, conceptually, a very interesting effect, and as this brief overview hopefully shows, there are many fascinating physics issues involved in both the theory and the experiments. From a practical standpoint, the main potential usefulness of teleportation is the possibility of working “offline”, taking the time to get a certain state or operation right before actually applying it. This possibility depends entirely on the practical realization of reliable quantum memories, which may not come to pass for quite some time yet. Also, for applications to quantum information processing much larger fidelities than those so far demonstrated will be necessary. Teleportation of non-optical systems, which has been proposed by a number of authors, but, so far, only realized in an NMR setting (Nielsen, Knill and Laflamme [1998]), may also be expected to be of substantial interest in the future.

References Badziag, P., Horodecki, M., Horodecki, P., Horodecki, R., 2000, Local environment can enhance fidelity of quantum teleportation, Phys. Rev. A 62, 012311.

5 On this question, which is already beyond the scope of this article, this author’s admittedly biased opinion is that the assumption that the output of a laser is in a coherent state with an unknown phase is, epistemologically, on a par with the assumption that a properly designed measurement apparatus, after interacting with a quantum system, definitely leaves it in one of a given set of possible outcomes, even if the specific outcome is not known. In this context, the “apparatus” is the lossy cavity, and the privileged states, in terms of their lifetimes, are precisely the coherent states (eigenstates of the photon annihilation operator). (As shown in Gea-Banacloche [1998b], atomic emission changes this slightly but not fundamentally.) This opinion appears to be shared by other authors (see, e.g., Wiseman and Vaccaro [2002]), while the privileged role of the coherent state decomposition has also been noted, from a somewhat different perspective, in van Enk and Fuchs [2002].

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E. Wolf, Progress in Optics 46 © 2004 Elsevier B.V. All rights reserved

Chapter 5

Intensity-field correlations of non-classical light by

H.J. Carmichael Department of Physics, University of Auckland, Private Bag 92019, Auckland, 1301, New Zealand

G.T. Foster Department of Physics and Astronomy, Hunter College, CUNY, New York, NY 10021, USA

L.A. Orozco and J.E. Reiner Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800, USA

and

P.R. Rice Department of Physics, Miami University, Oxford, OH 45056, USA

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(03)46005-8 355

Contents

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

357

§ 2. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

359

§ 3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

366

§ 4. Experiment in cavity QED . . . . . . . . . . . . . . . . . . . . . . . .

385

§ 5. Equal-time cross- and auto-correlations . . . . . . . . . . . . . . . . .

393

§ 6. Quantum measurements and quantum feedback . . . . . . . . . . . .

396

§ 7. Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . .

402

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

403

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

403

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§ 1. Introduction Studies of the fluctuations of light have occupied quantum optics since its beginnings. Experimental work in the field has followed two broad lines, the first focused on intensity fluctuations and the measurement of correlations between pairs of photon detections [particle aspect of light] (Brown and Twiss [1956], Kimble, Dagenais and Mandel [1977]), and the second is primarily concerned with squeezing experiments where the fluctuation variance of a quadrature amplitude of the optical field is measured [wave aspect of light] (Slusher, Hollberg, Mertz, Yurke and Valley [1985], Loudon and Knight [1987], Kimble and Walls [1987]). Until recently, these two lines of investigation remained separate. It is now possible, however, to combine them in a new approach that detects the fluctuations of an electromagnetic field by correlating its intensity and amplitude (Carmichael, Castro-Beltran, Foster and Orozco [2000], Foster, Orozco, Castro-Beltran and Carmichael [2000a]). The approach draws the particle and wave aspects of light together, and opens up a third-order correlation function of the electromagnetic field to experimental study. The new measurement strategy builds upon the relationship between quantum optical correlation functions and conditional measurements (Mandel and Wolf [1995]), and its physical interpretation is therefore illuminated through quantum trajectory calculations (Carmichael [1993]). Historically, it was the development of the intensity-intensity correlation technique of Hanbury Brown and Twiss (HBT) (Brown and Twiss [1956]) that provided the stimulus for a systematic treatment of optical coherence within the framework of quantum mechanics (Glauber [1963a, 1963b, 1963c]). A notable feature of the HBT approach is its reliance on a conditional measurement – i.e. data is collected on the cue of a conditioning photon count that identifies those times when an intensity fluctuation is in progress. In this way, the average fluctuation is recovered as a conditional evolution over time, and a sensitive probe of the nonclassicality of light is obtained. The standard squeezing measurement is not, by way of contrast, a conditional measurement. Through balanced homodyne detection (Yuen and Chan [1983a, 1983b]), it effectively measures the sub-Poissonian variance of a photon counting 357

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[5, § 1

distribution, after the photon counts have been integrated over many correlation times. The measurement is insensitive to fluctuations at low photon flux and the observed degree of squeezing is degraded by collection and detection inefficiencies. The measurement is resolved in the frequency domain and does not recover an evolution of the fluctuations over time. The intensity-field correlation function is measured through the conditional detection of the quadrature amplitude fluctuations of light. The measurement crosscorrelates the photocurrent of a balanced homodyne detector (BHD) with an initiating photon count in a natural extension of the HBT technique. It is extremely sensitive to the nonclassicality of light at low photon flux (weakly squeezed light) and, given sufficient detection bandwidth, resolves the fluctuations in time. For the case of Gaussian statistics, Carmichael, Castro-Beltran, Foster and Orozco [2000] showed that the full spectrum of squeezing is recovered from the Fourier transform of the time-resolved fluctuation. The measurement, like the HBT technique, is independent of detection efficiency, except for the inevitable efficiencydependence in the signal-to-noise ratio. To date, intensity-field correlations have been explored for the optical parametric oscillator (OPO) (Carmichael, Castro-Beltran, Foster and Orozco [2000]), in both theoretical (Carmichael, Castro-Beltran, Foster and Orozco [2000], Reiner, Smith, Orozco, Carmichael and Rice [2001]) and experimental (Foster, Orozco, Castro-Beltran and Carmichael [2000a], Foster, Smith, Reiner and Orozco [2002]) studies of cavity QED, and for a single two-level atom coupled to an OPO (Strimbu and Rice [2003]). On the theoretical side, connections have been made to fundamental questions in quantum measurement theory and statistical physics. Wiseman [2002], for example, has demonstrated a connection with weak measurements. Carmichael [2003] has shown that, in contrast to a conventional squeezing measurement, conditional homodyne detection distinguishes qualitatively between vacuum state squeezing and squeezed classical noise. Denisov, Castro-Beltran and Carmichael [2002] explored the time-reversal properties of the intensity-field correlations. They show that while the intensity-intensity correlation function is necessarily time symmetric, the intensity-field correlation function may be time asymmetric for non-Gaussian fluctuations. The time asymmetry indicates a breakdown of detailed balance. In related but earlier work, Yurke and Stoler [1987] proposed using intensityfield correlations between signal and idler channels to prepare and observe Fock states in the process of parametric down conversion. Recently, the tomographic reconstruction of a one-photon state was achieved working with an extension of their technique (Crispino, Giuseppe, Martini, Mataloni and Kanatoulis [2000], Lvovsky, Hansen, Aichele, Benson, Mlynek and Schiller [2001]). The reconstruc-

5, § 2]

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tion relies on a time-integrated correlation, since the time scales in parametric down conversion are too short for current technology to follow the fluctuation over time. Intensity-field correlations also arise, more indirectly, in various other contexts: Vyas and Singh [2000], Deng, Erenso, Vyas and Singh [2001] on the degenerate OPO, and Vogel [1991] for resonance fluorescence. The review is organized as follows. We begin, in Section 2, by presenting the general theoretical framework for the measurement of intensity-field correlations, including a discussion of the time-reversal properties of the correlation function. Section 3 illustrates the ideas with theoretical calculations for three specific quantum optical systems. The results of experiments in cavity QED are then presented in Section 4; there we give a thorough description of the experimental apparatus required. In Section 5 we review work on time-integrated intensity-field correlations in parametric down conversion. We finish, in Section 6, with an overview of the impact intensity-field correlations have made in the area of quantum measurement theory. § 2. Theory Figure 1 shows a schematic of the intensity-field correlator. It is based upon the HBT intensity correlator implemented in the modern “start”/“stop” scheme found, for example, in Foster, Mielke and Orozco [2000b]. The principal difference is that there is a balanced homodyne detector (BHD) in place of the second photon detector in what would normally be the “stop” channel; so it is appropriate to name this method as conditional homodyne detection (CHD). Operation of the

Fig. 1. Schematic of the intensity-field correlator. The homodyne current I (t) is sampled over a series of time windows, tj − τmax
t
tj + τmax , each centered on a “start” time tj .

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correlator proceeds as follows: within a few correlation times before and after each “start”, the homodyne current I (t) is digitized, recorded, and used to update a cumulative average; averaging Ns such samples reduces the shot noise so that the surviving signal is a conditional average of the quadrature amplitude fluctuations of the input optical field.

2.1. The intensity-field correlation function hθ (τ ) For a more detailed analysis of the measurement, we consider √ a general optical source with power bandwidth 2κ and output source-field 2κ bˆ (in units of the square-root of photon flux). In order to record a nonzero signal, the firing of the “start” detector must be biased towards the identification of quadrature amplitude fluctuations of a particular sign. To achieve this, a coherent offset of the sourcefield is generally needed (BS1 in fig. 1). The offset also carries an adjustable phase, allowing the free selection of the quadrature to be measured. The input field to the correlator is then expressed in terms of the source field as √ √
 (2.1) 2κ aˆ = 2κ bˆ + Aeiϑ , where Aeiϑ is determined by the complex amplitude of the offset. [A similar offset is used in some quantum state reconstruction schemes (Banaszek and Wódkiewicz [1996], Wallentowitz and Vogel [1996], Lutterback and Davidovich [1997]).] Sometimes the source-field has a non-zero mean amplitude, as is the case for the cavity QED system considered in Section 4. In such a case, the offset is not needed. A fraction η of the input light is now sent to the balanced homodyne detector, with the remaining fraction 1 − η going to the photon detector in the “start” channel (BS2 in fig. 1). The photon flux operator at the photon detector is thus given in terms of the photon number operator for the source field (for simplicity, free-field operators are neglected as they do not contribute to normal-ordered averages):  S = (1 − η)2κ aˆ †a. ˆ

(2.2)

The balanced homodyne detector samples the quadrature phase amplitude that is in phase with the local oscillator field (LO in fig. 1), with operator value     = 2 η2κ aˆ θ , aˆ θ ≡ 1 aˆ e−iθ + aˆ † eiθ , D (2.3) 2 where θ is the LO phase. The conditional homodyne photocurrent, averaged over the Ns “starts”, is then Hθ (τ ) =

 + τ ): : S(t)D(t + ξ(τ );  S

(2.4)

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: : denotes time and normal ordering, and ξ(t) is the residual local oscillator shot noise that is present because the ensemble average is taken over a finite number of samples only; its magnitude depends in the usual way on detection bandwidth and the number of samples Ns . For positive τ , Hθ (τ ) can be factorized in a straightforward way with the help of the quantum regression formula to give    θ ) + ξ(τ ), Hθ (τ ) = D(τ, (2.5) c where the subscript c denotes conditioning of the state at time t on the detection of a photon. This is one of the most powerful results of the intensity-field correlation function in its quantum mechanical formulation; it gives access to the conditional dynamics of the quadrature phase amplitudes of the field, similar to the manner in which the intensity-intensity correlation function gives the conditional dynamics of the intensity (Carmichael, Brecha and Rice [1991], Brecha, Rice and Xiao [1999]). For negative τ , a construction of the post-selected conditional dynamics may be made on the basis of Bayesian inference (Section 6.1). When the source field is small and non-classical, its fluctuations, a manifestation of the uncertainty principle, dominate over its steady-state amplitude. It is these fluctuations that are of interest, and therefore the input-field operator aˆ is conveniently decomposed as aˆ = α + a, ˆ with α = a ˆ = |α| eiφ , and ˆ the fluctuation of interest. We now substitute eqs. (2.2) and aˆ = bˆ ≡ bˆ − b (2.3) into eq. (2.4), and at the same time make the decomposition into a mean field plus fluctuation. In addition, for the present discussion we make the assumption, clearly valid for the case of Gaussian statistics, that third order moments of the field fluctuations vanish. The resulting correlation function in terms of the quadrature fluctuation aˆ φ = (aˆ e−iφ + aˆ † eiφ )/2 is:

 2:aˆ φ (0)aˆ θ (τ ): + ξ(τ ). (2.6) Hθ,φ (τ ) = η2κ 2|α| cos(φ − θ ) + |α|2 + aˆ † a ˆ The assumption of Gaussian statistics is not necessary, and as Denisov, CastroBeltran and Carmichael [2002] have shown, presumes detailed balance, which for some systems does not hold (see Section 2.3). It is only for this special case, though, that there is a direct and simple connection with the spectrum of squeezing. The maximum signal to noise ratio is obtained with the coherent intensity much ˆ If, however, larger than the incoherent intensity, |α|2  aˆ † a ˆ = bˆ † b. we choose the coherent offset in such a way that the coherent and incoherent intensities are the same, ˆ ˆ = bˆ † b, |α|2 = aˆ † a

(2.7)

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[5, § 2

√ although one gives up a little in signal-to-noise ratio (a factor of 2 ), one gains a new perspective in the discussion of nonclassical features in the correlation function. Here we leave the amplitude of the mean field arbitrary, but adjust its phase to be in phase with the local oscillator (φ = θ ). We then obtain a normalized corre√ lation function after dividing eq. (2.6) by η2κ 2|α| (Carmichael, Castro-Beltran, Foster and Orozco [2000]), hθ (τ ) = 1 +

2 1 + |α|2 /aˆ † a ˆ

ξ(τ ) :aˆ θ (0)aˆ θ (τ ): +√ . † aˆ a ˆ η2κ 2|α|

(2.8)

In the limit of negligible residual shot noise (Ns → ∞), we denote the correlation function by h¯ θ (τ ) = 1 +

2 1 + |α|2 /aˆ † a ˆ

:aˆ θ (0)aˆ θ (τ ): , aˆ † a ˆ

(2.9)

and the spectrum of squeezing (Collett and Gardiner [1984], Carmichael [1987]) may then be written as  ∞   S(Ω, θ ) = 4F (2.10) dτ cos(2πΩτ ) h¯ θ (τ ) − 1 , 0

2κaˆ †a ˆ

where F = = 2κ(|α|2 + aˆ † a) ˆ is the input field photon flux. Thus, h¯ θ (τ ) achieves a time-resolved measurement of the quadrature amplitude fluctuations of the squeezed electromagnetic field. Notice that the measurement is independent of detection and collection efficiencies, though the efficiency η does appear in eq. (2.8) as one of the factors affecting the single-to-noise ratio. The measured degree of squeezing also depends on the determination of the photon flux F . The technique is nevertheless less sensitive to efficiencies than traditional squeezing measurements (Bachor [1998]) since the propagation losses are taken into account by the normalization of h¯ θ (τ ). Under the assumed conditions of Gaussian statistics, h¯ θ (τ ) is necessarily symmetric in time. We may then write the Fourier pair:  ∞   dτ exp(i2πΩτ ) h¯ θ (τ ) − 1 , S(Ω, θ ) = 2F −∞ (2.11)  ∞ 1 h¯ θ (τ ) − 1 = dΩ exp(−i2π Ωτ )S(Ω, θ ). 4πF −∞ Notice that the photon flux plays a role, in inverse relationship, in the relative sizes of the spectrum of squeezing and the intensity-field correlation function. From this, it would seem that for large photon flux, nonclassical effects might be observed more readily in measurements of the spectrum of squeezing, and for low photon flux, in measurements of hθ (τ ). There is also a relationship between the

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time averaged h¯ θ (τ ) and the degree of squeezing at zero frequency, and between the frequency averaged spectrum of squeezing and h¯ θ (0):  ∞   dτ h¯ θ (τ ) − 1 , S(0, θ ) = 2F −∞ (2.12)  ∞ 1 h¯ θ (0) − 1 = dΩ S(Ω, θ ). 4πF −∞ 2.2. Classical bounds for hθ (τ ) Squeezing is directly related to a reduction in the variance of fluctuations in one of the field quadrature amplitudes. The squeezing manifests itself in the time domain through violations of classical bounds on the correlation function h¯ θ (τ ). Carmichael, Castro-Beltran, Foster and Orozco [2000] derived two such classical bounds whose derivation we review here. We begin from the observation that the fluctuation intensity may be written as a sum of the normal-ordered variances for the quadrature field amplitudes: 2 ˆ = :aˆ θ2: + :aˆ θ+π/2 :. aˆ † a

(2.13)

Combining eq. (2.9) with this result leads to an expression for h¯ θ (0) in the form h¯ θ (0) − 1 =

:aˆ θ2 : 2 . 2 1 + |α|2 /aˆ † a ˆ :aˆ θ2 : + :aˆ θ+π/2 :

(2.14)

In the classical case, both quadrature variances are greater than zero, so we may deduce both lower and upper bounds for hθ (0): 0
h¯ θ (0) − 1


2 1 + |α 2 |/aˆ † a ˆ

.

(2.15)

The upper bound, in particular, is quite different from the familiar bounds on the intensity-intensity correlation function. Generalizing to non-zero time delay, we have the Schwarz inequality :aˆ θ (0)aˆ θ (τ ): 2
:aˆ 2 (0)::aˆ 2 (τ ): = :aˆ 2 :2 , (2.16) θ θ θ which implies h¯ θ (τ ) − 1
or

2 1 + |α|2 /aˆ † a ˆ

h¯ θ (τ ) − 1
h¯ θ (0) − 1 .

|:aˆ θ2:| , aˆ † a ˆ

(2.17)

(2.18)

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[5, § 2

This second condition and the lower bound in eq. (2.15) are similar to the classical bounds associated with the definition of photon antibunching (Kimble, Dagenais and Mandel [1977], Walls [1979], Loudon [1980], Paul [1982]). For a classical field such that the intensity is much larger than the variance, one can prove in addition an inequality relating the intensity-field correlation function h¯ θ (0) and the intensity–intensity correlation function g (2) (0); one finds h¯ θ (0)
g (2) (0). (2.19) If there is an offset field α such that the intensity is equal to the variance, then the inequality is: h¯ θ (0)
2g (2) (0). (2.20) 2.3. Time reversal properties of hθ (τ ) The time symmetry of the cross-correlation of fluctuations about thermal equilibrium, B(t + τ )A(t) = B(t − τ )A(t), where A and B are thermodynamic quantities, has a central place in statistical physics; it provides the fundamental basis for the Onsager relations (Onsager [1931], Casimir [1945]). The symmetry follows from microscopic reversibility (A and B are assumed both symmetric or antisymmetric under time reversal), which requires that the equilibrium state be maintained through detailed balance (Tolman [1938]). In quantum optics, one is usually concerned with steady states away from equilibrium, where correlation functions of the light emitted by an open system are measured through photoelectric detection. The detected radiation field is outgoing and absorbed by the environment; its steady state is thus manifestly not symmetric under time reversal. In a situation like this, fluctuations about the steady state may exhibit a specific time order. The majority of studies in quantum optics have focused, nonetheless, on timesymmetric correlations. There are two main reasons for this. First, nonclassical phenomena such as photon antibunching and squeezing deal with autocorrelations, A(t + τ )A(t), which are symmetric by definition for a stationary process. Second, although detailed balance is not required by microreversibility away from equilibrium (Klein [1955], Tomita and Tomita [1973, 1974]), it may follow, nevertheless, from symmetry and boundary conditions (Graham [1971]). A laser, for example, maintains its steady state through detailed balance (Graham and Haken [1971]) in spite of the fact that it operates far from thermal equilibrium. The cross-correlation of field intensity and amplitude provides, in principle, for the observation of time asymmetric correlations. Concerning the requisite failure

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of detailed balance, Tomita and Tomita [1973, 1974] determined what is needed in the case of Gaussian fluctuations: there must exist “a coupling between more than one degrees of freedom, so that there can be a direction” in the nonequilibrium flux through the system. Such a coupling – between the atom(s) and the cavity field – is a central feature in cavity QED. In the case of Gaussian fluctuations, however, hθ (τ ) reduces to the autocorrelation of eq. (2.8), which is necessarily time symmetric. It follows that conditional detection of the kind considered can reveal time asymmetry only in a regime where the fluctuations are non-Gaussian (cross-correlating a “start” detection in one channel with homodyne detection in another provides wider possibilities). In this case a time asymmetric hθ (τ ) not only indicates a breakdown of detailed balance, it also provides direct evidence of non-Gaussian fluctuations. We might expect resonance fluorescence to provide the simplest example of a time asymmetric hθ (τ ); its fluctuations are non-Gaussian and a coupling between degrees of freedom enters through the optical Bloch equations. Quantum transitions, in resonance fluorescence, occur between two states only, however, which suggests that detailed balance has to hold, since it is the only sort of balance that can maintain a steady state (Klein [1955]). It is indeed readily shown that hθ (τ ) is symmetric in resonance fluorescence. The two-state restriction is lifted, on the other hand, for multiphoton scattering in cavity QED. In this context, Denisov, Castro-Beltran and Carmichael [2002] recently computed time-asymmetric intensity-field correlation functions which demonstrate the breakdown of detailed balance. Examples of their results are presented in Section 3.2.3. 2.4. Intensity-field correlations in classical optics In Chapter 8 of their celebrated book, Mandel and Wolf [1995] treat correlation functions of arbitrary order in the field, both even and odd orders. They develop Schwarz inequalities for cross-correlations of arbitrary order and show that in the case of Gaussian noise, the odd-order correlation functions are zero, the result we drew on in passing from eqs. (2.4) to (2.6). Moreover, when the field is quasi-monochromatic and the statistical ensemble characterizing the fluctuations is stationary – though not necessarily Gaussian – the odd-order correlations are again zero except at very high orders. The same authors treat quantum mechanical correlation functions of arbitrary order in Chapter 12 of their book, where they note that the odd-order correlation functions arise naturally in connection with nonlinear media. In these media the quantum expectation value of the intensity depends on odd-order correlation

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[5, § 3

functions (involving unequal numbers of creation and annihilation operators). They show again, however, that when the electromagnetic field is stationary and quasimonochromatic, the odd-order correlations must vanish unless the order is very large. These observations appear to contradict our results. The resolution is that the approach presented in fig. 1 for the intensity-field correlation strictly speaking uses four fields: two for the intensity detection and two for the homodyne detection, except that the presence of the strong local oscillator does not appear explicitly in normalized expressions. § 3. Examples The intensity-field correlation function has been calculated for three important sources of nonclassical light. We review the results in this section. The first is the Optical Parametric Oscillator (OPO) well below threshold (Carmichael, CastroBeltran, Foster and Orozco [2000]), where results for hθ (τ ) clarify how such a source of highly bunched light can nevertheless show quadrature squeezing. For the second, a cavity QED source, the intensity-field correlation captures the oscillatory exchange of excitation between the cavity mode and atoms, the normalmode or polariton oscillation; the oscillation is related to the spectrum of squeezing (Carmichael, Castro-Beltran, Foster and Orozco [2000]) and a discussion of its degradation through spontaneous emission is given (Reiner, Smith, Orozco, Carmichael and Rice [2001]). The third example is a two-level atom coupled to the intracavity field of an OPO, which shows a mixture of the behavior demonstrated in the first two examples (Strimbu and Rice [2003]). Various methods are available for calculating the intensity-field correlation function. Most directly, the two-time average in eq. (2.4) may be evaluated from a knowledge of the source master equation, dρ = Lρ, (3.1) dt using the quantum regression formula. Generally, a different formula applies for positive and negative τ (Carmichael [1999]), allowing for the time asymmetry of Section 2.3. We have  + τ ): : S(t)D(t √ η2κ  S     tr aˆ e−iθ eL|τ | (aρ ˆ ss aˆ † ) + c.c., τ  0, = aˆ † a (3.2) ˆ −1    tr (aˆ † a) ˆ eL|τ | (aˆ e−iθ ρss ) + c.c., τ
0, ˆ where aˆ is related through eq. (2.1) to the source quasimode b.

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367

Alternatively, a simulation of the conditional averaging process that yields the correlation function may be given within the framework of quantum trajectory theory. As is well known, the theory of quantum trajectories is formulated around the experimental data, viewed as a stochastic measurement record (Carmichael [1993]). For the detection scheme of fig. 1, the record comprises the continuous homodyne current, I (t), and the set of start times {tj }. The source quasimode is in a quantum state |ψREC (t), conditioned on this record. Realizations of I (t),{tj }, and |ψREC (t) obey a set of stochastic differential equations that may be simulated on a computer. By sampling an ongoing realization of I (t), one calculates the conditionally averaged photocurrent as Hθ (τ ) =

Ns 1  I (tj + τ ). Ns

(3.3)

j =1

To carry out this program, the explicit quantum stochastic process (unravelling of the density operator ρ) must be formulated in line with the principles introduced in Sections 8.4 and 9.4 of Carmichael [1993], generalized in this case to include the coherent offset of fig. 1 and to combine the continuous evolution under homodyne detection with the quantum jump conditioning, |ψ¯ REC (tj ) → a| ˆ ψ¯ REC (tj ), at the start times tj (the state |ψ¯ REC (t) is not normalized). Clearly, ˆ in time step dt, the probability of a start count is (1 − η)2κ(aˆ †a)(t) REC dt. Between starts, |ψ¯ REC (t) evolves according to the stochastic Schrödinger equation  
   S /ih¯ − 2κA e−iφ bˆ dt + √ηκ aˆ e−iθ dQt ψ¯ REC (t) , (3.4) d ψ¯ REC (t) = H S is the non-Hermitian source Hamiltonian. The source state is condiwhere H tioned through this equation on the ongoing realization of charge,  dQt = η2κ aˆ θ REC dt + dWt , (3.5) deposited in the homodyne detector output circuit; the Wiener increment dWt incorporates the shot noise. A simple filtering equation introduces a realistic detection bandwidth Γ : dI = −Γ (I dt − dQt ).

(3.6)

If spontaneous emission is present, it may be incorporated in the usual way through additional quantum jumps. The limit of weak excitation is a special case, since in this limit the correlation function is time symmetric and may be calculated from the quantum trajectory equations by a straightforward analytical method. We write  ψREC (t) =

2  n+{m}=0

 Cn,{m} (t) n, {m} ,

(3.7)

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where n denotes the photon number of the source quasimode, and {m} is the set of all other relevant quantum numbers (referring to the internal states of atoms in a cavity QED system, for example). Note that under the assumption of weak excitation, we may truncate the expansion at the level of two quanta. This is the minimal nontrivial truncation; one quantum is required to provide the “start” count, and at least one other is needed if there is to be a nontrivial conditional signal at the BHD. Note now that for weak excitation, the “start” counts are extremely infrequent on the time scale taken by the source to relax to its steady state. The time interval between one “start” and the next is then almost certain to be long enough for the steady state, ss  ψ REC =

2 

 ss n, {m} , Cn,{m}

(3.8)

n+{m}=0

to be reached. The approach to the steady state may be calculated from 1  d|ψ¯ REC  ¯ REC , = H S |ψ dt ih¯

(3.9)

where the terms proportional to aˆ and bˆ in eq. (3.4) are neglected as higher order contributions. The conditional state after each “start” is now obtained as  ψREC (t + ) ≡

1 

 Cn,{m} (tj+ ) n, {m} =

j

n+{m}=0

ss  a|ψ ˆ REC ss |aˆ † a|ψ ss  ψREC ˆ REC

,

(3.10)

and solving eq. (3.9) with this state as the initial condition yields  ψREC (tj + τ ) =

2 

 Cn,{m} (tj + τ ) n, {m} .

(3.11)

n+{m}=0

From eqs. (2.1), (3.8), and (3.11), we obtain 
ss   aˆ θ ss = Re C1,{m=0} + A eiϑ e−iθ ,      aˆ θ (tj + τ ) REC = Re C1,{m=0} (tj + τ ) + A eiϑ e−iθ ,

(3.12) (3.13)

and finally, taking the limit Ns → ∞ in eq. (3.3) (also Γ → ∞), the result for the normalized correlation function is θ (τ ) Re{[C1,{m=0} (tj + τ ) + A eiϑ ] e−iθ } H = . h¯ θ (τ ) ≡ √ ss Re[(C1,{m=0} + A eiϑ ) e−iθ ] η2κ aˆ θ ss

(3.14)

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Fig. 2. Schematic of the OPO. A classical drive E , of frequency 2ω, injects energy into a cavity which contains a medium that has a nonlinear susceptibility χ (2) . The output is a field at the subharmonic frequency ω.

3.1. Optical parametric oscillator Because of its simple nonlinearity, the process of parametric down conversion in a cavity has been the subject of extensive research in quantum optics. This process is the basis of the optical parametric oscillator (OPO), which is modelled (see fig. 2) by two modes of the electromagnetic field, with frequencies ωa and ωb , and a nonlinear interaction proportional to ih¯ (aˆ †2 bˆ − aˆ 2 bˆ † ). The Hamiltonian for the two coupled modes may be written as ih¯ χ †2 ˆ (aˆ b − aˆ 2 bˆ † ). (3.15) 2 Energy conservation requires that the frequencies are related, with ωb = 2ω, ωa = ω. The coupling χ between the modes is proportional to the second order nonlinear susceptibility of the medium, χ (2) . In addition to the interaction shown, the modes also couple to reservoirs with decay constants γa and γb to account ˆ The OPO for cavity loss, and there is a strong coherent drive E of cavity mode b. shows a point of instability as a function of the drive at E = Eth ≡ γa γb /χ ; below this threshold the subharmonic mode has zero mean amplitude, while for E > Eth a nonzero mean field is established and parametric oscillation sets in. The fluctuations in this system exhibit very large squeezing just below threshold (Collett and Gardiner [1984, 1985]). Conditions of low photon flux, well below threshold where the squeezing is small, are of particular interest from the point of view of the intensity-field correlations. Although the squeezing is small, the output spectrum is a Lorentzian squared (Collett and Loudon [1987]), a manifestation of squeezing-induced linewidth narrowing (Rice and Carmichael [1988]). The output intensity shows very large bunching as the photons are created in pairs, a condition that has been of interest for producing a conditional source of single photons for quantum cryptography. With regard to the intensity-field correlation, conditions of low photon flux are of particular interest because they lead to extremely large violations of the upper bound of eq. (2.15) (Carmichael, Castro-Beltran, Foster and Orozco [2000]). † H = h¯ ωa aˆ † aˆ + hω ¯ b bˆ bˆ +

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Fig. 3. Quantum trajectory simulation of CHD for the OPO: (i) X-quadrature amplitude (unsqueezed), ˆ = 2.0 × 10−4 (ii) Y -quadrature amplitude (squeezed); with intracavity photon number aˆ † a (E¯ = 0.02), η = 0.5, Ns = 10,000. The dashed lines are the classical bounds.

The OPO with normalized pump parameter E 1, (E ≡ E/Eth ) has quadrature variances and fluctuation intensity (Milburn and Walls [1981]): ¯ + E)/4, ¯ :(qˆX )2 : ≈ E(1

(3.16)

¯ − E)/4, ¯ :(qˆY )2 : ≈ −E(1

(3.17)

aˆ † a ˆ = :(qˆX )2 : + :(qˆY )2 : ≈ E¯2 /2.

(3.18)

and

The ratio :(qˆX,Y )2 :/aˆ † a ˆ which enters on the right-hand side of eq. (2.14) ¯ If E¯ 1, the upper bound in eq. (2.15) may be exceeded is of the order of 1/E. by orders of magnitude. Figures 3(i) and (ii) illustrate this prediction for broadband detection. Well below threshold, where the squeezing is small (8% at line center), the classical bounds are violated dramatically. A violation exists for both quadratures of the field. It is permitted because of the anomalous phase of the fluctuation in fig. 3(ii), where, although the BHD current sampling is triggered by photon counts, the averaged data records a fluctuation that is out of phase with the offset; surely trigger counts would be more probable at the times of in phase fluctuations. The anomalous phase allows the sum of the quadrature variances to be much smaller than the modulus of either taken individually, and hence leads to the large violation of inequality (2.15).

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Fig. 4. The intensity-field correlation at zero delay for the OPO: hX (0) − 1 (i) and hY (0) − 1 (ii) as a function of normalized pump parameter E¯ . The dashed lines are the classical bounds.

The results displayed in fig. 3 show that conditional homodyne detection is not simply an alternate method for the detection of squeezed light, but provides a completely different window on its nonclassicality. This is underlined by fig. 4, where the violation of inequality (2.15) is increasing for decreasing pump pa¯ CHD rameter, while the squeezing and photon flux both decrease. For small E, detects anomalously large fluctuations of the field amplitude which are isolated in time through the conditional measurement. It records only the real fluctuations associated with the rare two-photon pulses seen in direct photon detection. While the intensity–intensity correlation of the photon pulses is highly bunched and looks classical, with g (2) (0) ∼ 1/E¯2 , CHD resolves this correlation into quadrature amplitude components and uncovers the anomalous phase behavior at the level of the field amplitude. In related work, several authors have used the beating of a local oscillator with a signal field on a beam splitter to enhance the ability to measure nonclassical effects such as photon antibunching and squeezing. This includes work by Vogel [1991] on resonance fluorescence, and Vyas and Singh [2000] and Deng, Erenso, Vyas and Singh [2001] on the OPO. In the latter case, the output of the OPO mixed with the LO yields antibunched light, whereas it is highly bunched on its own. Neither of these schemes relies on a conditioned measurement. A scheme that does use a conditioned measurement to see antibunching in an OPO system has recently been proposed by Leach, Strimbu and Rice [2003]. Siddiqui, Erenso, Vyas and Singh [2003] discuss conditional measurements as probes of quantum dynamics and show that they provide different ways to characterize quantum fluctuations in a subthreshold degenerate OPO.

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Fig. 5. Schematic of the cavity QED system. A classical drive E at frequency ω injects energy into a single mode of the cavity with one or more two-level atoms coupled to the mode at rate g. The atomic polarization decay rate is γ⊥ and the cavity decay rate is κ.

3.2. Cavity QED We next consider a cavity QED system that consists of a single mode of the electromagnetic field interacting with a collection of two-level atoms (see fig. 5). Two spherical mirrors form an optical cavity that defines the field mode. A single or a few two level atoms are optimally coupled at rate g to the cavity mode. Dissipation occurs through decay of the field from the cavity at rate κ and decay of the atomic inversion at rate γ = 1/τ (τ is the radiative lifetime of the atomic transition) and polarization at rate γ⊥ . For purely radiative decay, γ = 2γ⊥ . The field E/κ drives the system through one of the mirrors and it is possible to detect the light that escapes from the cavity mode through the output mirror. The atom-cavity coupling rate is given by: 2 1/2 µ ω g= (3.19) 2h¯ ε0 V for cavity mode volume V , atomic transition frequency ω, and dipole moment µ. Work on Optical Bistability (OB) (Lugiato [1984]) produced a large amount of experimental and theoretical literature on the transmission properties of an optical cavity filled with two-level atoms. Two dimensionless numbers from the OB literature are useful for characterizing cavity QED systems: the saturation photon number n0 and the single atom cooperativity C1 . Defined as n0 = 2γ⊥ γ /3g 2 and C1 = g 2 /2κγ⊥ , they scale the influence of a photon and the influence of an atom in the system. The strong coupling regime of cavity QED n0 < 1 and C1 > 1 implies very large effects from the presence of a single photon and of a single atom. The Jaynes–Cummings Hamiltonian describes the interaction of a two-level atom with a single mode of the quantized electromagnetic field (Jaynes and Cummings [1963]),  = h¯ ωa σˆ z + h¯ ωc aˆ † aˆ − ih¯ g(σˆ + aˆ − aˆ †σˆ − ), H

(3.20)

5, § 3]

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373

where σˆ ± and σˆ z are the Pauli spin operators, or raising, lowering, and inversion operators of the atom, and aˆ † , aˆ are the raising and lowering operators for the field. The eigenstates of Hamiltonian (3.20) reveal the entanglement between the atom and the field. The spectrum has a first excited state doublet with states shifted by ±g from the uncoupled resonance. The equilibrium state of the atom-cavity system is significantly altered by the escape of a photon. The dynamic consist of a reduction (collapse) of the system state |ψ followed by a damped Rabi oscillation back to equilibrium. We are interested in the reduction of the equilibrium state of the cavity QED system after detecting a photon emitted from the cavity mode. Defining Aˆ θ ≡ (aˆ exp(−iθ ) + aˆ † exp(iθ ))/2, where aˆ is the annihilation operator for the cavity field and θ is the homodyne detector phase, we consider the quadrature amplitude, Aˆ 0◦ , in phase with the steady state of the field amplitude λ ≡ a ˆ = E/[κ(1+2C)]. We limit the discussion to the case where the cavity and laser are resonant with the atomic transition. For weak excitation, and assuming fixed atomic positions the equilibrium state to second order in λ is the pure state (Carmichael, Brecha and Rice [1991], Brecha, Rice and Xiao [1999]): 
√   |ψSS  = |0 + λ|1 + λ2 / 2 χβ|2 + · · · |G   + ς |0 + λςβ|1 + · · · |E + · · · , (3.21) where |G is the N atom ground state and |E is the symmetrized state for one atom in the excited state with all others in the ground state. We assume that all N atoms are coupled to the cavity mode with the same strength, g, with χ, β and ς derived from the master equation in the steady state (Carmichael, Brecha and Rice [1991]): √ N g0 λ 1 + 2C  ; ς =− χ = 1 − 2C1 ; β = (3.22)  1 + 2C − 2C1 γ⊥ where: C ≡ NC1 ;

C1 ≡

C1 . (1 + γ⊥ /κ)

(3.23)

After detecting the escaping photon, the conditional state is initially the reduced state a|ψ ˆ SS /λ, which then relaxes back to equilibrium. The reduction and regression is traced by (Carmichael, Brecha and Rice [1991], Brecha, Rice and Xiao [1999]):     |ψSS  → |0 + λ 1 + AF (τ ) |1 + · · · |G + · · · , (3.24)

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Intensity-field correlations of non-classical light

[5, § 3

where 4C1 C , 1 + 2C − 2C1



−(κ + γ⊥ )τ κ + γ⊥ cos Ω0 τ + F (τ ) = exp sin Ω0 τ , 2 2Ω0  1 Ω0 = g02 N − (κ − γ⊥ )2 . 4 A=−

(3.25) (3.26)

(3.27)

From eqs. (3.21) and (3.24) it is possible to see that after a photodetection, the quadrature amplitude expectation makes the transient excursion Aˆ 0◦ (τ ) → λ[1 + AF (τ )] away from its equilibrium value Aˆ 0◦  = λ. In the weak driving limit, which assumes up to two excitations in the steady state of the system, the conditional field measurement is:
 hθ (τ ) = 1 + AF (τ ) cos θ. (3.28) Thus, the correlation function hθ (τ ) measures the coefficient of the single photon state in eq. (3.24). It is usually a very small number and it is appropriate to talk of a field fluctuation at the sub-photon level. A is the relative change of the field inside the cavity caused by the escape of a photon (Carmichael, Brecha and Rice [1991], Reiner, Smith, Orozco, Carmichael and Rice [2001]). The limit of large N gives A ≈ −2C1 /(1 + γ⊥ /κ), showing the importance of the single atom cooperativity as the parameter that establishes the non-classicality of the field. The sign of A tells us that the fluctuation goes negative causing a reduction in the cavity field. Two dimensionless fields and intensities follow from the OB literature that allow to make contact with experiments: The intracavity field (intensity) with atoms √ in the cavity is given by x ≡ a/ ˆ n0 , (X ≡ aˆ † a/n ˆ 0 ), and the field (intensity) √ without atoms in the cavity y ≡ E/κ n0 , (Y = y 2 ); note that 2E 2 /κ is the input photon flux. 3.2.1. Low field, weak driving limit Figure 6 presents results from Reiner, Smith, Orozco, Carmichael and Rice [2001] for the intensity-field correlation function and the spectrum of squeezing for very low intensity and at most two excitations in the system. Both calculations are for a single atom maximally coupled using quantum trajectories. The size of the nonclassicality of h(τ ) is very large, and as is the case with the OPO, the size of squeezing is very small. Fluctuations are very rare, but they are very large compared to the mean. A single photon fluctuation is extremely large compared to a

5, § 3]

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375

Fig. 6. (i) Intensity-field correlation h(0◦ , τ ) from the quantum trajectory implementation of the conditioned homodyne detection for (g, κ, γ⊥ , Γ )/(2/π ) = (38.0, 8.7, 3.0, 100) MHz, X = 2.99 × 10−4 . (ii) Spectrum of squeezing calculated from the cosine Fourier transform of h(0◦ , τ ) in (i). The continuous thin line is the exact spectrum of squeezing.

saturation photon number of 0.01 and the system is driven to produce an intracavity intensity X ≈ 3 × 10−4 . The oscillations present are at a frequency given by the coupling constant g. The spectrum of squeezing is the so-called “vacuum Rabi” doublet (Carmichael, Brecha, Raizen, Kimble and Rice [1989]); the fluctuations develop as spontaneous Rabi oscillations. The anomolous phase yielding a dip rather than a peak at τ = 0 is responsible for the squeezing. The thin line in fig. 6(ii) is the spectrum of squeezing calculated directly from the quantum regression theorem. The solid line is the Fourier transform (see eq. 2.10) of fig. 6(i) which comes from averaging the photocurrent from a quantum trajectory simulation over 55,000 “starts”. Both approaches show the damped Rabi oscillations which precede and follow a photodetection. In the weak excitation limit, Rice and Carmichael [1988] derived an analytical expression for the spectrum of squeezing (thin line in fig. 6(ii)) which agrees with these results. Figure 7(i) from Reiner, Smith, Orozco, Carmichael and Rice [2001] shows the evolution of the field following the detection of a photon escaping through the cavity mode, and fig. 7(ii) shows the field evolution following the spontaneous emission of a photon out the side of the cavity. The collapse operation on the state |ψSS  of the type found in eq. (3.24) is the dynamical mechanism which describes these two results. These two distinct behaviors correspond fairly loosely to the regression to equilibrium observed in the step excitation in the field, fig. 7(i), and a step excitation in the atomic polarization, fig. 7(ii). Note the phase shift between the two responses. The steady state wavefunction determines the size of the steps. An undetected spontaneous emission produces the reduced state σˆ − |ψSS /λ, which sets up a completely different evolution as shown in fig. 7(ii).

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Intensity-field correlations of non-classical light

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Fig. 7. Regression of a cavity QED system back to steady state; N = 1, low intensity; (i) after the detection of a photon escaping out of the cavity mode; (ii) after the escape of a photon through spontaneous emission. The inset shows the sequence of events in terms of the cavity QED system and the detector. The parameters used are the same as in fig. 6.

Fig. 8. Quantum trajectory simulation of CHD for many-atom cavity QED. (i) X-quadrature amplitude (in phase with the mean field), the dashed lines are the classical bounds. Curve (ii) is the spectrum of squeezing obtained from the X-quadrature simulation.

Quantum trajectories allow calculation with more than one atom and even permit to include the effects of an atomic beam. This approach gives a more accurate picture of the process in the laboratory. Figure 8 from Carmichael, Castro-Beltran, Foster and Orozco [2000] illustrates a calculation of the conditional field applied to cavity QED. Figure 8(i) shows violations of the inequality (2.15), while squeezing is evident from both the anomalous phase of the oscillation and the calculation of the spectrum of squeezing from the Fourier transform of the correlation function (fig. 8(ii)). The quantum trajectory calculation takes into account a typical transit time for an atomic beam experiment, dipole coupling constant g = 3.77κ, ˆ = 1.5 × 10−4 , atomic decay rate of γ = 1.25κ, intracavity photon number aˆ † a η = 0.5, Ns = 20,000, and an overall detection bandwidth of Γ = 10κ. The re-

5, § 3]

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377

sults for many atoms show that the predictions for one atom hold at a reduced size in an atomic beam. 3.2.2. High field, outside the weak driving limit The weak field calculations of the previous section make it clear that in the strong coupling regime a cavity emission will always produce a negative shift in the field. The ratio of the probability for a spontaneous emission to the probability for a cavity emission from steady state is Pspont = 2NC1 . Pcavity

(3.29)

Thus, in the strong coupling regime it is more likely for an atom to spontaneously emit out the sides of the cavity than for the cavity to emit a photon. In the following we consider what happens in the likely event that a cavity photon follows a spontaneous emission. Figure 9 from Reiner, Smith, Orozco, Carmichael and Rice [2001] shows representative quantum trajectories calculated with two atoms in the cavity and allowing for more that two excitations in the system, so we are outside the weak driving limit. In fig. 9(i) the evolution starts with a spontaneous emission (A) out the side of the cavity, followed at (B) by a photon escaping through the cavity mode that is registered by the detector. The field jumps positive and changes curvature with the escaping cavity photon. The driving field (E/κ), atom-field coupling (g), and decay rates (κ, γ⊥ ) are such that the system is in a regime where the cavity field is bunched. Qualitatively, if there is a spontaneous emission event when the system contains few excitations, it returns the system to the steady state, as in fig. 6(i). If the spontaneous emission event happens while in the bunched regime, followed by a cavity emission, then there are probably more excitations in the system. With one of the atoms removed following the spontaneous emission, the probability for this energy to be in the cavity mode is increased. If there is a detection of a cavity photon soon after the spontaneous emission, then the system is in a regime where the intracavity field undergoes a large amplitude fluctuation, and the value of the cavity field is higher than the steady state value. This causes an upward jump in the expectation of the field. These types of events increase linearly with the number of atoms in the cavity, since the ratio of spontaneous emission events to cavity loss events is 2NC1 . The time evolution of the conditional field of the same system driven much harder shows multiple jumps, some from spontaneous emission and some from escapes through the mirror. The dynamics get very complicated; fig. 9(ii) shows

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Intensity-field correlations of non-classical light

[5, § 3

Fig. 9. A quantum trajectory simulation which shows the time evolution of the field back to steady state after the detection of a photon. (i) A spontaneous emission event followed by a cavity emission which starts the averaging process. (ii) A spontaneous emission event followed by many cavity emission events. The inset shows the sequence of events in terms of the cavity QED system and the detector. Both figures were prepared with √the following parameters: N = 2, X = 18.1, (g, κ, γ )/(2π ) = (38.0/ 2, 8.7, 3.0) MHz.

an example for illustration. The average value of the field resulting from the conditional fluctuations is much larger than the steady state in such cases. Figure 9 demonstrates the insight that can be gained by studying individual trajectories. In this case, when the system is undergoing a large fluctuation, the intracavity photon number increases. This provides for a larger excited state population and higher probability of spontaneous emission, typically followed by a series of cavity emissions. Quantum trajectories give us insight into the underlying physics of the system which might not be evident from direct numerical solution of the master equation. Here we can see how spontaneous emission produces an incoherent field that can degrade the non-classicality of the correlation function and the squeezing. The entire trajectory is a collection of events well separated in time of the type in fig. 9(i) and (ii). The average over many random realizations of these different events with an initial cavity emission setting the trigger at t = 0, recovers the conditioned field evolution. Figure 10 shows results for two atoms maximally coupled to the cavity mode with a drive that corresponds to a steady state intracavity intensity of n/n0 = 18, far from the weak driving limit. The coupling constant g produces a similar vacuum Rabi oscillation to that of fig. 6. The background that is visible in fig. 10(i) around τ = 0 comes from the spontaneously emitted photons (Reiner, Smith, Orozco, Carmichael and Rice [2001]). This background leads to a modification of

5, § 3]

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379

Fig. 10. (i) h0◦ (τ ) for N = 2 atoms beyond the low intensity regime. (ii) Squeezing spectrum from the correlation function in (i).

Fig. 11. Time asymmetry in the cross-correlation of the intensity and field amplitude of the forwards scattered light in cavity QED. Results for one atom and no external noise. All curves are for √ N g/κ = 8 and γ /κ = 1.25 with intracavity photon numbers aˆ † a ˆ = 10−4 (i), 10−3 (ii), 10−2 (iii), and 10−1 (iv).

the spectrum of squeezing, calculated from the symmetrized correlation function, as shown in fig. 10(ii). Comparing the spectra in fig. 6 to that of fig. 10, a positive peak centered at the LO frequency (Ω = 0) has appeared, corresponding to the higher rate of spontaneous emission. 3.2.3. Time asymmetry in cavity QED Figures 11, 12 and 13 present results from Denisov, Castro-Beltran and Carmichael [2002] for the time asymmetry (Section 2.3) of the correlation function in cavity

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Intensity-field correlations of non-classical light

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Fig. 12. Time asymmetry in the cross-correlation of the intensity and field amplitude of the forwards scattered light √ in cavity QED. Results for N  1 atoms and amplitude noise on the external field. All curves are for N g/κ = 8 and γ /κ = 1.25, Y = 13 and noise strength 2Υ 2 = 25 (i), 50 (ii), 80 (iii), and 120 (iv).

Fig. 13. Time asymmetry in the intensity-field correlation of the output field for many-atom cavity QED without external noise. The dotted lines are for emphasis of the asymmetry.

QED. Here they have not made any assumptions about the nature of the noise. First they look at the cavity QED system while increasing the strength of the external field in fig. 11 [(i)–(iv)], and compare results for increasing external noise in absorptive bistability in fig. 12 [(i)–(iv)]. The two sets of results are selected for the qualitatively similar development. They do not correspond to the same operating parameters, although the decay rates and coupling strengths are matched. In both the weak and strong excitation limits the correlation functions are time symmetric. Time asymmetry is limited to a transition region of non-Gaussian fluctuations. Note how the oscillation is inverted and much bigger in fig. 11 com-

5, § 3]

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381

pared with fig. 12. These distinctly nonclassical features are violations of the inequalities discussed earlier (see eqs. (2.15) and (2.18)). Figure 13 shows results for many atom cavity QED without external noise. These results come from a direct implimentation of the quantum regression formula. They are more similar to what happens in the laboratory (Foster, Orozco, Castro-Beltran and Carmichael [2000a], Foster, Smith, Reiner and Orozco [2002]). The results are averaged over 200 configurations of the five atoms most strongly coupled to a TEM00 standing-wave cavity mode [gj = g cos θj exp(−rj2 /w02 )] assuming a uniform spatial distribution of atoms: for eff = 11 atoms inside the mode waist, g0 /κ = 3.7, γ /κ = 1.25, and mean inN tracavity photon numbers aˆ † a ˆ ≈ 2.1 × 10−3 (i) and 7.3 × 10−3 (ii). The main deficiency of the approximation is an overestimate of the dephasing caused by atomic beam fluctuations. In spite of this, time asymmetry is found in qualitative agreement with the experimental results (figs. 3(a) and 4(b) of Foster, Orozco, Castro-Beltran and Carmichael [2000a]); in particular, the calculations reproduce the change in the sign of the asymmetry with increasing excitation strength. 3.3. Two-level atom in an optical parametric oscillator Strimbu and Rice [2003] have considered the intensity-field correlation function for a two level atom in a degenerate optical parametric oscillator. They show large violations of the Schwartz inequalities in hθ (τ ). We may view the system as an atom-cavity system driven by the occasional pair of correlated photons. 3.3.1. The physical system Consider a single two-level atom inside an optical cavity that also contains a material with a χ (2) nonlinearity. The atom and cavity are assumed to be resonant at frequency ω and the system is driven by light at 2ω. The system is shown in fig. 14. The interaction of the driving field with the nonlinear material produces

Fig. 14. Two-level atom inside a driven optical parametric oscillator. E is the driving field at frequency 2ω, g is the atom-field coupling, γ is the spontaneous emission rate out the sides of the cavity, and 2κ is the rate of intracavity intensity decay.

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light at the sub-harmonic frequency ω. This light consists of correlated pairs of photons, or quadrature squeezed light. In the limit of weak driving fields, these correlated pairs are created in the cavity and eventually two photons leave the cavity through the end mirror or out the side before the next pair is generated. The system Hamiltonian in the rotating wave and dipole approximations is
†  1 H = ih¯ E(aˆ †2 − aˆ 2 ) + ih¯ g(aˆ † σˆ − − aˆ σˆ + ) + hω (3.30) ¯ aˆ aˆ + 2 σˆ z , where g is the usual Jaynes–Cummings atom-field coupling defined in eq. (3.19). The effective two-photon drive E is proportional to the amplitude Ein (2ω) of a field at twice the resonant frequency of the atom (and resonant cavity) and the χ (2) of the nonlinear crystal in the cavity. We use standard techniques to treat dissipation with a Liouvillian describing loss due to the leaky end mirror at a rate κ and spontaneous emission out the side of the cavity at a rate γ⊥ = γ /2. 3.3.2. Discussion of the model Jin and Xiao [1993, 1994] considered the spectrum of squeezing and incoherent spectrum for this system. Clemens, Rice, Rungta and Brecha [2000] considered the incoherent spectrum in this system in the weak field limit, and found a variety of nonclassical effects. In the strong coupling regime, the spectrum consisted of a vacuum-Rabi doublet with holes in each sideband. Outside this regime spectral holes and narrowing were reported. These were attributed to quantum interference between various emission pathways, which vanishes when the number of intracavity photons and the number of pathways increases. As we are working in the weak driving field limit, we only consider states of the system with up to two quanta; basis states are denoted |−, n, |+, n where the first index denotes the state of the atom (− for ground state, and + for excited state) and the second index corresponds to the excitation of the field (n = number of quanta). In this case we describe the system by a conditioned wave function, which evolves via a non-Hermitian Hamiltonian and associated collapse processes (Carmichael [1993]). The state and Hamoltonian are given by ∞   ψc (t) = C−,n (t) e−iE−,n t |−, n + C+,n (t) e−iE+,n t |+, n,

(3.31)

n=0


2 
 HD = −iκ aˆ †aˆ − iγ⊥ σˆ + σˆ − + ih¯ E aˆ † − aˆ 2 + ih¯ g aˆ † σˆ − − aˆ σˆ + , and the collapse operators are √ Ccav. = 2κ a, √ Cspon.em. = γ σ− .

(3.32)

(3.33) (3.34)

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For an initial trigger detection in the transmitted field, the appropriate collapsed state is given by T ˆ SS  ψ = a|ψ . c |a|ψ ˆ SS |

(3.35)

In the weak field limit this becomes √ SS SS |+, 0 T 2 C−,2 |−, 1 + C+,1 ψ =

. c SS 2 SS 2 2|C−,2 | + |C+,1 |

(3.36)

Note there is no population in the ground state. As photons are created in the cavity in pairs, upon detection of a transmitted photon, the system contains one quanta, either as a cavity mode excitation (photon) or an internal excitation of the atom. In the weak field the probability of more than two quanta in the system initially is negligible; this is not the case for higher excitations. The certainty of the number of quanta is at the heart of all the nonclassical effects observed, these will vanish as the driving field increases. It is this driving of the system by the occasional pair of photons in an entangled state that creates most of the interesting effects. While this might be a difficult way to prepare such a state, by proper choice of g, κ, and γ⊥ , any superposition of |+, 0 and |−, 1 may be created. After the detection, the system evolves in time to T  ψ (τ ) = C CT (τ )|−, 1 + C CT (τ )|+, 0, (3.37) c −,1 +,0 where the superscript CT indicates a collapse associated with a photon detection in transmission. The appropriate initial conditions are √ SS 2 C−,2 CT (0) = , C−,1 (3.38) SS 2 SS 2 2|C−,2 | + |C+,1 | CT (0) = C+,0

SS C+,1 2

2

.

(3.39)

SS SS 2|C−,2 | + |C+,1 |

In terms of the one-photon probability amplitudes, √ CT SS CT (τ )C SS CT (τ ) cos θ C−,1 2 C−,1 (τ )C−,2 + C+,0 +,1

+ . hθ (τ ) = 1 + SS 2 SS 2 SS 2 SS 2 2|C−,2 | + |C+,1 | 2|C−,2 | + |C+,1 | (3.40) The first two terms are of order unity, while the third term is of order 1/F . For weak fields, this term can be arbitrarily large, in violation of the inequality in eq. (2.15).

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3.3.3. Results The intensity-field correlation function for the transmitted field of a two-level atom in an optical parametric oscillator in the weak field limit behaves essentially as a cavity QED system where an occasional pair of photons appears in the cavity and interacts with the system. The correlation function shows violations of the classical Schwartz inequality (2.15). Figure 15(i) shows results from Leach, Strimbu and Rice [2003] with a plot of hθ (τ ) for weak coupling (g/γ = 0.1, g/κ = 0.02), with cavity decay dom-

Fig. 15. Plot of hθ (τ ) as a function of γ τ for weak coupling g. (i) aˆ † a ˆ = 3.8 × 10−4 . (ii) aˆ † a ˆ = 2.0 × 10−2 . The dashed lines indicate the range allowed for classical fields. See text for other parameters.

Fig. 16. Plot of hθ (τ ) as a function of γ τ with strong coupling g and aˆ † a ˆ = 4.0 × 10−4 . The dashed lines indicate the range allowed for classical fields. See text for other parameters.

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inant over spontaneous emission (κ/γ = 5). It presents a large violation of the inequality (2.15). Figure 15(ii) plots hθ (τ ) for g/γ = 0.1 and g/κ = 1.0 with spontaneous emission dominant over cavity decay (κ/γ = 0.1); again there is a violation of inequality (2.15). Figure 16 shows results in the regime of strong coupling (g/γ = 5.0, g/κ = 10.0) where there is a further violation of inequality (2.15). The nonclassical behavior tends to go away as the driving field is increased (i.e. more photons are present) or as atoms are added to the system.

§ 4. Experiment in cavity QED The experimental investigation of the intensity-field correlation function in cavity QED (Foster, Orozco, Castro-Beltran and Carmichael [2000a], Foster, Smith, Reiner and Orozco [2002]) demonstrates the power of conditional homodyne detection. The measurement scheme detects the fluctuations of one quadrature of the cavity field as they happen in the laboratory. In this section we review the experimental results and present a description of the apparatus.

4.1. Cavity QED apparatus The cavity QED system consists of a beam of optically pumped Rb atoms traversing a driven high finesse Fabry–Perot cavity. At its heart is the optical cavity. Piezo-electric transducers are attached to the cavity mirrors to provide control over the cavity length. During measurements, a feedback loop holds it on the TEM00 resonance, where it supports a Gaussian standing wave mode with waist w0 = 21 µm and length l = 410 µm. A one-sided configuration is used with 300 ppm transmission at the output mirror and 10 ppm transmission at the input mirror. An effusive oven, 35 cm from the cavity, produces a thermal beam of Rb atoms in a chamber pumped by a large diffusion pump operated at typical pressures of 1 × 10−6 Torr. The oven is heated to ≈ 430 ± 0.1 K with the help of computer controlled feedback. Final collimation is provided by a 70 µm slit on the front of the cavity holder. The cavity is surrounded by a liquid-nitrogen-cooled Cu sleeve to reduce the background atomic vapor, as the presence of too much background destroys the observed correlations. The excitation source is a Verdi 5 pumped titanium sapphire (Ti:Sapph) laser, a modified Coherent 899-01. The primary laser beam is split into a signal beam and

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auxiliary beams for laser frequency locking, cavity locking, and optical pumping. All beams are on resonance with the 5S1/2 , F = 3 → 5P3/2 , F = 4 transition of 85 Rb at 780 nm. The atoms are optically pumped into the 5S 1/2 , F = 3, mF = 3 state in a 2.5 Gauss uniform magnetic field applied along the axis of the cavity using the appropriate circular polarization of the pumping light. The cavity is locked with a Pound–Drever–Hall scheme. During data collection, the laser beam traverses a chopper wheel which alternately passes the lock beam and opens the path from the cavity to the photon counting detectors at ≈ 1.1 kHz. Polarizing optics separate the signal from the lock. The signal beam is directed to the correlator. Between 50 and 85% of the signal emitted from the cavity is sent to the BHD. The remaining 50 to 15% of the signal goes to the avalanche photodiodes. The choice is guided by experimentally finding the best signal to noise ratio after averaging some 60,000 samples. The three rates governing the atom-cavity coupling, cavity decay, and atomic decay in the cavity QED apparatus are (g, κ, γ⊥ )/2π = (12, 8, 3) MHz, which yield C1 = 3 and n0 = 0.08, placing the experiment in the strong coupling regime of cavity QED. The measurements are carried out with an average intracavity field less than that of one photon. 4.2. Conditional homodyne detector Measurement of the intensity-field correlation requires a homodyne measurement of the signal to be made correlated with photon detections. A modified Mach– Zehnder interferometer was implemented to perform the measurement. Figure 17 shows a schematic of the interferometer and its integration with the cavity QED apparatus. Light enters the Mach–Zehnder interferometer, driving the cavity QED system on one arm and providing a local oscillator (LO) for the BHD on the other (Yuen and Chan [1983a, 1983b]). A fraction of the signal is directed to the BHD and the remainder is sent to the intensity detector (avalanche photodiode APD). The photocurrent from the BHD is proportional to the LO-selected quadrature amplitude of the signal field. Photon detections at the APD are correlated with the BHD photocurrent to measure the intensity-field correlation function via eq. (3.3). We discuss each component of the measurement in the following. 4.2.1. Mach–Zehnder interferometer The Mach–Zehnder interferometer is used to separate the laser into a local oscillator and a signal beam which, although they follow separate paths, maintain a

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Fig. 17. Schematic diagram of the cavity QED experimental setup.

constant relative phase. Control of the relative phase is critical for the measurement. It is achieved by adjusting the path difference of the two arms with a piezoactuated mirror and actively stabilizing the interferometer length with a feedback system. The stabilization uses a thermally stabilized He-Ne laser (λ = 633 nm) or diode laser (λ = 640 nm) locked using FM sidebands to an Iodine cell. The cavity QED system is transparent to the red wavelengths, but they form fringes at the Mach–Zehnder output. An edge filter separates the 780 nm and red wavelengths at the output. The MZ length is continually adjusted so such that it sits at a red fringe maximum or minimum. A phase change may be introduced by locking the length to different red fringes. In this way the IR phase can be adjusted in steps of δθIR = 146 ◦ = (180◦ × 633/780). There is also an optical path delay that can be mechanically adjusted to bring the IR and red in phase at a particular fringe. 4.2.2. Amplitude detectors The combined signal and LO field is directed to a pair of biased silicon photodiodes configured as a BHD. The AC coupled current from the photodiodes is amplified and subtracted to allow common mode rejection of local oscillator intensity noise (technical noise). The current from each detector first passes through a bias T which filters DC components below 100 kHz. The DC component provides a direct measure of the local oscillator current.

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4.2.3. Intensity detectors The intensity detectors are arranged as a photon correlator consisting of two avalanche photodiodes (APD) behind an unpolarized 50/50 beam splitter. The detectors have a quantum efficiency of 50%, a dark count rate of less than 50 Hz, and a 30 ns dead time. The detector electronics produce a TTL pulse for each photon detection, a copy of which goes to a counter that measures the photon count rate of each APD. These rates yield the mean intensity of the light emitted from the cavity after correcting for efficiencies and linear losses. 4.2.4. Correlator data acquisition The homodyne current is sampled with a digital oscilloscope triggered by photon detections registered at the APDs. The trigger is produced by the apparatus used elsewhere for intensity correlation measurements (Foster, Mielke and Orozco [2000b]). Instead of correlating the signals from the APDs, for the intensity-field correlation the two signals are combined in a logical OR. The digital oscilloscope samples the BHD photocurrent over a 500 ns window at 2 Gs/s with an 8 bit analog to digital (A/D) converter. It performs a summed average of the triggered samples. Typically up to 5 × 104 samples are taken.

4.3. Measurements The saturation photon number n0 in the experiment is less than one and hence the observed fluctuations are associated with the emission of a single conditioning photon. Figure 18 shows data taken at an intracavity intensity n/n0 = 0.30, cor-

Fig. 18. (i) Conditional field (unnormalized) at low intensity (n/n0 = 0.3) and (ii) its Fast Fourier Transform.

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Fig. 19. Plots (i) and (ii) are unnormalized homodyne averages with a phase difference of 146◦ .

responding to a mean intracavity photon number of 0.027. The data is in the low intensity regime. The trace on the left is the unnormalized correlation function H (τ ); it shows non-classical behavior, violating the bound of eq. (2.15), as the correlation has a minimum at τ = 0. On the right is its FFT, which in accord with eq. (2.10) is proportional to the spectrum of squeezing. The squeezing spectrum shows a dip at the vacuum Rabi frequency Ω0 . There is qualitative agreement with the prediction of fig. 6, although the data is not normalized as would be required for a quantitative comparison. The continuous line fitted to the FFT has the functional form predicted by the low intensity theory (Reiner, Smith, Orozco, Carmichael and Rice [2001]). The BHD measures the interference between the local oscillator and the emitted cavity field, which depends on their relative phase (eq. (2.6)). A comparison of the conditionally averaged AC coupled photocurrent for two different local oscillator phases is shown in fig. 19. When the relative phase is changed by 146◦ ≈ 180◦ (see Section 4.3.1) the sign of the interference changes. The normalization of these raw results is discussed in detail in the next section. Notice, however, that even without the normalization, the value of the field at τ = 0 in fig. 19(i) is clearly smaller than its steady-state value. This is further evidence of a non-classical field, since it violates the lower bound of eq. (2.15). This nonclassical feature demonstrates that the field fluctuations are anticorrelated in a similar way to the intensity in photon antibunching. Rather than seeing random field fluctuations, we see explicit evidence of the projection of the polarization field out of phase with the steady-state intracavity field.

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4.3.1. Normalization The correlation function defined by eq. (2.8) is normalized to the mean field; therefore, obtaining the proper normalization calls for precise knowledge of this field. As the detection system is AC coupled, the mean field must be determined in some indirect manner. In practice, the proper DC level and normalization has been determined by comparing the expected shot noise after averaging with the measured noise in the data. In this way, knowledge of the averaging procedure allows the normalized correlation function to be extracted. The noise amplitude for the normalized correlation function is given by  Bκ 1 , δh = (4.1) √ 2a ˆ η2κ 2Ns where, as in Section 2, Ns is the number of starts, κ is the cavity bandwidth, B is the detector bandwidth in units of the cavity bandwidth, η is the fraction of the output power sent to the BHD, and a ˆ is the mean intracavity field (no offset is used in this measurement). Assuming then that the data can be scaled with two constants, Ξ and Υ , such that hθ (τ ) = Ξ hexpt (τ ) + Υ,

(4.2)

the normalization of h(τ ) requires that Ξ hexpt (∞) + Υ = 1.

(4.3)

To determine Ξ we note that Ξ δhexpt = δh, and then assuming  that the coherˆ ), eq. (4.1) ent transmission dominates the incoherent transmission (a ˆ ≈ aˆ † a yields  1 B Ξ≈ (4.4) . 4δhexpt aˆ † aηN ˆ s Υ is then recovered from eq. (4.3). Aside from the reasonable assumption, this method determines the scaling from quantities measured in the experiment. The number of starts is recorded on the digital oscilloscope. The intracavity intensity and η are obtained from the measured flux at the APDs, and the detection bandwidth is determined by the 70 MHz low pass filter. The noise amplitude δhexpt is determined by taking the standard deviation of the unnormalized data. A second approach to the normalization of h(τ ) uses the knowledge that the normalized field correlation is the square root of the intensity correlation g (2) (τ ) in the weak field limit. This permits a DC level for the raw field measurement to be determined that properly scales the normalized intensity-field correlation

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function. The approach is less reliable for these particular measurements, however, because the data is not strictly taken in a weak field regime. Finally, one might determine the normalization by calculating the DC field expected from the measured photon flux. From the measured flux, the expected DC voltage level can be calculated. Adding this level and dividing the total by the same mean level normalizes the correlation data to give a long-time mean of unity. The difference between the first method of normalizing, on the basis of the expected noise, and the other two, is that it only includes the fraction of light directed to the BHD, without including the signal LO overlap, quantum efficiency, and additional losses. The first method of normalization was employed for the results presented here. As an example of the normalized results, fig. 20 shows hθ (τ ) for a larger intracavity intensity (n/n0 = 1.2). The dashed area in the figure marks the limits from the Schwartz inequalities in eqs. (2.15) and (2.18). The field is clearly non-classical. It is interesting to note that the intensity–intensity correlation function for an input intensity within 10% of that used to obtain fig. 20 shows only classical fluctuations, in the form of significant photon bunching (Foster, Orozco, Castro-Beltran and Carmichael [2000a]). Note that in comparison with figs. 18

Fig. 20. Measured normalized field correlation for an in-phase field in cavity QED. N = 13 and n/n0 = 1.2. The dashed region is classically allowed.

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and 19 a significant background has appeared. Qualitatively the correlation function agrees with that in fig. 10(i), where the stronger driving field causes many spontaneous emissions which interrupt the oscillatory evolution of the system back to the steady state. Reiner, Smith, Orozco, Carmichael and Rice [2001] study the effects of spontaneous emission in greater detail. 4.3.2. Spectrum of squeezing The Fourier transform of the normalized intensity-field correlation function, when multiplied by the source photon flux, gives the spectrum of squeezing (eq. (2.10)). Thus, a single time domain measurement of the fluctuation of the field quadrature amplitude yields the entire spectrum of squeezing in the frequency domain. The spectrum in fig. 21 is computed from the normalized data of fig. 20. Since the data contains noise, the time series was first symmetrized and then ordered in a one-dimensional array, with the positive times appearing first followed by the negative times in reverse order. The real part of the Fast Fourier Transform (FFT) is then taken, and multiplied by the time resolution of the data and twice the flux F (eq. (2.11)), as determined from the measured rates at the APDs and their measured efficiencies. There is squeezing below the standard quantum limit (or the shot noise level) at the vacuum Rabi frequency Ω0 . The magnitude of the squeezing at this fre-

Fig. 21. Squeezing spectra for the cavity system with n/n0 = 1.2 and source flux F = 10.9 × 106 photons/sec.

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quency is ≈ 5%. The positive peak at zero is primarily caused by the spontaneous emission noise (Reiner, Smith, Orozco, Carmichael and Rice [2001]) and is in qualitative agreement with the result displayed in fig. 10. The measurements of the cavity QED system show that the detection of a photon projects the system into a quantum state that evolves in time with a welldefined phase (relative to the mean field). The conditional homodyne detection uses this feature to observe the evolution of the field fluctuation; by triggering the collection of data from the BHD on a photon detection, one recovers the subphoton field fluctuation from the large shot noise background. The time-domain measurement provides the information required to construct the full spectrum of squeezing in an efficiency-independent manner.

§ 5. Equal-time cross- and auto-correlations We review in this section a series of theoretical and experimental papers that have studied conditional homodyne measurements at equal times, this means they do not look at the dynamics of the state as we have shown in previous sections, but they show very interesting results that demonstrate the quantum nature of conditional states of the electromagnetic field. The entanglement available on the output field produced in the parametric down-conversion process has been exploited thoroughly to perform studies of conditional quantum measurements. Yurke and Stoler [1987] proposed a cross correlation measurement, a conditional measurement of the quadrature of the electromagnetic field, as a way to measure the amplitude probability distributions for photon-number operator eigenstates. Ban [1996] studied the photon statistics of conditional output states with a photon counting detector at one output of a beam splitter and a homodyne detector at the second output of the beam splitter for different input fields. This treatment is equivalent to a measurement of h(0), the auto-correlation between the intensity and the field of a signal beam, and enables the author to study the Mandel Q factor of the conditional output state in detail. More recently Crispino, Giuseppe, Martini, Mataloni and Kanatoulis [2000], and Lvovsky, Hansen, Aichele, Benson, Mlynek and Schiller [2001] performed a cross correlation measurement and applied tomographic methods to reconstruct the Wigner distribution for a one photon state using conditional homodyne detection. Fiurasek [2001] studied the photon statistics conditioned on the homodyne detection in a similar system. Resch, Lundeen and Steinberg [2002a, 2002b] used intensity-field cross-correlations for measurements of conditional coherence in

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prepared quantum states and to implement a conditional phase switch at the single photon level.

5.1. Cross-correlations We generalize the definition of the unnormalized correlation function Hθ (τ ) in eq. (2.4) to allow two different modes, allowing us to cross correlate the intensity fluctuations on one mode with the field fluctuations of another mode: Hθ (τ )i,j =

j (t + τ ): : Si (t)D + ξ(t),  Si 

(5.1)

where the labels i and j refer to two different modes. The normalization proceeds in the same way as in Section 2 so we can now talk about the conditional field on mode j given a photon detection on mode i. This is useful in the case of the two entangled modes of a parametric down convertor (see fig. 22). The connection with squeezing given by eq. (2.10) for the auto-correlation hθ (τ ) is not valid in the case of the cross-correlation hθ (τ )i,j . 5.1.1. Proposal for measuring the amplitude probability of a Fock state Yurke and Stoler [1987] present a measurement strategy to obtain the probability distribution of an n photon Fock state. They suggest the highly correlated process of parametric down conversion where the number of photons counted by a photodetector in the idler beam during a coherence time could be used to gate a homodyne detector in the signal beam. The integrated output of the homodyne detector over a coherence time is only recorded when an m-photon wave packet enters its input port. In this way it is possible to map out the probability distribution for a field-amplitude component of a number-operator eigenstate.

Fig. 22. Simplified apparatus for measurement of the cross-correlation Hθ (0)i,j in parametric down conversion.

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Yurke and Stoler suggest that this conditional measurement could be used for the same purpose on other non-linear sources that produce non-classical light. 5.1.2. State reconstruction of a Fock state and quantum-optical catalysis Crispino, Giuseppe, Martini, Mataloni and Kanatoulis [2000] and Lvovsky, Hansen, Aichele, Benson, Mlynek and Schiller [2001] have reconstructed the quantum state of optical pulses containing single photons using the method of phase-randomized pulsed optical homodyne tomography. The method they apply is very similar, but we review here in more detail the experiment of Lvovsky, Hansen, Aichele, Benson, Mlynek and Schiller [2001]. They prepare a singlephoton Fock state |1, in a well-defined spatiotemporal mode, using conditional measurements on photon pairs born in the process of parametric down-conversion following the original suggestion by Yurke and Stoler [1987]. A single-photon counter is placed into one of the emission channels (labelled trigger in fig. 22) to detect photon pair creation events and to trigger the readout of a homodyne detector placed in the other (signal) channel. (See Hansen, Aichele, Hettich, Lodahl, Lvovsky, Mlynek and Schiller [2001] for a description of their detector.) The signal beam as shown in fig. 22 is not an optical beam in the traditional sense. The down-converted photons are in fact emitted randomly over a wide solid angle. The optical mode of the signal state is usable when a photon of a pair hits the trigger detector and is registered. Once the approximate Fock state is prepared, it is subjected to balanced homodyne detection (Yuen and Chan [1983a, 1983b]). A probability distribution of the phase-averaged electric field amplitudes is obtained with a non-Gaussian shape. The angle-averaged Wigner function is reconstructed from this distribution and shows a dip reaching classically impossible negative values around the origin of the phase space. This is a single mode intensity-field cross-correlation (a correlation integrated over the optical pulse), here presented as a conditional measurement. The evolution of the parametric down conversion process is very fast so they do not get the time dependence. Lvovsky and Shapiro [2002] have used the equal time intensity-field crosscorrelations to measure and characterize non-classical light. Another use of the cross-correlation between the intensity of one mode and the field of another is the work of Fiurasek [2001]. Instead of conditioning on the intensity fluctuations (photodetection) the paper proposes to use the homodyne signal to condition the photodetections. The photodetector only counts if the absolute value of the mea-

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Fig. 23. Simplified scheme of the experimental setup for H (0) of a Fock State with the help of an offset field.

sured idler quadrature lies inside a certain range. The conditioned generated signal is sub-Poissonian. As a follow up to their previous work Lvovsky and Mlynek [2002] use an intensity-field cross-correlator to measure the conditional field of a single photon. (See fig. 23 for a schematic of their apparatus.) They prepare the single photon through the parametric down conversion process with a conditional measurement and then use the Fock state as the “source” (see fig. 1) for the intensity-field correlator. Since the Fock state does not have a steady state amplitude, they add a coherent bias field to do the auto-correlation measurement. Lvovsky and Mlynek [2002] show that with the appropriate choice of offset, they can prepare and characterize a coherent superposition state t|0 + α|1 of the electromagnetic field by conditional measurements on a beam splitter. The single photon plays the role of a catalyst: it is explicitly present in both the input and the output channels of the interaction yet facilitates generation of a nonclassical state of light. Although their measurement does not give the full time dependent intensity-field correlation function, since they can obtain the value at τ = 0 only, from the Fourier Transform relations presented in Section 2 (see eq. (2.12)) it is possible to obtain the integrated spectrum of squeezing for this Fock state.

§ 6. Quantum measurements and quantum feedback The intensity-field correlation function has introduced a new way to analyze and study the non-classicality of the electromagnetic field. It does that mainly through the two Schwartz inequalities in eqs. (2.15) and (2.18), however, recent works by Wiseman [2002] and Carmichael [2003] show that it has further implications in the quantum theory of measurement through its relationship with the weak measurements of Aharonov, and in distinguishing qualitatively between vacuum state

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squeezing and the squeezing of classical noise. Carmichael [in press] has used the connection between the intensity-field correlation and the particle-wave aspects of light to further elucidate this important question. Conditional measurements and quantum feedback are intimately related (Wiseman [1994]). The last part of this section presents a proposal to use CHD in quantum feedback to modify the response of a cavity QED system. 6.1. Weak measurements We review the main ideas behind a weak measurement and follow Wiseman [2002] in developing the connection between hθ (τ ) and weak measurements. A weak measurement is one that minimally disrupts the system, while consequently yielding a minimal amount of information about the observable measured (Aharonov, Albert and Vaidman [1988]). For a given initial system state, the ensemble average of weak-measurement results is the same as for strong, i.e. projective measurement results. Where weak measurements are interesting is when a final as well as an initial state is specified. Here the final state is the result of a second measurement (a strong one), so that the ensemble average is taken over a postselected ensemble, in which the desired result for the final measurement was obtained. A homodyne measurement gives a current proportional to the expectation value of the quadrature operator and so conditions the system state. The null photon counting result also gives a continuous measurement that is weak, in the sense that the average change in the conditioned system in time δt is of order δt, as seen in quantum trajectories (Reiner, Smith, Orozco, Carmichael and Rice [2001]). However, unlike homodyne detection, sometimes the change is great; there is a quantum jump – a strong measurement result. Looking at the negative time side of the correlation function, it is possible to see that this is a weak value preselected by the system being in its stationary state and postselected on the photon detected at time τ = 0. The measurement of the correlation function as a function of time shows the dynamics of a weak value (quadrature of the electromagnetic field) over time. The strangeness of the weak values in this experiment is not surprising, since the conditions set by Aharonov, Albert and Vaidman [1990] are fulfilled. That is, the postselection is done on a rare event, the detection of one photon rather than zero photons. The formulation of the quantum trajectory must include the back action of the homodyne detection to be consistent when describing the times before the detection of a photon. Wiseman [2002] shows that hθ (τ ) reduces to a form as simple as that originally derived by Aharonov, Albert and Vaidman [1988].

398

Intensity-field correlations of non-classical light

[5, § 6

6.2. Vacuum state squeezing versus squeezed classical noise It is apparent from eqs. (2.9) and (2.10) that conditional homodyne detection offers a somewhat different view of squeezed light than the conventional squeezing measurement. Certainly, the latter is usually carried out in the frequency domain, but this difference is not important. In both cases we have a Fourier relationship between frequency and time of the sort given in eq. (2.11). The important difference is that in the conventional measurement, the relevant temporal correlation function is the direct autocorrelation of the current, i(t), from the balanced homodyne detector. This is calculated from a symmetrically ordered quantum average, rather than the normal-ordered average of eq. (2.9). The difference is a direct manifestation of conditioning, and from it, it follows that whereas a conventional homodyne measurement distinguishes quantitatively between vacuum state squeezing and squeezed classical noise, conditional homodyne detection distinguishes quantum from classical squeezing in a qualitative way (Carmichael [2003]). Physically, in any realistic example, squeezing is restricted to a finite bandwidth outside of which the field fluctuations are not squeezed. For vacuum state squeezing, the unsqueezed fluctuations appear as shot noise on i(t), the shot noise commonly being interpreted as a manifestation of the vacuum fluctuations of the measured field. If, in addition, this field carries broadband classical noise, the unsqueezed classical fluctuations also add to the fluctuations of i(t). Thus, in a conventional squeezing measurement, unsqueezed classical noise and shot noise contribute to the autocorrelation of i(t). In conditional homodyne detection by comparison, only the classical noise contributes; the shot noise is eliminated; through the conditional sample average the term ξ(τ ) in eq. (2.9) is reduced (in the ideal limit) to zero. To illustrate the comparison, we consider the squeezing of broadband classical noise, as depicted in fig. 24(i). Classical noise of bandwidth (halfwidth) Bc κ = 15κ is squeezed by a sub-threshold degenerate parametric oscillator over the bandwidth κ. Fluctuations on the squeezed quadrature of Eout are then measured using either conventional homodyne detection or conditional detection; the detection bandwidth is Bd κ = 25κ. Figures 24(ii) and (iii) show simulated results for the direct autocorrelation of the homodyne current, and thus provide the time-domain view of a conventional squeezing measurement. The presence of squeezing is evident from the broad negative dip, while fluctuations outside the squeezing bandwidth produce the narrow spike around τ = 0. In frame (ii), this spike is contributed to by both classical fluctuations and shot noise. Shot noise

5, § 6]

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Fig. 24. (i) Sketch of the degenerate OPO squeezer with broadband classical noise input. Under conventional homodyne detection of the field Eout [(ii) and (iii)], the autocorrelation of the homodyne current reveals nonsqueezed fluctuations over a wide bandwidth (central spike) and a narrow bandwidth of squeezed fluctuations (negative dip): for a classical noise bandwidth Bc κ = 15κ, detection bandwidth Bd κ = 25κ, and classical noise photon numbers (in the degenerate OPO cavity) of (ii) n¯ c = 0.2 and (iii) n¯ c = 0; the degenerate OPO squeezer is operated at 40% of threshold.

alone contributes in frame (iii), where the height of the spike is correspondingly reduced. For comparison, fig. 25 displays simulated results of conditional homodyne detection. In this case, there is no shot noise contribution to the central spike. The spike shrinks and eventually disappears as the classical noise level is decreased. A central spike in hθ (τ ) is therefore a qualitative indicator of the squeezing of classical, as opposed to vacuum state noise. Carmichael [2003] has also investigated to what extent stochastic electrodynamics is able to reproduce this distinction between classical and vacuum noise squeezing. He finds that the results of fig. 25 are reproduced, but with an additional background contribution generated by “starts” induced by the explicit vacuum noise (unphysical dark counts). In related work, triple correlations of the quadratures of the electromagnetic field have been used by Drummond and Kinsler [1995], Kinsler [1996], Chaturvedi and Drummond [1997], and Pope, Durmmond and Munro [2000] to investigate differences between the predictions of stochastic electrodynamics and quantum mechanics. They compare the former with the positive P representation, also focusing their attention on the parametric oscillator.

400

Intensity-field correlations of non-classical light

[5, § 6

Fig. 25. Broadband classical noise squeezing under conditional homodyne detection. The correlation function hY (τ ), with Y the squeezed quadrature, is plotted for the parameters of fig. 24 and classical noise photon numbers (in the degenerate OPO cavity) of (i) n¯ c = 0.2, (ii) n¯ c = 0.1, (iii) n¯ c = 0.05, and (iv) n¯ c = 0.

6.3. Application of hθ (τ ) to quantum feedback Most quantum feedback proposals use the BHD photocurrent to modify the drive acting on a quantum system. The goal of such feedback can vary from reducing out-of-quadrature noise (Tombesi and Vitali [1995]) to modifying the system dynamics (Wang and Wiseman [2001]). All these proposals rely on continuous feedback. Doherty and Jacobs [1999] showed that one can improve these schemes with knowledge of the conditioned state. A recent experiment by Smith, Reiner, Orozco, Kuhr and Wiseman [2002] shows the success of quantum feedback in a strongly coupled system through conditional intensity measurements. We consider adding feedback to the single atom cavity QED system (see Section 3.2 for theoretical details). A photon leaving the cavity initiates a fluctuation. Conditioning the BHD detectors to observe this fluctuation also creates the opportunity to apply feedback.

5, § 6]

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401

The dynamics of the system, once the detection of the photon is made, are governed by the regression to steady state of the field. The equations governing this regression are given in the Optical Bistability notation of Lugiato [1984] by x˙ = κy + gp − κx + Gx, p˙ = −gx − γ⊥ p,

(6.1)

where y is the normalized intracavity field in the absence of atoms (proportional to the drive), x the intracavity field in the presence of atoms, p is the atomic polarization, and G quantifies the strength of the field feedback. The intensity-field correlator measures the conditioned BHD photocurrent,    ic (t) = 2κη x(t) c + ξ(t), (6.2) which is proportional to the intra-cavity field and contains the shot noise ξ(t). We propose modulating the amplitude of the driving laser with the conditioned BHD photocurrent. This modifies the system Hamiltonian HS by adding the term Hfb = Gic (t)(aˆ − aˆ † ).

(6.3)

We improve this feedback by averaging away the shot noise contribution in eq. (6.3). Ns j =1 λic (tj + τ ) . Hfb (N, τ ) = (6.4) Ns

Fig. 26. Schematic of the feedback apparatus. The running average is fed back to reduce the contribution of the shot noise component.

402

Intensity-field correlations of non-classical light

[5, § 7

Fig. 27. Single atom cavity QED quantum conditioned field evolution: (i) without feedback, and (ii) with feedback. (g, κ, γ⊥ , Γ, G)/2π = (38.0, 8.7, 3.0, 100, 3.5) MHz.

Figure 26 presents a suggestion of how the intensity-field correlator in the experiment of Foster, Orozco, Castro-Beltran and Carmichael [2000a] would be modified to apply quantum feedback. Photons arrive at the avalanche photodiode at different times, t1 , t2 and t3 . These are well separated to show the reduction in shot noise with consecutive averages. Figure 27 shows the conditioned field evolution of a single maximally coupled stationary atom in cavity QED without and with the feedback protocol defined by eq. (6.3) set to enhance the vacuum Rabi oscillations of the system. The evolution has been maintained well beyond the limits set by (κ + γ⊥ )/2. The plot is the result of averaging 10,000 terms in eq. (6.4) to reduce the contribution from the shot noise.

§ 7. Conclusion and outlook Future studies may try to map out a full phase space picture of the conditional field as it evolves in time. This would involve sampling the field at various phases and performing some type of tomographic reconstruction to arrive at a quasiprobability distribution (Leonhardt [1997]). The conditional homodyne technique may have more general application to studying other sources both classical and non-classical since the information obtained through this intensity-field correlation approach complements and synthesizes that coming from intensity–intensity correlations (particle aspect) and squeezing (wave aspect).

5]

References

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Acknowledgements Work supported by NSF and NIST.

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Author index for Volume 46 A

Auld, B.A., 32 Aus der Au, J., 9, 18, 19, 35–37, 42, 43, 49, 50, 83, 86, 96 Avramopoulos, H., 32, 44

Abdolvand, A., 19, 37 Abrahams, E., 136 Acosta-Ortiz, S.E., 168 Aeschlimann, M., 96 Afifi, M., 275 Agate, B., 37 Agnesi, A., 35 Agostini, P., 97 Agranovich, V.M., 204, 208, 215 Agrawal, G.P., 45, 46 Aguilo, M., 19 Aharonov, Y., 397 Aichele, T., 358, 393, 395 Aka, G., 18, 19, 37 Aktsipetrov, O.A., 205 Albert, D., 397 Alexander, J., 36 Alfano, R.R., 17, 34, 200, 204 Alphonse, G.A., 39 Anderson, P.W., 136, 217 Andreadakis, N.C., 39 Andrejco, M.J., 15 Andreyev, O., 96 Angelow, G., 9, 18, 19, 36–38, 51, 64 Angert, N.B., 17 Anndo, N., 293 Antoine, P., 96, 97 Antsygina, T.N., 146 Aoshima, T., 290 Apolonski, A., 94 Aron, A., 19, 37 Arsenieva, A.D., 127, 180–182, 185 Arya, K., 217 Arzate, N., 204 Asaka, A., 285, 286 Asaki, M.T., 33, 63 Aschwanden, A., 9, 38, 39, 90, 97 Asom, M.T., 8, 12, 33, 49–51 Audebert, P., 96, 97

B Bachor, H.-A., 344, 362 Backus, S., 33, 92 Bado, P., 32 Badziag, P., 323, 324 Baggett, J.C., 90, 96 Bahar, E., 160 Balachandran, R.M., 200 Balcou, P., 96, 97 Balembois, F., 19, 34, 37 Baltuska, A., 9, 89, 92 Ban, M., 393 Banaszek, K., 360 Bar-Joseph, I., 15, 49 Barabanenkov, Yu.N., 120 Baraldi, L., 38 Bardroff, P.J., 318 Barnes, T.H., 249 Barnum, H., 323 Barr, J.R.M., 32, 33 Barry, N.P., 35 Bartels, A., 36 Barthelemy, A., 18, 36 Bartolini, L., 279 Barton, J.S., 255 Barty, C.P.J., 33, 96 Bass, F.G., 119, 137, 161 Bass, M., 18 Basu, S., 32 Bauch, A., 95 Bauer, M., 96 Beach, R.J., 35, 41–43 Beaudoin, Y., 54 Becker, M.F., 12, 72 405

406

Author index for Volume 46

Becker, P.C., 7, 62, 89 Beckmann, P., 119, 124, 161 Beddard, T., 36 Beenakker, C.W.J., 200 Beheim, G., 290, 299 Beigang, R., 18, 35 Benedict, G., 7 Benett, W.J., 43 Bennett, C.H., 313, 316–319, 323, 324, 326 Bennett, J.M., 231 Benson, O., 358, 393, 395 Bente, E., 36, 37 Berger, G.A., 200 Berginc, G., 131, 195 Bergman, K., 35 Berry, M.V., 122, 124 Biegert, J., 9, 89, 93 Biegman, J.F., 264 Bilinsky, I.P., 52 Birman, J.L., 217 Biswal, S., 18, 36 Bland, S.W., 37 Bloembergen, N., 204, 208 Bloom, D.M., 8, 32, 33, 44, 73 Boag, A., 156 Böhm, M., 36 Boiko, A., 19, 37 Bona, G.L., 38, 42 Born, M., 165, 228 Borodin, N.I., 17 Bosch, T., 286 Boschi, D., 329, 331 Bouma, B.E., 36 Bourennane, M., 325 Bourliaguet, B., 36 Bourrely, C., 131, 195 Bouvier, M., 32 Bouwmeester, D., 314, 332, 336 Bowen, W.P., 344 Bowers, J., 39, 75, 77 Boyd, G.D., 8, 12, 33, 49–51 Bozhevolnyi, S.I., 205, 218, 219 Brabec, T., 33, 79, 80, 94, 97 Bradley, D.J., 7 Branca, S., 329, 331 Brangaccio, D.J., 245, 251 Brassard, G., 313, 316–319 Braun, A., 18, 36 Braun, B., 9, 11, 18, 35, 49, 50, 52, 73, 74

Braunstein, S.L., 316, 320, 321, 327, 344–346, 349 Brecha, R.J., 361, 373–375, 382 Brekhovskikh, L.M., 137 Brennen, G.K., 325 Briegel, H.-J., 324 Broschat, S., 162 Brouwer, P.W., 200 Brovelli, L.R., 14, 18, 34–36, 42, 46, 50, 51, 65, 66, 82, 85 Brown, C.T.A., 37 Brown, F., 203 Brown, G.C., 127, 129, 130, 148 Brown, G.S., 119, 124, 155, 231 Brown, R.H., 357 Brownell, M., 38 Brun, A., 19, 34, 37 Bruning, J.H., 245, 251, 255, 264 Brunner, F., 9, 19, 37, 38, 90, 96 Buccafusca, O., 38 Buchler, B.C., 344 Burk, M., 36 Burns, D., 36, 37 Burow, R., 256, 263, 267 Bursukova, M.A., 19 Butterworth, S.D., 36 Buzek, V., 315 Byer, R.L., 32 C Cadilhac, M., 129 Calsamiglia, J., 328, 329 Calvo-Perez, O., 162 Cao, H., 200 Carlisle, C.B., 292 Carlson, N.W., 43 Carmichael, H.J., 357, 358, 361–363, 365–367, 369, 373–379, 381, 382, 385, 389, 391–393, 396–399, 402 Carminati, R., 217, 221, 232 Carolan, T.A., 255 Carrig, T.J., 18 Casimir, H.B.G., 364 Cassanho, A., 18, 35 Castro-Beltran, H.M., 357, 358, 361–363, 365, 366, 369, 376, 379, 381, 385, 391, 402 Cavallari, M., 90 Cavalleri, A., 96

Author index for Volume 46 Celli, V., 120, 127, 129, 130, 133, 134, 143, 148, 150, 152, 155, 157–159, 179, 197, 198, 229 Cerf, N.J., 347 Cerullo, G., 34, 54, 90 Cesar, S.L., 121 Chai, B.H.T., 18, 33–35 Chaikina, E.I., 170, 178, 197–201, 224, 226, 232 Chan, C.H., 156, 229–231 Chan, V.W., 357, 386, 395 Chan, Z., 205 Chang, C.H., 168 Chang, L.-W., 285 Chang, M.-W., 291, 292 Chang, R.K., 204, 208 Chang, R.P.H., 200 Chang, T.Y., 15, 49 Chang, Y., 36 Chang, Y.-S., 291, 292 Chase, L.L., 17, 18 Chaturvedi, S., 399 Chebbour, A., 291 Chee, J.K., 32 Chefles, A., 326 Chen, J., 253, 258, 260, 264, 265, 267, 274 Chen, J.S., 161 Chen, M.F., 124 Chen, Y., 9, 36, 64 Chen, Y.-K., 76 Chen, Z., 204 Chenais, S., 19, 37 Cheng, Z., 9, 89, 92 Chernikov, S.V., 36 Chew, W.C., 156, 229, 230 Chien, C.Y., 54 Chien, P.-Y., 285, 291, 292 Chilla, J.L.A., 92 Chin, M.A., 76 Chiu, T.H., 8, 12, 33, 34, 39, 46, 49–51 Cho, S.H., 9, 36, 37, 64 Chou, C.W., 344 Chou, H.-T., 230 Christov, I.P., 34, 63 Chu, K.C., 92 Chuang, I.L., 313, 314, 325, 342 Chuang, Y.-H., 299 Chudoba, C., 19, 37, 38 Chung, J., 34 Cirac, J.I., 324, 325

407

Clemens, J.P., 382 Coe, J.S., 32, 54 Collett, M.J., 362, 369 Collin, R.E., 160 Collings, B.C., 35 Collins, R.J., 10, 12 Conlon, P.J., 34 Contag, K., 37 Coopersmith, M., 127, 129, 148 Corke, M., 287, 291 Corkum, P.B., 94, 96, 97 Cornet, A., 255 Corvi, M., 208 Couderc, V., 18, 36 Courjaud, A., 19, 37 Coutaz, J.L., 204, 208 Creamer, D.B., 124 Creath, K., 245 Crépeau, C., 313, 316–319 Crispino, M., 358, 393, 395 Crosby, P., 38 Crust, D.W., 53 Cruz, C.H.B., 7, 62, 89 Culshaw, B., 270, 291 Cummings, F.W., 372 Cundiff, S.T., 35, 94, 95 Cunningham, J.E., 33–35, 37, 50, 51, 54, 77 Curley, P.F., 17, 32, 33, 79 D Dagenais, M., 357, 364 Dainty, J.C., 170, 172 Danailov, M.B., 19 Dändliker, R., 282, 296, 298 Dandridge, A., 246 Dangel, R., 38 Dapkus, P.D., 51 Dardenne, K., 18 Darmanyan, S.A., 208 Davidovich, L., 360 Davies, D.E.N., 270, 287, 291 Davies, J.I., 37 Dawson, M.D., 36, 37 de Almeida, N.G., 338 de Groot, P., 264, 269, 281, 282, 286 De Maria, A.J., 12 De Martini, F., 341 de Riedmatten, H., 334 De Silvestri, S., 9, 34, 54, 89, 90, 93

408

Author index for Volume 46

de Souza, E.A., 35, 50, 51 Deck, R.T., 204, 215 Dekorsky, T., 36 Delfyett, P.J., 33, 34, 39 DeLong, K.W., 92 DeMartini, F., 329, 331 Demidovich, A.A., 19 den Boef, A.J., 283, 299 den Outer, P., 197, 198 Deng, H., 359, 371 Denisov, A., 358, 361, 365, 379 Depine, R.A., 158, 168 DeSanto, J.A., 119, 232 Deutsch, D., 316 Dhanjal, S., 39, 97 Diaz, F., 19 Diddams, S.A., 94 Diels, J.-C., 7, 10, 75 Dienes, A., 75, 92 Diening, A., 18 DiGiovanni, D., 39 Dill III, C., 11 DiMauro, L.F., 97 DiVincenzo, D.P., 317, 323, 324, 326 Dixon, R.W., 49 Doherty, A.C., 400 Donegan, M.M., 342 Dörr, F., 91 Downer, M.C., 204 Drake, J.M., 200 Drescher, M., 94, 97 Driscoll, T.J., 90 Drummond, P.D., 399 Druon, F., 19, 37 Duling III, I.N., 15 Dummer, R.S., 197, 198 Dunlop, A.E., 72, 93, 94 Dür, W., 324, 325 Durfee III, C.G., 92 Dymott, M.J.P., 34–36, 38 Dysthe, K.B., 124 E Economou, G., 287 Eibl, M., 332, 336 Eichenberger, D.J., 49 Eiju, T., 249, 256, 263 Einstein, A., 318 Ekerdt, J.G., 204

Ekert, A.K., 314, 324 El-Shenawee, M., 160 Ell, R., 19, 37 Elliot, J., 52 Elson, J.M., 156, 231 Elssner, K.-E., 256, 263, 267 Emanuel, M.A., 35, 41–43 Erenso, D., 359, 371 Erhard, S., 37, 43 Escamilla, H.M., 198–201, 224, 226, 233 Evans, J.M., 33 F Fairman, P.S., 264, 269, 271–273 Falcone, R.W., 96 Falcoz, F., 34 Fan, T.Y., 18, 32, 41 Farias, G.A., 121, 204 Farrant, D.I., 264 Feldstein, M.J., 218 Feng, S., 127, 180–182, 185, 200 Fercher, A.F., 295 Ferencz, K., 36, 63, 64, 89 Ferguson, A.I., 8, 17, 32–37, 73 Ferguson, J.F., 8, 12, 33, 34, 49–51 Fermann, M.E., 15, 54, 79 Ferray, M., 96 FerriDeCollibus, M., 279 Feugnet, G., 54, 77 Fischer, E., 299 Fitch, M.J., 342 Fittinghoff, D.N., 92 Fitzgerald, R.M., 160 Fitzwater, M.A., 160 Fiurasek, J., 393, 395 Flannery, B.P., 155 Fleischer, S.B., 15 Flood, C.J., 33, 34 Florez, L.T., 33, 34, 39 Fluck, R., 9, 11, 18, 35, 36, 49–52, 63 Foley, J.T., 221 Ford, H.D., 255 Fork, R.L., 7, 12, 62, 72, 75–77, 89 Fornetti, G., 279 Förster, E., 96, 97 Foster, G.T., 357–359, 362, 363, 366, 369, 376, 381, 385, 388, 391, 402 Fourmaux, S., 96, 97 Franson, J.D., 342

Author index for Volume 46 Franze, B., 249, 282 Freilikher, V.D., 130, 146, 150, 180, 181, 185, 188, 195, 196, 200, 201, 224 Freischlad, K., 271 Freitas, B.L., 43 French, P.M.W., 33–36 Freund, C.H., 264 Freundorfer, A.P., 291 Friberg, A.T., 129, 137, 170, 172 Fritsch, K., 290 Fuchs, C.A., 344–346, 349, 350 Fujii, S., 291 Fujimoto, J.G., 9, 13, 19, 32, 33, 36–38, 52, 53, 64, 65, 78 Fujimoto, J.O., 36 Fuks, I.M., 119, 121, 137, 161 Fukui, M., 204, 208 Funaba, T., 293 Fung, A.K., 124, 161 Fürbach, A., 92 Furusawa, A., 344–346, 349 Furusawa, K., 90, 96 G Gabbe, D.R., 17 Gäbel, K.M., 36 Gale, G.M., 90 Gallagher, J.E., 245, 251 Gallatin, G.M., 286 Gallmann, L., 9, 18, 36–38, 51, 64, 65, 90, 92, 93 Galvanauskas, A., 15 Gao, J., 37 García, N., 232 García-Guerrero, E.E., 197, 232, 233 Garcia-Molina, R., 158 Gardiner, C.W., 362, 369 Garduno-Mejia, J., 36 Garmash, V.M., 17 Garmire, E., 51 Garvey, D., 33, 63 Gaume, R., 19, 37 Gauthier, J.C., 96, 97 Gavalda, Jna., 19 Gavrilenko, V.I., 204 Gayen, S.K., 17 Gea-Banacloche, J., 348–350 Geindre, J.P., 96, 97 Genack, A.Z., 200

409

Georges, P., 19, 34, 37 Gerstenberger, D.C., 8, 32, 44, 73 Geusic, J.E., 17 Gharbi, T., 273 Giacomini, S., 341 Gibson, A.F., 49 Giesen, A., 18, 34, 36, 37, 43 Gifford, M., 33 Giles, I.P., 270 Gilliand, Y., 264 Gini, E., 11, 39, 52, 97 Gires, F., 59 Gisin, N., 334 Giunti, C., 298 Giuseppe, G.D., 358, 393, 395 Gland, J., 96 Glauber, R.J., 357 Gmitter, T., 39 Godil, A.A., 32, 33 Goh, K.W., 344 Goldblatt, N., 54, 77 Golding, P.S., 38 Golovkina, V.N., 205 Gomes, A.S.L., 200 Goodberlet, J., 32 Goodman, J.W., 171, 174, 181, 183, 184, 187 Goossen, K.W., 34 Gopinath, J.T., 19, 38 Gordon, J.P., 7, 62, 72, 75–77, 89 Gorecki, C., 291 Gori, F., 221 Gottesman, D., 313, 325, 342 Gouaux, F., 286 Gouveia-Neto, A.S., 33 Graf, M., 18, 36 Graf, T., 37 Graf, Th., 36 Graham, R., 364 Grange, R., 38 Grangier, P., 347 Grant, R.S., 53, 92 Grasbon, F., 35 Grasborn, F., 94 Grawert, F., 19, 38 Gray, P.F., 170 Gredeskul, S.A., 146 Greene, B.I., 7, 75 Greffet, J.-J., 130, 162, 217, 219, 221, 232 Greivenkamp, J.E., 245, 254, 255, 264 Griebner, U., 19

410

Author index for Volume 46

Griffiths, R.B., 316 Grillon, G., 96, 97 Grosshans, F., 347 Grygier, R.K., 204, 215 Grzanna, J., 256, 263, 267 Gu, Z.-H., 125, 170, 172, 197, 198, 203, 221, 232 Güell, F., 19 Guo, T., 96 H Ha-se, J., 251 Haberl, F., 15, 54 Hache, F., 90 Haensch, T.W., 94 Haible, P., 249 Haiml, M., 38, 51, 52, 87 Haken, H., 364 Hall, G.E., 91 Hall, J.L., 94, 95 Haller, M., 127, 129, 130, 148 Hamano, K., 292 Hanato, H., 150 Hand, D.P., 255 Hanna, D.C., 32, 33, 36, 96 Hänsch, T.W., 94, 95 Hansen, H., 358, 393, 395 Hansen, P.B., 39 Harder, C., 18, 19, 36, 37 Hardy, L., 329, 331 Hargrove, L.E., 12 Hariharan, P., 256, 263, 266, 277 Häring, R., 9, 11, 38, 39, 52, 90, 97 Harmer, A.L., 17 Harrison, J., 11, 18 Harter, D., 15 Haus, H.A., 9, 13, 15, 19, 32, 36, 37, 53, 64–66, 70, 72, 73, 76, 78 Hayden, P., 316 He, Z., 249 Hecht, E., 165 Hefferle, P., 91 Heine, C., 64 Heine, F., 18, 35 Heinzmann, U., 97 Helbing, F.W., 19, 37, 93–95 Hellwarth, R.W., 10 Henkmann, J., 82

Henrich, B., 18, 35 Hentschel, M., 94, 97 Henyey, F.S., 124 Heritage, J.P., 39, 92 Hernández-Walls, R., 170 Herriott, D.R., 245, 251 Hetterich, M., 37 Hettich, C., 395 Heumann, E., 18 Heynau, H., 12 Hibino, K., 269, 271–273 Higuchi, K., 283 Hill, N.R., 129, 143 Hillery, M., 315 Hinsch, K.D., 255 Hirlimann, C., 89 Ho, P.-T., 32 Ho, S.T., 200 Hoenders, B.J., 124 Hofer, M., 15, 39, 54, 79 Hollberg, L.W., 357 Hollins, R.C., 122 Holzwarth, R., 94, 95 Hong, C.K., 327 Hönninger, C., 9, 13, 18, 19, 34–38, 43, 46, 49, 50, 52, 87, 88, 90 Hoogland, S., 39, 97 Horne, M.A., 324 Horodecki, M., 322–324 Horodecki, P., 322–324 Horodecki, R., 322–324 Horvath, C., 35 Hotate, K., 246, 249, 285, 299–301 Hotz, D.F., 199 Hou, A.S., 32 Hövel, R., 37, 43, 51 Hu, H.Z., 295 Huang, C., 204 Huang, C.-P., 33, 34, 63 Huang, J.-R., 255 Huber, G., 18, 35, 52 Huelga, S.F., 326 Hug, K., 282 Hughes, D.W., 32, 33 Hugi, J., 39 Hulin, D., 96, 97 Hummel, S.G., 51 Huxley, J.M., 32 Hyde, S.C.W., 35

Author index for Volume 46 I Iaconis, C., 92, 93 Iblisdir, S., 347 Ichimura, T., 293 Iizuka, K., 291 Imai, M., 291 Imai, Y., 291 Imoto, N., 316 Ina, H., 281, 290, 293 Innerhofer, E., 9, 37, 38, 90, 96 Ippen, E.P., 7, 9, 13, 15, 19, 32, 36–38, 46, 49, 53, 64, 65, 75, 76, 78 Ishida, Y., 33, 34 Ishii, Y., 245, 248, 251, 253, 258–260, 262, 265, 267, 273, 274, 277, 279, 281, 286, 288, 293, 295, 296, 299, 301, 302 Ishimaru, A., 120, 136, 137, 142, 156, 161, 165, 230 Islam, M.N., 15, 49 Itatani, T., 36 Ito, M., 245 Ittner, T., 299 Ivanov, M.Y., 96 Iwata, K., 292 Izatt, J.A., 33 J Jackson, D.A., 287, 291 Jackson, D.R., 137, 142 Jackson, J.D., 165 Jacobs, B.C., 342 Jacobs, K., 400 Jacobson, J., 32 Jaggard, D.L., 122 Jakeman, E., 122, 124 Jamasbi, N., 36 James, D.F.V., 221 James, R., 291 Jan, W.Y., 35, 50, 51 Jaynes, E.T., 372 Jenkins, G.M., 174, 175 Jensen, T., 18 Jenssen, H.P., 17, 18, 35 Jeong, T.M., 36 Jha, S.S., 203, 204, 208 Ji, W., 46 Jiang, W., 39, 75, 77 Jin, S., 382

411

Joannes, L., 255 Johannsen, I., 18, 36 John, S., 200, 219 Johnson, A.M., 32, 72 Johnson, J.T., 150, 230 Jonathan, D., 325, 326 Jones, D.J., 94 Jones, J.D.C., 255, 287 Jones, R., 35 Jong, D.-T., 285 Joost, H., 255 Jordan, D.L., 122 Joschko, M., 46 Jost, M., 35 Jozsa, R., 313, 316–319 Juhasz, T., 8, 73 Jung, I.D., 9, 13, 14, 18, 34–36, 49–51, 58, 63, 65, 66, 82–85 K Kachoyan, B.J., 133, 157, 229 Kafka, J.D., 59 Kahn-Harari, A., 18 Kakuma, S., 286 Kam Wa, P.L., 18, 34 Kamatani, O., 299 Kammler, M., 96 Kamp, M., 14, 34, 35, 42, 50, 51, 65, 66, 82, 85 Kanatoulis, H., 358, 393, 395 Kane, C., 180 Kane, D.J., 92 Kaneko, Y., 38 Kang, E.C., 36 Kanzieper, E., 150 Kapp, D.A., 155 Kapteyn, H.C., 33, 34, 63, 92, 96 Kapusta, O.I., 205 Kar, A.K., 46 Karlsson, A., 325 Karszewski, M., 37, 43 Kärtner, F.X., 9, 13, 14, 18, 19, 34–38, 46, 49–52, 58, 63–66, 73, 74, 82–86 Kato, J., 253, 260, 264, 285 Kawakita, K., 291 Kawanishi, T., 159, 195 Kean, P.N., 9, 33, 53, 54, 77

412

Author index for Volume 46

Keller, U., 8, 9, 11–14, 18, 19, 32–39, 41–46, 49–52, 54, 58, 63–66, 72–74, 77, 80–90, 92–97 Kellner, T., 18, 35 Kemp, A.J., 37 Kempe, M., 200 Kenney-Wallace, G.A., 91 Kersey, A.D., 287, 291 Khuri-Yakub, P.T., 8, 32, 44, 73 Kienberger, R., 94, 97 Kigre Inc., 18 Kikuta, H., 292 Kilburn, I.J., 38 Kim, B.G., 51 Kim, J., 338 Kim, M.J., 124, 170, 172 Kim, Y.-H., 337 Kimble, H.J., 321, 325, 344–346, 349, 357, 364, 375 Kimmitt, M.F., 49 Kimura, T., 245 Kingsley, S.A., 291 Kinsler, P., 399 Kisel, V.E., 37 Kishner, S., 281, 282 Kisliuk, P., 10 Kitoh, M., 292 Kivelson, M.G., 124 Klein, M.J., 364, 365 Kleineberg, U., 97 Klimov, I., 37, 38 Klopp, P., 19 Kmetec, J.D., 36 Knight, P.L., 357 Knill, E., 329, 337, 339, 342, 350 Knotts, M.E., 125, 152, 153, 163, 167, 170, 172, 178 Knox, W.H., 32, 33, 35, 37, 49–51, 54, 77 Koashi, M., 316 Kobayashi, K., 36, 37, 284 Kobayashi, S., 245, 281, 290, 293 Kobayashi, T., 9, 90 Koechner, W., 18 Koliopoulos, C., 271 Kolodziejski, L.A., 46 Kong, J.A., 132 König, F., 19, 37 Koontz, E.M., 46

Kopf, D., 9, 18, 19, 34–37, 41–43, 49–51, 63, 74, 82, 83, 86 Kornelis, W., 93 Kourogi, M., 290 Krainer, L., 9, 19, 36–38, 51, 52 Krause, A., 301 Krausz, F., 9, 33–36, 63, 64, 79, 80, 89, 92, 94, 97 Kravtsov, Yu.A., 120, 122 Kretschmann, E., 204 Krumbügel, M.A., 92 Krupke, W.F., 17 Kuang, L., 205 Kubecek, V., 35, 36 Kubo, R., 182 Kubota, T., 291 Kück, S., 18 Kuga, Y., 120, 136, 221, 230 Kuhr, S., 400 Kuizenga, D.J., 71 Kuizenga, K.J., 12, 72 Kuleshov, N.V., 18, 19, 37 Kulik, S.P., 337 Kumar, N., 200 Kumkar, M., 9, 38, 90 Kurz, H., 36 Kutovoi, S.A., 17 Kuzmin, A.N., 19 Kway, W.L., 17, 18 Kwiat, P.G., 329 L Laflamme, R., 329, 337, 339, 342, 350 Lagatsky, A.A., 18, 19, 37 Lagendijk, A., 120, 136, 200, 222 Lai, G., 262 Lai, S.T., 8, 73 Lam, P.K., 344 Landford, N., 53 Langford, N., 37 Langlois, P., 46 Lanker, M., 34 Laporta, P., 34–36 Laptev, V.V., 17 Larkin, K.G., 271 Lawandy, N.M., 200 Leach, J., 371, 384 Lebert, R., 36

Author index for Volume 46 Lecomte, S., 9, 37 Lederer, M.J., 19, 37, 38 Lee, B., 284, 285 Lee, C.-T., 285 Lee, C.H., 32, 204, 208 Lee, H.-W., 338 Lee, H.P., 122 Lee, J.K., 231 Lee, K.S., 122 Lee, P.A., 180 Lee, S., 204 Lefaucheur, J.L., 18, 34 Lefort, L., 36 Lei, C., 96 Leistner, A.J., 264 Lemoff, B.E., 33 Lenz, G., 15 Lenzner, M., 9, 35, 89, 92 Leonelli, R., 36 Leonhardt, U., 402 Leskova, T.A., 128, 130, 132, 139, 150, 158, 162, 170, 180, 181, 183, 188, 197, 202–206, 210, 213–215, 221, 224, 225, 232, 233 Leuchs, G., 19, 36, 37 Leung, D.W., 325 Levine, B.M., 170 Lewenstein, M., 96, 97 Leyva-Lucero, M., 205, 206, 210, 213, 214 L’Huillier, A., 96, 97 Li, K.D., 8, 32, 33, 44, 73 Li, L., 168, 231 Li, X.F., 96 Li, Z., 161 Licciardello, D.C., 136 Lichtenstein, N., 19, 37 LiKamWa, P., 33 Lim, D., 204 Lim, S.P., 122 Lin, N., 122 Lincoln, J.R., 34 Ling, J.D., 32 Linz, A., 17 Liszka, E.G., 155 Liu, C.-H., 200 Liu, H., 19, 37 Liu, J., 260, 285 Liu, J.M., 32 Liu, K.X., 33, 92

413

Liu, L.Y., 32, 53 Liu, X., 36 Lodahl, P., 344, 395 Loesel, F.H., 35, 36, 43 Logan, R.A., 76 Lombardi, E., 341 Lompré, L.A., 96 Long, M., 17 Longhi, S., 34–36 Loo, K.M., 204 Lord Rayleigh, 129 Loudon, R., 357, 364, 369 Louradour, F., 18, 36 Lu, J.Q., 130, 132, 155, 170, 172, 191, 192, 195–198, 205, 206, 224 Lugiato, L.A., 372, 401 Luna, R.E., 168–170, 172 Lundeen, J.S., 393 Luntz, G., 38 Luther-Davis, B., 19, 37 Lütkenhaus, N., 328, 329 Lutterback, L.G., 360 Lutz, R.C., 87 Luysberg, M., 87 Lvovsky, A.I., 358, 393, 395, 396 M Mabuchi, H., 325 Macaskill, C., 133, 157, 229 Macías, D., 232 Maciejko, R., 36 Macomber, S.H., 286 Madrazo, A., 129, 159, 196, 221 Magni, V., 34, 54 Maier, N., 299 Mainfray, G., 96 Major, A., 37 Mak, P., 38 Maker, G.T., 8, 32, 33, 73 Malacara, D., 245 Malacara, Z., 245 Malcolm, G.P.A., 17, 32, 33 Malyshkin, V., 128, 180, 181, 183, 232 Manabe, T., 302 Mandel, L., 222, 327, 357, 364, 365 Mandelbrot, B.B., 121 Mandel’shtam, L.I., 137 Manhart, S., 298

414

Author index for Volume 46

Mann, A., 327 Manus, C., 96 Maradudin, A.A., 119–121, 124, 125, 127–134, 136, 137, 139–143, 148–152, 154–162, 168–170, 179–181, 183, 188, 191, 192, 194–198, 202–206, 210, 213, 214, 220, 221, 224, 225, 227–229, 232, 233 Marchesi, M., 35 Marcikic, I., 334 Marcos, H.M., 17 Maret, G., 120, 136 Margheri, G., 298 Marron, J.C., 293 Martin, D., 52, 97 Martínez, A., 170 Martinez, O.E., 62, 72, 77, 92 Martinez-Niconoff, G., 232 Martini, F.D., 358, 393, 395 Maruyama, T., 283, 284 Marvin, A., 129, 130, 143, 150 Marvin, A.M., 158, 159 Masazumi, N., 293 Massons, J., 19 Mataloni, P., 358, 393, 395 Mateos, X., 19 Matrosov, V.N., 17 Matsnev, S.Y., 17 Matsuda, K., 249 Mattle, K., 329, 332, 336 Matuschek, N., 9, 18, 35, 37, 49–51, 58, 64, 65, 92, 93 Maurer, R., 298 Mayer, A.P., 202 Maystre, D., 129, 169, 208, 218 Maytorena, J.A., 208 McCarthy, M.J., 33 McClung, F.J., 10 McConnell, G., 37 McCoy, J.J., 155 McGurn, A.R., 120, 125, 127, 128, 131–134, 136, 140, 143, 148–151, 154, 155, 179–181, 183, 188, 191, 194, 197, 198, 204, 213, 215, 232 McIntosh, J.W., 33 Meinlschmidt, P., 255 Melchior, H., 11, 52, 97 Mellish, P.M., 34 Mellish, R., 33, 35, 36

Méndez, E.R., 120, 124, 125, 127, 130–132, 134, 137, 139–142, 148–152, 154, 155, 158, 160, 168, 170, 172, 176–179, 188, 197–201, 205, 206, 210, 213, 214, 224–226, 232, 233 Mendoza, B.S., 204, 208, 209 Mendoza-Suarez, A., 158 Merkel, K., 256, 263, 267 Mertz, J.C., 357 Meyer, J., 36 Meyn, J.-P., 18, 52 Michel, T., 124, 125, 131, 132, 136, 150–155, 167, 170, 172, 197, 198, 213 Michel, T.R., 159, 163, 170, 179 Michielssen, E., 156 Mielke, S.L., 359, 388 Mikhailov, V.P., 18 Mikulla, B., 36 Mikulyak, R.M., 76 Milburn, G., 370 Milburn, G.J., 320, 329, 337, 339, 342 Milder, D.M., 162 Millar, R.F., 129 Miller, A., 18, 33, 34 Miller, B.I., 15, 49 Miller, D.A.B., 8, 12, 33, 49–51 Milosevic, N., 97 Mindl, T., 91 Minkov, B.I., 34 Mirtchev, T., 36 Misirpashaev, T.Sh., 200 Missaggia, L.J., 52 Mitchell, D.E., 204 Mitschke, F.M., 15, 53 Mlynek, J., 358, 393, 395, 396 Mnatzakanian, S., 284, 285 Mochán, W.L., 208, 209 Mocker, H.W., 12 Mohebi, M., 36 Mohr, S., 19, 37 Mollenauer, L.F., 15, 53, 89 Mølmer, K., 348 Monguzzi, A., 54 Monro, T.M., 90, 96 Mooradian, A., 11 Morf, R., 64 Morgan, C.J., 254 Morgner, U., 9, 19, 36–38, 64

Author index for Volume 46 Morier-Genoud, F., 9, 13, 18, 19, 35–37, 39, 43, 46, 49, 51, 52, 58, 64, 83, 86–88, 97 Morris, R.C., 17 Morzov, V.N., 76 Moser, M., 9, 11, 13, 18, 19, 34–37, 41–43, 46, 49–52, 83, 86–88 Moszkowski, S.A., 124 Mougel, F., 18, 19, 37 Moulton, P.F., 9, 17 Mourou, G., 19, 37, 54 Mourou, G.A., 18, 36 Moussa, M.H.Y., 338 Mudaliar, S., 231 Mukohzaka, N., 249 Muñoz-López, J., 232, 233 Munro, W.J., 399 Murakami, A., 302 Murao, M., 325, 326 Murata, K., 253, 258, 260, 264 Murnane, M.M., 33, 34, 63, 92, 96 Myers, J.F., 17 N Naganuma, K., 34 Nagata, R., 292 Nakagawa, T., 36, 37 Nakajima, T., 260, 285 Nakamura, T., 265, 267, 274 Nakano, H., 33 Nakatani, N., 286 Nam, C.H., 36 Nara, M., 284, 285, 291 Nassim, A.-K., 255 Nassim, K., 275 Nathel, H., 33, 34 Navarrete, A.G., 170 Nebel, A., 35 Nees, J., 18, 19, 36, 37 Negrete-Regagnon, P., 170, 178 Negus, D.K., 33, 54, 77 Nelson, L.E., 15 Neviere, M., 204, 208 New, G.H.C., 13, 65 Newkirk, H.W., 17 Nielsen, M.A., 314, 350 Niemz, M.H., 35 Nieto-Vesperinas, M., 119, 128, 129, 137, 151, 180, 181, 183, 221, 232 Nisoli, M., 9, 89, 93, 94

415

Niwa, N., 246 Noack, F., 36 Noginov, M.A., 35 Norris, B., 49 Norris, T.B., 92 Notni, G., 301 Novikov, I.V., 162, 168, 169, 232 Novikova, N.N., 205 Nuss, M.C., 32, 72 O Ober, M.H., 15, 39, 54, 79 Occhionero, G., 279 Odagiri, Y., 293 O’Dell, E.W., 17 O’Donnell, K.A., 120, 124, 125, 134, 137, 139–142, 149, 150, 152, 153, 163, 167, 170–172, 176–179, 185–187, 205, 212–216 Offrein, B.J., 38 Ogilvy, J.A., 119, 124, 142 Ogura, H., 150, 159, 195, 196 Ohba, R., 286 Ohde, N., 274, 288 Ohtsubo, J., 290, 302 Ohtsuka, Y., 290 Oka, K., 290 Okada, K., 262, 269, 273 Okazaki, H., 283, 284 Okhrimchuk, V.G., 17 Okugawa, T., 300 Olague, G., 232 Olsson, N.A., 45, 46 Olszak, A., 255, 277 Ong, T.T., 155, 159 Onodera, R., 251, 259, 262, 265, 267, 274, 277, 279, 281, 286, 288, 293, 295, 296, 299 Onsager, L., 364 Oreb, B.F., 256, 263, 264, 269, 271–273 Orozco, L.A., 357–359, 362, 363, 366, 369, 374–378, 381, 385, 388, 389, 391–393, 397, 400, 402 Ose, T., 262, 269 Otani, Y., 290 Otto, A., 204 Ou, Z.Y., 327 Ozrin, V.D., 120

416

Author index for Volume 46

P Paasschens, J.C.J., 200 Páez, G., 275 Pak, K., 156, 221, 229–231 Pan, C.-L., 299 Pan, C.L., 291 Pan, J.-W., 332, 336 Pan, N., 52 Pang, G., 200 Pang, Y., 34 Papetti, F., 279 Parks, R.E., 203 Paschotta, R., 9, 11, 13, 14, 18, 19, 36–39, 43, 46, 49, 51, 52, 66, 80, 81, 85–88, 90, 96, 97 Pascual, M., 221 Pastur, L.A., 146 Pathak, R., 35 Patorski, K., 255, 277 Paul, H., 364 Paulus, G.G., 94 Payne, S.A., 17, 18 Pedersen, K., 205 Pedrini, G., 259 Peng, G.D., 203 Pennacchio, C., 35 Peres, A., 313, 316–319 Pessot, M.A., 8, 73 Pestryakov, E.V., 17 Petek, H., 37 Petermann, K., 18 Peterson, O.G., 17 Petit, R., 129 Petricevic, V., 17, 34 Petrov, V., 19, 36 Phillion, D.W., 275 Phillips, M.W., 33 Phu, P., 156 Pic, E., 204 Piché, M., 33, 54, 77 Pieterse, J.-W.J., 59 Pincemin, F., 219 Ping, Q., 253, 264 Pittman, T.B., 342 Plenio, M.B., 325, 326 Podlipensky, A.V., 18 Podolsky, B., 318 Pohlmann, U., 18 Polhemus, C., 277

Politch, J., 282 Pollack, M.A., 12 Polli, D., 90 Pollock, C.R., 18, 34 Polzik, E.S., 344–346, 349 Ponce, M.A., 125, 197, 198 Ponomarenko, S.A., 229 Pope, D.T., 399 Popescu, S., 322, 329, 331, 341 Popov, Y.M., 76 Poppe, A., 94 Porath, R., 96 Pradhan, P., 200 Prasad, A., 35, 42, 43 Prasankumar, R.P., 38 Press, W.H., 155 Prewitt, A., 122 Priori, E., 94 Prongué, D., 296, 298 Pshenichnikov, M.S., 9, 89, 92 Puech, K., 36 Pujol, M.C., 19 Pustilnik, M., 130, 188, 195, 196, 200, 201, 224 Q Qian, L., 36 Quail, J.C., 204 Quartel, J.C., 232 R Rachafi, S., 275 Raizen, M.G., 375 Ralph, T.C., 344 Ramakrishnan, T.V., 136 Ramaswamy, M., 33 Ranka, J.K., 94 Raybaut, P., 19, 37 Raybon, G., 76 Razvalyaev, V.N., 17 Read, K., 96 Reali, G.C., 35 Reck, M., 340 Reffert, S., 52 Reichert, J., 95 Reid, D.T., 36 Reider, G.A., 94, 97

Author index for Volume 46 Reiner, J.E., 358, 366, 374, 375, 377, 378, 381, 385, 389, 392, 393, 397, 400 Reinisch, R., 204, 208 Resch, K.J., 393 Rhee, J.-K., 92 Rice, P., 371, 384 Rice, P.J., 375 Rice, P.R., 358, 361, 366, 369, 373–375, 377, 378, 381, 382, 389, 392, 393, 397 Rice, S.O., 137, 172 Richardson, D.J., 90, 96 Richman, B., 92 Ripin, D.J., 19, 38 Riris, H., 292 Rischel, C., 96, 97 Rizvi, N.H., 33, 34 Roberts, J.S., 39, 97 Robl, T., 32 Roentgen, P., 42 Rosen, M.D., 96 Rosen, N., 318 Rosenberg, A., 17 Rosenfeld, D.P., 245, 251 Rosich, P.K., 158 Roskos, H., 32, 49, 50 Ross, G.W., 96 Rothe, A., 299 Rothrock, D.A., 121 Rousse, A., 96, 97 Ruane, M.F., 38 Ruddock, I.S., 7 Rudnick, J., 204, 208 Rudolph, T., 348 Ruffing, B., 35 Ruiz-Cortés, V., 125, 170, 178, 197, 198, 232 Rungta, P.K., 382 Rupp, T., 93 Rußbüldt, P., 36 Ryabtsev, G.I., 19 Rytov, S.M., 122 S Sagehashi, I., 246 Saichev, A.I., 120 Saillard, M., 218, 219 Sakai, M., 284 Sakane, I., 9, 90 Sakuta, H., 262, 269 Sala, K.L., 91

417

Saleh, B.E.A., 40, 247, 286 Salieres, P., 96, 97 Salin, F., 19, 33, 37, 54, 77 Salvadé, Y., 282 Sam, R.C., 17 Sanchez, A., 41 Sánchez-Gil, J.A., 127, 130, 148, 160, 181, 188, 195, 196, 202, 220, 224 Sanders, B.C., 348 Sandoz, P., 273 Sangani, H., 156 Sansone, G., 9, 89, 93 Sant, A.J., 170, 172 Sartania, S., 9, 89, 92 Sarukura, N., 33 Sasaki, O., 283, 284, 302 Sasazaki, H., 284 Sasnett, M.W., 40 Sato, A., 269, 273 Sauer, N., 15, 49 Sauvain, E., 200 Schaer, S.F., 36, 43 Schaich, W.L., 208 Scharte, M., 96 Schenkel, B., 9, 89, 93 Scheps, R., 17 Scherer, N.F., 218 Scheuer, V., 9, 18, 19, 35–38, 51, 58, 64 Schibli, T.R., 19, 38, 46, 64 Schiffer, R., 121 Schiller, S., 358, 393, 395 Schlüter, F., 255 Schmidt, A.J., 33, 79 Schmidt, O., 96 Schnabel, R., 344 Schneider, S., 91 Schön, S., 38, 51 Schönhense, G., 96 Schott Glass Technologies, 17, 18 Schroeder, K.S., 293 Schult, R.L., 124 Schulz, P.A., 32 Schwider, J., 245, 256, 263, 267 Sciarrino, F., 341 Scrinzi, A., 97 Scully, M.O., 321, 343, 344, 348, 349 Seas, A., 34 Sebban, S., 96, 97 Seckold, J.A., 264 Seeber, W., 18, 36

418

Author index for Volume 46

Seelig, E.W., 200 Seilmeier, A., 32 Sennaroglu, A., 34 Sentenac, A., 161, 162 Sergeev, A.B., 76 Sergienko, A.V., 329 Serreze, H.B., 17 Servagent, N., 286 Servin, M., 245 Sha, W.L., 200 Shagam, R.N., 245 Shah, J., 45 Shank, C.V., 7, 10, 62, 75, 89 Shannon, D.C., 8, 32, 44, 73 Shapiro, B., 181, 184 Shapiro, J.H., 395 Sharp, R.C., 52 Shchegrov, A.V., 130, 132, 137–139, 150, 197, 220, 221, 224, 225, 227–229, 232, 233 Shcherbitsky, V.G., 18, 37 Shcheslavskiy, V., 36 Shen, J., 159 Shen, Y.R., 207, 208 Sheng, P., 120, 136, 143, 217 Sheppard, C.J.R., 122, 232 Sheridan, J.A., 33 Shestakov, A.V., 17, 34, 35 Shi, Z., 51 Shigematsu, Y., 293 Shih, Y., 329, 337 Shin, R.T., 230 Shirakawa, A., 9, 90 Shmelev, A.B., 119 Shor, P.W., 326 Sibbett, W., 9, 33, 36, 37, 53, 54, 77 Siddiqui, S., 371 Siders, C.W., 96 Siegman, A.E., 12, 34, 40, 43, 55, 71, 72 Siegner, U., 45, 87 Silberberg, Y., 15, 39, 72, 73 Silvestri, S.D., 9, 89, 94 Simeonov, S., 136, 232 Simon, H.J., 204, 205 Simonsen, I., 128, 130, 159, 188, 196, 203, 232, 233 Simpson, W.M., 72 Singh, S., 359, 371 Sipe, J.E., 204, 208 Siyuchenko, O.G., 17

Sizer II, T., 32, 44 Sizmann, A., 19, 36, 37 Skidmore, J.A., 35, 41–43 Sleeper, A.M., 203 Slusarev, V.A., 146 Slusher, R.E., 357 Smith, L.K., 17, 18 Smith, P.G.R., 96 Smith, R.A., 162 Smith, S.D., 46 Smith, W.P., 358, 366, 374, 375, 377, 378, 381, 385, 389, 392, 393, 397, 400 Smolin, J.A., 317, 323, 324, 326 Smolyaninov, I.I., 218, 219 Smythe, R.A., 264 So, V.C.Y., 204, 208 Soccolich, C.E., 15, 49 Sodnik, Z., 299 Sokolowski-Tinten, K., 96 Solarz, R.W., 43 Sole, R., 19 Song, D., 325 Song, J., 156, 229 Sorbello, G., 36 Sørensen, J.L., 344–346, 349 Sorokin, E., 18, 35 Sorokina, I.T., 18, 35 Sosnowski, T.S., 92 Soto-Crespo, J.M., 129, 137, 151 Soubret, A., 131, 195 Spälter, S., 36 Specht, P., 87 Spence, D.E., 9, 33, 54, 77 Spielmann, C., 9, 33–36, 63, 64, 79, 80, 89, 92, 94 Spielmann, Ch., 35, 97 Spinelli, L., 54, 77 Spizzichino, A., 119, 161 Spock, D.E., 52 Spolaczyk, R., 256, 263, 267 Spühler, G.J., 11, 19, 35–38, 43, 51, 52, 63 Squier, J., 33, 54, 77 Stagira, S., 9, 89, 93, 94 Stalder, M., 18 Stark, J.B., 35 Stegeman, G.I., 204, 208 Steinberg, A.M., 393 Steinmeyer, G., 9, 18, 37, 64, 65, 72, 90, 92–95 Stenger, J., 72, 93, 94

Author index for Volume 46 Stenholm, S., 318 Stentz, A., 94 Stern, E.A., 204, 208 Stetser, D.A., 12 Stingl, A., 34–36 Stock, M.L., 15 Stoddart, A.J., 161 Stoffel, N., 39 Stoffregen, B., 184 Stolen, R.H., 53, 62, 89 Stoler, D., 358, 393–395 Stone, A.D., 180 Stone, J., 39 Stormont, B., 37 Stover, J.C., 173, 176, 231 Strand, T., 284, 285 Strek, W., 19 Strimbu, C.E., 358, 366, 371, 381, 384 Strojnik, M., 275 Strutt, J.W., 129 Stryland, E.W.V., 7 Stulz, L.W., 39 Su, Z.B., 217 Sucha, G., 15 Suda, H., 284 Südmeyer, T., 9, 37, 38, 90, 96 Sugaya, T., 36, 37 Sullivan, A., 92 Sun, X., 122 Sunderman, E.R., 15, 49 Suominen, K.-A., 328 Surrel, Y., 254, 275 Sutherland, J.M., 35 Sutter, D.H., 9, 18, 35, 37, 51, 58, 64, 65, 72, 92–94 Suzuki, T., 283, 284, 302 Svelto, O., 9, 34, 35, 43, 89, 93 Svirko, Y.P., 36 Sweetser, J., 92 Symul, T., 344 Szipöcs, R., 9, 18, 34, 35, 63, 64, 89 T ’t Hooft, G.W., 33, 54, 77 Taccheo, S., 34–36 Taft, G., 34, 63 Takahashi, K., 283 Takahashi, N., 159, 196, 286

Takahashi, T., 302 Takahashi, Y., 274, 288 Takano, S., 36 Takasaka, M., 9, 90 Takatsuji, T., 269 Takeda, M., 251, 281, 290, 292, 293 Takeda, T., 290 Takiguchi, Y., 204 Tamura, K., 15 Tan-no, N., 293 Tanahashi, A., 290 Tanbun-Ek, T., 76 Tapié, J.-L., 54 Tatam, R.P., 255 Tatarskii, V.I., 122, 124, 125, 142 Tatarskii, V.V., 124, 125 Tatsuno, K., 253, 258 Taylor, J.R., 33–36 Teich, M.C., 40, 247, 286 Teisset, C.Y., 37, 38 Telle, H.R., 72, 93–95 Tempea, G., 36, 89, 94 Terhal, B.M., 326 Teukolsky, S.A., 155 Thalmann, R., 296, 298 Thoen, E.R., 46 Thomas, S., 38 Thorndike, A.S., 121 Thorpe, A., 36 Thorsos, E., 162 Thorsos, E.I., 119 Tien, A.-C., 92 Tilsch, M., 35, 51, 58, 64 Tisch, J.W.G., 93 Titov, A.N., 19 Tittel, W., 334 Tiziani, H., 259 Tiziani, H.J., 249, 282, 299 Tobey, R., 96 Toigo, F., 129, 143 Tolman, R.C., 364 Tomaru, T., 37 Tombesi, P., 400 Tomita, H., 364, 365 Tomita, K., 364, 365 Tomlinson, W.J., 62, 89 Tong, Y.P., 34–36 Torizuka, K., 19, 36–38 Torre, R., 171, 205, 212–216

419

420

Author index for Volume 46

Torrungrueng, D., 230 Toth, C., 96 Tournois, P., 59 Tran, P., 132, 133, 152, 155–157, 197, 198, 229 Treacy, E.B., 62 Trebino, R., 92 Treps, N., 344 Tribillon, G., 273, 291 Tropper, A.C., 39, 97 Trunov, V.I., 17 Truong, G., 36 Tsang, L., 120, 136, 156, 168, 221, 229–231 Tschudi, T., 9, 18, 19, 35–38, 51, 58, 64 Tsuda, S., 35, 50, 51 Tsujiuchi, J., 262, 269, 273 Tsunoda, Y., 253, 258 Tutov, A.V., 202 Tveten, A.B., 246 Twiss, R.Q., 357 U Udem, T., 94, 95 Uehira, I., 301 Uemura, S., 19, 37, 38 Uiberacker, M., 97 Uitert, L.G.V., 17 Uschmann, I., 96, 97 Uttam, D., 270, 291 V Vaccaro, J.A., 326, 350 Vaidman, L., 319, 328, 397 Vail’ev, P., 39, 75 Valdmanis, J.A., 7, 32, 72, 75–77 Valley, J.F., 357 Valster, A., 35, 36 van Albada, M.P., 120, 136, 200 van de Hulst, H.C., 165 van der Poel, C.J., 35 van der Ziel, J.P., 76 van Driel, H.M., 33, 34 van Enk, S.J., 319, 325, 349, 350 van Loock, P., 344, 345 Van Stryland, E.W., 33 Vasconcelos, E.F., 121 Vasil’ev, P.P., 76

Vedral, V., 325, 326 Vetterling, W.T., 155 Viana, B., 19, 37 Villas-Bôas, C.J., 338 Villoresi, P., 94 Vincent, P., 208 Vitali, D., 400 Vivien, D., 18, 19, 37 Vogel, W., 359, 360, 371 Vohnsen, B., 218, 219 von der Linde, D., 96 von Hoegen, M.H., 96 Voronovich, A.G., 119, 121, 124, 162 Vozzi, C., 9, 89, 93 Vry, U., 295 Vyas, R., 359, 371 W Wagenblast, P.C., 19, 38 Wagner, R.L., 156, 229 Wait, J.R., 158 Walker, A.C., 46 Walker, D.R., 33, 34 Walker, J.A., 34 Walker, S.J., 32, 44 Wallace, R.W., 8, 32, 44, 73 Wallenstein, R., 35 Wallentowitz, S., 360 Walling, J.C., 17 Wallis, R.F., 191, 192 Walls, D.F., 357, 364, 370 Walmsley, I.A., 92, 93 Walpole, J.N., 52 Walsh, C.J., 264 Walther, H., 94 Wang, C.-L., 299 Wang, H.S., 18, 34 Wang, J., 32, 400 Wang, Q.H., 200 Wang, W., 218 Wang, X., 205 Wang, Y., 205 Wang, Z.L., 150, 159, 195, 196 Ward, B.K., 264 Warnick, K.F., 230 Warren, R.E., 292 Wasik, G., 19, 37 Watanabe, Y., 292

Author index for Volume 46 Watson, J.G., 204 Watts, D.G., 174, 175 Watts, M.L., 59 Weber, E.R., 87 Wei, Z., 9, 89 Weihs, G., 340 Weiner, A.M., 15, 91 Weinfurter, H., 327, 329, 332, 336, 340 Weingarten, K.J., 8, 9, 18, 32–38, 41, 42, 44, 49–51, 63, 73, 74, 82 Weiss, S., 19, 37 Werner, R.F., 322 West, C.S., 125, 137, 139–141, 149, 171, 172, 179, 185–187, 205, 214–216 Westerwalbesloh, Th., 97 White, A.D., 245, 251 White, W.E., 92 Wickramasinghe, H.K., 299 Wiemann, C., 96 Wiersma, D.A., 9, 89, 92 Wiersma, D.S., 200 Wiesner, S.J., 316 Williams, C.C., 299 Williams, C.J., 325 Wilson, K.R., 96 Windeler, R.S., 93, 94 Winebrenner, D.P., 137, 142 Wintner, E., 18, 33, 35, 79 Wise, F., 34, 36 Wise, F.W., 35 Wiseman, H.M., 350, 358, 396, 397, 400 Wódkiewicz, K., 360 Wolf, E., 165, 221, 222, 227, 228, 357, 365 Wolf, P.E., 120, 136 Wombell, R.J., 232 Wong, R., 291 Wooters, W.K., 313, 316–319 Wootters, W.K., 314, 317, 323, 324, 326 Wright, J.A., 124 Wu, M.C., 76 Wu, R.Q., 204 Wu, S.C., 124 Wyant, J.C., 245, 251 X Xiao, M., 361, 373, 382 Xu, L., 35, 36, 94

421

Y Yakovlev, V., 97 Yamagishi, J., 302 Yamaguchi, H., 285 Yamaguchi, I., 253, 260, 264, 285, 292 Yamamoto, H., 290 Yamamoto, Y., 245 Yan, L., 32 Yanovsky, V.P., 34, 35 Yariv, A., 43 Yatagai, T., 262 Ye, J., 95 Yeh, P., 43 Yen, R., 89 Yokota, M., 285, 286 Yonemura, M., 253 Yoran, N., 328 Yoshida, T., 283 Yoshimura, T., 293 Yoshino, T., 274, 284–286, 288, 291 Yoshizawa, T., 290 Youngquist, R.C., 287 Yuen, H.P., 357, 386, 395 Yurke, B., 357, 358, 393–395 Yurkevich, I., 130, 180, 181, 185, 188, 195, 196, 200, 201, 224 Z Zatti, S., 298 Zavelani-Rossi, M., 90 Zayats, A.V., 218, 219 Zayhowski, J.J., 11, 18 Zbinden, H., 334 Zehnder, O., 38 Zeilinger, A., 314, 324, 329, 332, 336, 340 Zeller, S.C., 38 Zhang, G., 11, 18, 34–36, 41–43, 51, 52, 58, 63, 82, 221 Zhang, T.C., 344 Zhang, Z., 36, 37 Zhang, Z.Q., 200 Zhao, Y.G., 200 Zhitnyuk, V.A., 17 Zhou, F., 33 Zhou, J., 33, 34, 63 Zhou, L.B., 205 Zhou, X., 325

422 Zhu, X., 53 Zierau, W., 221 Zimmermann, E., 282 Zimmermann, M., 95 Zirngibl, M., 39 Zoller, P., 324, 325

Author index for Volume 46 Zou, L.-F., 168 Zou, Y., 259 Zubairy, M.S., 321, 343, 344, 348, 349 Zukowski, M., 324 Zurek, W.H., 314 Zyuzin, A.Yu., 200

Subject index for Volume 46 A

E

acousto-optic modulator 43 Anderson localization 217 autocorrelation function 123, 212

Einstein–Podolsky–Rosen state 318, 342, 343 electrooptic modulator 43 entangled state 322–325, 339, 373 entanglement swapping 324

B Baker–Hausdorff formula 320 Bell inequality 322, 323 – operator basis 315 – -state measurement 327, 328–330, 332, 333, 335, 336 Bethe–Salpeter equation 145, 185, 187 Bloch sphere 322 Bragg reflector, saturable 51 Brewster angle 42, 62 Brillouin scattering 301

F Fabry–Perot cavity 385 – structure 59 Fock state 358, 396 – – , amplitude probability of 394 – – , state reconstruction of 395 fractal surface 121 frequency-resolved optical gating 92 Fresnel reflection coefficient 138 G

C

Gaussian aperture 199 – beam 164, 171 – pupil function 171 – random process 122–125, 172, 184, 211 – – surface 122, 124, 125 – statistics 184 Green’s theorem 210 group velocity dispersion 68, 70, 81

cavity QED 358, 360, 365, 366, 368, 379–381, 384, 393, 402 – – , experiments in 385–393 – – , time symmetry in 379 chirped mirror 63 CNOT quantum logic gate 326, 330 correlation function 361 – – , angular intensity 128, 180–189 – – , Gaussian 129, 151, 153, 177, 179 – – , third order 357 – intensity-field 360, 365, 366, 369, 395–397 – – -intensity 357, 364, 391, 402 cross-spectral density 222

H Hanbury Brown–Twiss 357 Hankel function 132 Haus’s master equations 66, 82 homodyne detection 344, 367, 397, 398 – – , balanced 357–360, 395 – – , conditional 359, 386, 402 Huygens’ principle 131

D detailed balance 365 differential reflection coefficient 128, 202 Dirichlet boundary condition 151, 160, 188 – surface 229 down conversion 329

I interferometer, Fabry–Perot 247 – , Fizeau 265, 266, 270 423

424

Subject index for Volume 46

– , Gires–Tournois 58–60 – , holographic 301 – , Hong–Ou–Mandel 327 – , laser-diode 251, 252 – , Mach–Zehnder 291, 386, 387 – , Michelson 291 – , optical 245 – , phase conjugate 301 – , Twyman–Green 250, 275, 277, 283, 288 interferometry – , feedback 285 – , Fizeau 264, 268, 269, 273 – , heterodyne 286, 287, 294, 295, 297 – , holographic 246 – , laser–diode 245, 246, 250, 251 – , – – wavelength scanning 268 – , phase-conjugate 246 – , phase-modulating 284 – , phase-shifting 251, 253, 259, 260, 277, 278 J Jaynes–Cummings coupling 384, 385 – – Hamiltonian 372, 382 K Kerr effect 52, 54, 69 – – , nonlinear 9 – lens 53, 54 – nonlinearity 52, 54 Kirchhoff approximation 119, 161, 230, 233 L laser – , color center 14, 15, 49 – , diode 8 – , homogeneously broadened 12 – , ion doped fiber 15 – , mode locked fiber 15 – , Nd:glass 43 – , pulse mode-locked dye 7, 75, 76 – , Q-switched 44 – , semiconductor 39, 76, 77 – , solid-state 7–11, 14, 15, 39, 42, 46, 49, 76, 77 – , Ti:sapphire 9, 40, 64, 78, 79, 85 M Mandel Q factor 393

Maxwell’s equation 164, 206 mode locking, active 11, 12 – – , Kerr lens 9 – – , passive 11–13, 52, 74, 77, 87 – – , Q-switched 8, 10 – – , saturable absorber 13, 14, 65 – – , soliton 65, 81, 82, 85, 86, 88 moiré pattern 277, 280 multiple scattering from rough surface 119, 120, 133, 136, 189 N Neumann boundary condition 151 – Liouville iterative scheme 155 – – series solution 161 – surface 229 no-cloning theorem 314 O Onsager relation 364 optical bistability 372, 401 – Bloch equations 365 – modulator 43 – parametric oscillator 321, 343, 358, 366, 369 – – , two-level atom in 381 P parametric down conversion 358 – oscillation 96 Pauli spin operators 373 photon antibunching 364 power spectrum, Gaussian 139, 140, 154, 156, 157, 196, 202 – – of surface roughness 124 – – , West–O’Donnell 140, 141, 149, 196, 203, 214–216 Poynting vector 165 P representation, positive 399 pulse propagation in dispersive medium 54–65 Q Q-switching 10–12, 43, 46, 49, 52 – instability 12, 13, 49, 50, 87 quadrature fluctuation 361 quantum computation 325 – – with linear optics 329 – error-correcting code 324

Subject index for Volume 46 – information theory 313 – jump 367 – key distribution 333 – logic gate 326 – network 325 – repeater 324 – teleportation (see teleportation) – trajectory 378 – well absorber 50 R Rabi frequency 375 – oscillation 373, 375 random phase screen 199 – surface 119, 120, 202 – – , characterization of 121–126 – – , one-dimensional 125, 195 – – , two-dimensional 125 Rayleigh equation 129–131, 138, 159, 203, 220 – method 129 resonance fluorescence 359, 371 S saturable absorber 8, 11–13, 39, 44–48, 52, 68, 74, 77, 79, 80, 82, 87, 88 – – , antiresonant Fabry–Perot 49 – – mirror, dispersive 51 – – mirror, semiconductor 9, 51, 96 – – , semiconductor 45, 47, 49 scattering S-matrix 127, 182 Schrödinger equation, stochastic 367 Schwarz inequality 363, 365 – – , violation of 384 second harmonic generation 96, 204, 205, 211 self-amplitude modulation 44, 77 – -phase modulation 69, 70 Sellmeier equations 56

425

speckle pattern 136, 172, 187 squeezed state 321, 342 – – , two-mode 343 squeezing 357, 369, 371, 398, 399, 402 – , degree of 358, 363 – parameter 343 – , spectrum of 375, 376, 382, 392, 396 – , vacuum state 358, 398 Stokes matrix 153, 165, 167–169 – vector 165–167 sub-Poissonian variance 357 surface plasmon polariton 214, 217 – – – assisted enhanced backscattering 150 – – –, Green’s function of 143 – – –, multiply-scattered 147, 205 T teleportation 313, 314 – , continuous-variable 319, 321, 342–350 – , imperfect 321 – , multi-dimensional 318 – of discrete variables 326–342 tomographic reconstruction 358 two-photon absorption 46 U ultrashort pulse generation 7 V vacuum Rabi doublet 375, 382 – – oscillation 384, 385, 389, 402 velocimetry, laser diode 290 W Wiener fringes 228 – Khintchine theorem 293, 301 Wolf effect 221 – shift 227

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.

427

1– 36 37– 83 85–143 145–197

428 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

429

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

430

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

431

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

432 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

433

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

434

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

435

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

436

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

Cumulative index – Volumes 1–46∗ 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.

437

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

438

Cumulative index – Volumes 1–46

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 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. Christov, I.P.: Generation and propagation of ultrashort optical pulses

6, 53 33, 319 35, 61 18, 257 36, 129 36, 1 27, 227 16, 357 36, 245 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, 353 41, 97 16, 289 21, 287 41, 1 32, 203 41, 283 37, 345 41, 97 13, 69 29, 199

Cumulative index – Volumes 1–46 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 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 Essiambre, R.-J., G.P. Agrawal: Soliton communication systems

439

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 33, 203 20, 1 10, 165 37, 1 43, 433 12, 163 14, 161 31, 189 38, 1 7, 359 21, 355 16, 233 37, 185

440

Cumulative index – Volumes 1–46

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 Giacobino, E., see Reynaud, S.

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, 353 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, 309 18, 1 13, 169 17, 85 30, 1

Cumulative index – Volumes 1–46 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.

441 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., 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 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

40, 77 28, 87 46, 241 35, 145

Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, P., B. Roizen-Dossier: Apodisation Jaeger, G., A.V. Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection Jamroz, W., B.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.

5, 247 3, 29 42, 277 38, 419 20, 325 38, 343 9, 179

442

Cumulative index – Volumes 1–46

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. Klein, M.C., see Flytzanis, C. 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 29, 321 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 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

14, 47 11, 123 41, 483 16, 119 6, 1 16, 1 39, 373 8, 343 41, 97

42, 93

5, 287 38, 263 40, 271 35, 61 21, 69 41, 419

Cumulative index – Volumes 1–46 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 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 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

443

43, 295 33, 129 40, 115 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, 115 11, 305 23, 113 22, 145 21, 1 15, 77 8, 373 46, 115 38, 263 40, 271 30, 261 36, 1 27, 227 17, 279 7, 231 19, 45 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201

38, 165

444

Cumulative index – Volumes 1–46

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 Noethe, L.: Active optics in modern large optical telescopes

41, 97 25, 1 34, 249 23, 113 24, 1 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

Padgett, M.J., see Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., see Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., see Brunner, W. Pegis, R.J.: The 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

34, 249 34, 249 25, 191 44, 1 34, 183 15, 139 44, 85 7, 299 42, 93 39, 63 42, 325 46, 353 35, 61 38, 165 22, 197 30, 87 22, 197 24, 165 33, 319 29, 65 39, 291 32, 1 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67 37, 57 41, 97 18, 127

Cumulative index – Volumes 1–46

445

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.

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

Qiao, Y., see Psaltis, D.

31, 227

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. Ronchi, L., see Wang Shomin 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.

27, 315 34, 159 45, 53 31, 227 16, 289

28, 181 46, 353 31, 321 30, 1 29, 321 46, 353 14, 89 8, 239 19, 281 3, 29 25, 279 35, 1 13, 69 24, 39 4, 145 15, 77 29, 321 4, 199 14, 195

446

Cumulative index – Volumes 1–46

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. 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 Schwider, J.: Advanced evaluation techniques in interferometry Schwider, J., see Schulz, G. Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework Sergienko, A.V., see Jaeger, G. Sharma, S.K., 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 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.

29, 65 28, 87 6, 259 26, 1 41, 419 36, 49 33, 389 21, 355 35, 197 14, 195 17, 163 25, 349 13, 93 28, 271 13, 93 10, 89 16, 413 42, 277 39, 213 46, 115 30, 87 44, 143 27, 227 31, 189 33, 203 15, 245 10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355 13, 169 39, 213 27, 109 42, 219 31, 263 5, 145 37, 345 20, 325

Cumulative index – Volumes 1–46

447

Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy Subbaswamy, K.R., see Mills, D.L. Suhara, T., see Nishihara, H. Sundaram, B., see Milonni, 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.

9, 73 2, 1 19, 45 24, 1 31, 1 12, 1 21, 287 8, 133

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 Torre, A.: The fractional Fourier transform and some of its applications to optics Torre, A., see Dattoli, G. Tripathi, V.K., see Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering Turunen, J., M. Kuittinen, F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.

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 43, 531 31, 321 13, 169 2, 131 40, 343 17, 239

Upatnieks, J., see Leith, E.N. Upstill, C., see Berry, M.V. Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids

6, 1 18, 257 19, 139

Vampouille, M., see Froehly, C. 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.

20, 63 6, 259 22, 77 1, 289 15, 245 37, 57 42, 219

448

Cumulative index – Volumes 1–46

Vernier, P.J.: Photoemission Vlad, V.I., D. Malacara: Direct spatial reconstruction of optical phase from phase-modulated images Vogel, W., see Welsch, D.-G.

14, 245

Walmsley, I.A., see Raymer, M.G. Wang, B.C., see Glesk, I. Wang Shomin, 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 Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Turunen, J.

28, 181 45, 53 25, 279 14, 89 29, 293 34, 333 4, 241 13, 267 27, 161

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

33, 261 39, 63

39, 63 10, 89 17, 163 27, 161 31, 263 40, 1 1, 155 10, 137 28, 1 33, 389 40, 343

22, 271 6, 105 8, 295 28, 87 28, 87 32, 145 41, 97 45, 119 11, 77 23, 221 32, 61

40, 271

Cumulative index – Volumes 1–46 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.

449 38, 263 32, 203 18, 204 45, 119 22, 77