PROGRESS IN OPTICS VOLUME 4 9 E. Wolf
Preface v Chapter 1. Gaussian apodization and beam propagation, Virendra N. Mahajan (El Segundo, CA and Tucson, AZ, USA) 1 1. Introduction 3 2. Theory 5 2.1. Pupil function 5 2.2. Point-spread function 5 2.3. Optical transfer function 5 3. Aberration-free Gaussian pupil 7 3.1. Pupil function 7 3.2. Point-spread function 1 0 3.3. Optical transfer function 1 5 4. Defocused Gaussian pupil 1 6 4.1. Pupil function 1 6 4.2. Point-spread function 1 7 4.3. Axial irradiance 2 1 4.4. Optical transfer function 3 1 5. Strehl ratio, aberration tolerance, and Zernike-Gauss polynomials 34 5.1. Strehl ratio 34 5.2. Depth of focus 3 5 5.3. Balanced aberrations 3 9 5.4. Zernike-Gauss polynomials 40 5.5. Strehl ratio for primary aberrations 43 6. Balancing of defocus aberration with spherical aberration or astigmatism 5 4 6.1. Focused beam 5 4 6.2. Collimated beam 62 7. Aberrated Gaussian pupil 66 7.1. Spherical aberration 66 7.2. Symmetry properties 66 7.3. Line of sight 70 8. Weakly-truncated Gaussian pupils and beams 7 1 8.1. Pupil function 7 1 8.2. Point-spread function 7 2 8.3. Radius of curvature of the propagating wavefront 77 8.4. Collimated beam 78 8.5. Beam focusing and waist imaging by a lens 8 0 8.6. Optical transfer function 8 4 8.7. Strehl ratio, aberration balancing, and orthogonal polynomials 85
8.8. Beam characterization and measurement 8 8 8.9. Nonparaxial Gaussian beams 90 9. Conclusions 9 1 Acknowledgements 9 2 Appendix 9 2 References 9 3 Chapter 2. Controlling nonlinear optical processes in multi-leve l atomic systems, Amitabh Joshi and Min Xiao (Fayetteville, AR, USA) 97 1. Introduction 99 2. Modified linear and nonlinear optical properties in multi-level atomic systems 104 2.1. Absorption reduction and dispersion enhancement 106 2.2. Enhanced nonlinearity in EIT systems 11 6 3. Enhanced four-wave mixing processes with induced atomic coherence 126 3.1. Three-level atomic systems 127 3.2. Four-level atomic systems 13 0 4. Controlled optical bistability and optical multistability with three-level atoms inside a n optical cavity 13 5 4.1. Theoretical calculation 13 7 4.2. Controllable optical bistability 140 4.3. Controllable optical multistability 143 4.4. Controllable direction of hysteresis cycle in optical bistability/optical multistability 14 6 4.5. Dynamic hysteresis in optical bistability 14 9 5. Controlled optical switching in three-level atomic systems 15 2 5.1. Controlled switching between bistable states 154 5.2. All-optical switching controlled by coupling laser beam 15 5 5.3. Controlled optical switching in four-level N-type atomic system 159 6. Controlled optical instability with three-level atoms inside an optical cavity 16 2 6.1. Theoretical calculation 164 6.2. Experimental measurements 167 7. Summary and outlook 170 Acknowledgement 172 References 172 Chapter 3. Photonic crystals, Henri Benisly (Orsay, France) and Claud e Weisbuch (Palaiseau, France and Santa Barbara, CA, USA) 177 Nomenclature 179 Main variables and notations 180 1. Introduction 180 2. Basics of periodic dielectric media and theoretical tools 184 2.1. 1D Bragg structure 184 2.2. Density of states, local DOS 19 1 2.3. Generalizing to 2D and 3D periodic crystals 194 2.4. Basics of band calculation 199 2.5. The discovery of PBG 20 1 2.6. Bloch waves 208 2.7. Theoretical tools : a brief overview 209
3. Three-dimensional photonic crystals 21 4 3.1. Introduction 21 4 3.2. Fabrication of 3D photonic crystals 21 6 3.3. Measurements 22 2 3.4. Metallodielectric systems and microwave regime 225 4. Two-dimensional photonic crystals 22 9 4.1. Theoretical aspects 22 9 4.2. 2D photonic crystal with vertical guidance 23 0 4.3. 2D realizations 23 8 4.4. Applications and characterization methods 246 5. Photonic crystal fibers 27 1 Issues of fibre world 27 1 The playing field of photonic crystal fibers 27 1 Single-mode fiber 27 5 True PC guidance 276 High -Delta fiber 278 Special dispersion 27 8 Special applications: nonlinear optics, supercontinuum generation, etc 28 0 Brief overview of PCF modeling tools 28 3 6. Conclusion and perspectives 28 3 Acknowledgements 28 8 References and special issues 28 8 Special issues of regular joumals on photonic crystals, in chronological order 28 8 References 28 9 Chapter 4. Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield, Christian Brosseau (Brest, France ) and Aristide Dogariu (Orlando, FL, USA) 31 5 1. Introduction 31 7 2. A brief survey of the interplay between polarization concepts and geometry 32 0 2.1. From Bartholinus (1669) to Stokes (1852) : polarization is a geometric property of light 32 1 2.2. From Stokes (1852) to Poincar6 (1892) : polarization is an electromagnetic propert y of light 32 3 2.3. From Poincare (1892) to Wolf (1954): polarization is a statistical property of light 32 5 2.4. And now: geometric algebra is back 32 9 3. Density matrix and the convexity property of the states of polarization 33 2 3.1. Polarization and the density matrix 33 3 3.2. Scalar invariants 33 5 3.3. The convex set of polarization states 33 5 3.4. Summary of geometric algebra 33 7 4. Polarization of a plane wave 33 8 4.1. Density matrix and degree of polarization 339 4.2. Some important consequences and concrete applications 344 5. Polarization of an arbitrary wave 34 9 5.1. Density matrix and polarization descriptors 350 5.2. Some applications 36 1 6. Postscript 363
7. Summary and prospects 36 4 Acknowledgements 36 7 Appendix A : Lie groups in polarization optics 36 8 Al.SU(2). 36 8 A.2. SU(3) 37 0 Appendix B : Madison convention for the density matrix of massive spin-1 particles 37 1 Appendix C : Degree of polarization of a field consisting of a superposition of an ensembl e of evanescent waves of random amplitude and of black-body radiation 374 References 376 Chapter 5. Quantum cryptography, Miloslav Dusek (Olomouc, Czech Republic), Norbert Lütkenhaus (Erlangen, Germany) and Martin Hendryc h (Castelldefels, Barcelona, Spain) 38 1 1. Ciphering 38 3 1.1. Introduction, cryptographic tasks 38 3 1.2. Asymmetrical ciphers (public-key cryptography) 38 5 1.3. Symmetrical ciphers (secret-key cryptography) 38 7 1.4. Vemam cipher, key distribution problem 38 9 2. Quantum key distribution 39 1 2.1. The principle, eavesdropping can be detected 39 1 2.2. Quantum measurement 39 1 2.3. Quantum states cannot be cloned 39 2 2.4. Protocol BB84 39 3 2.5. Eavesdropping, intercept-resend attack 395 3. Some other discrete protocols for QKD 397 3.1. Two-state protocol, B92 397 3.2. B92 protocol with a strong reference pulse 397 3.3. Six-state protocol 39 8 3.4. SARG protocol 399 3.5. Decoy-state protocols 399 3.6. Entanglement-based protocols 400 4. Experiments 403 4.1. QKD with weak laser pulses 403 4.2. Entanglement-based protocols 409 5. Technology 41 3 5.1. Light sources 41 3 5.2. Detectors 41 8 5.3. Quantum channels 42 2 6. Limitations 42 3 6.1. Transmission rate 42 3 6.2. Limit on the distance 42 4 6.3. Quantum repeaters 424 7. Supporting procedures 42 5 7.1. Estimation of leaked information 42 5 7.2. Error correction for classical bit strings 425 7.3. Privacy amplification for classical bit strings 426 7.4. Advantage distillation for classical bit strings 42 8 7.5. Authentication of public discussion 428
8. Security 429 8.1. Attacks on ideal protocols 430 8.2. Secure key rates from classical three-party correlations 432 8.3. Bounds on quantum key distribution 434 8.4. Security proofs 436 8.5. Specific attacks 439 8.6. Results 442 8.7. Side channels and other imperfections 445 9. Prospects 446 Acknowledgements 446 References 446 Chapter 6. Optical quantum cloning, Nicolas J. Cerf (Bruxelles, Belgium) and Jaromfr Fiuräsek (Olomouc, Czech Republic) 45 5 1. Introduction and history 457 1.1. The no-cloning theorem 457 1.2. Beyond the no-cloning theorem 459 1.3. Quantum cloning without signaling 463 1.4. Content of this review 464 2. Overview of quantum cloning machines 466 2.1. Universal cloning machines 466 2.2. Pauli and Heisenberg cloning machines 468 2.3. Phase- and Fourier-covariant cloning machines 469 2.4. Group-covariant cloning machines 47 1 2.5. High-d state-dependent cloning machines 472 2.6. Cloning a pair of orthogonal qubits 473 2.7. Entanglement cloning machines 474 2.8. Real cloning machines 474 2.9. Highly-asymmetric cloning machines 475 2.10. Continuous-variable cloning machines 476 2.11. Probabilistic cloning machines 47 8 2.12. Economical cloning machines 479 3. One-to-two quantum cloning as a CP map 480 3.1. Isomorphism between CP maps and operators 480 3.2. Covariance condition 482 3.3. Cloning as a semidefinite programming problem 483 3.4. Double-Bell ansatz 48 5 3.5. Heisenberg cloning machines 489 3.6. Three special cases of Heisenberg cloners 492 4. N-to-M universal quantum cloning 495 4.1. Optimal cloning transformation 495 4.2. Optimality proof for 1 -a M cloning of qubits 500 4.3. Universal asymmetric quantum cloning 50 1 4.4. Universal-NOT gate 503 5. Universal cloning of photons 504 5.1. Amplification of light 504 5.2. Symmetrization 51 4 5.3. Universal asymmetric cloning of photons 51 8 5.4. Cloning of orthogonally polarized photons 522
6. Phase-covariant cloning of photons 525 6.1. Phase-covariant cloning of qubits 525 6.2. Phase-covariant cloning of qudits 528 6.3. Optical phase-covariant cloning 529 6.4. Experimental 140-3 phase-covariant cloning 532 7. Cloning of optical continuous variables 53 3 7.1. Cloning of coherent states 53 3 7.2. Cloning by phase-insensitive amplification 535 7.3. Experimental cloning of coherent states 53 8 7.4. Gaussian distribution with finite width 539 7.5. Cloning of conjugate coherent states 540 8. Conclusions 54 1 Acknowledgements 54 2 References 54 2 Author index for Volume 49 54 7 Subject index for Volume 49 57 1 Contents of previous volumes 57 5 Cumulative index - Volumes 1-49 587