Nonlinear Optics and Quantum Electronics

MAX SCHUBERT BERND WILHELMI Friedrich-Scfailler-Universität Jena

A Wiley-Interscience Publication

John Wiley & Sons New York / Chichester / Brisbane / Toronto / Singapore

Contents

Notation and Symbols xv

PART I GENERAL CONCEPTS AND METHODS OF NONLINEAR OPTICS 1

Chapter 1. Electromagnetic Fields. Classical Description 3

1.1. Electromagnetic fields in vacuo 3 1.1.1. Maxwell's equations 4 1.1.2. Expansion of the radiation field in modes 4

1.1.2.1. Modes of a cavity 5 1.1.2.2. Modes of propagating waves 8

1.1.3. Radiation fields in real resonators 10 1.1.3.1. Quality of the resonator 11 1.1.3.2. Plane Fabry-Perot resonator 12

1.2. Electromagnetic fields with sources 16 1.2.1. Microscopic relations 16 1.2.2. Macroscopic relations 17 1.2.3. Effective fields and field corrections \ 23

1.3. Relationship between polarization and field strength 27 1.3.1. Basic equations 27

1.3.1.1. Linear polarization 28 1.3.1.2. Nonlinear polarization 29

1.3.2. Susceptibilities in the frequency domain 31 1.3.2.1. First-order susceptibility 32 1.3.2.2. Nonlinear optical susceptibilities 35 1.3.2.3. Second-order susceptibilities for

monochromatic fields 36 1.3.2.4. Higher-order susceptibilities for

monochromatic fields 41 1.3.2.5. Spatial symmetry of susceptibilities 45 1.3.2.6. Time-reversal symmetry 55 1.3.2.7. Overall permutation symmetry 58 1.3.2.8. Manley-Rowe relations 63

vüi CONTENTS

1.4. Wave propagation in nonlinear optical media 67 1.4.1. Fourier transformation 68 1.4.2. Monochromatic plane waves 69 1.4.3. Monochromatic light beams 77 1.4.4. Waves with slowly varying amplitudes 78 1.4.5. Interaction processes in resonators 82 1.4.6. Excitation of plane-wave modes 84

References 89

Chapter 2. The Quantized Free Radiation Field 90

2.1. The quantization of the free radiation field 91 2.1.1. General procedure 91 2.1.2. Quantization of the field expanded in plane waves 96

2.2. The photon field 100 2.2.1. The one-mode field of linearly polarized photons 100 2.2.2. The total field 104

v 2.2.3. Photons of fixed angular momentum and parity 110 2.3. Properties of typical field states 114

2.3.1. Pure states of the field 115 2.3.1.1. Photon-number states 115 2.3.1.2. Coherent states 117 2.3.1.3. Eigenstates of the electric field strength 120 2.3.1.4. Summary of the physical interpretation 124

2.3.2. Mixed states of the field 127 References 133

Chapter 3. Interaction Between Radiation and Matter 134

3.1. Foundations of the interaction between radiation and matter 135

3.1.1. The interaction operator 135 3.1.2. The determination of physically relevant

quantities 141 3.2. Concepts and representative approximation methods 143

3.2.1. The dipole approximation 143 3.2.2. The rotating-wave approximation 148

3.2.2.1. Basic concept 148 3.2.2.2. Application to two-photon processes 150

3.2.3. The use of susceptibilities in the interaction Hamiltonian 153

3.2.4. The influence of dissipative systems on optical phenomena 154

3.2.4.1. Atomic system interacting with a dissipative system 154

CONTENTS ix

3.2.4.2. Description of dissipation by ensemble averages 164

3.2.4.3. Atomic systems interacting with dissipative systems and radiation 169

3.3. Basic one-phöton processes 170 3.3.1. Emission and absorption of one photon by an

atomic system 170 3.3.2. Line-broadening effects 175

3.3.2.1. Natural line width 176 3.3.2.2. Homogeneous and inhomogeneous line

broadening 179 References 183

Chapter 4. Semiclassical Description of Nonlinear Optics 184

4.1. Relationship between polarization and field strength 185 4.2. The time-dependent material response 188 4.3. Susceptibilities in the frequency domain 191 4.4. Susceptibilities of loss-free atomic systems 193 4.5. Susceptibilities of atomic systems with losses 202 4.6. Direct evaluation of transition probabilities 214 4.7. Equation of motion for measurable quantities 219 4.8. Description of two-level systems in analogy to the

Bloch equations 228 References 235

Chapter 5. Statistical and Coherence Properties of the Radiation Field and Their Measurement 236

5.1. Photodetection on the basis of the external photoeffect 236 5.1.1. Photoelectric counting by a photocell. The

photon-counting distribution 237 5.1.2. Photoelectric counting by several photocells.

Joint probabilities „ 245 5.2. Correlation functions and coherence properties 247

5.2.1. Classical correlation functions 247 5.2.2. Quantum correlation functions and their

application 252 References 262

Chapter 6. Nonstationary Processes 263

6.1. Pulse propagation through dispersive linear optical media 264 6.2. Generation of light pulses in dispersive nonlinear optical

media 269 6.3. Pulse propagation through nonresonant nonlinear optical

media 272

CONTENTS

6.3.1. Pulse propagation through nondispersive nonlinear optical media 272

6.3.2. Pulse propagation through dispersive nonlinear optical media. Solitary pulses 275

References 278

PART II EFFECTS AND PROCESSES OF NONLINEAR OPTICS 279

Chapter 7. Nonlinear One-Photon Processes in Lasers 281

7.1. Continuously running laser processes 281 7.1.1. Nonlinear constitutive equations of the active

laser medium 282 7.1.2. Interaction of the active medium with the

resonator field 286 7.2. Influence of fluctuations on laser processes 298

7.2.1. Equations of motion containing fluctuation forces 299 7.2.2. Density-operator equation and Fokker-Planck

equation for the laser field 308 7.3. Properties of the laser output radiation 317

7.3.1. Phase noise, amplitude noise, and line width 317 7.3.2. Photon distributions and correlation functions 327

References 334

Chapter 8. Nonlinearities in Transient One-Photon Processes 335

8.1. Nonstationary semiclassical equations 335 8.2. Quasistationary excitation 339 8.3. Transient excitation of atomic systems with

negligible relaxation 343 8.3.1. Atomic response for negligible relaxation 343 8.3.2. Observation of oscillations in the

occupation-number inversion 350 8.4. Transient excitation of atomic systems with relaxation 352

8.4.1. Atomic response affected by relaxation 352 8.4.2. Observation of damped optical nutation and free

polarization decay 355 8.4.3. Photon echoes and stimulated photon echoes 362

8.5. Shaping of very short light pulses. Self-induced transparency 367

8.5.1. Area theorem for inhomogeneously broadened absorbers 368

8.5.2. Distortionless pulses 370 8.5.3! Experimental investigation of

self-induced transparency 378

CONTENTS

8.6. Shaping of light pulses 379 8.6.1. Pulse shaping by two-level systems 379 8.6.2. Nonlinear filtering by saturable absorption and

gain depletion 382 8.6.3. Combined action of nonlinear filters in passively

modelocked cw-pumped dye lasers 385 References 386

Chapter 9. Nonlinearities and Quantum Phenomena in Transient One-Photon Processes 389

9.1. Transient fluorescence of single atoms 389 9.1.1. Basic equations 389 9.1.2. Calculation of atomic correlation functions 394 9.1.3. Energy and power of fluorescence 403 9.1.4. Intensity correlation functions 410

9.2. Photon antibunching of fluorescent light 412 9.3. Three-wave mixing and light diffraction by induced

transient gratings 419 9.3.1. Energy density of fluorescence 420 9.3.2. Self-diffraction and photon echo 425

9.3.2.1. Resonant excitation of homogeneously broadened transitions 426

9.3.2.2. Nonresonant excitation of homogeneously broadened transitions 427

9.3.2.3. Inhomogeneously broadened transitions 428 9.3.2.4. Evaluation of relaxation parameters 430

9.3.3. Diffraction of probe pulses 431 9.3.3.1. Homogeneously broadened transitions 432 9.3.3.2. Inhomogeneously broadened transitions 433 9.3.3.3. Evaluation of relaxation parameters 433

9.3.4. Experimental observations 433 9.4. Superfluorescence 439

References 445

Chapter 10. Multiphoton Absorption and Emission 447

10.1. Basic phenomena 447 10.2. Transition probability of multiphoton absorption , 452

10.2.1. Semiclassical treatment 452 10.2.2. Quantum-theoretical treatment 454 10.2.3. Transition probability in solids 457

10.3. Attenuation of the electromagnetic field 462 10.3.1. The decrease of the number of photons in one

radiation mode 462

XU CONTENTS

10.3.2. Wave attenuation 469 10.3.2.1. Attenuation of coherent waves 470 10.3.2.2. Attenuation of fluctuating waves 475

10.4. Multiphoton ionization 482 10.5. Measurement of intensity correlation functions by

two-photon fluorescence 487 10.6. Two-photon emission and two-photon lasing processes 490

' 10.6.1. Basic phenomena 490 10.6.2. Self-sustained light generation in two-photon

lasers 493 10.6.3. Coherence properties of radiation generated by

two-photon emission. Nonclassical light 495 References 501

Chapter 11. Generation of Harmonics and Sum and Difference Frequencies. Parametric Amplification and Oscillation 504

11.1. Amplitude equations for two and three interacting light waves 505 11.1.1. Interaction of two light waves 505 11.1.2. Interaction of three light waves 511

11.1.2.1. Sum and difference frequencies 512 11.1.2.2. Parametric amplification 513

11.2. Quantum fundamentals of the processes 515 11.2.1. Frequency conversion 515 11.2.2. Parametric processes 519

11.3. Material parameters and applications 524 li.3.1. The model of the anharmonic oscillator and

its application 524 11.3.2. Phase matching and focusing 531 11.3.3. Mode-structure effects 537 11.3.4. Up-conversion and third-harmonic and

higher-harmonic generation 539 11.3.5. Parametric amplification, oscillation,

and fluorescence 541 11.4. Coherence properties 544

11.4.1. Coherence behavior with unaltered pump field 546 11.4.2. General treatment 548 References 551

Chapter 12. Stimulated Raman Scattering 553

12.1. Stimulated Raman scattering by polarizable molecules 554 12.1.1. Classical model for the interaction of radiation

with molecules 554

CONTENTS хш

12.1.2. Quantum description of the vibrational Raman effect 564

12.1.3. Behavior of Stokes and anti-Stokes waves in a medium 572 12.1.3.1. Amplification and generation of the

Stokes wave 572 12.1.3.2. Amplification and generation of the

anti-Stokes wave 577 12.1.4. Specific Raman scattering processes and

applications 581 12.1.4.1. Ordinary (off-resonance) vibrational

Raman effect 581 12.1.4.2. Inverse Raman effect 583 12.1.4.3. Active Raman scattering 585 12.1.4.4. Comparison of various methods 593

12.2. Stimulated Raman scattering by phonons and polaritons 593 12.2.1. Phonons and polaritons 594 12.2.2. Interaction of the external radiation field with

phonons and polaritons 602 12.2.3. Specific processes and applications 607

12.2.3.1. Amplification and generation of Stokes, phonon, and polariton waves 607

12.2.3.2. Coherence properties 613 12.2.3.3. Investigation and exploitation of

material properties 616 12.3. Stimulated Brillouin scattering 618

12.3.1. Fundamentals of thermal and stimulated Brillouin scattering 619

12.3.2. Applications 623 12.4. Spin-flip processes and stimulated Raman scattering 627

12.4.1. Fundamentals of spin-flip processes 627 12.4.2. Applications 628 References 630

i

Chapter 13. Optical Bistability 632 13.1. Intrinsic dispersive optical bistability in resonators 635 13.2. Nonlinear optical media for bistable devices 641 13.3. Transient response of bistable devices 644 13.4. Experimental studies 650

References 653

Chapter 14. Nonlinear Optical Phase Conjugation 656 14.1. Properties of phase-conjugated fields 656 14.2. Nonlinear optical mechanisms for phase conjugation 659

XIV CONTENTS

14.2.1.. Mechanisms connected with a change of the state of the medium 659

14.2.2. Mechanisms without change of the state of the medium 661

14.3. Applications 666 References 669

Appendix A. Compilation of Quantum-Theoretical Definitions and Relations 671

A.l Dirac formulation of quantum theory 671 A.1.1. States, dynamical variables, observables 671

A.l.1.1. The physical meaning of the basic quantities 672

A.l.1.2. Mathematical properties of state vectors and linear operators 674

A.1.2. Description of the physical measurement 680 A.1.3. Construction of the vector space of a

physical system 682 A.l .4. The temporal behavior of a physical system 683 A.l .5. The density operator 685 A.1.6. Aspects of quantum field theory 688

A.2. Treatment of basic physical problems 694 A.2.1. The interaction picture 694 A.2.2. Time-dependent perturbation theory 696 A.2.3. Transition probabilities and rates 697 A.2.4. Eigenvalue problem of the position operator 699 A.2.5. Occupation-number representation of

atomic systems 700 A.2.5.1. Description on the basis of boson

operators 700 A.2.5.2. Description on the basis offermion

operators , 703 References 707

General References 709

Index 711