Preface ..................................................... xiii
1 Introduction and Review ..................................... 1
1.1 Introduction to Ultrashort Laser Pulses ................. 1
1.2 Brief Review of Electromagnetics ........................ 4
1.2.1 Maxwell's Equations ............................... 4
1.2.2 The Wave Equation and Plane Waves ................. 6
1.2.3 Poynting's Vector and Power Flow .................. 8
1.3 Review of Laser Essentials ............................. 10
1.3.1 Steady-State Laser Operation ..................... 10
1.3.2 Gain and Gain Saturation in Four-Level Atoms ..... 15
1.3.3 Gaussian Beams and Transverse Laser Modes ........ 17
1.4 Introduction to Ultrashort Pulse Generation Through
Mode-Locking ........................................... 22
1.5 Fourier Series and Fourier Transforms .................. 25
1.5.1 Analytical Aspects ............................... 25
1.5.2 Computational Aspects ............................ 28
Problems ................................................... 30
2 Principles of Mode-Locking ................................. 32
2.1 Processes Involved in Mode-Locking ..................... 32
2.2 Active Mode-Locking .................................... 33
2.2.1 Time-Domain Treatment ............................ 34
2.2.2 Frequency-Domain Treatment ....................... 40
2.2.3 Variations of Active Mode-Locking ................ 43
2.3 Passive Mode-Locking Using Saturable Absorbers ......... 44
2.3.1 Saturation Model ................................. 47
2.3.2 Slow Saturable Absorber Mode-Locking ............. 50
2.3.3 Fast Saturable Absorber Mode-Locking ............. 54
2.4 Solid-State Laser Mode-Locking Using the Optical
Kerr Effect ............................................ 57
2.4.1 Nonlinear Refractive Index Changes ............... 57
2.4.2 Self-Amplitude Modulation, Self-Phase
Modulation, and Group Velocity Dispersion ........ 58
2.4.3 Additive Pulse Mode-Locking ...................... 60
2.4.4 Kerr Lens Mode-Locking ........................... 64
2.4.5 Mode-Locking Solutions ........................... 75
2.4.6 Initiation of Mode-Locking ....................... 81
Problems ................................................... 83
3 Ultrafast-pulse Measurement Methods ........................ 85
3.1 Terminology and Definitions ............................ 85
3.2 Electric Field Autocorrelation Measurements and
the Power Spectrum ..................................... 88
3.3 Electric Field Cross-Correlation Measurements and
Spectral Interferometry ................................ 91
3.3.1 Electric Field Cross-Correlation ................. 92
3.3.2 Spectral Interferometry .......................... 93
3.3.3 Application: Optical Coherence Tomography ........ 96
3.4 Intensity Correlation Measurements ..................... 99
3.4.1 Correlation Measurements Using Second-Harmonic
Generation ....................................... 99
3.4.2 Experimental Procedures ......................... 108
3.4.3 Correlation Measurements Using Two-Photon
absorption ...................................... 110
3.4.4 Higher-Order Correlation Techniques ............. 111
3.5 Chirped Pulses and Measurements in
the Time-Frequency Domain ............................. 112
3.6 Frequency-Resolved Optical Gating ..................... 118
3.6.1 Polarization-Gating FROG ........................ 119
3.6.2 Self-Diffraction FROG ........................... 122
3.6.3 Second-Harmonic-Generation FROG ................. 124
3.6.4 Frequency-Resolved Optical Gating Using
Temporal Phase Modulation ....................... 125
3.6.5 Signal Recovery from FROG Traces ................ 126
3.7 Pulse Measurements Based on Frequency Filtering ....... 130
3.7.1 Single-Slit Approaches .......................... 131
3.7.2 Double-Slit Approach ............................ 134
3.8 Self-Referencing Interferometry ....................... 135
3.8.1 Time-Domain Interferometry of Chirped Pulses .... 135
3.8.2 Self-Referencing Spectral Interferometry ........ 137
3.9 Characterization of Noise and Jitter .................. 139
Problems .................................................. 144
4 Dispersion and Dispersion Compensation .................... 147
4.1 Group Velocity Dispersion ............................. 147
4.1.1 Group Velocity Definition and General
Dispersion Relations ............................ 147
4.1.2 General Aspects of Material Dispersion .......... 151
4.2 Temporal Dispersion Based on Angular Dispersion ....... 155
4.2.1 Relation Between Angular and Temporal
Dispersion ...................................... 155
4.2.2 Angular Dispersion and Tilted Intensity
Fronts .......................................... 159
4.3 Dispersion of Grating Pairs ........................... 161
4.4 Dispersion of Prism Pairs ............................. 166
4.5 Dispersive Properties of Lenses ....................... 173
4.6 Dispersion of Mirror Structures ....................... 177
4.6.1 The Gires-Tournois Interferometer ............... 178
4.6.2 Quarter-Wave Stack High Reflectors .............. 180
4.6.3 Chirped Mirrors ................................. 182
4.7 Measurements of Group Velocity Dispersion ............. 186
4.7.1 Interferometric Methods ......................... 187
4.7.2 Frequency-Domain Intracavity Dispersion
Measurements .................................... 190
4.8 Appendix .............................................. 191
4.8.1 Frequency-Dependent Phase Due to Propagation
Through a Slab: Alternative Derivation .......... 191
4.8.2 Impedance Method for Analysis of Dielectric
Mirror Stacks ................................... 192
Problems .................................................. 195
5 Ultrafast Nonlinear Optics: Second Order .................. 198
5.1 Introduction to Nonlinear Optics ...................... 198
5.2 The Forced Wave Equation .............................. 201
5.2.1Frequency-Domain Formulation ..................... 202
5.2.2 Time-Domain Formulation ......................... 203
5.3 Summary of Continuous-Wave Second-Harmonic
Generation ............................................ 204
5.3.1 Effect of Phase Matching ........................ 207
5.3.2 Phase Matching in Birefringent Media ............ 209
5.3.3 Focusing Effects in Continuous-Wave SHG ......... 215
5.4 Second-Harmonic Generation with Pulses ................ 220
5.4.1 SHG in the Quasi-Continuous-Wave Limit .......... 220
5.4.2 Ultrashort-Pulse SHG ............................ 221
5.4.3 Quasi-Phase Matching ............................ 228
5.4.4 Effect of Group Velocity Walk-off on SHG-Based
Pulse Measurements .............................. 233
5.5 Three-Wave Interactions ............................... 237
5.5.1 Sum Frequency Generation ........................ 240
5.5.2 Difference Frequency Generation ................. 244
5.5.3 Optical Parametric Amplification ................ 245
5.6 Appendix .............................................. 253
5.6.1 Spatial Walk-off and Pulse Fronts in
Anisotropic Media ............................... 253
5.6.2 Velocity Matching in Broadband Noncollinear
Three-Wave Mixing ............................... 254
Problems .................................................. 256
6 Ultrafast Nonlinear Optics: Third Order ................... 258
6.1 Propagation Equation for Nonlinear Refractive Index
Media ................................................. 258
6.1.1 Plane Waves in Uniform Media .................... 260
6.1.2 Nonlinear Propagation in Waveguides ............. 261
6.1.3 Optical Fiber Types ............................. 264
6.2 The Nonlinear Schrodinger Equation .................... 266
6.3 Self-Phase Modulation ................................. 270
6.3.1 Dispersionless Self-Phase Modulation ............ 270
6.3.2 Dispersionless Self-Phase Modulation with
Loss ............................................ 273
6.3.3 Self-Phase Modulation with Normal Dispersion .... 274
6.3.4 Cross-Phase Modulation .......................... 275
6.4 Pulse Compression ..................................... 276
6.5 Modulational Instability .............................. 283
6.6 Solitons .............................................. 286
6.7 Higher-Order Propagation Effects ...................... 291
6.7.1 Nonlinear Envelope Equation in Uniform Media .... 292
6.7.2 Nonlinear Envelope Equation in Waveguides ....... 295
6.7.3 Delayed Nonlinear Response and the Raman
Effect .......................................... 296
6.7.4 Self-Steepening ................................. 306
6.7.5 Space-Time Focusing ............................. 308
6.8 Continuum Generation .................................. 310
Problems .................................................. 313
7 Mode-Locking: Selected Advanced Topics .................... 316
7.1 Soliton Fiber Lasers: Artificial Fast Saturable
Absorbers ............................................. 316
7.1.1 The Figure-Eight Laser .......................... 317
7.1.2 Energy Quantization ............................. 322
7.1.3 Soliton Sidebands ............................... 324
7.2 Soliton Mode-Locking: Active Modulation and Slow
Saturable Absorbers ................................... 328
7.2.1 Harmonically Mode-Locked Soliton Fiber Lasers ... 328
7.2.2 The Net Gain Window in Soliton Mode-Locking ..... 330
7.3 Stretched Pulse Mode-Locking .......................... 337
7.3.1 Stretched Pulse Mode-Locked Fiber Laser ......... 337
7.3.2 Dispersion-Managed Solitons ..................... 340
7.3.3 Theoretical Issues .............................. 342
7.4 Mode-Locked Lasers in the Few-Cycle Regime ............ 344
7.5 Mode-Locked Frequency Combs ........................... 347
7.5.1 Comb Basics ..................................... 347
7.5.2 Measurement Techniques .......................... 350
7.5.3 Stabilization of Frequency Combs ................ 354
7.5.4 Applications .................................... 356
Problems .................................................. 360
8 Manipulation of Ultrashort Pulses ......................... 362
8.1 Fourier Transform Pulse Shaping ....................... 362
8.1.1 Examples of Pulse Shaping Using Fixed Masks ..... 364
8.1.2 Programmable Pulse Shaping ...................... 369
8.1.3 Pulse-Shaping Theory ............................ 376
8.2 Other Pulse-Shaping Techniques ........................ 386
8.2.1 Direct Space-to-Time Pulse Shaping .............. 386
8.2.2 Acousto-optic Dispersive Filters ................ 390
8.3 Chirp Processing and Time Lenses ...................... 394
8.3.1 Space-Time Duality .............................. 394
8.3.2 Chirp Processing ................................ 397
8.3.3 Time Lens Processing ............................ 399
8.4 Ultrashort-Pulse Amplification ........................ 405
8.4.1 Amplification Basics ............................ 406
8.4.2 Special Issues in Femtosecond Amplifiers ........ 411
8.5 Appendix .............................................. 416
8.5.1 Fresnel Diffraction and Fourier Transform
Property of a Lens .............................. 416
8.5.2 Wave Optics Model of a Grating .................. 418
Problems .................................................. 420
9 Ultrafast Time-Resolved Spectroscopy ...................... 422
9.1 Introduction to Ultrafast Spectroscopy ................ 422
9.2 Degenerate Pump-Probe Transmission Measurements ....... 426
9.2.1 Co-polarized Fields: Scalar Treatment ........... 426
9.2.2 Vector Fields and Orientational Effects ......... 431
9.3 Nondegenerate and Spectrally Resolved Pump-Probe:
Case Studies .......................................... 439
9.3.1 Femtosecond Pump-Probe Studies of Dye
Molecules ....................................... 440
9.3.2 Femtosecond Pump-Probe Studies of GaAs .......... 444
9.4 Basic Quantum Mechanics for Coherent Short-Pulse
Spectroscopies ........................................ 451
9.4.1 Some Basic Quantum Mechanics .................... 451
9.4.2 The Density Matrix .............................. 456
9.5 Wave Packets .......................................... 460
9.5.1 Example: Semiconductor Quantum Wells ............ 461
9.5.2 Molecules ....................................... 462
9.6 Dephasing Phenomena ................................... 469
9.6.1 Linear Spectroscopies ........................... 469
9.6.2 Models of Dephasing ............................. 475
9.6.3 Measurement of Dephasing Using Transient
Gratings ........................................ 481
9.6.4 Two-Dimensional Spectroscopy .................... 494
9.7 Impulsive Stimulated Raman Scattering ................. 499
Problems .................................................. 505
10 Terahertz Time-Domain Electromagnetics ................... 507
10.1 Ultrafast Electromagnetics: Transmission Lines ...... 507
10.1.1 Photoconductive Generation and Sampling ...... 507
10.1.2 Electro-optic Sampling ....................... 513
10.2 Ultrafast Electromagnetics: Terahertz Beams ......... 516
10.2.1 Generation and Measurement of Terahertz
Pulses ....................................... 517
10.2.2 Terahertz Spectroscopy and Imaging ........... 527
Problems ................................................. 531
References .................................................. 533
Index ....................................................... 563
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