1.1

Ultrafast Optics 1

1.1.1

High-Power Applications 1

1.1.1.1
1.1.1.2
1.1.1.3
1.1.1.4
1.1.1.5
1.1.1.6
1.1.1.7
1.1.1.8
1.1.1.9
1.1.1.10

Power, Peak Power, and Pulse Duration 1
Pulse Energy 2
Fluence 2
High-Power Lasers 2
Average Power and Repetition Rate 3
Intensity and Field Amplitude of CW Light 3
Peak Intensity and Beam Size 4
Gaussian Beams and Gaussian Pulses 5
Atomic Units 5
Nonlinear Optics and Strong Field Physics 6

1.1.2

High-Speed Imaging 7

1.1.2.1
1.1.2.2
1.1.2.3

Framing Camera 8
Streak Camera 9
Pump–Probe Technique 10

1.1.3

Timescale of Electron Dynamics: The New Frontier 11

1.1.3.1

Atomic Unit of Time 11

1.2

Attosecond Light Pulses 12

1.2.1

Mathematical Description of Attosecond Optical Pulses 13

1.2.1.1
1.2.1.2
1.2.1.3
1.2.1.4

Time Domain 13
Temporal Phase and Chirp 14
Frequency Domain 15
Time-Bandwidth Product 16

1.2.2

Propagation of Attosecond Pulse in Linear Dispersive

Media 17
1.2.2.1
1.2.2.2

Index of Refraction and Scattering Factor 17
Photoabsorption Cross Section and

Transmission 19

1.2.2.3
1.2.2.4
1.2.2.5
1.2.2.6
1.2.2.7

Gas Medium 19
Thin Film 19
Spectral Phase 20
Carrier-Envelope Phase 21
Group Velocity Dispersion and Group Delay

Dispersion 22

1.2.2.8
1.2.2.9

Pulse Broadening and Compression 23
GVD of Filters 23

1.3.1
1.3.2

Pulse Compression by Perturbative Harmonic Generation 27
High-Order Harmonic Generation 29

1.3.2.1
1.3.2.2
1.3.2.3

Attosecond Pulse Train 29
Three-Step Model 31
Singe Isolated Attosecond Pulses 32

1.3.3

Measurement of Attosecond Pulse Duration 34

1.3.3.1
1.3.3.2
1.3.3.3
1.3.3.4

Response of the Gas Photocathode 34
Momentum Streaking 34
Time to Momentum Conversion 35
Time Resolution 37

1.4

Properties of Attosecond XUV Pulses 38

1.4.1
1.4.2

Pulse Energy 38
Divergence Angle 39

1.4.2.1
1.4.2.2

XUV Mirrors at Glancing Incidences 39
Multilayer XUV Mirrors 40

1.4.3

Challenges and Opportunities in Attosecond Optics 40

2.2.1
2.2.2
2.2.3
2.2.4
2.2.5

Gaussian Beam in Free Space 49
Gaussian Beam Focusing 51
Aberration of Focusing Mirrors 52
Spherical Aberration of Focusing Lenses 53
Nonlinear Medium 54

2.2.5.1
2.2.5.2

Optical Kerr Effect 54
B Integral 54

2.2.5.3
2.2.5.4

Kerr Lens and Self Focusing 54
Optical Damage 55

2.3

Laser Pulse Propagation 56

2.3.1
2.3.2

Wavelength Bandwidth 56
Propagation in Linear Dispersive Medium 56

2.3.2.1
2.3.2.2
2.3.2.3
2.3.2.4

Sellmeier Equation 57
Second-Order Approximation 58
Group Velocity Dispersion 58
High-Order Dispersions 59

2.4

Mirrors 59

2.4.1
2.4.2

Metal Mirrors 60
Dielectric Mirrors 60

2.4.2.1
2.4.2.2
2.4.2.3

High-Energy Mirrors 60
Broadband Mirrors 61
Broadband High-Energy Mirrors 61

2.4.3

Chirped Mirrors with Negative GDD 62

Prism Pairs 62

2.5.1
2.5.2
2.5.3
2.5.4
2.5.5
2.5.6
2.5.7

Phase Delay 63
Group Delay Dispersion 64
Single Glass Slab 64
Two Slabs and Prism Pairs 65
Brewster’s Angle Configuration 66
Effects of the Second Prism 67
Double Pass Configuration 67

Grating Pairs 68

2.6.1
2.6.2
2.6.3
2.6.4
2.6.5

Phase Matching 69

Phase 70

Group Delay Dispersion 70
Optical Pulse Compressor 70
Optical Pulse Stretcher 71

Laser Pulse Propagation in Nonlinear Media 71

2.7.1
2.7.2

Self-Phase Modulation 71
Photonic Crystal Fiber 73

2.7.2.1

Highly Nonlinear Fiber 73

2.7.3

Hollow-Core Fibers 74

Femtosecond Oscillator 75

2.8.1
2.8.2

Ti:Sapphire Crystals 76
Principle of Mode Locking 76

2.8.2.1
2.8.2.2
2.8.2.3

Longitudinal Modes 76
Mode Locking 77
Pulse Picker 77

2.8.3

Kerr Lens Mode Locking 78

2.8.3.1

Stability Range of a Laser Cavity 79

Chirped Pulse Amplifiers 79

2.9.1

Configurations 79

2.9.1.1
2.9.1.2

Multipass Amplifier 79
Regenerative Amplifier 80

2.9.2

Gain Narrowing 80

2.9.2.1
2.9.2.2
2.9.2.3

Gain Cross Section 80
Gain Narrowing 81
Effects of the Seed Pulse Bandwidth 82

2.9.3

Gain Narrowing Compensation 83

2.9.3.1
2.9.3.2

Spectral Shaping 83
Optical Parametric Chirped Pulse Amplification
84

Pulse Characterization

2.10.1

FROG
2.10.1.1
2.10.1.2
2.10.1.3
2.10.1.4

84
Autocorrelators 84
FROG Trace 85
Phase Retrieval 86
Principal Component Generalized Projection

2.10.2

Multiphoton Intrapulse Interference Phase Scan

87

2.10.2.1
2.10.2.2
2.10.2.3
2.10.2.4

Setup 88
Principle 88
Experimental Approach 89
High-Order Phases 90

2.11 Few-Cycle Pulses

90

2.11.1
2.11.2

Chirped Mirror Compressor 90
Adaptive Phase Modulator 91

2.11.2.1
2.11.2.2
2.11.2.3

Zero-Dispersion Stretcher 91
Spatial Light Modulator 91

MIIPS for Compressing Pulses from Hollow-Core

Fibers 92
White-Light Chirp Compensation 93
FROG Measurements 94

2.11.2.4
2.11.2.5

3.1

Introduction 101

3.1.1

Definition of Carrier-Envelope Phase 101

3.1.1.1
3.1.1.2
3.1.1.3

Linearly Polarized Field 101
Circularly Polarized Field 103
Elliptically Polarized Field 103

3.1.2

Physics Processes Sensitive to Carrier-Envelope Phase 103

3.1.2.1
3.1.2.2

Sub-Cycle Field Strength Variation 103
Sub-Cycle Gating 104

3.2.1

Effects of Group and Phase Velocity Difference 104

3.2.1.1
3.2.1.2
3.2.1.3

Group and Phase Velocity 104
Gouy Phase and Carrier-Envelope Phase 105
Index of Refraction 106

3.2.2

Prism-Based Compressor 107

3.3

Carrier-Envelope Phase in Laser Oscillators 108

3.3.1

Carrier-Envelope Phase Offset Frequency 109

3.3.1.1
3.3.1.2

Carrier-Envelope Phase Change Rate 109
Carrier-Envelope Offset Frequency 109

3.3.2

Stabilization of Offset Frequency 111

3.3.2.1

Measuring f0 by f-to-2f Interferometers 111

3.4.1
3.4.2

Oscillator Configuration 112
f-to-2f Interferometer 113

3.4.2.1
3.4.2.2
3.4.2.3

White-Light Generation 113
Setup 114
Beat Signal 115

3.4.3

Locking the Offset Frequency 115

3.4.3.1

Phase Detector and Proportional Integral

Control 115

3.4.3.2

Stability of the Locked f0 116

3.4.4

Noise of the Interferometer 117

3.4.4.1

Error in Measuring f0 117

3.4.4.2
3.4.4.3

Interferometer Locking 118
Noise Spectrum 119

Measurement of the Carrier-Envelope Phase of Amplified Pulses 119

3.5.1

Single Shot f-to-2f Interferometry 121

3.5.1.1
3.5.1.2

Interferometer Setup 121
Fourier Transform Spectral Interferometry 122

3.5.2

Precisions of the Carrier-Envelope Phase Measurement 123

3.5.2.1

Experimental Determination of the
Carrier-Envelope Phase–Energy Coupling
Coefficient 123
Explanation of the Carrier-Envelope Phase–Energy
Coupling 125

3.5.2.2

3.5.3

Two-Step Model 126

3.5.3.1
3.5.3.2
3.5.3.3

Filamentation in Sapphire Plate 126
White-Light Generation 128
Frequency Phase of White Light, Nonlinear Phase,
and Carrier-Envelope Phase 129
Group Delay 130
Carrier-Envelope Phase Measurement Error 130

3.5.3.4
3.5.3.5

Carrier-Envelope Phase Shift in Stretchers and Compressors 132

3.6.1

Carrier-Envelope Phase Shift Introduced by Grating-Based
Compressors 132

3.6.1.1
3.6.1.2
3.6.1.3

Carrier-Envelope Phase 132
Beam Pointing 134
Grating Separation 134

3.6.2

Carrier-Envelope Phase Shift Introduced by Grating-Based
Stretcher 135

3.6.2.1

Pulse Duration 136

Stabilization of the Carrier-Envelope Phase in CPA 137

3.7.1

Using the Compressor 137

3.7.1.1
3.7.1.2

Frequency Response of the PZT Mount 138
Frequency Response of the f-to-2f Interferometer
and of the PZT 139
Carrier-Envelope Phase Locking 140

3.7.1.3

3.7.2

Using the Stretcher 140

3.7.2.1

Dependence of Carrier-Envelope Phase on the
Effective Grating Separation 142

3.7.2.2

Compensation of Slow Carrier-Envelope Phase
Drift 143
Effects of the Oscillator f-to-2f Stability 143

3.7.2.3

3.8

Controlling of the Stabilized Carrier-Envelope Phase 145

3.8.1
3.8.2

Carrier-Envelope Phase Staircase 145
Phase Sweeping 145

3.9

Carrier-Envelope Phase Measurements after Hollow-Core Fibers 146

3.9.1
3.9.2
3.9.3

Experimental Setup 147
Carrier-Envelope Phase Stability 150
Energy to Carrier-Envelope Phase Coupling
Coefficient 151

3.10.1
3.10.2

Carrier-Envelope Phase Stability 152
Carrier-Envelope Phase Error Introduced by the Zero-Dispersion
Stretcher 153
Compensate the Carrier-Envelope Phase Shift Introduced
by the 4f System 154

3.10.3

3.11 Power Locking for Improving Carrier-Envelope Phase Stability 156

3.11.1
3.11.2
3.11.3
3.11.4

Feedback Loop 156
Pockels Cell 157
Power Stability 158
Carrier-Envelope Phase Stability 158

4.1

Three-Step Model 165

4.1.1

Recombination Time 168

4.1.1.1

Graphic Solutions and Kramers–Henneberger
Frame 168

4.1.1.2

Numerical Solutions and Fitting

Functions 169

4.1.2
4.1.3
4.1.4

Return Energy 170
Long and Short Trajectories 171
Chirp of Attosecond Pulses 172

4.1.4.1
4.1.4.2
4.1.4.3
4.1.4.4

Short Trajectory 174
Long Trajectory 174
The General Case 175
High-Order Chirp 175

4.2.1

The Keldysh Theory 176

4.2.1.1
4.2.1.2

Volkov States 176
Fermi’s Golden Rule and Photoionization
Rate 177

4.2.1.3

Keldysh Parameter 178

4.2.2
4.2.3

PPT Model 180
ADK Model 183

4.2.3.1
4.2.3.2

Cycle-Averaged Rate 184
Cycle-Averaged Rate of an Elliptically
Polarized Field 184

4.2.3.3

Saturation Ionization Intensity 185

4.2.4

Attosecond Electron and Photon Pulses 185

4.2.4.1
4.2.4.2

Returning Electron Pulse 185
Attosecond Pulse Train and High-Order
Harmonics 186

Cutoff Photon Energy 186

4.3.1

Saturation Field and Intensity 187

4.3.1.1
4.3.1.2
4.3.1.3
4.3.1.4
4.3.1.5
4.3.1.6

Sech Square Pulse 188
Definition of Ionization Saturation 188
ADK Rate 189
Circularly Polarized Pulses 189
Linearly Polarized Fields 191
Saturation Intensity for Linearly Polarized

Field 192

4.3.1.7

Ionization Probability 193

4.3.2

Cutoff due to Depletion of the Ground State 194

4.3.2.1
4.3.2.2
4.3.2.3

Ionization Potential 195
Pulse Width 196
Wavelength of the Driving Laser 197

Free Electrons in Two-Color Laser Fields 199

4.4.1

Equation of Motion 199

4.4.1.1

Return Time 201

4.4.2
4.4.3

Return Energy 202
Two-Color Gating 203

Polarization Gating 204

4.5.1

Electrons in Elliptically Polarized Laser Fields 205

4.5.1.1
4.5.1.2
4.5.1.3
4.5.1.4

Laser Field 205
Equations of Motion 206
Transverse Displacement 207
Quantum Diffusion 208

4.5.2

Isolated Attosecond Pulse Generation 208

4.5.2.1
4.5.2.2
4.5.2.3
4.5.2.4
4.5.2.5
4.5.2.6

Principle of the Polarization Gating 208
Laser Field 210
Fields inside the Polarization Gate 211
Electron Trajectories 213
Polarization Gate Width 214
Optics for Creating Laser Pulse for

Polarization Gating 215
Upper Limit of Laser-Pulse Duration 217

4.5.2.7

5.1.1

Approximations 223

5.1.1.1
5.1.1.2
5.1.1.3
5.1.1.4
5.1.1.5
5.1.1.6
5.1.1.7

Dipole Radiation and Dipole Moment 223
Single Active Electron Approximation 224
Electric Dipole Approximation 225
Strong Field Approximation 225
Continuum-State Wave Function 226
Total Wave Function 226
Dipole Moment 226

5.1.2

Continuum Wave Packet 227

5.1.2.1

Analytical Approach to Solve the Schrödinger
Equation 228

5.1.2.2
5.1.2.3

Solution of the Differential Equation 229
Conservation of Canonical Momentum 230

5.1.3

Saddle-Point Approach 231

5.1.3.1

One-Dimensional Saddle Point
Approximation 232
3D Saddle-Point Method 233

5.1.3.2

5.1.4

Dipole Moment for Linearly Polarized Driving Laser 236

5.1.4.1
5.1.4.2

Laser Field 236
Momentum and Action 237

5.1.5
5.1.6

Dipole Transition Matrix Element 238
Coulomb Corrections 240

5.1.6.1
5.1.6.2
5.1.6.3

Correction to the Recombination Term 240
Correction to the Ionization Step 241
Matrix Element 241

5.2

Temporal Phase of Harmonic Pulses 242

5.2.1
5.2.2

Intrinsic Dipole Phase 243
Gaussian Analysis of the Temporal Phase 244

5.2.2.1
5.2.2.2
5.2.2.3

Laser Pulses 244
High Harmonic Pulses 245
High Harmonic Spectrum 247

5.2.3

Experimental Results 247

5.2.3.1
5.2.3.2
5.2.3.3

Using 40 fs Lasers 247
Numerical Simulation Results 248
Few-Cycle Driving Laser 248

5.3

Effects of Molecular Orbital Symmetry 250

5.3.1

Experimental Results 251

5.3.1.1

Ellipticity Control 251

5.3.1.2
5.3.1.3

High Harmonic Cutoff 252
Ellipticity Dependence 253

5.3.2

Numerical Simulations 253

5.3.2.1
5.3.2.2
5.3.2.3

Bonding Orbital and Antibonding Orbital 254
Simulation Results 256
Role of Interference 256

5.4

Polarization Gating Revisit 258

5.4.1

SFA for Polarization Gating 258

5.4.1.1
5.4.1.2

Single Atom Response 258
Propagation Effects 260

5.4.2

Results of Simulations 261

5.4.2.1

Double Attosecond Pulses Generated with

Multicycle NIR Lasers 261

5.4.2.2

Isolated Attosecond Pulse Generated with
Few-Cycle NIR Lasers 262
Effects of Carrier-Envelope Phase 265

5.4.2.3

5.5

Complete Reconstruction of Attosecond Burst 267

5.5.1

Approximations 267

5.5.1.1
5.5.1.2

Strong Field Approximation 267
Single Active-Electron Approximation 268

5.5.2

Ionization in Two-Color Field 268

5.5.2.1
5.5.2.2
5.5.2.3
5.5.2.4

XUV Field 268
Photoelectron Wave Packet 269
Effects of Dipole Matrix Elements 270
Photoelectron Wave Packet Produced by the
Two-Color Field 271
Time Delay between the Two Fields 272

5.5.2.5

5.5.3
5.5.4

Saddle Point Approximation 273
FROG-CRAB Trace 275

5.5.4.1
5.5.4.2
5.5.4.3
5.5.4.4

Electron Phase Modulator 275
FROG-CRAB Trace 275
Dipole Correction 276
Central Momentum Approximation 277

6.1

Wave-Propagation Equation 282

6.1.1

Wave Equations for the Total Fields 282

6.1.1.1
6.1.1.2

Maxwell Equations 282
Wave Equation for Electric Field 283

6.1.2

Wave Equations for High-Harmonic Fields 283

6.1.2.1

Monochromatic Driving Laser 284

6.1.3

Linearly Polarized Fields 284

6.1.3.1

Paraxial Approximation 285

Phase Matching for Plane Waves 285

6.2.1

Perfect Phase Matching in Lossless Media 286

6.2.1.1
6.2.1.2

Plasma Dispersion 287
Pressure (Plasma) Gradient Gas Target 289

6.2.2

Effect of Absorption 290

6.2.2.1

Absorption Limit 290

6.2.3
6.2.4

Maker Fringes 292
Rule of Thumb for Optimizing XUV
Photon Flux 293
Effects of Intensity Distribution in the Propagation Direction 294

6.2.5

6.2.5.1

Quasiphase Matching 296

Phase Matching for Gaussian Beams 296

6.3.1

On-Axis Phase Matching without Plasma and Gas
Dispersion 297
On-Axis Phase Matching without Neutral Gas Dispersion 299
Off-Axis Phase Matching 300

6.3.2
6.3.3

Phase Matching for Pulsed Lasers 301

6.4.1

Wave Equation 301

6.4.1.1
6.4.1.2
6.4.1.3

Beams with Axial symmetry 302
Retarded Coordinate 303
Plane Waves 303

6.4.2
6.4.3
6.4.4
6.4.5
6.4.6

Paraxial Wave Equation in the Frequency Domain 304
Carrier-Envelope Phase 305
Propagation of Few-Cycle Pulses 305
Integral Approach 307
Calculating the Electric Field in the Far-Field 309

Compensating the Chirp of Attosecond Pulses 310

6.5.1

Numerical Simulation Method 311

6.5.1.1
6.5.1.2
6.5.1.3

NIR Laser Field 311
Single-Atom Response 312
Macroscopic Attosecond Signal 313

6.5.2

Simulation Results 314

6.5.2.1
6.5.2.2
6.5.2.3
6.5.2.4
6.5.2.5
6.5.2.6

Ground-State Depletion 314
Gated XUV Spectrum 314
Modulation in the Single-Atom Spectrum 315
Comparison with the Semiclassical Results 316
Chirp of Attosecond Pulses 316
Chirp Compensation 318

Phase Matching in Double-Optical Gating 320

6.6.1
6.6.2

Principle of Double-Optical Gating 321
Major Factors 322

6.6.2.1
6.6.2.2
6.6.2.3

Intrinsic Phase of Isolated Attosecond Pulses 323
On-Axis Phase Matching 324
Pressure Gradient 326

6.6.3

Experimental Results 327

6.3.3.1
6.3.3.2

Experimental Setup 327
Gating Optics 327

6.6.4

Gas-Target Location 329

6.6.4.1
6.6.4.2

Argon Gas 329
Neon Gas 329

6.6.5

Gas Pressure 330

6.6.5.1
6.6.5.2

Argon Gas 330
Neon Gas 333

6.7

Summary 333

Problems 333
References 334

Review Articles
Phase Matching

334
334

Polarization Gating

335

Double-Optical Gating

335

Dipole Phase

336

7.1.1.1
7.1.1.2
7.1.1.3

Bessel Functions 340
Narrow Annular Aperture 342
On Axis 343

7.1.2
7.1.3

Transverse Variation 343
Field Distribution in the Propagation Direction 344

7.1.3.1

Gouy Phase 345

7.2

Detection Gas 346

7.2.1

Effects of Spin–Orbit Coupling and Inner
Shells 346
Maximum Pressure 346

7.2.2

7.3

Electron Time-of-Flight Spectrometer 349

7.3.1

Field-Free TOF 350

7.3.1.1
7.3.1.2
7.3.1.3

Energy Resolution 351
Retarding Potential 351
Time-Resolution Measurement 352

7.3.2

Magnetic Bottle 353

7.3.2.1
7.3.2.2
7.3.2.3
7.3.2.4
7.3.2.5
7.3.2.6
7.3.2.7
7.3.2.8
7.3.2.9
7.3.2.10

Parallelization of the Trajectories 353
Acceptance Angle 355
Energy Resolution 355
Adiabaticity Parameter 355
Transition Region 356
Transverse Magnification 356
Overall Considerations 356
Construction of the Magnetic Bottle 357
Experimental Energy Resolution 358
Retarding Potential 358

7.3.3

Position-Sensitive Detector 358

7.3.3.1

Experimental Determination of the Energy

Resolution 361

7.3.3.2
7.3.3.3

Setup 361
Energy Resolution Calibration 362

7.3.4

Velocity Map Imaging 363

7.4

Measurement of Temporal Width of a Single Harmonic Pulse 364

7.4.1

Sidebands 366

7.5.1

Reconstruction of Attosecond Beating by Interference of
Two-Photon Transition Experiments 368

7.5.1.1

Spectral Phase and Harmonic Emission Time 370

7.5.2

Transition-Matrix Element in XUV Field 371

7.5.2.1
7.5.2.2
7.5.2.3
7.5.2.4
7.5.2.5

Fermi’s Golden Rule 371
First-Order Approximation 371
Dipole Approximation 372
Absorption Cross Section 372
Neon Atom 372

7.5.3

Transitions in XUV and IR Fields 373

7.5.3.1

Attosecond Pulse Train Generated with One-Color
Driving Field 373

7.5.3.2
7.5.3.3

Sideband Intensity Oscillation 374
Two-Color Driving Field 375

7.6

Complete Reconstruction of Attosecond Bursts 376

7.6.1

CRAB Trace 377

7.6.1.1
7.6.1.2

Temporal-Phase Gate 379
Reconstruction Algorithm 379

7.6.2

Linearly Polarized Dressing Laser Field 380

7.6.2.1
7.6.2.2
7.6.2.3
7.6.2.4
7.6.2.5

Energy Shift 381
Phase and Laser Field 381
Ponderomotive Shift 381
NIR Laser Intensity 382
Observation Angle 383

7.6.3

Attosecond Pulse Train 384

7.6.3.1
7.6.3.2

Tm ¼ 2Ttr 384
Attosecond Pulses near the Cutoff Region 384

7.6.4

Perturbative Regime of CRAB 384

7.6.4.1

Attosecond Pulse Train Generated with One-Color
Lasers 385
Attosecond Pulse Train Generated with Two-Color
Lasers 386

7.6.4.2