List of Symbols .............................................. xvii
Introduction .................................................... 1
Part I The Electromagnetic Field ................................ 5
1. Geometrical Optics ........................................... 7
1.1. General Aspects ............................................ 7
1.2. Ray Transfer Matrices ...................................... 9
1.2.1. One-Dimensional Optical Systems ..................... 9
1.2.2. Matrix Elements and Liouville's Theorem ............ 19
1.2.3. Misaligned Optical Systems ......................... 29
1.2.4. Two-Dimensional Optical Systems .................... 34
1.2.5. Rotation and Tilt .................................. 38
1.2.6. The ABCD Law for the Radius of Curvature ........... 47
1.2.7. Eigensolutions and Eigenvalues ..................... 50
1.3. Optical Resonators and Ray Transfer Matrices .............. 52
2. Wave Optics ................................................. 57
2.1. Huygens' Principle and Kirchhoff-Integral ................. 57
2.2. Diffraction ............................................... 61
2.2.1. Rectangular Aperture ............................... 61
2.2.2. Circular Aperture .................................. 67
2.3. Collins-Integral .......................................... 71
2.3.1. One-Dimensional Optical Systems .................... 71
2.3.2. Two-Dimensional Optical Systems .................... 73
2.4. Collins-Integral and Vanishing Ray Matrix Elements ........ 75
2.4.1. Imaging Condition (B=0) ............................ 75
2.4.2. Fourier Transformation (A=0) ....................... 76
2.5. Gaussian Beams ............................................ 80
2.5.1. Gaussian Beams in One-Dimensional Optical
Systems ............................................ 80
2.5.2. Elliptical Gaussian Beams .......................... 91
2.6. Intensity Moments and Beam Propagation .................... 96
2.6.1. Stigmatic and Simple Astigmatic Beams .............. 96
2.6.2. Generalized Astigmatic Beams ...................... 102
2.6.3. Beam Quality ...................................... 107
2.7. Coherence ................................................ 110
2.7.1. Temporal Coherence ................................ 110
2.7.2. Spatial Coherence ................................. 117
2.8. Diffraction Theory of Optical Resonators ................. 134
2.8.1. Integral-Equation for the Electric Field
Distribution ...................................... 134
2.8.2. The Gaussian Beam as a Fundamental Resonator
Mode .............................................. 136
2.9. Plane Wave Presentation of Diffraction ................... 138
2.10.Diffraction-Free Beams ................................... 143
2.11.Beam Propagation in Anisotropic Crystals ................. 150
3 Polarization ................................................ 153
3.1. General Aspects .......................................... 153
3.2. Jones Matrices ........................................... 156
3.2.1. Definition ........................................ 156
3.2.2. Matrices for Rotated Polarizing Optics ............ 161
3.2.3. Combination of Several Polarizing Optics .......... 163
3.3. Eigenstates of Polarization .............................. 167
3.4. Polarization in Optical Resonators ....................... 169
3.4.1. Eigenstates of the Roundtrip Jones Matrix ......... 169
3.4.2. Polarization and Diffraction-Integrals ............ 170
3.5. Depolarizers ............................................. 171
3.6. Momentum and Angular Momentum of a Beam .................. 173
3.6.2. The Poynting Vector of Structured-Beams ........... 176
3.6.3. Angular Momentum .................................. 178
Part II Basic Properties of Optical Resonators ................ 187
4 The Fabry Perot Interferometer .............................. 189
4.1. General Aspects .......................................... 189
4.2. The Fabry Perot Interferometer ........................... 191
4.2.1. Passive Fabry Perot Interferometer ................ 191
4.2.2. Applications of FPIs .............................. 197
4.2.3. Fabry Perot Interferometer with Gain - The
Laser Resonator ................................... 199
4.3. Optical Coatings ......................................... 204
4.3.1. Coating Design Matrix Method ...................... 204
4.3.2. Quarter Wavelength Systems ........................ 209
4.3.3. Coating Methods and Materials ..................... 214
4.3.1. Part HI Passive Open Resonators ................... 217
5. Stable Resonators .......................................... 219
5.1. General Aspects .......................................... 219
5.2. Unconfined Stable Resonators ............................. 221
5.2.1. Transverse Mode Structures ........................ 222
5.2.2. Resonance Frequencies ............................. 233
5.2.3. The TEMoo-Mode .................................... 235
5.2.4. Higher Order Modes ................................ 242
5.2.5. Focusability and Beam Quality ..................... 250
5.3. Aperture Limited Stable Resonators ....................... 259
5.3.1. One Aperture Limited Mirror ....................... 261
5.3.2. Two Aperture Limited Mirrors ...................... 267
5.4. Misalignment Sensitivity ................................. 271
5.4.1. One Aperture Limited Mirror ....................... 273
5.3.1. Two Aperture Limited Mirrors ...................... 277
6. Resonators on the Stability Limits ......................... 281
6.1. Resonators with g,g2=l ................................... 281
6.2. Resonators with One Vanishing g-Parameter ................ 285
6.3. The Confocal Resonator ................................... 287
7. Unstable Resonators ........................................ 295
7.1. General Aspects .......................................... 295
7.2. Geometric-Optical Description of Unstable Resonators ..... 296
7.2.1. Beam Propagation .................................. 296
7.2.2. Focusability ...................................... 303
7.3. Diffraction Theory ....................................... 311
7.3.1. Mode Structure, Beam Quality, and Losses .......... 311
7.3.2. Applications of Unstable Resonators ............... 317
7.4. Misalignment Sensitivity ................................. 317
7.6. Unstable Resonators with Homogenous Output Coupling ...... 328
7.7. Unstable Resonators with Graded Reflectivity Mirrors ..... 330
7.7.1. Resonator Properties .............................. 330
7.7.2. Production of VRMs ................................ 334
7.7.3. Laser Performance of VRM Unstable Resonators ...... 337
8. Resonators with Internal Lenses ............................ 341
8.1. Resonators with Internal Lenses .......................... 341
8.2. Resonators with Polarizing Optics ........................ 344
8.2.1. The Twisted Mode Resonator ........................ 346
8.2.2. Resonators with Variable Output Coupling .......... 347
8.2.3. The Pockels Cell Resonator ........................ 349
8.2.4. Resonators with Radially Birefringent Elements .... 351
8.2.5. Resonators with Azimuthally Birefringent
Elements .......................................... 353
8.2.6. Resonators with Radial-Azimuthally
Birefringent Elements ............................. 355
8.2.7. Compensation of Radial-Azimuthal Birefringence .... 358
Part IV Open Resonators with Gain ............................. 365
9 The Active Medium ........................................... 367
9.1. General Aspects .......................................... 367
9.2. Effective Length of a Resonator .......................... 368
9.3. Amplification and Efficiencies ........................... 370
9.4. The Laser Equations ...................................... 374
9.5. Line Broadening and Hole Burning ......................... 382
9.5.1. Homogeneous and Inhomogeneous Line Broadening ..... 382
9.5.2. Spatial Hole Burning .............................. 386
9.6. Spectral Gain Distribution and Frequency Pulling ......... 388
9.7. The Spectral Lincwidth of Laser Modes .................... 391
10. Output Power of Laser Resonators .......................... 393
10.1.Output Power of Stable Resonators ........................ 393
10.1.1.Linear Resonators ................................. 393
10.1.2.Folded Resonators without Beam Overlap ............ 402
10.1.3.Folded Resonators with Beam Overlap ............... 403
10.1.4.Ring Resonators ................................... 408
10.2.Output Power of Unstable Resonators ...................... 410
11.Influence of Gain on Mode Structure and Loss ............... 413
11.1.General Aspects .......................................... 413
11.2.Stable Resonators ........................................ 414
11.2.1.Fundamental Mode Operation ........................ 414
11.2.2.Transverse Multimode Operation .................... 422
11.3.Unstable Resonators ...................................... 425
11.3.1 Mode Structure and Loss ........................... 425
11.3.2 Optimum Extraction Efficiency ..................... 427
11.4.Mode Structure and Steady State Condition ................ 431
12.Q-switched Laser Resonators ................................ 433
12.1.General Aspects .......................................... 433
12.2.Rate Equations for Q-switching ........................... 436
12.2.1.Inversion Densities ............................... 436
12.2.2.Energy, Pulse Duration and Peak Power ............. 438
12.3.Optimum Output Coupling .................................. 441
12.4.Repetitive Q-switching .............................. 443
13.Resonators with Variable Internal Lenses ................... 451
13.1.General Aspects .......................................... 451
13.1.1.Thermal Lensing in Solid State Lasers ............. 451
13.1.2.Ray Transfer Matrices ............................. 454
13.2.Stable Resonators ........................................ 457
13.2.1.Fundamental Mode Operation ........................ 457
13.2.2.Transverse Multimode Operation .................... 460
13.2.3.Beam Radii, Divergence, and Beam Quality .......... 465
13.2.4.Output Power and Beam Quality ..................... 467
13.2.5.Output Power in Fundamental Mode Operation ........ 473
13.2.6.Spherical Aberration .............................. 476
13.3.Unstable Resonators ...................................... 488
13.3.1.Beam Propagation .................................. 488
13.3.2.Positive Branch Confocal Unstable Resonators ...... 490
13.3.3.Rod-Imaging Unstable Resonator .................... 494
13.3.4.Near Concentric Unstable Resonator ................ 497
13.3.5.Beam Quality and Focusing ......................... 500
14.Resonators with Several Active Media ....................... 505
14.1.General Aspects .......................................... 505
14.2.Output Power and Efficiency .............................. 507
14.2.1.Oscillator Arrangement ............................ 507
14.2.2.Oscillator-Amplifier Arrangement .................. 508
14.3.Multirod Solid State Lasers .............................. 509
14.3.1.The Equivalent g-Diagram .......................... 509
14.3.2.Beam Quality and Output Power ..................... 511
14.3.3.Multirod Resonators with Variable
Reflectivity Mirrors .............................. 515
15.Misalignment Sensitivity of Output Power ................... 517
15.1.General Properties ....................................... 517
15.2.Stable Resonators in Multimode Operation ................. 519
15.2.1.Without Thermal Lensing ........................... 519
15.2.2.With Thermal Lensing .............................. 523
15.3.Stable Resonators in Fundamental Mode Operation .......... 529
15.4.Unstable Resonators ...................................... 531
15.4.1.Without Thermal Lensing ........................... 531
15.4.2.With Thermal Lensing .............................. 535
16.Resonators with Nonlinear Elements ......................... 539
16.1.General Aspects .......................................... 539
16.2.Intracavity Second Harmonic Generation ................... 540
16.2.1.Basic Properties of SIIG .......................... 540
16.2.2.Ffficiency of Intracavity Second Harmonic
Generation ........................................ 548
16.2.3.Phase Mismatch, Axial Modes, and Conversion
Efficiency ........................................ 552
16.2.1.xiv.Contents
16.2.4 Resonator Configurations .......................... 554
16.3.Intracavity Third Harmonic Generation .................... 558
16.3.1.General Properties of Third Harmonic Generation ... 558
16.3.2.Properties of Intracavity Third Harmonic
Generation ........................................ 563
16.4.Resonators with Phase-Conjugate Mirrors .................. 566
16.4.1.General Properties of a Phase-Conjugate Mirror .... 566
16.4.2.Optical Resonators with a Phase-Conjugate
Mirror ............................................ 567
16.4.3.Phase-Conjugate Resonators using SBS .............. 574
Pan V Special Resonator Concepts .............................. 583
17.Prism Resonators ........................................... 585
17.1.Porro Prism Resonator .................................... 585
17.2.Corner Cube Prism Resonator .............................. 591
18.Fourier Transform Resonators ............................... 595
18.1.Self-Filtering Unstable Resonators ....................... 595
18.2.Stable Fourier Transform Resonators ...................... 600
19.Hybrid Resonators .......................................... 605
19.1.General Aspects .......................................... 605
19.2.Unstable-Stable Resonators ............................... 606
19.3.Waveguide Resonators ..................................... 607
19.3.1.Motivation ........................................ 607
19.3.2.Eigenmodes of Hollow Waveguides ................... 609
19.3.3.Properties of Waveguide Resonators ................ 622
19.3.4.Properties of Slab Waveguide Lasers ............... 638
19.3.5.Solid State Waveguide Lasers ...................... 643
20.Resonators for Annular Gain Media .......................... 647
20.1.Characteristics of Annular Gain Lasers ................... 647
20.2.Stable Resonators with Toric Mirrors ..................... 649
20.2.1.Transverse Mode Structure ......................... 649
20.2.2.Beam Quality ...................................... 651
20.3.Herriot Cell Resonators .................................. 654
20.4.Unstable Resonators ...................................... 658
20.4.1.Toric Unstable Resonators ......................... 658
20.4.2.Azimuthally Unstable Resonators ................... 660
20.4.3.Spherical Unstable Resonators ..................... 663
21.Ring Resonators ............................................ 665
21.1.General Properties of Ring Resonators .................... 665
21.2.Unstable Ring Resonators ................................. 670
21.3.Nonplanar Ring Resonators ................................ 673
22 Single Mode Laser Resonators ............................... 67S
22.1.Axial Mode Spectrum of Lasers ............................ 675
22.2.Axial Mode Selection with Intracavity Elements ........... 677
22.3.Axial Mode Selection in Coupled Resonators ............... 679
22.4.Resonators for Homogeneously Broadened Lasers ............ 682
22.5.Micro-optical Resonators ................................. 684
Part VI Measurement Techniques ................................ 687
23.Measurement of Laser Parameters ............................ 689
23.1.Measurement of Losses, Gain, and Efficiency .............. 689
23.1.1.Findlay-Clay Analysis ............................. 689
23.1.2.Delay-Time Analysis ............................... 697
23.1.3.Measurement of Diffraction Losses ................. 701
23.1.4.Measurement of the Saturation Intensity ........... 703
23.2.Measurement of Thermal Lensing ........................... 704
23.2.1.Focusing of an Expanded Probe Beam ................ 705
23.2.2.Deviation of a Collimated Probe Beam .............. 707
23.2.3.Change in Laser Properties ........................ 708
24.Measurement of Laser Beam Parameters ....................... 713
24.1.Measurement of Beam Quality .............................. 713
24.1.1.The Beam Propagation Factor ....................... 713
24.1.2.ISO Standardized Methods .......................... 715
24.1.3.Beam Attenuation .................................. 722
24.1.4.Beam Quality Analyzers ............................ 722
24.2.Measurement of Polarization .............................. 724
24.3.Measurement of the Phase ................................. 727
References .................................................... 731
Index ......................................................... 787
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