Norman Hodgson and Horst Weber

Fundamentals, Advanced Concepts and Applications

With 502 Figures

Springer

Contents

List of Symbols and Abbreviations xv

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 28 1.2.4 Two-Dimensional Optical Systems 31 1.2.5 Rotation and Tilt 34 1.2.6 The ABCD Law of Geometrical Optics 42 1.2.7 Eigensolutions and Eigenvalues 46

1.3 Optical Resonators and Ray Transfer Matrices 48

2 Wave Optics 53

2.1 Huygens' Principle and Kirchhoff Integral 53 2.2 Diffraction 57

2.2.1 Rectangular Aperture 57 2.2.2 Circular Aperture 63

2.3 Collins-Integral 67 2.3.1 One-Dimensional Optical Systems 67 2.3.2 Two-Dimensional Optical Systems 69

2.4 Collins-Integral and Vanishing Ray Matrix Elements 71 2.4.1 Imaging Condition (B=0) 71 2.4.2 Fourier Transformation (A=0) 72

2.5 Gaussian Beams 76 2.5.1 Gaussian Beams in One-Dimensional Optical Systems 76 2.5.2 Elliptical Gaussian Beams 87

2.6 Intensity Moments and Beam Propagation 92

viii Contents

2.6.1 Stigmatic and Simple Astigmatic Beams 92 2.6.2 Generalized Astigmatic Beams 98 2.6.3 Beam Quality 103

2.7 Diffraction Theory of Optical Resonators 105 2.7.1 Integral-Equation for the Electric Field Distribution 105 2.7.2 The Gaussian Beam as a Fundamental Resonator Mode . . . . 107

2.8 Diffraction Free Beams 109

3 Polarization 115

3.1 General Aspects 115 3.2 Jones Matrices 118

3.2.1 Definition 118 3.2.2 Matrices for Rotated Polarizing Optics 123 3.2.3 Combination of Several Polarizing Optics 124

3.3 Eigenstates of Polarization 128 3.4 Polarization in Optical Resonators 130

3.4.1 Eigenstates of the Roundtrip Jones Matrix 130 3.4.2 Polarization and Diffraction-Integrals 131

3.5 Depolarizers 132

Part II Basic Properties of Optical Resonators 135

4 The Fabry Perot Resonator 137

4.1 General Aspects 137 4.2 The Fabry Perot Interferometer 139

4.2.1 Passive Fabry Perot Interferometer 139 4.2.2 Applications of FPIs 145 4.2.3 Fabry Perot Interferometer with Gain - Laser Resonator . . . . 147

4.3 Optical Coatings 152 4.3.1 Coating Design Matrix Method 152 4.3.2 Quarter Wavelength Systems 157 4.3.3 Coating Methods and Materials 161

Part III Passive Open Resonators 163

5 Stable Resonators 165

5.1 General Aspects 165 5.2 Unconfined Stable Resonators 167

5.2.1 Transverse Mode Structures 168 5.2.2 Resonance Frequencies 178 5.2.3 The TEMoo-Mode 180 5.2.4 Higher Order Modes 187

Contents ix

5.2.5 Focusability and Beam Quality 194 5.3 Aperture Limited Stable Resonators 203

5.3.1 One Aperture Limited Mirror 205 5.3.2 Two Aperture Limited Mirrors 210

5.4 Msalignment Sensitivity 214 5.4.1 One Aperture Limited Mirror 216 5.3.2 Two Aperture Limited Mrrors 220

6 Resonators on the StabUity Limits 223

6.1 Resonators with gig2=l 223 6.2 Resonators with One Vanishing g-Parameter 227 6.3 The Confocal Resonator 230

7 Unstable Resonators 237

7.1 Geometric-Optical Description of Unstable Resonators 238 7.2.1 Beam Propagation 238 7.2.2 Focusability 245

7.3 Diffraction Theory 253 7.3.1 Mode Structure, Beam Quality, and Losses 253 7.3.2 Applications of Unstable Resonators 259

7.4 Misalignment Sensitivity 259 7.5 Off-Axis Unstable Resonators 265 7.6 Unstable Resonators with Homogeous Output Coupling 270 7.7 Unstable Resonators with Graded Reflectivity Mirrors 271

7.7.1 Resonator Properties 271 7.7.2 Production of VRMs 275 7.7.3 Laser Performance of VRM Unstable Resonators 278

8 Resonators with Internal Optical Elements 281

8.1 Resonators with Internal Lenses 281 8.2 Resonators with Polarizing Optics 284

8.2.1 The Twisted Mode Resonator 286 8.2.2 Resonators with Variable Output Coupling 287 8.2.3 The Pockels Cell Resonator 289 8.2.4 Resonators with Radially Birefringent Elements 291 8.2.5 Resonators with Azimuthally Birefringent Elements 293 8.2.6 Resonators with Radial-Azimuthally Birefringent Elements . 295

Part IV Open Resonators with Gain 301

9 The Active Medium 303

9.1 General Aspects 303

X Contents

9.2 Effective Length of a Resonator 304 9.3 Amplification and Efficiencies 306 9.4 The Laser Equations 310 9.5 Line Broadening and Hole Burning 317

9.5.1 Homogeneous and Inhomogeneous Line Broadening 317 9.5.2 Spatial Hole Burning 321

9.6 Spectral Gain Distribution and Frequency Pulling 322 9.7 The Spectral Linewidth of Laser Modes 325

10 Output Power of Laser Resonators 327

10.1 Output Power of Stable Resonators 327 10.1.1 Linear Resonators 327 10.1.2 Folded Resonators without Beam Overlap 336 10.1.3 Folded Resonators with Beam Overlap 337 10.1.4 Ring Resonators 342

10.2 Output Power of Unstable Resonators 344

11 Influence of Gain on Mode Structure and Loss 347

11.1 General Aspects 347 11.2 Stable Resonators 348

11.2.1 Fundamental Mode Operation 348 11.2.2 Transverse Multimode Operation 356

11.3 Unstable Resonators 359 11.3.1 Mode Structure and Loss 359 11.3.2 Optimum Extraction Efficiency 361

11.4 Mode Structure and Steady State Condition 365

12 Resonators with Variable Internal Lenses 367

12.1 General Aspects 367 12.1.1 Thermal Lensingin Solid State Lasers 367 12.1.2 Ray Transfer Matrices 369

12.2 Stable Resonators 372 12.2.1 Fundamental Mode Operation 372 12.2.2 Transverse Multimode Operation 375 12.2.3 Beam Radii, Divergence, and Beam Quality 380 12.2.4 Output Power and Beam Quality 382 12.2.5 Output Power in Fundamental Mode Operation 388 12.2.6 Spherical Aberration 390

12.3 Unstable Resonators 397 12.3.1 Beam Propagation 397 12.3.2 Positive Branch Confocal Unstable Resonators 399 12.3.3 Rod-Imaging Unstable Resonator 403 12.3.4 Near Concentric Unstable Resonator 406 12.3.5 Beam Quality and Focusing 409

Contents xi

13 Resonators with Several Active Media 413

13.1 General Aspects 413 13.2 Output Power and Efficiency 415

13.2.1 Oscillator Arrangement 415 13.2.2 Oscillator-Amplifier Arrangement 416

13.3 Multirod Solid State Lasers 417 13.3.1 The Equivalent g-Diagram 417 13.3.2 Beam Quality and Output Power 419 13.3.3 Multirod Resonators with Variable Reflectivity Mirrors . . . 422

14 Misalignment Sensitivity of the Output Power 423

14.1 General Properties 423 14.2 Stable Resonators in Multimode Operation 425

14.2.1 Without Thermal Lensing 425 14.2.2 With Thermal Lensing 429

14.3 Stable Resonators in Fundamental Mode Operation 435 14.4 Unstable Resonators 437

14.4.1 Without Thermal Lensing 437 14.4.2 With Thermal Lensing 441

15 Resonators with Internal Nonlinear Elements 445

15.1 General Aspects 445 15.2 Intracavity Second Harmonic Generation 446

15.2.1 Basic Properties of SHG 446 15.2.2 Efficiency of Intracavity Second Harmonic Generation . . . 454 15.2.3 Phase Mismatch, Axial Modes, and Conversion Efficiency 458 15.2.4 Resonator Configurations 459

15.3 Resonators with Phase-Conjugate Mirrors 462 15.3.1 General Properties of a Phase-Conjugate Mirror 462 15.3.2 Optical Resonators with a Phase-Conjugate Mirror 464 15.3.3 Phase-Conjugate Resonators using SBS 470

Part V Special Resonator Concepts 483

16 Prism Resonators 485

16.1 Porro Prism Resonator 485 16.2 Corner Cube Prism Resonator 491

17 Fourier Transform Resonators 495

17.1 Self-Filtering Unstable Resonators 495 17.2 Stable Fourier Transform Resonators 500

xii Contents

18 Hybrid Resonators 505

18.1 General Aspects 505 18.2 Unstable-Stable Resonators 506 18.3 Waveguide Resonators 507

18.3.1 Motivation 507 18.3.2 Eigenmodes of Hollow Waveguides 509 18.3.3 Properties of Waveguide Resonators 522 18.3.4 Properties of Slab Waveguide Lasers 538

19 Resonators for Annular Gain Media 543

19.1 Characteristics of Annular Gain Lasers 543 19.2 Stable Resonators with Toric Mirrors 545

19.2.1 Transverse Mode Structure 545 19.2.2 Beam Quality 547

19.3 Herriot Cell Resonators 550 19.4 Unstable Resonators 554

19.4.1 Toric Unstable Resonators 554 19.4.2 Azimuthally Unstable Resonators 556 19.4.3 Spherical Unstable Resonators 559

20 Ring Resonators 561

20.1 General Properties of Ring Resonators 561 20.2 Unstable Ring Resonators 566 20.3 Nonplanar Ring Resonators 569

21 Single Mode Resonators 571

21.1 Axial Mode Spectrum of Lasers 571 21.2 Axial Mode Selection with Intracavity Elements 573 21.3 Axial Mode Selection in Coupled Resonators 575 21.4 Resonators for Homogeneously Broadened Lasers 578

Part VI Measurement Techniques 581

22 Measurement of Laser Head Parameters 583

22.1 Measurement of Losses, Gain, and Efficiency 583 22.1.1 Findlay-Clay Analysis 583 22.1.2 Delay-Time Analysis 591 22.1.3 Measurement of Diffraction Losses 595 22.1.4 Measurement of the Saturation Intensity 597

22.2 Measurement of Thermal Lensing 598 22.2.1 Focusing of an Expanded Probe Beam 599 22.2.2 Deviation of a Collimated Probe Beam 601

Contents хш

22.2.3 Change in Laser Properties 602

23 Measurement of Laser Beam Parameters 605

23.1 Measurement of Beam Quality 605 23.1.1 The Beam Propagation Factor 605 23.1.2 ISO Standardized Methods 606 23.1.3 Measurement of Beam Waist and Far Field Divergence . . . 609 23.1.4 Beam Quality Analyzers 610 23.1.5 Determination of Beam Diameters 611 23.1.6 Beam Attenuation 613

23.2 Measurement of Polarization 614

References 619