Optical Resonators Fundamentals, Advanced Concepts and Applications

Springer-Verlag London Ltd.

Norman Hodgson and Horst Weber

Optical Resonators Fundamentals, Advanced Concepts and Applications

With 502 Figures

, Springer

Dr. rer. nat Norman Hodgson Carl Zeiss Inc., Humphrey Instruments, 2992 Alvarado Street, San Leandro, CA 94577-0700, USA

Prof. Dr. Ing. Horst Weber Optisches Institut, Technische Universität Berlin, Strasse des 17 Juni 135, 10623 Berlin, Germany

ISBN 978-1-4471-3597-5 ISBN 978-1-4471-3595-1 (eBook) DOI 10.1007/978-1-4471-3595-1

British Library Cataloguing in Publication Data Hodgson, Norman

Optical resonators : fundamentals, advanced concepts and appl ications l.Optical resonance 2.Lasers l.Title II.Weber, Horst

Library ofCongress Cataloging-in-Publication Data A catalog reeord for this book is available from the Library ofCongress

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© Springer-Verlag London 1997

Originally published by Springer-Verlag London Limited in 1997. Softcover reprint of the hardcover 1 st edition 1997

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Preface

Since its first demonstration in 1960, the laser has found widespread application in diverse areas induding medicine, materials processing, optical communications and information technology. The number of engineers and scientists working on lasers or in laser related fields is continuously increasing as new applications for this exciting technology are being discovered This also means that more and more people need to gain a detailed knowledge of lasers and their characteristics.

The basic understanding of the properties of lasers and their radiation requires knowledge ofthe physics of optical resonators. The laser beam characteristics as well as efficiency and sensitivity against misalignment are determined mainly by the resonator. Despite this important role optical resonators play in laser engineering, most publications treat them either on a too basic and incomplete level or in the form of a theoretical presentation that is only useful to academics. The result is that very often an engineer or physicist confronted with a laser resonator problem will have difficulties finding information in scientific publications unless he is able to derive his own equations or successfully link the publication's results to his own unique problem.

It is for this reason that we decided to write this overview on optical resonators which covers basics as well as the latest research results. Although the emphasis was put on application and laser engineering problems, the book should also satisfy readers seeking a more thorough background in the field. The first part, entitled "The Electromagnetic Field", provides the reader with the theoretical background necessary for the mathematical description of resonators. We tried to keep the mathematical level as low as possible, e.g. the Kirchhoff Integral is derived in an empirical way instead of using the common approach of applying Green's theorem to the wave equation. Ray transfer matrices in geometrical optics as well as basic and advanced concepts in diffraction theory and beam propagation are presented here. However, it is not necessary to work oneself through this part to make use of the rest of the book. All succeeding parts can be used without having read the theoretical part. But the reader who seeks a better understanding of the derivation and applicability of the presented equations will get help here. Anyone new to the field of lasers and laser resonators should certainly go through the theory part to get familiar with the general mathematical concepts of optics.

The outline of the book was chosen such that the subject matter becomes more specialized with proceeding chapters. We will start in Part TI with the Fabry-Perot Interferometer to discuss the basic resonator properties such as loss, gain, threshold, and line width. The following part will deal with passive (no active medium)

vi Preface

resonators. Here we deal with linear stable and unstable resonators which represent probably 95% of all resonators currently used in lasers. Leaving the active medium out of the treatment is the classical approach to the subject since the gain generally only perturbs the physical properties ofthe resonator rather than completely changing them. The influence of the medium on the resonator properties will be discussed in Part IV. This part also reviews the physics of laser emission and presents output power calculation models as well as the effects of gain on the mode structure.

A collection of special resonator concepts is presented in Part V. These concepts are either only used in a limited number of applications or laser designs, or might play an important role in the near future. Resonator schemes such as prism resonators, Fourier transform resonators, hybrid resonators, and resonators for annular gain media fall into this category. We also included the ring resonator into this part although some readers might argue that it deserves its own part since it is a widely used scheme and probably more important than any other resonator presented here.

A collection of major measurement techniques is given in Part VI. This will help the practicing engineer to make a detailed analysis ofhis laser system. Among others, techniques for measuring gain, losses, and beam quality are invaluable for anyone designing and working with laser systems.

A detailed reference list will help the reader to get more information on a preferred subject. Wehave included the titles of the publications as a help, and publications which give a good review or are a must to read are referred to in the text. We certainly do not claim completeness but to the best of our knowledge we have covered as many publications as possible. The references are listed in their chronological order to give the reader a feeling for the historical development in the specific area.

We hope that this monograph will help you to get more insight into optical resonators and assist you in analyzing and solving the problems you are facing as a laser engineer or physicist. We also hope that, after having worked with this book, you will love resonators and lasers as much as we do!!

We are gratefully indebted to Douglas J. Golding of Cogent Light Technologies, Inc. and Dr. Christopher L. Petersen of Carl Zeiss, Inc. for helping to improve the presentation in all parts and for checking the derivations and equations. Wholehearted thanks are addressed to Dr. William L. Nighan of Spectra Physics, and Prof. Dr. Ralf Menzel of the University Potsdam, Germany, for many helpful discussions on intracavity second harmonic generation and phase-conjugate resonators, and to Herbert Gross of Carl Zeiss überkochen, Germany, for sharing his knowledge on waveguide resonators. We also wish to thank Ingeborg Woll scheid for drafting the majority of the figures, and Kathleen M. MilIar of Humphrey Instruments, Inc. for taking time from her busy schedule to proofread the final manuscript.

Our special thanks are due to Imke Mowbray, Christopher Greenwell, and Nicolas Pinfield of Springer-Verlag London Ltd for their support and assistance in preparing this book

September 1996 üakland, CA Berlin, Germany

Dr. rer. nat. Norman Hodgson Prof. Dr. Ing. Horst Weber

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=O) . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.4.2 Fourier Transformation (A=O) ....................... 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 ofSeveral Polarizing Optics .............. 124

3.3 Eigenstates ofPolarization .............................. 128 3.4 Polarization in Optical Resonators ........................ 130

3.4.1 Eigenstates ofthe 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 Tbe 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 ofFPIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Misalignment Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 5.4.1 One Aperture Limited Mirror . . . . . . . . . . . . . . . . . . . . . . . . 216 5.3.2 Two Aperture Limited Mirrors . . . . . . . . . . . . . . . . . . . . . . . 220

6 Resonators on tbe Stability Limits ....................... 223

6.1 Resonators with gl&=l .................................. 223 6.2 Resonators with One Vanishing g-Parameter . . . . . . . . . . . . . . . . . . 227 6.3 Tbe 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 ofUnstable 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 ofVRMs .............................. 275 7.7.3 Laser Performance of VRM Unstable Resonators. . . . . . . . . 278

8 Resonators witb 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 Tbe Pockels Cell Resonator. . . . . . . . . . . . . . . . . . . . . . . . . 289 8.2.4 Resonators with Radially Birefringent Elements. . . . . . . . . . 291 8.2.5 Resonators with Azimutbally Birefringent Elements ...... 293 8.2.6 Resonators with Radial-Azimuthally Birefringent Elements . 295

Part IV Open Resonators witb Gain . . . . . . . . . . . . . . . . . . . . . . . . . . 301

9 Tbe 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 ofUnstable Resonators. . . . . . . . . . . . . . . . . . . . . . 344

11 Inßuence 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 Lensing in 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 Aspeets ...................................... 413 13.2 Output Power and Effieieney. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

13.2.1 Oseillator Arrangement ........................... 415 13.2.2 Oseillator-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 Refleetivity 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 Aspeets ...................................... 445 15.2 Intraeavity Seeond Harmonie Generation ................... 446

15.2.1 Basic Properties of SHG . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 15.2.2 Effieieney ofIntraeavity Seeond Harmonie Generation ... 454 15.2.3 Phase Mismateh, Axial Modes, and Conversion Effieieney 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 Optieal 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

xü Contents

18 Hybrid Resonators 505

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

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

19 Resonators for Annular Gain Media ..................... 543

19.1 Charaeteristies of Annular Gain Lasers . . . . . . . . . . . . . . . . . . . . . 543 19.2 Stable Resonators with Torie Mirrors ...................... 545

19.2.1 Transverse Mode Strueture . . . . . . . . . . . . . . . . . . . . . . . . . 545 19.2.2 Beam Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

19.3 Herriot Cell Resonators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 19.4 Unstable Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

19.4.1 Torie Unstable Resonators .. . . . . . . . . . . . . . . . . . . . . . . . 554 19.4.2 Azimuthally Unstable Resonators. . . . . . . . . . . . . . . . . . . . 556 19.4.3 Spherieal Unstable Resonators . . . . . . . . . . . . . . . . . . . . . . 559

20 Ring Resonators ..................................... 561

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

21 Single Mode Resonators ..... . . . . . . . . . . . . . . . . . . . . . . . . . . 571

21.1 Axial Mode Speetrum ofLasers .......................... 571 21.2 Axial Mode Seleetion with Intraeavity Elements. . . . . . . . . . . . . . 573 21.3 Axial Mode Seleetion in Coupled Resonators. . . . . . . . . . . . . . . . 575 21.4 Resonators for Homogeneously Broadened Lasers ............ 578

Part VI Measurement Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

22 Measurement ofLaser Head Parameters. . . . . . . . . . . . . . . . . . 583

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

22.2 Measurement ofThermal Lensing . . . . . . . . . . . . . . . . . . . . . . . . . 598 22.2.1 F oeusing of an Expanded Probe Beam . . . . . . . . . . . . . . . . 599 22.2.2 Deviation of a Collimated Probe Beam . . . . . . . . . . . . . . . . 601

Contents xiii

22.2.3 Change in Laser Properties . . . . . . . . . . . . . . . . . . . . . . . . . 602

23 Measurement of Laser Beam Parameters ................. 605

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

23.2 Measurement ofPolarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614

References ............................... . . . . . . . . . . . . . . . 619

Index ................................................. . 653

List of Symbols and Abbreviations

Abbreviations

ADP AD*P AIP Al Al20 3 Ar Au Banana BBO BeO BPP C2Cl3F3 C2F6 cc CDA CD*A CCl4 CO2 Cr CS2 Cu cw ccw FPI FR FWHM GGG GRM GSGG H20 ReNe HfD2 HSURIA

ICSHG IEEE

ISO KDA KDP KD*P KrF KTP

ammonium dihydrogen phosphate deuterated ammonium dihydrogen phosphate American Institute of Physics aluminum alumina, aluminum oxide argon gold barium sodium niobate beta barium borate beryllia, beryllium oxide beam parameter product freon 113 hexafluorethane complex conjugate cesium dihydrogen arsenate deuterated cesium dihydrogen arsenate tetra carbon chloride carbon dioxide chromium carbon disulfide copper continuous wave, clockwise counterclockwise Fabry Perot Interferometer F araday rotator full width half maximum gadolinium gallium gamet graded reflectivity mirror gadolinium scandium gallium gamet water helium neon hafnium dioxide half symmetric unstable resonator with intra-cavity axicon intracavity second harmonic generation The Institute of Electrical and Electronics Engineers, Inc. International Standardization Organization pottasium dihydrogen arsenate pottasium dihydrogen phosphate deuterated potassium dihydrogen phosphate krypton fluoride potassium titanyl phosphate

xvi

LAP LBO LH LiNb03 LiSAF LSB MgF2 N~AlF6 NBUR NCUR Nd OC OSA PBUR POM QR RH RIUR SBS SRS SF6 SFUR Si02 SPIE

T~05 Ti02 TFR VRM XeCI XeF YAG YAP YLF YV04 ZnS Zr02

Symbols

List of Symbols and Abbreviations

L-arginine phosphate lithium triborate left handed lithium niobate lithium scandium fluoride lanthanum scandium borate magnesium fluoride cryolite negative branch unstable resonator near concentric unstable resonator neodyrnium output coupler Optical Society of America positive branch unstable resonator 3-methyl-4-nitropyridine-l-oxide quartz rotator right handed rod imaging unstable resonator stimulated Brillouin scattering stimulated Raman scattering sulfur hexafluoride self-filtering unstable resonator silicon dioxide Society of Photo-Optical Instrumentation Engineers tantalum pentoxide titanium dioxide tightly folded resonator variable reflectivity mirror xenon chloride xenon fluoride yttrium aluminum gamet yttrium aluminum perovskite yttrium lithium fluoride yttrium vanadium oxide (vanadate) zinc sulphide zirconium dioxide

aperture width, aperture radius lateral shift of probe beam inner radius of annular gain medium cross sectional area ray transfer matrix element 2x2 ray transfer submatrix cross sectional area of laser beam cross sectional area of beam in active medium

List of Symbols and Abbreviations

A.m b b b b bi B B B_ c Co

cpn C C C Cp d,cJo,dl>rlz,dJ ,d4A d

cJo d,dJ,rlz

~ D D Dr D. D e E, Eo, E(x,y), E(r,cl))

~ E, E(x,y) f, fl> f2 f f(v) F gJ,& g gJ* ,g2* &, &(v) &~

Uth &I G G Go h h,hl>h2 H H...

nonnalization constant, waveguide modes aperture height radius of laser rod outer radius of annular gain medium proportionality factor nonnalized coefficients ray transfer matrix element 2x2 ray transfer submatrix nonnalization constant, waveguide modes speed of light in homogeneous medium speed oflight in vacuum (=3x108 m/s) mode expansion coefficient proportionality constant ray transfer matrix element 2x2 ray transfer submatrix mode expansion vector, waveguide beam diameter wall thickness of annular medium center thickness ofbirefringent element distances mode expansion coefficient ray transfer matrix element refractive power

xvü

refractive power for radially polarized light refractive power for azimuthally polarized light 2x2 ray transfer submatrix propagation direction unit vector electric field (scalar) far field electric field (vector) focallength repetition rate line shape function finesse of FPI g-parameters of resonator mirrors 1,2 g-parameter of resonator mirror g-parameters with intemallenses small-signal gain coefficient small-signal gain small-signal gain at the laser threshold gain coefficient for SBS gain factor equivalent g-parameter small-signal gain factor Planck constant (=6.626xl0-J4 Js) distance from lens surface to principal plane magnetic field (amplitude) Hermite polynomial of order m

XVlll List of Symbols and Abbreviations

H I,r-,J-,Io I(z) Is I SE I J, k k k,ko k K U ~

~zz L Leff Li Lo LOl>L02 Lop!

L* L' Lp, m m m M,Mo,MI>M2 ~,~.,~y M MP n n, 00, nl> n2 nA Il; ns n.(r) ne(r) N No Neff Ncq p p p P, PI> P2 P P

magnetie field (veetor) intensity of eleetromagnetie field intensity of eleetromagnetie field at eoordinate z saturation intensity intensity of spontaneous emission unity matrix Bessel funetion of order Q real number Boltzmann eonstant (=1.381xl0·23 JIK) wavenumber wave veetor diffraetion integral operator length azimuthal index for eireularly symmetrie modes geometrieallength of optieal axis in zig-zag slab length effeetive resonator length distanee geometrieal resonator length distanee from mirror to beam waist optieal resonator length effeetive resonator length with intemallenses effeetive waveguide length Laguerre polynomial of order p,Q mode number (integer) mass slope in Findlay-Clay diagram magnifieation beam propagation faetor ray transfer matrix Jones matrix mode number (integer) index of refraetion index of refraetion of air index of refraetion of eoating i index of refraetion of substrate index of refraetion for radial polarization index of refraetion for azimuthal polarization number of folds in folded resonators density of atomslionslmoleeules effeetive Fresnel number equivalent Fresnel number integer radial index for eireularly symmetrie modes gas pressure power degree of polarization second order moment matrix

List of Symbols and Abbreviations

Pj Pth Pclcctr Po.-P out.ma>< q q q, q)o 'h Q Q-l

r,r)or2 ro r,rx,ry R R,.,.,. ~t R)oR2 R,R)oR2,~,Rl·,R2 • Ra R Rxx,~,~,Ryy R s S S t t tD T, T(v) T Tij TIIIIIX U UM4 V

V V,V)oV2,VJ,V4 Vs VD Voo Vo1 Vtot W

Ws

Wo

W

propagation matrix for coatingj threshold power for SBS phaseconjugation electrical pump power output power of laser resonators maximum output power of laser resonators mode number (integer) number of photons beam parameter of Gaussian beam cavity qu~ity complex beam matrix radial coordinate radius of mirror vertex amplitude reflectance intensity reflectance maximum reflectance of FPI

xix

optimum reflectance for maximum laser power reflectance of mirror 1,2 radius of curvature of wavefront center reflectance ofvariable reflectivity mirror radius of curvature (electric field) elements of 2x2 curvature matrix (electric field) 2x2 curvature matrix (electric field) length shift sensitivity stack matrix for optical coatings amplitude transmission (electric field) time delay time intensity transmission temperature transition matrix from coating i to coatingj maximum transmission voltage quarter wave voltage (pockels cell) ray vector Verdet constant loss factor (=1-10ss) loss factor due to scattering loss factor due to diffraction mode volume of TEMoo mode mode volume in medium with beam overlap total mode volume in medium beamradius,profileradius radius of gain profile waist radius pump rate cartesian coordinate distance in the x direction factor characterizing line broadening

xx List of Symbols and Abbreviations

lXum

~'~1>~2 ~ ~ ~um Y Y Y Y YI>Y2,Y3 r,rl,r2 11,l1 x,l1 y 11 IX 11cj> I1v I1D I1L I1n I1N I1P c1cctr I1Pout I1t I1V I1x & & &v €

€o

TI TI cxcit T1

List of Symbols and Abbreviations

A Ao Aq ~ 1.1 1.10 v Vo v hom Vinhom

vq Vq* 1t

P, PI' P2 o Ocff

0 0

0*

't

't*

'tB ()

()

()

()n

41, cflh 412 X XI X2 1Jr lfm Cl

wavelength center wavelength wavelength ofaxial mode of order q eigenvalue, p-th waveguide resonator mode matrix eigenvalue

XXI

permeability ofvacuum (=1.257xl0-10 Vs/(Am» light frequency center frequency of atomic transition homogeneous line width inhomogeneous line width frequency ofaxial resonator mode of order q axial mode frequency in resonators with gain = 3.141592 ... radius of curvature of surfaces cross section of stimulated emission effective cross section of stimulated emission cross section of stimulated emission at line center collision cross section lifetime decay time due to atomic collisions phonon lifetime (Brillouin scattering) spherical coordinate (angle) full angle of divergence (86.5% power content) phase field of waveguide eigenmode of order n phase induced by retarder atomic susceptibility real part of atomic susceptibility imaginruy part of atomic susceptibility phase field of free space eigenmode of order m angular beat frequency angular frequency