Optical Resonators: Fundamentals, Advanced Concepts and Applications

666 Pages·1997·32.363 MB·English

by  Dr. rer. nat. Norman Hodgson, Prof. Dr. Ing. Horst Weber (auth.)

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
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
permitted under the Copyright, Designs and Patents Act 1988, this publication may only be
reprodueed, stored or transmitted, in any form or by any means, with the prior permission in writing of
the publishers, or in the ease of reprographie reproduction in aeeordanee with the terms oflicences
issued by the Copyright Licensing Ageney. Enquiries coneerning reproduction outside those terms
should be sent to the publishers.
© Springer-Verlag London 1997
Originally published by Springer-Verlag London Limited in 1997.
Softcover reprint of the hardcover 1 st edition 1997
The use of registered names, trademarks, ete. in this publieation does not imply, even in the absence of
a specifie statement, that such names are exempt from the relevant laws and regulations and therefore
free for general use.
The publisher makes no representation, express or implied, with regard to the aeeuracy of the
information eontained in this book and eannot aecept any legal responsibility or Iiability for any errors
or omissions that may be made.
Typesetting: Camera ready by authors
69/3830-543210 Printed on acid-free paper
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 Dr. rer. nat. Norman Hodgson
üakland, CA Prof. Dr. Ing. Horst Weber
Berlin, Germany
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