Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts and Applications

801 Pages·2005·13.665 MB·English

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

Springer Series in
108
OPTICAL SCIENCES
Founded by H.K .V . Lotsch
Editor-in-Chief: W.T. Rhodes, Atlanta
Editorial Board: T. Asakura, Sapporo
T.W. Hansch, Garching
T. Kamiya, Tokyo
F. Krausz, Vienna and Garching
B. Monemar, Linkoping
H. Venghaus, Berlin
€5. Weber, Berlin
H. Weinfurter, Munich
Springer Series in
OPTICAL SCIENCES
The Springer Series in Optical Sciences, under the leadership of Editor-in-Chief William T. Rhodes, Georgia
Institute of Technology, USA, provides an expanding selection of research monographs in all major areas of
optics: lasers and quantum optics, ultrafast phenomena, optical spectroscopy techniques, optoelectronics,
quantum information, information optics, applied laser technology, industrial applications, and other
topics of contemporary interest.
With this broad coverage of topics, the series is of use to all research scientists and engineers who need
up-to-date reference books.
The editors encourage prospective authors to correspond with them in advance of submitting a manu-
script. Submission of manuscripts should be made to the Editor-in-Chief or one of the Editors. See also
http://springeronline.com/series/624
Editor-in-Chief
William T. Rhodes
Georgia lnstitute of Technology
School of Electrical and Computer Engineering
Atlanta, GA USA
303jz-0~~0,
E-mail: [email protected]
Editorial Board
Toshimitsu Asakura Bo Monemar
Hokkai-Gakuen University Department of Physics
Faculty of Engineering and Measurement Technology
1-1, Minami-26, Nishi 11. Chuo-ku Materials Science Division
Sapporo, Hokkaido 064-0926, Japan Linkoping University
E-mail: [email protected] 58183 Linkoping, Sweden
E-mail: [email protected]
Theodor W. Hansch
Max-Planck-Institut fur Quantenoptik
Hans-Kopfermann-Strasse Herbert Venghaus
1
Garching, Germany Heinrich-Hertz-lnstitut
85748
E-mail: fiir Nachrichtentechnik Berlin GmbH
[email protected]
Einsteinufer
Takeshi Kamiya Berlin,3 G7 ermany
10587
Ministry of Education, Culture, Sports E-mail: [email protected]
Science and Technology
National Institution for Academic Degrees
Otsuka, Bunkyo-ku Horst Weber
3-29-1
Tokyo 112-0012, Japan Technische Universitat Berlin
E-mail: [email protected] Optisches lnstitut
Strasse des 17. Juni
Ferenc Krausz Berlin, Germ1a3n5y
10623
Vienna University of Technology E-mail: [email protected]
Photonics Institute
Gusshausstrasse
Wien, Austr2ia71 387 Harald Weinfurter
1040
E-mail: Miinchen
[email protected] Ludwig-Maximilians-Universitat
and Sektion Physik
Max-Planck-lnstitut fur Quantenoptik Schellingstrasse
4111
Hans-Kopfermann-Strasse I 80799 Miinchen, Germany
Garching, Germany E-mail:
85748 [email protected]
Norman Hodgson Horst Weber
Laser Resonators and
l
Beam Propagation
Fundamentals, Advanced Concepts
and Applications
Second Edition
With 587 Figures
-
Springer
a
Dr. rer. nat. Norman Hodgson Prof. Dr. Ing. Horst Weber
Coherent, Inc. Optisches Institut
5100 Patrick Henry Drive Technische UniversitPt Berlin
Santa Clara. CA 95054 Strasse des 17. Juni 135
USA 10623 Berlin
norman.hodgson @coherent.com Germany
[email protected]
Library of Congress Control Number: 2005923403
ISBN- 10: 0-387-40078-8 e-ISBN 0-387-251 10-3 Printed on acid-free paper.
ISBN-] 3: 978-0387-40078-5
0 2005 Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY
1001 3, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec-
tion with any form of information storage and retrieval, electronic adapeation, computer software, or by
similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they
are not identified as such, not to be taken as an expression of opinion as to whether or not they are
is
subject to proprietary rights.
Printed in the United States of America. (MVY)
9 8 7 6 5 4 3 2 1
springeronline.com
Preface
Since its first demonstration in 1960, the laser has found widespread application in diverse
areas including 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 new applications for this exciting technology are being
as
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
of the physics of optical resonators. The laser beam characteristicsa s 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 laser 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, beam propagation and
coherence are presented here. However, it is not necessary to work oneselfthrough 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.
vi Preface
The outline of the book was chosen such that the subject matter becomes more specialized
with proceeding chapters. We will start in Part I1 with the Fabry-Perot Interferometer to
discuss the basic resonator properties such loss, gain, threshold, and line width. The
as
following part will deal with passive (no active medium) 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 of the
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, the effects of gain on the mode
structure, as well as pulsed operation of laser resonators via Q-switching or mode-locking..
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, resonators for annular gain media, and micro-
resonators 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 of his laser system. A mong 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. We have 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 laser 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 laser
resonators and laser beams as much as we do!!
This book represents a revised version of “Optical Resonators”, published by Springer-
Verlag in 1996. Obsolete material has been deleted and new information has been added.
In particular, the following areas have been expanded or added:
In Part I, “The electromagnetic field”, we have added an overview of temporal and
spatial coherence and their mathematical description incorporating advanced
mathematical concepts such as the global degree of coherence and the Wigner function.
In addition, angular momentum of beams and the beam propagation in anisotropic
crystals are treated as well.
Preface vii
In IV,” Open Resonators with Gain”, a chapter on Q-switched laser resonators has
Part
been added. The chapter on nonlinear optical elements has been expanded to include
third harmonic generation.
In Part V,” Special Resonator Concepts”, a paragraph on micro-optical resonators has
been included.
In VI, Measurement Techniques”, the chapter on beam quality measurement has
Part “
been rewritten to reflect the IS0 standard IS0 11146:1999. We have also added a
paragraph on the experimental determination of the phase.
In addition to these expansions, we have updated the reference lists for all chapters and
incorporated new material in many chapters. This book would not have been possible
without the many contributions that have appeared in the scientific literature. No reference
list can ever be complete, and at this point we apologize to any of our colleagues whose
work has not been acknowledged or adequately represented. We also would like to thank
those readers that have pointed out errors in the first version of this book.
Our special thanks are due to Kathleen M. Millar for proof-reading the original manuscript
and to Dr. Keith Murdoch, Dr. Lukas E. Hunziker, Dr. Edward J. Reed, all of Coherent, Inc.
and Dr. Hanna J. Hoffman of Spectra-Physicsf or proof-reading selected chapters ofthe final
manuscript. We also thank Dr. Hans Koelsch, Margaret Mitchell and Virginia Lipscy of
Springer-Verlag New York, Inc. for their support and assistance in preparing this book.
July 2004
Belmont, CA Dr. rer. nat. Norman Hodgson
Berlin, Germany Prof. Dr. Ing. Horst Weber
Contents
List ofSymbols ................................................. 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 Waveoptics ............................................... 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=O) ................................. 75
2.4.2 Fourier Transformation (A=O) ............................. 76
2.5 Gaussian Beams ............................................ 80
2.5.1 Gaussian Beams in One-Dimensional Optical Systems .......... 80
2.5.2 Elliptical Gaussian Beams ................................ 91
X Contents
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
2.1 1 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,B earns ................... 176
3.6.3 Angular Momentum ..................................... 178
Part I1 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
Contents xi
Part I11 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 TEM,- 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
5.3.2 Two Aperture Limited Mirrors ............................ 277
6 Resonators on the Stability Limits ............................ 281
6 .I Resonators with g.g. =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
7.1 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
7.5 Unstable Resonators in Off-Axis Geometry ....................... 323
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