Table Of ContentInstabilities and Chaos
in Quantum Optics"
NATO ASI Series
Advanced Science Institutes Series
A series presenting the results of activities sponsored by the NA TO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The series is published by an international board of publishers in conjunction with the
NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics New York and London
C Mathematical Kluwer Academic Publishers
and Physical Sciences Dordrecht, Boston, and London
o Behavioral and Social Sciences
E Applied Sciences
F Computer and Systems Sciences Springer-Verlag
G Ecological Sciences Berlin, Heidelberg, New York, London,
H Cell Biology Paris, and Tokyo
Recent Volumes in this Series
Volume 170-Physics and Applications of Quantum Wells and Superlattices
edited by E. E. Mendez and K. von Klitzing
Volume 171-Atomic and Molecular Processes with Short Intense Laser Pulses
edited by Andre D. Bandrauk
Volume 172-Chemical Physics of Intercalation
edited by A. P. Legrand and S. Flandrois
Volume 173-Particle Physics: Cargese 1987
edited by Maurice Levy, Jean-Louis Basdevant, Maurice Jacob,
David Speiser, Jacques Weyers, and Raymond Gastmans
Volume 174-Physicochemical Hydrodynamics: Interfacial Phenomena
edited by Manuel G. Velarde
Volume 175-Superstrings
edited by Peter G. o. Freund and K. T. Mahanthappa
Volume 176-Nonlinear Evolution and Chaotic Phenomena
edited by Giovanni Gallavotti and Paul F. Zweifel
Volume 177-lnstabilities and Chaos in Quantum Optics II
edited by N. B. Abraham, F. T. Arecchi, and L. A. Lugiato
Series B: Physics
I nstabi I ities and Chaos
in Quantum Optics II
Edited by
N. B. Abraham
Bryn Mawr College
Bryn Mawr, Pennsylvania
F. T. Arecchi
University of Florence
and National Institute of Optics
Florence, Italy
and
L. A. Lugiato
Turin Polytechnic
Turin, Italy
Springer Science+Business Media, LLC
Proceedings of a NATO Advanced Study Institute on
Instabilities and Chaos in Quantum Optics,
held June 28-July 7, 1987,
in II Ciocco, Italy
Library of Congress Cataloging in Publication Data
NATO Advanced Study Institute on Instabilities and Chaos in Quantum Optics
(1987: II Ciocco, Italy)
Instabilities and chaos in quantum optics II.
(NATO ASI series. Series B, Physics; v. 177)
Proceedings of a NATO Advanced Study Institute on Instabilities and Chaos in
Quantum Optics, held in II Ciocco, Italy, June 28-July 7,1987.
"Published in cooperation with NATO Scientific Affairs Division."
Includes bibliographical references and index.
1. Quantum optics—Congresses. 2. Lasers—Congresses. 3. Masers—Con-
gresses. 4. Chaotic behavior in systems—Congresses. 5. Nonlinear opt ics -
Congresses. I. Abraham, N. B. (Neal B.) II. Arecchi, F. T. III. Lugiato, L. A. (Luigi A.),
1944- . IV. North Atlantic Treaty Organization. Scientific Affairs Division. V.
Title. VI. Series.
QC46.15.N35 1987 535 8-12479
ISBN 978-1-4899-2550-3 ISBN 978-1-4899-2548-0 (eBook)
DOI 10.1007/978-1-4899-2548-0
© Springer Science+Business Media New York 1988
Originally published by Plenum Press, New York in 1988
Softcover reprint of the hardcover 1st edition 1988
All rights reserved
No part of this book may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical, photocopying,
microfilming, recording, or otherwise, without written permission from the Publisher
PREFACE
This volume contains tutorial papers from the lectures and seminars
presented at the NATO Advanced Study Institute on "Instabilities and Chaos
in Quantum Optics", held at the "Il Ciocco" Conference Center,
Castelvecchio Pascoli, Lucca, Italy, June 28-July 7, 1987. The title of
the volume is designated Instabilities and Chaos in Quantum Optics II,
because of the nearly coincident publication of a collection of articles on
research in this field edited by F.T. Arecchi and R.G. Harrison
[Instabilities and Chaos in Quantum Optics, (Springer, Berlin, 1987) 1.
That volume provides more detailed information about some of these topics.
Together they will serve as a comprehensive and tutorial pair of companion
volumes.
This school was directed by Prof. Massimo Inguscio, of the Department
of Physics, University of Naples, Naples, Italy to whom we express our
gratitude on behalf of all lecturers and students. The Scientific Advisory
Committee consisted of N.B. Abraham of Bryn Mawr College; F.T. Arecchi of
the National Institute of Optics in Florence and the University of
Florence, and L.A. Lugiato of the Politechnic Institute of Torino. The
school continues the long tradition of Europhysics Summer Schools in
Quantum Electronics which have provided instruction and training for young
researchers and advanced students working in this field for almost twenty
years.
In addition to the support from the NATO ASI program, support was also
received from the following organizations:
u.S. National Science Foundation
Consiglio Nazionale delle Ricerche, Italy
Settore di Fisica Atomica e Molecolare del GNSM
Universita di Napoli
Universita degli Studi di Pisa
Istituto Nazionale di Ottica
Bryn Mawr College
Lambda-Physik, Gmbh, Gottingen
European Office of the U.S. Office of Naval Research (for a special
session on transverse effects in optical bistability and
instabilities)
dB Electronic, Milan
Officine Galileo, Firenze
European Physical Society (Quantum Electronics Division)
IBM (Italy)
Ente Nazionale Energie Alternative (ENEA, Italy)
Coherent, Inc.
Laser Optronics S.R.L. (Italy)
Elicam S.R.L. (Italy)
MicroControle-Nachet (Italy)
v
We are grateful for the expert administrative help of Giovanna
Inguscio, Iva Arecchi, and Anna Chiara Arecchi in the management of the
meeting.
We wish to thank all of the lecturers fo·r the clarity of their
presentations and to especially thank the contributors to this volume who
have helped to enhance its tutorial value and the speed of its production.
The school was followed by an International Workshop on Instabilities,
Dynamics and Chaos of Nonlinear Optical Systems which welcomed over 120
experts in the field to an intense three-day presentation of their latest
research results. The participants in the school who had been prepared by
their ten days of study received the added benefit of presentations from
and interaction with many other scholars who are contributing to the rapid
growth of the field. We are pleased to be able to supplement the tutorial
section of this volume with a report of the presentations at the workshop
which includes mention of many of their latest results, descriptions of new
areas of study, and suggestions of areas where further progress is needed
and/or expected. The "Meeting Report" includes many references to where
these new results can be found in the research literature.
N.B. Abraham
F.T. Arecchi
L.A. Lugiato
December, 1987
vi
CONTENTS
Laser (and Maser) Instabi~ities
25 YEARS OF LASER INSTABILITIES 1
L.A. Lugiato, L.M. Narducci, J.R. Tredicce, and D.K. Bandy
SHIL'NIKOV CHAOS IN LASERS ... 27
F.T. Arecchi
INSTABILITIES IN FIR LASERS . . . . . . . . • . . . . . . . . . . . .. 41
C.O. Weiss
ANALYSIS OF INSTABILITY AND CHAOS IN OPTICALLY PUMPED
THREE LEVEL LASERS . . . 53
R.G. Harrison, J.V. Moloney, J.S. Uppal and W. Forysiak
THEORY AND EXPERIMENTS IN THE LASER WITH SATURABLE
ABSORBER 69
E. Arimondo
GAS LASER INSTABILITIES AND THEIR INTERPRETATION . . . . . . . . . .. 83
L .. W. Casperson
EXPERIMENTAL STUDIES OF INSTABILITIES AND CHAOS IN SINGLE-MODE,
INHOMOGENEOUSLY BROADENED GAS LASERS ••. 99
N.B. Abraham, M.F.H. Tarroja and R.S. Gioggia
MULTISTABILITY AND CHAOS IN A TWO-PHOTON MICROSCOPIC MASER 123
L. Davidovich, J.M. Raimond, M. Brune and S. Haroche
BISTABLE BEHAVIOR OF A RELATIVISTIC ELECTRON BEAM IN A MAGNETIC
STRUCTURE (WIGGLER). . . . . . . . . . . . . . . . . . . 139
R. Bonifacio and L. De Salvo Souza
C~assica~ and Quantum Noise
PUMP NOISE EFFECTS IN DYE LASERS . . . . . . . . . . • 147
M. San Miguel
QUANTUM CHAOS IN QUANTUM OPTICS: LECTURES ON THE QUANTUM DYNAMICS
OF CLASSICALLY CHAOTIC SYSTEMS . . . . • . 163
R. Graham
vii
Transverse Effects
TRANSVERSE EFFECTS IN NONLINEAR OPTICS 193
J.V. Moloney
SPATIAL CHAOS IN BISTABLE OPTICAL ARRAYS . • • . . . . . . . . . • . . . 219
W.J. Firth
INSTABILITIES IN PASSIVE OPTICAL SYSTEMS: TEMPORAL AND SPATIAL
PATTERNS . ...•. . • . . 231
L.A. Lugiato, L.M. Narducci, R. Lefever, and C. Oldano
Dynamics in Optical Bistability and Nonlinear Optical Media
IKEDA DELAYED-FEEDBACK INSTABILITIES 247
H.M. Gibbs, D.L. Kaplan, F.A. Hopf, M. LeBerre, E. Ressayre and A. Tallet
EXPERIMENTAL INVESTIGATION OF THE SINGLE-MODE INSTABILITY IN OPTICAL
BISTABILITY . . • 257
A.T. Rosenberger, L.A. Orozco and H.J. Kimble
DYNAMICS OF OPTICAL BISTABILITY IN SODIUM AND TRANSIENT BIMODALITY . . . 265
W. Lange
OPTICAL BISTABILITY: INTRODUCTION TO NONLINEAR ETALONS GaAs ETALONS AND
WAVEGUIDES; REGENERATIVE PULSATIONS ............... 281
H.M. Gibbs
SPATIAL AND TEMPORAL INSTABILITIES IN SEMICONDUCTORS • . . • . . . . . . 297
I. Galbraith and H. Haug
FOUR-WAVE MIXING AND DYNAMICS 311
W.J. Firth
Methods of Analysis in Nonlinear Dynamics
BIFURCATION PROBLEMS IN NONLINEAR OPTICS . . . . . . . . . . • . • . . . 321
P. Mandel
STRANGE ATTRACTORS: ESTIMATING THE COMPLEXITY OF CHAOTIC SIGNALS • • • 335
R. Badii and A. Politi
METHODS OF ADIABATIC ELIMINATION OF VARIABLES IN SIMPLE LASER MODELS . . 363
G.L. Oppo and A. Politi
Meeting report:
INSTABILITIES, DYNAMICS AND CHAOS IN NONLINEAR OPTICAL SYSTEMS . . . . . 375
N.B. Abraham, E. Arimondo, and R.W. Boyd
Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 393
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
viii
25 YEARS OF LASER INSTABILITIES
L.A. Lugiato
Dipartimento di Fisica, Politecnico di Torino, Torino, Italy
L.M. Narducci, J.R. Tredicce
Physics Departtnent, Drexel University, Philadelphia, Pa. 19104
and
D.K.Bandy
Physics Departtnent, Oklahoma State University, Stillwater, Ok. 74078
The purpose of these lectures is to provide an introduction to the field of laser
instabilities and an overview of one of the most popular theoretical models: the
homogeneously broadened, unidirectional ring laser system. We discuss both multimode
and single-mode operation, the possible steady states and their stability properties. In the
process, we identify some valuable features of the plane-wave Maxwell-Bloch equations and
single out some of their shortcomings. We conclude this survey with an outline of current
attempts at removing the remaining open problems with an extension of the plane-wave
theory and the inclusion of transverse effects.
1. Introduction
Contrary to what the title of these lectures may suggest, the laser is much older than just
25 years. As I learned in the well known textbook by O. Sveltol (the reader may wish to
confmn this with his or her own eyes), the fIrst recorded mention of the laser dates back
almost 2000 years ago in the writing of Pliny the Elder (Gaius Plinius Secundus ) who
reported that .. Laser. .. inter eximia naturae dona numeratum plurimus compositionibus
inseritur .. (The laser is numbered among the most remarkable gifts of nature, and lends itself
to a variety of applications; from Historiae Naturalis). The laser was a plant that used to
grow on the shores of modem day Lybia. It was a popular ingredient in Roman cuisine, a
celebrated tonic and a powerful aid to cure the wounds of enemy arrows and to remove the
sting of poisonous insects. With the advance of the Sahara desert to the shores of the
Mediterranean sea, the laser became extinct. It was discovered again in 1960. Its modem
version is not well known for its culinary virtues, although, under appropriate conditions, it
does find medical applications. What makes it interesting from our point of view, however,
is its unusual propensity for producing radiation that varies in intensity even under steady or
nearly steady pumping conditions. This feature, in fact, caught people's attention from the
very early days of the modem era of the laser. The appearance of spiking action, as this
pulsing phenomenon is usually called, was observed in maser systems even before the
discove(Y of the laser2, but it became a virtually universal feature of solid state optical
devices:;. Random spiking, for example, is almost a signature of ruby and neodymium
lasers, while beautifully re~ular undamped oscillations have been produced with neodymium
lasers in clad optical fibers .
It would be only natural to expect that a c.w. pumped laser should produce a steady
output, and indeed many lasers can be made to operate in a very stable way. Often,
