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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 B
Computer and Systems Sciences Ecological Sciences Cell
Biology
Plenum Publishing Corporation New York and London
Kluwer Academic Publishers Dordrecht, Boston, and London
Springer-Verlag Berlin, Heidelberg, New York, London, 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
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 op
t i cs - 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.
QC446.15.N35 1987 535 88-12479
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.
vi
December, 1987
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 . . .
R.G. Harrison, J.V. Moloney, J.S. Uppal and W. Forysiak
THEORY AND EXPERIMENTS IN THE LASER WITH SATURABLE ABSORBER
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 ••.
N.B. Abraham, M.F.H. Tarroja and R.S. Gioggia
MULTISTABILITY AND CHAOS IN A TWO-PHOTON MICROSCOPIC MASER L.
Davidovich, J.M. Raimond, M. Brune and S. Haroche
BISTABLE BEHAVIOR OF A RELATIVISTIC ELECTRON BEAM IN A MAGNETIC
STRUCTURE (WIGGLER). . . . . . . . . . . . . . . . . . .
R. Bonifacio and L. De Salvo Souza
C~assica~ and Quantum Noise
PUMP NOISE EFFECTS IN DYE LASERS . . . . . . . . . . • M. San
Miguel
QUANTUM CHAOS IN QUANTUM OPTICS: LECTURES ON THE QUANTUM DYNAMICS
OF CLASSICALLY CHAOTIC SYSTEMS . . . . • .
R. Graham
193
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 W.J. Firth
Methods of Analysis in Nonlinear Dynamics
311
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
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,
Fig. 1.1 Schematic representation of the energy flow between atoms
and field according to the rate equations.
however, and even in spite of the most elaborate precautions, the
output acquires a time-dependent behavior. In this case, we say
that the laser developes an instability. Some unstable behaviors
can be understood as the result of optical interference between
nearly indepedent modes of operation, others have a considerably
more complicated origin. Quite generally, however, these dynamical
effects are interesting because they represent a spontaneous
departure from a state of time translational symmetry, induced by
the nonlinearity of the interaction between radiation and matter.
In addition, these temporal behaviors tend to undergo significant
qualitative changes, as one varies the control parameters of the
system, and offer in this way valuable clues on the internal
mechanisms that make a laser work.
Some of the earliest theoretical models of laser action focused on
the energy exchanges between a collection of inverted two-level
atoms and the cavity field (Fig. 1.1). In their simplest versionS,
the rate equations couple the population difference D = N2 - Nl,
between the two active levels to the photon number n, according to
the nonlinear system of equations
dD 1 <it = - 2WnD -~ - Do)
dn I -=WnD--n dt Tc
(1.1 a)
(1.1 b)
where W is proportional to the absorption and stimulated emission
rate; T1-l is the rate of approach of the population difference to
its equilibrium value, Do, under the action of both spontaneous
emission and the external pump; T c is the escape time of the
electromagnetic energy out of the optical cavity.
The rate equations (1.1) predict that, above a certain pump level
(the threshold inversion) the laser begins to operate, as evidenced
by the growth of the photon number n from a very small initial
value. On the other hand, these equations predict also that the
output intensity can only approach a constant value after a
transient that can either involve a monotonic or, at most, a damped
oscillatory evolution. Thus, at least in the simple form given by
Eqs. (1.1), the rate equations are unable to describe undamped
oscillations. In fact, within the context of the plane-wave model,
one is forced to conclude that this failure is related to the
absence of coherence in the coupling between radiation and matter.
This conclusion was partially revised as a result of a very recent
theoretical study of the ring laser that included the possibility
of transverse variations of the field and of the atomic variables6.
In the plane-wave approximation, however, it seems inescapable that
one must look at a deeper level than the rate equations if one
wants to trace the origin of unstable behaviors.
Actually this conclusion was re§ognized in some of the earliest
contributions to this problem7,8. Grazyuk and Oraevsky , in 1964,
discovered that the coherent single-mode laser equations (see
sections 3 of these lecture notes), a simplified predecessor of the
more modem Maxwell-Bloch theory, predict the appearance of
instabilities and undamped pulsations under well defmed
mathematical conditions that involve the unsaturated gain of the
active medium and the cavity and atomic relaxation rates. By
coherent laser equations we mean a description built on the basic
premise that the interaction between light and matter involves the
coupling between an optical wave and the atomic dipoles, and that
the evolution of the radiated field is determined by a macroscopic
polarization, in agreement with Maxwell's electrodynamics.
2
Subsequent investigations of the laser equations revealed the
appearance of unstable phenomena in the form of periodic pulsations
even under multimode operating conditions lO. In fact, a modern
study of the multimode instability ll confmned not only the
existence of periodic oscillations, but also of considerably more
complicated patterns which are highly suggestive of deterministic
chaos.
An important advance was recorded in 1975 by Haken 12 who noted the
I<.xistence of an isomorphism between the single-mode laser
model and the Lorenz equations 1:.;. The Lorenz equations,
originally derived to simulate the onset of convective hydrodynamic
instabilities, were already well known at the time in the
mathematical literature as the source of deterministic chaos14.
Thus, the isomorphism between the laser and the Lorenz equations
determined at once that the laser itself could also be the source
of chaotic behavior. In fact, the chaotic instability is, by far,
the most common type of unstable behavior for a resonant
single-mode laser model. Over a restricted range of parameters,
however, periodic oscillations can also be obtained as shown in
Ref. 15.
A different typ~ of unstability, the so-called phase instability,
was discovered by Hendow and Sargent16 and independently in Ref.
17. This is also a multimode instability, analogous in a way to the
one discovered by Risken and NummedallOa,b and by Graham and Haken
lOc, but its dynamical origin lies in the loss of stability of the
field phase rather than its amplitude. The experimental
confirmation of this phenomenon18 showed good quantitative
agreement between the theoretical predictions and the observed
phenomenology.
The emphasis of our discussion, so far, has been with homogeneously
broadened models, i.e. with laser systems whose active atoms have
all the same transition frequency. Very interesting regular and
chaotic output oscillations, however, have also been observed in
inhomogeneously broadened lasers by Casperson19,whose pioneering
contributions had a significant influence on the modern revival of
interest in this subject, and by Abraham and collaborators20. This
subject will be reviewed by Professors Casperson and Abraham in
this volume and will not be discussed further in our
lectures.
In closing this brief introduction we must mention that, strictly
speaking, the plane-wave approximation is only a first cut at the
description of real laser systems. For a long time, however, and by
general consensus this approach has been regarded as adequate in
capturing the essential features of laser dynamics. There are
reasons to believe, on the other hand, that the plane-wave
approximation may be too restrictive and, in fact, perhaps even
unsuitable for a close qualitative match between theory and
experiments. One of the aims of these lectures is to offer
arguments in favor of the need for a more accurate theoretical
description.
For the purpose of producing a balanced presentation and for
pedagogical reasons, we devote our attention mainly to the
development of a theory of the homogeneously broadened laser in the
plane-wave approximation; we discuss the main predictions
concerning the steady state and the unstable behaviors for both
single and multimode configurations, and identify some of the major
unsolved issue of the Maxwell-B~h theory. In our concluding section
we review the highlights of a recent generalization that includes
transverse effects, and compare the most significant differences
between the conclusions of these two approaches.
2. The Maxwell-Bloch Theory of a Ring Laser - Multimode
operation
The conceptual foundations of the Maxwell-Bloch theory of the laser
rest on the self-consistent loop sketched in Fig. 2.1. An incident
electromagnetic field interacts with a collection of microscopic
dipoles and creates a macroscopic polarization. This, in turn, acts
as the source of a radiated field which interacts again with the
microscopic dipoles, etc. Energy is fed into the active medium by
an external source, the pump, and is extracted from the interaction
volume in the form of electromagnetic energy radiated by the
macroscopic polarization.
The mathematical basis for this model is provided by the classical
wave equation for the slowly varying complex amplitude of the
electromagnetic field and by the Schroedinger
3
Fig. 2.1 Schematic representation of the self-consistent approach
for the description of the interaction of light and matter.
equation for a collection of two-level systems. This mix of
dassical and quantum equations is the essence of the semiclassical
description of the laser21. Its self-consistency derives from the
indentification of Maxwell's macroscopic polarization as a suitable
ensemble average of the quantum mechanical dipole operator over the
collection of atoms.
The irreversible decay of the atomic variables is simulated by
phenomenological relaxation terms, while the presence of a lossy
resonant cavity around the active medium is accounted for by
appropriate boundary conditions. Most traditional theories of the
laser, at this point, add the assumption that the cavity field
behaves as a plane wave, to a good approximation. This assumption,
long held as an acceptable way to tame the complexity of real
resonators, is beginning to be viewed with some suspicion6. It is
possible, in fact, that some of the fine, and not so fme, prints of
laser dynamics may be more sensitive to the shape of the cavity
field than one had expected. For the sake of simplicity, however,
and in order to illustrate the advantages and drawbacks of this
approach, we concentrate mainly on the plane-wave theory of the
laser.
In these lectures we consider a ring cavity of the type shown in
Fig. 2.2, with a total round-trip length A, two partially
reflecting mirrors (mirrors 1 and 2) and additional ideally
reflecting surfaces to complete the loop. This resonator is assumed
to operate in a unidirectional mode with the help of a
non-reciprocal element such as a Faraday isolator.
The optical cavity is characterized by an infinite number of
equispaced resonances at the frequencies IXn = n 21tC/A (n =
O,±1,±2, ... ). The active medium has a transition frequency COA
and is assumed to have a homogeneously broadened lineshape with a
width 11.' For convenience of terminology, the empty cavity mode
whose frequency COc lies closest to the atomic transition frequency
is called the resonant mode; all other modes are labelled
collectively as off-resonant modes. From now on, the frequency of
the off-resonant modes will be measured relative to COc.
The Maxwell-Bloch equations for the ring laser are
i)F 1 i)F -+--=-aP ()z c at
: = -'YJ FD + (l+iSAC)P}
(2.1a)
(2.1c)
where F is the slowly varying complex amplitude of the cavity
field, scaled to the square root of saturation intensity, P is the
normalized amplitude of the macroscopic polarization, and D is the
population difference between the upper and lower levels; a denotes
the small signal gain constant per unit length, 3AC the frequency
detuning between the atomic transition frequency and the resonant
mode, in units of the atomic linewidth 11. [ i.e. 3AC = (coA
COc)/YJ.]' and 111 is the decay rate of the atomic population.
Equations (2.1) are supplemented
4
F(O,t) = R F(L,t - M) (2.2)
where L is the length of the active medium, R is the reflectivity
coefficient of mirrors 1 and 2 and ~t = (A - L)/c is the
propagation time of light through the empty segment of the cavity.
Note that, in this form, the Maxwell-Bloch equations and boundary
condition fully account for propagation effects, including the
delay associated with the round trip time of the radiation.
2.1 Steady State
The possible stationary states of Eqs (2.1) have the form
F(z,t) = FsI(z) e-ili!ll
0= FsPsI + (1 +~)P sl
(2.3a)
(2.3b)
(2.3c)
(2Aa)
(2Ab)
O=-l(F*P +F P*)+D -1 (2Ac) 2 sl sl sl sl st
SQ is the unknown difference between the carrier frequency of the
steady state laser field and the resonant cavity mode, and the
parameter ~ is defmed as
~ = S - '6QJy AC .L
(2.5)
The atomic variables can be calculated at once in terms of the
stationary field profile with the result
Fig. 2.2
s st 1 + ~2 + IF (z)12 sl
(2.6a)
sl
(2.6b)
Schematic representation of the ring cavity. The active medium has
length L, while the ring's length is A.
5
where FSI (z) is the solution ofEq. (2.4a) subject to the boundary
condition
F st(O) = R Fst(L) exp[ion(A-L)/c) (2.7)
Several facts are immediately obvious. The steady state
polarization and the field envelope are generally out of phase with
respect to each other by an amount that depends on the detuning
parameter and the position of the opemting laser line. In
resonance, however, PSI and Fst have zero relative phase
difference. The steady state population saturates at high intensity
levels in the sense that Dst(z) -7 0 as IFst(z)1 -7 00. Note, also,
that the level of saturation of the population difference depends
on the detuning parameter, through d, with the largest degree of
saturation corresponding, as expected, to the resonant case.
In order to calculate the longitudinal proftle of the steady state
field and its output value we substitute Eq. (2.6a) into E~p.4a)
and, at the same time, represent the field amplitude in terms of
its modulus and phase
i8(z) F iz) = p(z) e (2.8)
In this way Eq. (2.4a) takes the form
dp=a p dz 1 + d2 + p2
(2.9a)
(2.9b)
The two coupled equations (2.9) can be combined to yield the first
integral of the problem
In(P(z») = _1- [e(z) - e(o) - IDz] p(O) d c
(2.10)
while Eq. (2.9a) can be integmted at once to give
{l + d2) In(P(z») +.!. (p2(z) - p2(0») = fJ:Z p(O) 2
(2.11)
The boundary condition (2.7), expressed in terms of the field
modulus and phase yields the two constraining relations
p(O) = R peL)
C
(2.12a)
(j = 0,±I,±2, .. ) (2.12b)
which show that, in principle, the boundary conditions can be
satisfied by more than one possible solution. This result is
important because it suggests the possibility of coexisting steady
states and mode-mode intemction.
The output laser intensity can be calculated at once from Eq.
(2.11) after setting z = L and using the boundary condition (2.
12a). The result is
p2(L) = _2_ (aL - (l+d2) lin Ri) (2.13) l-R2
The unknown operating frequency or, equivalently, the value of d
follows by setting z = L in Eq. (2.10) and using the boundary
condition (2. 12b). The required result is
or
6
1 A-L on ) InR= - (- on -+ 21tj --L d c c
s: ( c lin RI) c lin RI s: 21tc . un 1+-- =--u +-J
A A AC A 'Y.l
(214)
(2.15)
J(\ 11 11 Un) U n+1 U n+ 2
UJA
Schematic representation of the multiple stationary solutions. The
solid vertical lines represent three adjacent cavity modes. The
dashed lines labelled j=O. 1. and 2 indicate the position of three
steady state solutions; ro A denotes the position of the center of
the atomic line.
The quantity c lIn RI/A represents the decay rate of the cavity
field, while 21tC/A is the frequency spacing between adjacent
cavity resonances (the free spectral range). After introducing the
symbols
c lIn Rl K=--,
(j) KaAC+ a y j &1. = COL - OlC = l.J:
J Y +K J.
(j=O,±I.±2 •... ) (2.17)
where the index j is a label for the multiple solutions. Equation
(2.17) is the well known mode-pulling formula, written in a
slightly unconventional way. In most textbooks Eq. (2.17) is
usually given in the form
(j) (OlC+ a j) y + OlA K COL = 1 J.
y+K J.
(2.18)
which shows that the operating laser frequency is the weighted
average of the atomic resonant frequency and the frequency of one
of the cavity modes.
The steady state analysis cannot predict which of the possible
solutions, j = O,±I,±2, ... , is actually excited under given
conditions; it can, however, identify the range of allowed values
of j by requiring that p2(L) remain positive, i.e. that the j-th
solution satisfy the threshold condition
(aL)tbrJ = (1+.12) lIn RI (219)
Figure 2.3 illustrates schematically how various steady state
solutions are organized in relation to the various empty cavity
resonances. if their respective gain is sufficiently large to
overcome the cavity losses.
One may also inquire into the longitudinal profile of the field in
steady state. This can be done with the help ofEq. (2.11) after
setting p(O) = R p(L). and calculating p(L) from Eq. (2.13). The
resulting transcendental equation for p(z) can be solved easily by
numerical means.
Depending on the value of the gain. aL. and of the free spectral
range. we can distinguish two significantly different situations.
as illustrated in Figs. 2.4 a.b. In the first case. the intermode
spacing is large enough that. at most. only one nontrivial steady
state solution can exist for every value of the detuning parameter
aAC• In the second case. more than one steady state can satisfy the
lasing condition for selected values of the detuning.
The physical meaning of Fig (2.4a) is unambiguous. If one performs
a detuning scan. beginning with the resonant configuration. aAC =
O. the output intensity is maximum, at first; on increasing the
value of aAC' the intensity decreases until, eventually, the laser
is driven below threshold. A further increase of the detuning
parameter causes the next steady state to
7
Fig.2.4a
5.--------------------------,
-3 -1
:;
0 -3 -1 ~ ... c 3
Fig. 2.4b Same as Figure 2.4a with an intermode spacing of 3 (in
units of 11-)
go above threshold; the corresponding output intensity then
increases monotonically until B AC equals a full free spectral
range of the ring cavity (at/"h). Note that the carrier frequency
of the operating stationary state changes as the laser switches
from the steady state j = 0 to j = 1. This is a trivial consequence
of the fact that, at some point during the detuning scan, the laser
is driven below threshold and then begins to operate again in
correspondence to a different cavity resonance. The situation is
less transparent in the case of Fig (2.4b). At the beginning of the
detuning scan, with B AC = 0, there is no ambiguity as to the
steady state configuration of the laser. However, when bAC becomes
sufficiently large. two coexisting steady states (or more,
depending on the parameters of the problem) may develop. At this
point we may envision two possible scenarios:
(i) the lasing mode retains control of the laser process well into
the coexistence region or until, perhaps, its losses overcome the
available gain;
(ii) the two coexisting steady states compete with one
another.
In case (i) we can anticipate the appearance of discontinuous jumps
of the steady state intensity and frequency, in addition to
hysteresis upon reversing the direction of the scan; in case (ii)
the competing steady states have different frequencies and can be
expected to produce undamped beats at a frequency approximately
equal to the intermode spacing. The correcrness of this conjecture
can be verified only with the help of additional considerations on
the stability of the steady state. We now turn our attention to
this aspect of the problem.
2.2 The linear stability analysis
The discussion of the previous section makes no mention of the
actual physical realizability of the various steady states. This
aspect of the problem can be addressed with a study of the
stability properties of a given configuration. The stability of a
steady state can be probed in the usual way, as with all nonlinear
dynamical systems, by perturbing the dynamical variables around a
steady state configuration, and by following the evolution of the
perturbations according to their linearized equations of motion.
The complex exponential
8
rate constants, A, of the infinitesimal perturbations provide the
required information about the stability of the system. If Re A is
negative for all the degrees of freedom, the steady state of
interest is stable. If even one of the real parts of the rate
constants acquires a positive value, the system is unstable and
departs exponentially from the steady state, in response to the
applied perturbation.
The general stability analysis of the Maxwell-Bloch equations (2.1)
is a rather difficult problem that has been solved exactly only a
short time ag023 . The main source of complications is the spatial
dependence of the steady state field and of the atomic variables
which makes even the linearized equations quite complicated to
analyze.
A useful mathematical limit that is responsible for significant
simplifications in the theoretical analysis of the Maxwell-Bloch
equations is the so-called uniform field limit. Originally invented
by Bonifacio and Lugiato in their early studies of optical
bistability24, the uniform field limit is characterized by the
requirements
with
(2.20a)
(2.20b)
The uniform field limit is also known as "the mean field limit" in
the literature. We find the former label to be more precise in
describing our physical situation. At first sight one may be
tempted to view the uniform field limit as a trivial
oversimplification of the problem because if one continues to lower
the unsaturated gain per pass, eventually the laser will be driven
below threshold. This limit, however, prescribes the simultaneous
reduction of both the unsaturated gain and the transmission losses,
so that non trivial operating conditions can be obtained, depending
on the value of the gain parameter C.
The reason for the chosen terminology is not difficult to
visualize: if the gain per pass is very small, a wave entering the
active medium at a given loop experiences a very small
amplification leading to a negligible longitudinal variation of
both the field amplitude and the atomic variables. By reducing the
transmission losses one can also insure that the laser will
continue to operate as much above threshold as desired. Naturally,
as one approaches the uniform field conditions, the duration of the
transient evolution into steady state becomes progressively longer,
as the field requires more and more round trips in order to reach
its asymptotic configuration. It is obvious that for the uniform
field limit to make sense the only losses of the resonator must
arise from the finite transmittivity of the mirrors because any
fixed residual loss of a different origin would unavoidably force
the laser below threshold.
While the uniform field limit sets no upper bound to the strength
of the cavity field and to the level of the atomic saturation, it
is less obvious to what extent this approximation can provide a
useful description of a realistic laser (leaving aside the issue of
the plane-wave assumption). Recent studies11 ,17 have shown that
the uniform field limit is a remarkably robust approximation, in
the sense that the exact solutions of the Maxwell-Bloch equations
(2.1) are in good qualitative and even quantitative agreement with
the corresponding approximate solutions of the uniform field model
even for unsaturated gain parameters <XL of about 1 and
transmittivity coefficients of about 0.2. It is this favorable
aspect of the approximation that makes it an extremely useful tool
for the study of the dynamical aspects of the Maxwell-Bloch
equations.
We carry out the linear stability analysis in two steps: first, we
derive an appropriate set of modal equations in the uniform field
limit, and then linearize these equations and derive the
characteristic equation for the rate constants.
2.2a The modal equations
In order to introduce the notion of cavity modes even in the
presence of transmission losses, we define a new set of space-time
coordinates25
9
and the new field and atomic variables
Fl (z',t') = F(z' ,t) exp(f lnR)
Pl(z',t') =P(z',t') exp(f lnR)
Dl(z' ,t') = D(z' ,t')
aPl { . } at' = - 11. FlDl+ (1+lSAC) P 1
where the symbol
aL 2C = IInRI
(2.21a)
(2.21b)
(2.22a)
(2.22b)
(2.22c)
(2.23a)
(2.23b)
(2.23c)
(2.24)
represents the gain parameter. The main purpose of the
transformations (2.21) and (2.22) is to produce new boundary
conditions that are of the standard periodicity type2S: i.e.
(2.25)
and which allow the decomposition of the dynamical variables into a
Fourier series, as commonly done in ordinary vibration problems.
Thus, we let
( Fl(Z' 't'»)
The wave numbers ~ are selected in such a way that
k = 27tc n (0+1 +2 ) n L' n= ,- .- , ... (2.27)
With this choice, the boundary condition (2.25) is satisfied
automatically. The modal functions of the ring cavity in the new
reference system are
1 ik"z' u~=-e ~~
and satisfy the orthogonality relation L
(u (z'),u (z'» = Jdz' u*(z') u (z') = ~ (2.29) n m n m n,m o
The construction of the equations of motion for the modal
amplitudes fn(t'), lPn(t'), cln(t') is now a simple matter, which
involves substituting Eqs. (2.26) into Eqs. (2.23) and taking into
account the orthonormality of the cavity modal functions. The
result is a complicated infinite set of equations which is exactly
equivalent to the original Maxwell-Bloch equations (2.23) [or
(2.1)].
In the uniform field limit the modal equations become
df .....!. = i&Q f - K (f + 2Cp ) de n n n
df" .....!. = -i&Q r- -K (f*+ 2Cp*) de n n n
dp* L - - _n = _ y { f' d* + [1 -i (~ - &Q - a )] pO} dt' .L n'
non' AC n n
n'
dd ~ dt~ = i andn- ~I{ - ~ £.J (r:, Pn+n'+ fn' P:'-n) + dn-
Sn,o}
n'
(2.30a)
(2.30b)
(2.3Oc)
(2.3Od)
(2.30e)
where the symbols ~n and <Xn represent frequencies scaled to
Y.L' and where the field and atomic amplitudes are defmed according
to relations of the type
-icJc"I' -ia I' x (t') = x (t') e == x (t') e • (2.31)
n n n
Finding the stationary states of Eqs. (2.30) looks like a very
difficult task. Instead, close inspection of Eqs. (2.30) shows that
the possible steady states can be represented by the simple
formulae I7
f! = [2C _ (l+A~)]1I2~ . n J n,J (2.32a)
(2.32b)
(2.32c)
where
(2.33)
and
J 1 + K/y 1 .L .L
(2.34)
11
2C> 1 + ll~ (2.35) J
is satisfied for more than one value of j. This result agree with
Eq. (2.19) after application of the unifonn field condition.
Thus each steady state is characterized by five non-zero Fourier
amplitudes leading to single- mode operation with a uniform field
profile in steady state and to a purely harmonic structure both in
space and time.
2.2bThe characteristic eigenvalues of the linearized
equations
We begin the linear stability analysis by setting
x (t') = x(j)S .+ Sx (t') n n D,j n
d (t') = d(j)S + Sd (t') n n n,O n
(2.36a)
(2.36b)
where xn stands for fn' fn *, Pn' or Pn *. Substitution of Eqs.
(2.36) into Eqs. (2.30) leads to an infinite dimensional linear
system of equations for the fluctuation variables aXn and &In.
The remarkable aspect of this problem is that the infmite system
breaks up into separate blocks of five equations each where the
fluctuation variables afn+i, aPn+j' afj_n*' apj_n. and &In for
fixed values of the steady state index j and the modal inaex n are
couple1i to one another, but not to other variables, according to
the equations17
df. ~:J = i 50j Sfn+f K: «ifn+/ 2C (ipn+j) (2.37a)
df. ~= - i 50. (if: - K «if.* + 2C ap~ ) dt' J J-n J-n J-n
(2.37b)
dPn+j = _ 'Y {Ii) Sd + (if .doG)+ [1 + i (ll.- ~ )] Bp +.} dt' .1 J
n n+J J n n J
(2.37c)
dp~ {.Ii). 6) -. } -t!!. = - 'Y t:' (id + Sf.* do + [1 - i (ll.+ ex
)] (ip. dt' .1 J n J-n J n J-D (2.37d)
ddn = i ex Sd - 'Y {- 1. (t.j) • (ip . + p~ (if.* + f~) (ip~). +
dt' n n .1 2 J n+J J J-n J J-n
+p~)(if .)+Sd} J n+J n (2.37e)
In order to solve Eqs. (2.37) we introduce the ansatz
(ifn+j(t') (ifO). n+J
(if.. (t') <5t.0)* . J-n J-n
(iPn+j(t') At' q,(0). = e n+J (2.38)
(ip.* (t') J-n q,~0).
J-D
12
and obtain a fIfth-degree characteristic equation for A. of the
fann 5
~ A(a Iv ) (My l =0 ~ 1 n".L .L i=O
(2.39)
where the coefficients Ai are complicated but explicit expressions
that depend on the stationary state parameters, as well as the
frequency <In of the n-th Fourier mode. The steady state that is
being probed, is stable if and only if the real parts of all five
eigenvalues Anj are negative for all values of n. The appearance of
a positive real part of an eigenvalue tor a given value of n is an
immediate indication that the selected steady state in unstable
against a small perturbation. The instability manifests itself with
the growth of sidebands at frequencies ±<In/y.L. Hence the field
amplitude FI(z',t') departs from its unifann stationary
configuration and develops a space-time structure.
A numerical study17 of the fifth-degree equation (2.39) shows that
for every value of CJ.r/Y.L, three of the five eigenvalues have
large and negative real parts. They correspond to the evolution of
the linearized equations that are most closely related to the
atomic dynamics. These eigenvalues, labelled "atomic eigenvalues"
for convenience, can be ignored for the purpose of instabilities
studies. The remaining two eigenvalues, for every value of n, can
be identified as being responsible for the linearized evolution of
the field amplitude and phase. The telltale sign that motivates
this identification is that the resonant phase eigenvalue has a
zero real part because of the marginal stability of the phase
variable in steady state for the resonant mode.
While the only physically meaningful eigenvalues are the ones for
which ex.,. is an integer multiple of aIt it is convenient to
regard this parameter as a continuous variable and to plot Re A. as
a continuous function of ex.,.. Two typical results are shown in
Figs. 2.5. Here we plot the two largest real parts of the
linearized ei~envalues as functions of the mode frequency an'
viewed as a continuous variable, for ~ AC=O and for two values of
the unsaturated gain. In the case shown in Fig. 2.5a all the
complex rate constants have negative real parts so that the steady
state is stable for the chosen value of the gain. Figure 2.5b
corresponds to a larger value of the unsaturated gain. In this
case, the cavity modes at frequency UJ = ±3 <Xl are unstable
(the figure displays only half of the symmetric plot). This implies
that an arbitrary perturbation at these frequencies grows
exponentially during the linear regime, and the cavity field
becomes a superposition of the steady state solution, oscillating
at frequency roc, and of the growing sidebands whose frequencies
are roc± UJ. Hence, the total output intensity shows the
characteristic beat pattern due to the interference of these
frequency components and the laser displays self-pulsing.
To learn about the nonlinear evolution, in this case, one has to
solve the full Maxwell-Bloch equations as done for example in Ref.
11 and 17. The result of a calculation of this type, corresponding
to the parameters chosen in Fig. 2.5b, shows periodic oscillations
with a frequency that departs very little from UJ even in the
nonlinear regime.
o.o!\' Re'k/Yl
6 9 ttn/Yl 12
Fig. 2.5a The two largest real parts of the eigenvalues of the
linearized equations are plotted as functions of an (in units ofy.J
viewed as a continuous variable for aL=O.8, R=O.95, a 1=3ll'
'lJr.'Yl =1.5 and /)AC=U. underthe conditions of this simulation
the steady state is staHle. The line marked (a) denotes the
amplitude eigenvalue; the line marked (p) denotes the phase
eigenValue.
13
0.05
Re'A.ly.l
-0.10 0 3 6 9 cxn/y.l 15
Fig. 2.5b Same as Figure 2.5a with aL=2.0. For sufficiently high
values of the gain, the real part of the amplitude eigenvalue
becomes positive and the steady state becomes unstable by
developing sidebands at a±3= ±3a\.
For larger values of the gain it is possible that more than one
sideband be unstable at the same time. In this case the nonlinear
dynamics can become very complicated and even develop the
characteristic signatures of deterministic chaos 11.
When SAC is either zero or small as compared with the free spectral
range of the cavity, the appearance of unstable behavior is
triggered by the positive real part of the amplitude eigenvalue.
This situation usually requires that the unsaturated value of the
gain be at least a factor of 10 larger than the threshold value for
ordinary laser action. This is a typical feature of the amplitude
instabilities in the plane-wave Maxwell-Bloch model, one that
appears to have no readily identifyable match in experimental
situations. For this reason it has remained a major stumbling block
against a straightforward interpretation of the observed behaviors
and a comparison with these simple theoretical models.
We now tum our attention to a situation such as shown in Fig. 2.4b,
where multiple steady states can exist for certain values of the
detuning parameter. The linear stability
Fig.2.6a
14
Re'A.lY.l
o.ook----------------l
p
o 2 4
For increasing values of the detuning parameter the phase
eigenvalue eventually develops a positive real part. In this case
the detuning is not sufficiently large to create an instability.
The parameters are aL=O.5, R=O.95, u\=31-1' ru/h =0.8, and
°AC=O·7.
0.05,---------------,
Fig.2.6b Same as Fig. 2.6a with 0Ac=1.2.
analysis [Le. the solution of the characteristic equation (2.39)]
shows that the lasing mode retains control of the laser emission
well into the region of coexistence of multiple steady states when
the ratio 1'I1/'y.L is sufficiently smaller than unity, while
competition between coexisting steady states and self-pulsing is
more typical of situations where 'Y1I/'y.L is closer to unity. In
both cases the mechanism for the emergence of an instability can be
quite different from the one described above, especially if the
unsaturated gain is so small than an amplitude instability cannot
emerge.
In Figures 2.6 we show again the two largest parts of the
linearized eigenvalues plotted as functions of the mode frequency
<Xn viewed as a continuous variable. For small values of the
detuning, the lasing mode is stable (Fig. 2.6a); for larger values
of 5AC' in the region of coesistence of multiple steady states, one
of the real parts of the eigenvalues (the one corresponding to the
cavity frequency <Xl in this figure) becomes positive, and the
lasing mode develops an instability.
At this point the subsequent evolution of the system is determined
by the stability properties of the coexisting stationary solution.
If the coexisting solution is stable, the laser operation is
transferred from the unstable to the stable state, with a
discontinuous change of the asymptotic output intensity and
operating frequency. If, on the other hand, the coexisting solution
is itself unstable, undamped pulsations develop (the system cannot
fmd stable fIXed points). A schematic illustration of the behavior
of the intensity and operating frequency as a function of the
detuning parameter is shown in Fig. 2.7 for the case in which the
coexisting mode is stable. The self pulsing solutions corresponding
to YU/'y.L of the order of unity tend to be simple periodic
oscillations with some distortion due to the nonlinearity of the
problem. An example is given in Ref. 17.
Direct inspection of the behavior of the linearized eigenvalues in
Figs. 2.5b and 2.6b shows immediately that, while both kinds of
unstable behaviors are of the multimode type, in the sense that
they involve the running laser mode and at least a pair of
sidebands, the type of eigenvalues whose real parts become positive
is different in these two cases. In Fig. 2.5b the unstable
eigenValue is associated with the linear response of the field
amplitude; in the case of Fig. 2.6b the unstable eigenvalue is
connected with the field phase. For this reason, it makes sense to
label the first type of behavior "amplitude instability" and the
second "phase instability" .
We note, in closing, that for sufficiently large values of both the
gain and the detuning parameters, both amplitude and phase
instabilities can develop. This situation is more complicated and
it has not been analyzed in any detail, as far as we know.
'1m. We Frequency
Fig. 2.7a Schematic behavior of the output intensity as a function
of the detuning parameter
Freq. Offset
-- Detuning
Fig. 2.7b Schematic behavior of the laser operating frequency as a
function of the detuning parameter
15
3. The single-mode laser equations
The sin!!le-mode laser model is the oldest to have displayed the
appearance of unstable behavior8,'J,I'2. In dealing with this
aspect of the instability problem two questions come immediately to
mind:
i) how can one talk about modes in a lossy resonator, and
ii) how does the single-mode model follow from the exact
Maxwell-Bloch equations?
We have already answered the first question in Section 2 when we
showed how to transform the boundary conditions (2.2) into the
standard periodicity condition (2.25). The resulting infinite set
of coupled mode equations for the Fourier amplitudes fn(t'),
Pn(t'), etc. are quite removed from the single-mode model and an
arbitrary truncation of the set offers no clues as to the physical
conditions under which this limit is a valid approximation.
Sufficient conditions for the validity of the single-mode model are
brought about by the following steps. First, we impose the uniform
field limit (2.20), whence the modal equations for the field and
atomic amplitudes take the form given by Eqs. (2.30). Next, we
require that all the modal amplitudes fn' Pn' and <In, with n '"
0, vanish identically at t' = O. Finally we impose the limit
~-+O c
(3.1)
whose effect is to move every cavity mode, except for the resonant
one, so far away from the atomic gain line that any active role on
their part is effectively excluded.
Under these conditions the Maxwell-Bloch equations reduce to the
single-mode model
df dt' = - 1( (f + 2Cp) (3.2a)
:. =-YPd+p) (3.2b)
dd (h'=-~I(-fp+d-l) (3.2c)
where, for simplicity, we have made the further assumption of
resonance between the center of the atomic gain line and the only
remaining cavity mode. The single-mode laser equations, also kno\yn
as the Haken-Lorenz model, are unstable under the following two
conditions'J,12
16
i)
ii)
20
2C
10
(3.3a)
(3.3b)
Fig. 3.1 Instability boundaries of the single-mode laser model are
plotted in the plane of the parameters K"fy.J. and 2C for (a)
II{'Y.l =2.0, (b) 'YII/'Y.!. =1.0, (c) 'YuI'Y.l =0.1. The optimum
(i.e. the most unstable) configuration corresponds to 1CI'Y.l =3
and 1cthr = 9.
Fig. 3.2
o 40 80
Time evolution of the output intensity according to Eqs. (3.2).
When 2C exceeds the second threshold value, the laser output often
develops undamped chaotic pulsations. The parameters chosen in this
simulation are 2C=15.0, Iiry1. =0.5, lCIY1. =4.
The bad cavity condition is a constraint for the cavity linewidth
relative to the sum of the atomic decay rates, and should not be
interpreted literally as an indication that the cavity design must
be of poor optical quality (low finesse). It is true in fact, that
for most optical lasers, the atomic linewidth 11. is so large, that
Eq. (3.3a) can only be satisfied by a cavity with a very high
damping rate. However, as discussed for example in Professor C.O.
Weiss' lectures, most far infrared lasers are characterized by very
narrow atomic linewidths, so that the actual quality of the optical
cavity, in this case, does not have to be poor in absolute terms. A
useful graphical display of Eqs (3.3b) is shown in Fig. 3.1 which
displays the domain of instability of a single-mode laser in the
space of the parameters 2C and KI"fJ.. For each value of the ratio
~t"Y1.' the solid curves represent the instability boundary of the
laser. It is clear that, even under the most favorable conditions,
the ratio Kl11. must be sufficiently larger than unity, and that
the gain must exceed the ordinary laser threshold value by about a
factor of 10. Note that the limiting boundary corresponding to
111'11. ~ 0 is very close to curve (c) of Fig. 3.1, so that very
small values of 11(11. do not help significantly in lowering the
instability threshold.
It is interesting to summarize some of the dynamical aspects of the
Haken-Lorenz model. For low values of the unsaturated gain, the
laser operates stably and approaches its steady state
monotonically; for higher values of the gain , the approach to
steady state is characterized by damped relaxation oscillations.
Eventually, above the instability threshold (often called the
second laser threshold), undamped pulsations set in. Usually they
take the form of erratic oscillations as shown in Fig. 3.2, but for
selected values of the parameters, periodic behaviors can also
developI5. These are common when 111'11. is smaller than about 0.2.
In Fig. 3.3 we show a periodic solution of the Haken-Lorenz
equations whose electric field amplitude executes symmetric
oscillations in the positive and negative direction. This
Fig.3.3a
+10
f
t
-10
Time dependence of the electric field amplitude for 2C=12.0,
YII'Y1. =0.14, 1CIY1. =4. This is an example of a symmetric
solution.
17
+o.S
-10 +10
I -o.S . Fig. 3.3b Projection of the phase space trajectory shown
in Fig. 3.3a in the (f,d) plane.
type of solution is called symmetric for reasons that are
especially clear if one inspects the corresponding phase-space
portrait (Fig. 3.3b). A small change of 1(1'11. brings about a
symmetry breaking transition (Fig. 3.4), leading to unequal
positive and negative excursions of the electric field, as shown
more clearly by the phase portrait (Fig. 3.4b).
It is interesting to note that the intensity patterns corresponding
to Figs. 3.3a and 3.4a would lead one to believe that a period
doubling bifurcation has taken place. In fact, this is not the
case. Because the electric field amplitude is not directly
observable. however, one
+10
f
-10
Fig. 3.4a Time dependence of the electric field amplitude for 2C=
12.0, YlI/y.L =0.17, Kly.L =4. This is an example of an asymmetric
solution.
+O.S
-10-+-~----W-------L--
-o.S
Fig. 3.4b Projection of the phase space trajectory shown in Fig.
3.4a in the (f,d) plane.
18
must ask how a symmetry breaking transformation can be identified
in a traditional measurement. The answer lies in the application of
heterodyne mixing techniques with a stable reference source.
Consider, in fact, the superposition of the signal of 'interest
with a coherent reference wave. The total electric field at the
detector takes the form
-iClll -i0l0l I\ol(t) = f(t) e + A e + c.C. (3.4)
where A and roo are the amplitude and the carrier frequency of the
reference signal, respectively. The photocmrent produced by a
standard square-law detector is given by
{ -i (00 - 000> 1 i (00 - 00 ) I} \t) = coost x IAI2+ If(t)12+
f(t) A * e + f(t)* A e 0 (35)
The power spectrum of i(t) contains a delta-function contribution
at zero frequency, due to the constant background intensity IAI2,
the homodyne power spectrum of If(t)12 in the low frequency range,
and finally the spectrum of the electric field f(t) centered at the
frequency difference leo-rool of the two optical carriers. The main
difference between a symmetric and an asymmetric solution is that
the former has a zero average value for the electric field, while
the latter does not. Hence, the heterodyne spectrum of a symmetric
solution will display symmetric frequency components around
lro-rool , but no spectral power at leo-rool, while the
distinguishing signature of an asymmetric solution will be the
appearance of a line at the frequency leo-rool, in addition to
symmetric sidebands.
Further changes of 'YU''Y.L produce real bifurcations and
eventually chaos. An example of a period doubled asymmetric
solution is shown in Fig. 3.5. A period-5 window in a chaotic
region is shown in Fig 3.6.
+10
f
i
-10
Fig. 3.5a Time dependence of the electric field amplitude for 2C=
12.0, 'Yll/lL =0.19, 1(/'Y.L =4. lbis is an example of an
asymmetric solution of period 2.
+0.5
-10-+---'-----\\t-----'-----J4-l
Fig. 3.5b Projection of the phase space trajectory shown in Fig.
3.5a in the (f,d) plane.
19
-0.5
-10 -\i----L.----W------'---+H-fI1
Fig . 3.6 Projection of the phase space trajectory of a period-5
solution in the (f.d) plane.
4. A brief overview of the experimental situation
Detailed and up to date accounts of the experimental advances in
the study of laser instabilities are given elsewhere in this
volume. Here, we provide an infonnal survey of a few experimental
facts for the purpose of highlighting some of the successful
predictions and some of the open problems of the Maxwell-Bloch
theory.
4.1 Multimode instabilities
An unambiguous experimental verification of the multimode
instabilities of the amplitude type is still not available.
According to the theory, the threshold gain for amplitude
instabilities is very high so that effects which are ignored by the
present formulation are likely to become important well before the
instability threshold is reached. A very interesting type of
multimode instability was discovered by Hillman, Krasinski, Boyd
and Stroud26 in 1984 and has been made the subject of renewed
experimental efforts by the Rochester27, and Brown University28
groups. The most interesting features of the experiments discussed
in Ref. 27 can be summarized as follows.
An Argon-pumped dye laser begins to lase tinder resonance
conditions with about 2 Watts of Argon pump power. At a pump level
of about 5.5 Watts the operating laser line disappears, to be
replaced in an apparently discontinuous way by two symmetric
sidebands that move progressively further apart at higher pump
power. When the experiment is performed out of resonance, a new
frequency component appears at the instability threshold, and
pushes the operating laser line away from it with the result that,
again, bichromatic emission is observed. The two monochromatic
components of the laser output intensity are spaced by enormous
amounts, of the order of 100 A or more. The power spectra
calculated from numerical solutions of the Maxwell-Bloch equations
l1 provide a very poor representation of the observed data because,
first of all, the experimental instability thresholds, even under
resonant conditions, appear to be much lower than the ones
predicted theoretically and, second, because the operating laser
line never disappears in the theoretical simulations. If the
instability reported in Refs. 26 and 27 is of the Risken-Nummedal
type, it would seems that substan~~l theoretical revisions will be
needed. In addition, some other current experimental facts suggest
that bidirectional propagation may also playa role, so that an
extension of the current models to include bidirectional
propagation is indicated.
A very interesting theoretical proposal29 suggests that the origin
of the observed multimode instability could lie in the complicated
band structure of the ground state of the dye molecules. An
important favorable feature of this work is that it predicts a
lower instability threshold than the standard two-level models. It
remains to be seen if the dynamical consequences of this
theoretical suggestion will also agree with the experimental
results.
MuItimode phase instabilities, unlike their amplitude counterparts,
appear to have received an adequate experimental verification 18
with a homogeneously broadened C02
20
laser. Detuning scans at low total gas pressure and low cavity
losses have provided good qualitative confirmation of the type of
theoretical predictions displayed in Fig. 2.4a under experimental
conditions such that the separation between adjacent cavity modes
is large enough that at most only one steady state at a time is
possible for all values of BAC' The laser is set into a resonant
configuration ( B AC=O) by adjusting the end mirrors until one
obtains the largest output intensity. Upon increasing the detuning
parameter, the output intensity decreases monotonically until the
losses overcome the gain, and the output intensity drops to zero. A
further increase in BAC pushes the next cavity mode under the gain
curve, and the laser intensity begins to grow again toward its
maximum value, which signals the reappearance of the resonance
condition. This situation corresponds to the expected behavior of a
homogeneously broadened laser under conditions where unique values
of the intensity and operating frequency exist for each setting of
the detuning parameter.
At higher pressure in the active volume and higher cavity losses,
the separation between adjacent resonances can be made comparable
to or smaller than the gain linewidth. In this case, more than one
cavity mode can fulfill the threshold condition for laser action
for selected values of BAC [See Fig. 2-4b] and, correspondingly,
more than one steady state solution becomes available. If, under
these conditions, one again adjusts the laser in a resonant
configuration and then increases the detuning parameter, the output
intensity, at first, decreases steadily and then undergoes a sudden
jump to a value comparable to its maximum level. An accompanying
change in operating frequency is revealed by the transient
modulation of the output intensity during the switching process
whose frequency matches the free spectral range of the resonator. A
reversal of the detuning scan shows hysteresis as predicted by the
theory. We note, in addition, that a C02 laser is characterized by
a very small value of 111'11. which, according to the theory,
should favor the appearance of discontinuous transitions and
hysteresis, rather than competition and undamped pulsation. This is
just what one finds experimentallyl!!. A word of caution should be
advanced when comparing these results with the theoretical
predictions because the laser used in the work of Ref. 18 is of the
Fabry-Perot type. It is still rewarding to find a good qualitative
match between theory and experiments.
4.2 Single-mode instabilities
The Haken-Lorenz model is predicted to undergo unstable
oscillations under stringent experimental conditions8,9,12: the
cavity linewidth must be larger than the sum of the atomic
relaxation rates and the unsaturated gain must exceed its laser
threshold value by at least a factor of 10. If we consider that for
most lasers it is difficult to exceed the threshold gain by more
than about a factor of two or three, it is easy to see why gain
values such as required by the single-mode theory are usually
inaccessible.
Much to their credit, Weiss30, Lawandy31, and their respective
collaborators understood that a possible tactic for testing the
single-mode laser theory was to use far infrared (FIR) transitions
as the active lasers lines. FIR lasers are characterized by very
high gain and line widths that are much smaller than their optical
counterparts. As a result, the bad cavity condition can be
satisfied without unusually large cavity losses, and a very high
level of inversion can be produced. A potentially serious drawback
of these types of lasers is that they must be pumped by another
laser source whose coherence is likely to introduce multilevel
dyn~mical effects in the active medium whose effect may be hard to
sort out experimentally32.
While it is important to keep this caveat in the back of one's
mind, it is tempting to suggest that the experiments described in
Refs. 29 [see especially Ref. 29h] may have shown some of the
characteristic features of the Haken-Lorenz model: high instability
thresholds, chaos, periodic oscillations for small values of
111'11., bifurcations etc. Clearly a detailed qualitative
comparison will have to wait until the effects due to the coherence
of the pump laser have been clarified.
5. Laser instabilities and transverse effects
The plane-wave Maxwell-Bloch model discussed in the previous
sections is a formulation flexible enough to provide detailed
insights into the working mechanism of a
21
laser. Yet, for all the important contributions that this model has
made to laser physics, it has also shown a consistent pattern of
quantitative and often also qualitative disagreement with the
experimental facts, especially under unstable conditions.
The Maxwell-Bloch equations do predict dynamic instabilities, but
the assertion that most of these phenomena can occur only for very
high values of the pump parameter is a major obstacle against the
interpretation of the experimental results in terms of this
theoretical framework.
An additional problem arises with the unconditional stability of
the rate equation limit. Accepting this result forces the
conclusion that all laser instabilities are manifestations of
atomic coherence, a difficult proposition to hold in the face of
the behavior of Ruby, Nd: Y AG, CO2 and other lasers which are well
known to develop instabilities, and for which the validity of the
rate equation description appears to be well justified.
Indications that not all is well with the plane-wave approximation
came from, among other sources, the lack of qu!ntitative agreement
between the predictions of the plane-wave studies of optical
bistability 3 and the results of the absolute measurements by
Kimble and collaborators under steady state conditions34.
Additional s~ong disagreements surfaced between the dynamical
predictions of the plane-wave theory 5 and the observed pulsation
pattems36.
In fact, we have known for some time that transverse effects Cjln
have a strong influence on the stationary and dynamical behavior of
driven systems37. Moreover, it is likely that deviations from the
plane-wave predictions may even be more pronounced when the optical
resonator contains an active rather than a passive system.
With this in mind, and following the footsteps of earlier attempts
at improving the descriptions based on the plane-wave
approximation38, an investigation was launched to include several
important physical asgects and design features of laser resonators
which are normally excluded from consideration :
i) the diffraction caused by the fmite cross section of the cavity
field and by its transverse variations of amplitude and
phase;
ii) the wavefront distortion introduced by curved optical surfaces;
iii) the lack of transverse uniformity in the population
inversion.
The Maxwell-Bloch equations, generalized to include diffraction, an
arbitrary pump profile, and new boundary conditions to reflect the
presence of curved mirrors, have been analyzed in steady state and
used to investigate the linear response of the laser to
infmitesimal perturbations around the steady state. With the help
of a suitable extension of the uniform field limit, the main
results reported in Ref. 6 can be summarized as follows:
a)
b)
c)
d)
22
the steady state has a single longitudinal and transverse mode
structure; this is a direct generalization of the corresponding
statement in the plane-wave limit;
depending on the geometry of the resonator and the parameters of
the active medium, single or multiple steady state solutions may
exist; this is also true in the plane-wave limit;
the instability threshold can be very close to the ordinary laser
threshold, depending on the parameters of the system. With regard
to the instability threshold one of the most influential parameters
is the ratio of the transverse dimension of the pumped medium to
the beam waist. When this ratio is large, the instability threshold
is small, and viceversa. This result is a significant improvement,
relative to the plane-wave theory, where the instability threshold
is largely independent of the geometrical parameters of the
resonator;
instabilities persist in the rate equation limit and even under the
full adiabatic elimination regime. This result stands in striking
contrast with the behavior of the Maxwell-Bloch equations which are
always stable under these conditions.
These facts, taken together, give encouraging indications that the
low instability thresholds observed experimentally may be related
to the nonuniform transverse profile of the field and of the atomic
variables, and that control of these instabilities may be achieved
with appropriate tailoring of the geometrical and pump parameters
of the system
Acknowledgements
This work was partially supported by a NATO travel grant and by the
European Economic Community (EEC) twinning project on Dynamics of
Nonlinear Optical Systems.
References
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Verlag, Berlin, 1982. 15. L.M. Narducci, H. Sadiky, L.A. Lugiato,
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(1982); (b) S.T. Hendow
and M. Sargent, TIl, J. Opt. Soc. Am. !l2. 84 (1985). 17. L.M.
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Phys.
Rev. An. 1842 (1986). 18. J.R. Tredicce, L.M. Narducci, D.K. Bandy,
L.A. Lugiato and N.B. Abraham, Opt.
Comm. jQ, 435 (1986). 19. (a) L.W. Casperson, IEEE J. Quantum
Electron. OE-14, 756 (1978); (b) L.W.
Casperson, Phys. Rev. A21, 911 (1980); (c) L.W. Casperson, Phys.
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L.W. Casperson, J. Opt. Soc. Am.!l2. 993 (1985).
20. (a) J. Bentley and N.B. Abraham, Opt. Comm. 41, 52 (1982); (b)
M. Maeda andN.B. Abraham, Phys. Rev. £UQ, 3395 (1982); (c) N.B.
Abraham, T. Chyba, M. Coleman, R.S. Gioggia, N. Halas, L.M. Hoffer,
S.N. Liu, M. Maeda and J.C. Wesson, in
23
Laser Physics. J.D. Harvey and D.P. Walls, eds., Springer Lecture
Notes in Physics, Vol. 182, Springer Verlag, Berlin, 1983, p. 107.;
(d) R.S. Gioggia and N.B. Abraham, Phys. Rev. Lett. Sl, 650 (1983);
(e) R.S. Gioggia and N.B. Abraham, Opt. Comm. £L 278 (1983); (0
R.S. Gioggia and N.B. Abraham, Phys. Rev. A29, 1304 (1984), (g)
M.P.H. Tarroja, N.B. Abraham, D.K. Bandy, and L.M. Narducci, hys.
Rev. A34, 3148 (1986).
21. This philosphy was developed during the very early days of the
Laser. See, for example, (a) H. Haken and H. Sauermann, Z. Phys.
ill, 261 (1963); (b) A.N. Oraevsky, Molecular Oscillators, Nauka,
Moskow, 1964; (c) W.E. Lamb, Jr., Phys. Rev. 134, 1429 (1964); (d)
P.T. Arecchi and R. Bonifacio, IEEE I. Quantum Electron. DE-I, 169
(1965); (e) Ya.I. Khanin, Dynamics ofOuantum Oscillators, Soviet
Radio, Moscow, 1975.
22. L.A. Lugiato, in Progress in Optics, Vol. XXI, E. Wolf, ed.,
North Holland, Amsterdam, 1984, p. 69.
23. (a) L.A. Lugiato, L.M. Narducci, and M.F. Squicciarini, Phys.
Rev. A34, 3101 (1986); (b) R.R. Snapp, Ph.D. dissertation,
University of Texas at Austin, 1986 (unpublished).
24. (a) R. Bonifacio and L.A. Lugiato, Lett. Nuovo Cimento, 21, 505
(1978); (b) R. Bonifacio and L.A. Lugiato, Lett. Nuovo Cimento, 21,
517 (1978).
25. (a) L.A. Lugiato, Opt. Comm.ll, 108 (1980); (b) L.A. Lugiato,
Z. Phys. B41, 85 (1980).
26. L.W. Hillman, J. Krasinsky, R.W. Boyd and C.R. Stroud, Jr.,
Phys. Rev. Lett. ll, 1605 (1984).
27. (a) L.W. Hillman, R.W. Boyd, J. Krasinsky and C.R. Stroud, Jr.,
in Optical Bistability 2, C.M. Bowden, H.M. Gibbs and S.L. MacCall,
eds., Plenum Press, New York, 1984, p. 305; (b) L.W. Hillman, J.
Krasinky, K. Koch, and C.R. Stroud, Jr., J. Opt. Soc. Am. B2, 211
(1985); (c) C.R. Stroud, Jr., K. Koch, and S. Chakmakjian, in
Optical Instabilities, R.W. Boyd, M.G. Raymer and L.M. Narducci,
Cambridge University Press, Cambridge, 1986, p. 274; (d) C.R.
Stroud, Jr., K. Koch, S. Chakmakjian and L.W. Hillman, in Optical
Chaos, J. Chrostowski and N.B. Abraham, eds., SPIE Vol. 667, SPIE,
Bellingham, 1986, p. 47.; (e) S. Chakmakjian, K. Koch and C.R.
Stroud, Jr., Digest of the International Workshop on Instabilities,
Dynamics and Chaos in Nonlinear Optical Systems, ETS Editrice,
Pisa, 1987, p. 119.
28. N.M. Lawandy, R.S. Afzal and W.S. Rabinovich, private
communication. 29. Fu Hong and H. Haken, Digest of the
International Workshop on Instabilities,
Dynamics and Chaos in Nonlinear Optical Systems, ETS Editrice,
Pisa, 1987, p. 121. 30. (a) C.O. Weiss and W. Klische, Opt. Comm. ~
413 (1984); (b) C.O. Weiss and W.
Klische, Opt. Comm. n, 47 (1984); (c) C.O. Weiss, J. Opt. soc. Am.
B2, 137 (1985); (d) C.O. Weiss, W. Klische, P.S. Ering and M.
Cooper, Opt. Comm. ~ 405 (1985); (e) W. Klische and C.O. Weiss,
Phys. Rev. A31, 4049 (1985); (0 E. Hogenboom, W. Klische, C.O.
Weiss and A. Godone, Phys. Rev. Lett . .5.5" 2571 (1985); (g) C.O.
Weiss, P. Spiezeck, H.R. Telle and H. Li, Opt. Comm. ~ 193 (1986);
(h) C.O. Weiss, K. Siemsen and T.Q. Wu, Digest of the International
Workshop on Instabilities, Dynamics and Chaos in Nonlinear Optical
Systems, ETS Editrice, Pisa, 1987, p.105.
31. (a) N.M. Lawandy and G.A. Knopf, IEEE I. Quantum Electron.
QE:..l.6, 701 (1980); (b) N.M. Lawandy, IEEE I. Quantum Electron.
QIH.B" 1992 (1982); (c) N.M. Lawandy, J. Opt. Soc. Am. Bl. 108
(1985).
32. (a) M.A. Dupenuis, R.R.E. Salomaa, and M.R. Siegrist, Opt.
Comm. & 410 (1986); (b) S.C. Mehendale and R.G. Harrison, opt.
Comm. ®, 257 (1986); (c) j. Pujol, F. Laguana, R. Corbalan, R.
Vilaseca, Digest of the International Workshop on Instabilities,
Dynamics and Chaos in Nonlinear Optical Systems, ETS Editrice,
Pisa, 1987, p. 109; (d) R.O. Harrison, J.V. Moloney, J.S. Uppal,
Digest of the International Workshop on Instabilities, Dynamics and
Chaos in Nonlinear Optical Systems, ETS Editrice, Pisa, 1987, p.
116.
33. R. Bonifacio and L.A. Lugiato, Opt. Comm..l2. 172 (1976); see
also Ref. 22. 34. (a) D.E. Grant and H.I. Kimble, Opt. Lett. L 353
(1982); (b) D.E. Grant and H.J.
Kimble, Opt. Comm. 44,415 (1983); (c) A.T. Rosenberger, L.A. Orozco
and H.I. Kimble, Phys. Rev. A28. 2569 (1983).
35. L.A. Lugiato, L.M. Narducci, D.K. Bandy and C.A. Pennise, Opt.
Comm.~, 281 (1982).
24
36. L.A. Orozco, A.T. Rosenberger and R.I. Kimble, Phys. Rev. Lett.
a 2547 (1984). See also lectures by Professor A.T. Rosenberger in
this volume.
37. See for example: (a) R.I. Ballagh, I. Cooper, M.W. Hamilton,
W.J. Sandle and D.W. Warrington, Opt. Comm. n, 143 (1981); (b) P.D.
Drummond, IEEE I. Quantum Electron. QE-17, 301 (1981); (c) W.I.
Firth and E.M. Wright, Opt. Comm. ~ 223 (1982); (d) I.V. Moloney
and H.M. Gibbs, Phys. Rev. Lett. !!i, 1607 (1982); (e) L.A. Lugiato
and M. Milani, Z. Phys. B50, 171 (1983).
38. (a) L.W. Casperson, IEEE I. Quantum Electron. QE-lO, 629
(1974); (b) L.W. Casperson, I. Opt. Soc. A. QQ. 1373 (1976); (c) G.
Stephen and M. Trumper, Phys. Rev. A28, 2344 (1983); (d) V.S.
ldiatulin and A.V. Uspenskii, Qpt. Acta 21, 773 (1974); (e) L.A.
Lugiato and M. Milani, Opt. Comm. 44, 57 (1983); (t) L.A. Lugiato,
R.I. Horowicz, G. Strini and L.M. Narducci, Phys. Rev. A30. 1366
(1984).
25
and Dept. of Physics - University of Firenze - Italy
The onset of deterministic chaos in lasers is studied by
referring
to low dimensional systems, in order to isolate the characteristics
of
chaos from the random fluctuations due to the coupling with a
thermal
reservoir. For this purpose, attention is focused on single mode
homogeneous line lasers, whose dynamics is ruled by a low number
of
coupled variables. In the examined cases, experiments and
theoretical
model are in close agreement. In particular I describe Shil'nikov
chaos,
how it can be characterized, and the strong resulting coupling
between
nonlinear dynamics and statistical mechanics.
1. COHERENCE AND CHAOS IN LASERS
Quantum optics from its beginning was considered as the physics of
coherent and intrinsically stable radiation sources. Lamb's
semiclassical theory /1/ showed the role of the e.m. field in the
cavity in ordering the phases of the induced atomic dipoles, thus
giving rise to a macroscopic polarization and making possible a
description in terms of very few collective variables. In the case
of a single mode laser and a homogeneous gain line this means just
five coupled degrees of freedom, namely, a complex field amplitude
E, a complex polarization P, and a
population inversion N. A corresponding quantum theory, even for
the
simplest laser model, does not lead to a closed set of
equations,
however the interaction with other degrees of freedom acting as
a
thermal bath (atomic collisions, thermal radiation) provides
truncation
of high order terms in the atom-field interaction /2,3,4/. The
problem
may be reduced to five coupled equations (the so-called
Maxwell-Bloch
equations) but now they are affected by noise sources to account
for the
coupling with the thermal bath /5/. As they are stochastic, or
Langevin,
equations, the corresponding solution in closed form refers to a
suitable weight function or phase space density. In any case the
average
motion matches the semiclassical one, and fluctuations playa
negligible
27
role if one excludes the bifurcation points where there are changes
of stability in the stationary branches. Leaving out the peculiar
statistical phenomena which characterize the threshold points and
which suggested a formal analogy with thermodynamic phase
transitions /6/ the main point of interest is that a single mode
laser provides a highly stable or coherent radiation field.
From the point of view of the associated information, the standard
interferometric or spectroscopic measurements of elassical optics,
relying on average field values or on their first order correlation
functions, are insufficient. In order to characterize the
statistical features of quantum optics it was necessary to make
extensive use of photon statistics /7,8/.
From a dynamical point of view, coherence is equivalent of having a
stable fixed point attractor and this does not depend on details of
the nonlinear coupling, but on the number of relevant degrees of
freedom. Since such a number depends on the time scales on which
the output field is observed, coherence becomes a question of time
scales. This is the reason why for some lasers coherence is a
robust quality, persistent even in presence of strong
perturbations, whereas in other cases coherence is easily destroyed
by the manipulations common in the laboratory use of lasers, such
as modulation, feedback or injection from another laser.
Here I give a general presentation of low dimensional chaos in
lasers. For a more complete approach to the problem, I refer to a
recent monograph on the subject /9/.
We focus on those situations in quantum optics which permit close
comparison between experiments and theory. By purpose I do not
tackle the vast class of inhomogeneously broadened lasers, where it
is extremely difficult to derive close correspondences between
experiments and theory because of the large number of coupled
degrees of freedom.
If we couple Maxwell equations with Schrodinger equations for N
atoms confined in a cavity, and expand the field in cavity moC'es,
keeping only the first mode which goes unstable, this is coupled
with the collective variables P and ~ describing the atomic
polarization and population inversion as follows (E being the
slowly varying mode ampli tude) :
E = - kE + gP
011 0
For simplicity we consider the cavity frequency at resonance with
the atomic resonance, so that we can take E and P as real variables
and we have three coupled equations. Here k, tL' 111 are the loss
rates for field, polarization and population, respectively; g is a
coupling constant and Ao is the population inversion which would be
established
28
by the pump mechanism in the atomic medium in the absence of
coupling. While the first equation comes from Maxwell's equations,
the other two imply the reduction of each atom to a two-level atom
resonantly coupled with the field.
The presence of loss rates means that the three relevant degrees of
freedom are in contact with a "sea" of other degrees of freedom. In
principle, Eqs. (1) could be deduced from microscopic equations by
statistical reduction techniques /5/.
The similarity of the Maxwell-Bloch equations (1) to the Lorenz
equations /10/ would suggest the easy appearence of chaotic
instabilities in single-mode, homogeneous-line lasers. Indeed the
Lorenz model is a suggestive example of the general fact that a
nonlinear coupling of at least three dynamical degrees of freedom
may induce instabili ties in the" motion, which in such cases
becomes irregular. However time scale considerations rule out the
full dynamics for most of the available lasers. The Lorenz
equations have damping rates within one order of magnitude. In
contrast, in most lasers the three damping rates are wildly
different from one another.
The following classification has been introduced /11/
Class A (e.g., He-Ne, Ar+, Kr+, dye): 1 .. "~ ~II >'>
k.
Class B
The two last equations can be solved at equilibrium (adiabatic
elimination procedure) and one single nonlinear field equation
describes the laser. N=l means a fixed point attractor, hence
coherent emission.
(e.g. , ruby, Nd, CO2 ): 1'.J. ';>;;. Ie. ~ a'" Only the
polarization is adiabatically eliminated and the dynamics is ruled
by two rate equations for the field and population. N=2 allows also
for periodic oscillations.
Class C The complete set of eqs. (1) has to be used, hence
Lorenz-like chaos is feasible, whenever 'Y "~ 'I' ~ k.
OJ. fJ"
We have carried out a series of experiments on the birth of
deterministic chaos in CO2 lasers (Class B). In order to increase
by at least 1 the number of degrees of freedom, we have tested the
following configurations.
(i) Introduction of a time dependent parameter to make the system
non autonomous /12/. Precisely, an electro-optical modulator
modulates the cavity losses at a frequency near the proper
oscillation frequency ..Q. provided by a linear stability analysis,
which for a CO2 laser happens to lie in the 50-100 KHz range,
making it easy to take an accurate set of measurements.
ii) Injection of signal from an external laser detuned with respect
to main one, choosing the frequency difference near the above
mentioned
Jr.L • With respect to the external reference the laser field has
two
29
quadrature compQnent~ whiCh repre3ent two dynamie~l v~~iQblee.
Hence we reach N = 3 and observe chaos /11/.
(iii) Use a bidirectional ring, rather than a Fabry-Perot cavity
/13/. In the latter case the boundary conditions constrain the
forward and backward waves, by phase relations at the mirrors, to
act as a single standing wave. In the former case the forward and
backward waves have just to fill the total ring length with an
integer number of wavelengths but there are no mutual phase
constraints, hence they act as two separate variables. Furthermore,
when the field frequency is detuned wi th respect to the center of
the gain 1 ine, a complex population grating arises from
interference of the two counter-going waves, and as a result the
dynamics becomes rather complex, requiring N ~ 3 dimensions.
(iv)Add an overall feedback, besides that provided by the mirrors,
by modulating the losses with a signal provided output intensity
/14/. If the feedback has a time comparable with the population
decay time, it provides equation sufficient to yield chaos.
cavity by the
constant a third
Notice that while methods (i), (ii) and (iv) require an external
device, (iii) provides intrinsic chaos. In any case, since
feedback, injection and modulation are currently used in laser
applications, the evidence of chaotic regions cautions against
optimis'tic trust in the laser coherence.
Of course, the requirement of three coupled nonlinear equations
does not necessarily restrict the attention to just Lorenz
equations. In fact none of the explored cases i) to iv) corresponds
to Lorenz chaos.
2. SHIL'NIKOV CHAOS, THE METHOD OF RETURN TIME, AND FLUCTUATION
ENHANCEMENT
Of the whole phenomenology explored in the past years, I select a
single topic of particular relevance. I report experimental
evidence of quasi homoclinic behavior characterized by pulses with
regular shapes but chaotic in their time sequence /15/. The
regularity in the shape means that the points at any Poincare
section are so closely packed that extremely precise measurements
of their position would be required to yield relevant features of
the motion. On the contrary, return times to a Poincare section
close to the unstable point display a large spread, due to the
sensitive dependence of the motion upon the intersection
coordinate. Based on such a consideration, we introduce the spread
in return times as the specific indicator of homoclinic chaos. Our
experimental data yield iteration maps of return times in close
agreement with those arising from the theory of Shil'nikov chaos
/16,17/. Thus, the test introduced in Ref. 15 appears as the most
direct one to characterize chaotic dynamical situations associated
with pulses almost equal in shape but having fluctuating occurrence
times. Furthermore, at variance with the theory, the experimenta