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
1. O. Svelto, Principles of Lasers, Plenum Press, New York, 1982. 2. See for example G. Makhov, C. Kikuchi, J. Lambe and R.W. Terhune, Phys. Rev .
.l.Q2, 1399 (1958). 3. For some early studies see: (a) R.J. Collins, D.F. Nelson, A.L. Schawlow, W. Bond,
C.G.B. Garret and W. Kaiser, Phys. Rev. Lett. i. 303 (1960); (b) D.F. Nelson and W.S. Boyle, Appl. Opt. 1. 181 (1962); (c) T.N. Zubarev and A.K. Sokolov, Sov. Phys. Doklady 2, 1006 (1965).
4. E. Snitzer and C.G. Young, in ~ Vol. 2, A.K. Levine, ed., Marcel Dekker, New York, 1968, p. 191.
5. For a modem review of the laser rate equations, see Ref. 1, chapter 5. 6. L.A. Lugiato, F. Prati, L.M. Narducci, P. Ru, J.R. Tredicce and D.K. Bandy, Phys.
Rev. A, submitted for publication. 7. Some of the early maser studies dealing with unstable behaviors include: (a) K.Y.
Khaldre and R.V. Khokhlov, Izv. Vyss. Uchebn. Zaved. Radiofiz. (USSR) 1, 60 (1958); (b) A.G. Gurtovnick, Izv. Vyss. Uchebn. Zaved. Radiofiz. (USSR) 1. 83 (1958); (c) A.N. Oraevsky, Radio. Eng, Electron. Phys. (USSR)~, 718 (1959).
8. The first instability studies of homogeneously broadened lasers include: (a) A.V. Uspensky, Radio Eng. Electron. Phys. (USSR)'&' 1145 (1963); (b) A.V. Uspenskii, Radio. Eng. Electron. Phys. (USSR) 2. 605 (1964); (c) V.V. Korobkin and A.V. Uspenskii, Sov. Phys. JETP ll, 693 (1964); (d) H. Haken, Z. Phys . .l.2.Q., 327 (1966); (e) H. Risken, C. Schmid and W. Weidlich, Z. Phys.1.2!. 337 (1966).
9. (a) A.Z. Grazyuk and A.N. Oraevsky, in Ouantum Electronics and Coherent Light, P.A. Miles, ed., Academic Press, New York, 1964; (b) A.Z. Grazyuk and A.N. Oraevsky, Radio. Eng. Electron. Phys. (USSR) 2, 424 (1964).
10. (a) H. Risken and K. Nummedal, J. Appl. Phys. 32. 4662 (1968); (b) H. Risken and K. Nummedal, Phys. Lett. 26A, 275 (1968); (c) R. Graham and H. Haken, Z. Phys. ill. 420 (1968).
11. L.A. Lugiato, L.M. Narducci, E.V. Eschenazi, D.K. Bandy and N.B. Abraham, Phys. Rev. A32, 1563 (1985).
12. H. Baken, Phys. Lett. 53A, 77 (1975). 13. E. Lorenz, J. Atm. Sci. 2Q, 130 (1963). 14. For an extensive discussion of the Lorenz chaos, see: C.T. Sparrow, The Lorenz
Equations: Bifurcations. Chaos and Strane;e Attractors, Springer Verlag, Berlin, 1982. 15. L.M. Narducci, H. Sadiky, L.A. Lugiato, and N.B. Abraham, Opt. Comm.ll, 370
(1985). 16. (a) S.T. Hendow and M. Sargent, TIl, Opt. Comm. 43, 59 (1982); (b) S.T. Hendow
and M. Sargent, TIl, J. Opt. Soc. Am. !l2. 84 (1985). 17. L.M. Narducci, J.R. Tredicce,L.A. Lugiato, N.B. Abraham and D.K. Bandy, 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. Rev. A23, 248 (1981); (d) L.W. Casperson, J. Opt. Soc. Am.!l2. 62 (1985); (e) L.W. Casperson, J. Opt. Soc. Am.!l2. 73 (1985); (f) 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