Pulsating instabilities and chaos in lasers

Prog. Quant. Electr. 1985, Vol. 10, pp. 147-228

0079-6727/85 $0.00 + .50 Copyright ©1986. Pergamon Press Ltd.

Printed in Great Britain. All rights reserved.

PULSATING

INSTABILITIES AND CHAOS IN LASERS

ROBERT G . HARRISON* a n d D H R U B A J. BISWASt *Physics Department, Heriot-Watt University, Edinburgh EH144AS, U.K. $M.D.R.S., Physics Group, Bhabha Atomic Research Centre, Bombay 400085, India

CONTENTS 1. Introduction 1.1. General overview 1.2. Overview of laser instabilities 1.2.1. Hydrodynamic systems - - Lorenz equations 1.2.2. Laser systems - - Maxwell-Bloch equations

147 147 153 153 154

2. Mechanism of Single Mode Instability 2.1. Mode splitting 2.1.1. Spontaneous mode splitting 2.1.2. Induced mode splitting 2.2. Threshold for laser instabilities

157 158 158 159 161

3. Single Mode Homogeneously Broadened Systems 3.1. Optically pumped far infrared lasers 3.1.1. Resonant pumping 3.1.2. Detuned pumping 3.2. Near resonantly pumped mid infrared systems

167 168 168 172 174

4. Single Mode Inhomogenously Broadened Systems 4.1. He-Xe laser 4.2. He-Ne laser at 3.39#m

177 179 185

5. Single Mode Lasers with External Control Parameter 5.1. Modulation of an external field or population inversion 5.1.1. Modulated external field 5.1.2. Modulation of inversion 5.1.3. Experimental observation of instabilities in lasers with external modulation 5.2. Single mode laser with injected signal: Theory and experimental observations 5.3. Lasers with saturable absorbers

189 189 190 192 192 198 206

6. Multimode Laser 6.1. Multiaxial mode systems 6.2. Two mode systems 6.3. Transverse mode systems

206 209 210 212

7. Conclusions

221

8. References

223

1. I N T R O D U C T I O N

1.1. General Overview Instabilities in laser emission notably in the form of spontaneous coherent pulsations have been observed almost since the first demonstration of laser action. Indeed the first laser operated in 1960 by Maiman ") produced noisy spiked output even under condition of quasi-steady excitation and provided the early impetus for studies of such effects. Subsequent theoretical efforts towards understanding these phenomena have up to recently been at a modest level, due in part to the wide scope of alternative areas of fertile investigation provided by lasers during this period. However, in the last few years there has been a rapidly growing resurgence of interest in this area. This has been motivated by a major reappraisal of classical dynamical systems which has been stimulated by recent profound mathematical discoveries. It is now clear that many physical systems containing some form of nonlinearity, such as the laser, may exhibit pulsating 147 JPQE I0 :3-A

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R.G. HARRISONand D. J. BISWAS

instabilities and chaos, the behaviour of which is deterministic. Further, the discovery that chaos evolves through particular routes with well defined scenarios and that such routes are universal has stimulated experimenters to search for physical systems that exhibit these properties. These phenomena have now been observed in areas as diverse as fluid flow, chemical reactions, population ecology and superconducting devices. Perhaps the most recent newcomer to the field, the laser provides a nearly ideal model system for such investigation and already a wide range of these phenomena have been observed. Along with a proliferation of new observations many of the instabilities known to occur in these systems are now being reinterpreted in lieu of our new insight in this area. The behaviour we describe here is called deterministic since it is completely determined by the set of equations and initial conditions describing the system. In this sense deterministic chaos is then the unpredictable quasi-random behaviour displayed by some nonlinear systems governed by such equations. Turbulence is an extreme form of this behaviour. It is important to stress that this behaviour does not implicitly depend on the influence of external sources of noise or an infinite number of degrees of freedom or the uncertainty associated with quantum mechanics; the actual source of irregularity is the property of the nonlinear system. In considering deterministic behaviour one is tempted into the misconception that such behaviour must be regular since successive states evolve continuously from each other. However as early as 1892 it was shown by Poincare that particular mechanical systems where time evolution is governed by Hamiltonian equations could display chaotic behaviour. The subsequent, much later, discovery by Lorentz in 1963 ~z~that even a simple set of three coupled, first order, nonlinear differential equations can lead to completely chaotic trajectories is recognised today as a landmark in the early development of this field. This work is indeed fundamental to our understanding of laser instabilities. The temporal evolution of a system can be represented by the trajectory of a point in the phase space of its dynamical variables. In dissipative systems, such as the laser, different initial points will often evolve onto a subset of the phase space, termed an attractor. In elucidating the behaviour of a nonlinear system represented in phase space it is instructive to consider a familiar dynamical system such as a periodically forced pendulum in a frictional environment. A dynamical system in general is characterised by the fact that the rate of change of its variable is given as a function of the value of the variable at that time. The space defined by the variables is called the phase space. The pendulum's behaviour can be described by the motion of a point in a two-dimensional phase space whose coordinates are the position and velocity of the pendulum. In more complicated systems involving many variables the dimension of the phase space will however be considerably larger. If an initial condition of a dynamical system is picked at random and allowed to evolve for a long time, the system, after all the transients have died out, will eventually approach a restricted region of the phase space called an attractor. A dynamical system can have more than one attractor in which case different initial conditions lead to different types of long-time behaviour. The simplest attractor in phase space is a fixed point; the system here is attracted towards this point and stays there. This is the case for a simple pendulum in the presence of friction; regardless of its initial position, the pendulum will eventually come to rest in a vertical position. Let us now consider the case of the pendulum under the influence of an external periodic driving force. The system is then fully nonlinear in contrast to the case of the simple pendulum and leads to strikingly different behaviour. Irrespective of the initial conditions the pendulum always ends up making a periodic motion. The limitor attractor of the motion here is a periodic cycle called a limit cycle. However, when the magnitude of the driving force exceeds a certain critical value, the periodic motion of the pendulum breaks down into more complex chaotic patterns which never repeat itself. This motion represents a third kind of attractor in phase space called a chaotic or strange attractor. All these different forms of attractor are illustrated in Fig. 1,

Pulsating instabilitiesand chaos in lasers

149

0 .Y TJXEDPOINT A CONSTANTSOLUTION

LIMIT CYCLE

A SIMPLEP[RIO01¢ SOLUTION

SIRA~GE MTRACTOR A CHAOTICSOLUTION

FIG. 1. Differentform of the attractor in the phase space (N. B. Abraham,Laser Focus, May, 1983). the strange attractor is for the Lorenz-Haken system described in Section 1.2. A trajectory on a chaotic attractor exhibits most of the properties intuitively associated with random functions, although no randomness is ever explicitly added. The equations of motion are purely deterministic; the random behaviour emerges spontaneously from the nonlinear system. Over short times it is possible to follow the trajectory of each point, but over longer periods small differences in position are greatly amplified thus making the predictions of longterm behaviour impossible. Irratic and aperiodic temporal behaviour of any of the systems variables implies a corresponding continuous spectrum for its Fourier transform which is therefore also a further signature of chaotic motion. However, other factors, including noise, can also lead to continuous spectra, and distinguishing chaos from noise is one of the major problems of the field. Hence, although time series, power spectra and, as discussed below, routes to chaos collectively provide strong evidence of deterministic behaviour, further signatures are nevertheless required for its full characterisation and in discriminating it from stochastic behaviour. Here analysis of trajectories of a point in the phase of its dynamical variables is required. However this has posed the problem that for a system with N degrees of freedom it seemed that it would be necessary to measure N independent variables; an awesome task quite impossible for complex systems. To this end considerable effort has been expended in recent years by mathematicians to develop practical techniques to extract specific finite dimensional information from the limited output provided by an experiment; typically the time record of a specific physical observable (one variable of the system, say V). Based on embedding theorems(3 s) for almost every observable and time delay T the m dimensional portraits constructed from the vectors {V(tk), V(tk + T ) . . . . V(tk + (m -- 1)T}; k = 1. . . . . . ~ , will have many of the same properties as one constructed from measurements of the N independent variables, ifm 1> 2N + 1. In practice mis increased by one at a time until additional structure fails to appear in the phase portrait when an extra dimension is added from which the dimensionality of the attractor is determined. Furthermore finite noise level in an experiment which tends to obscure the details of the trajectory on an attractor, may be isolated in the reconstruction of the attractor by embedding in progressively higher dimension. From these reconstructions, Lyapunov exponents may be determined which measure the average rate of exponential separation or contraction of nearby points on the attractor. These measure intrinsically dynamical properties, unlike power spectra, and provide an unambiguous distinction between noisy and chaotic motion. A positive Lyapunov exponent implies exponential separation of nearby trajectories, the signature of deterministic chaos, while a negative exponent implies exponential contraction, which, in the absence of a positive exponent implies an attracting fixed point. A zero exponent implies motion on a limit cycle. The number of Lyapunov exponents of a particular type provides the dimensions of the manifold on which the attractor lies. Details of these methods are to be found in Refs 3-14. In view of these quite recent developments in the

150

R.G. HARRISONand D. J. BISWAS

analysis of nonlinear dynamical behaviour their full implementation is only now beginning to be felt in tlie processing of experimental data on laser systems. Indeed with the exception of two reports (10,1 ~)experiments so far have concentrated on measurements of time series, power spectra and the identification of routes to chaos in these systems. The recent discoveries that deterministic chaos proceeds via a limited number of specific routes on varying a control parameter of the nonlinear system is of profound importance. This behaviour is universal in that such routes are not restricted to a particular model description of a particular physical system. Rather, nonlinear physical systems in all branches of science which may be formally described by the same set of mathematical equations will give solutions which evolve in time in an identical way through one or other routes to chaotic motion. The unique effect of unification between many separate scientific disciplines brought about by these discoveries is only now beginning to be felt. There are at least three common routes in which a nonlinear system can become chaotic if an external control parameter is varied. These are commonly referred to as period doubling, intermittency and two frequency scenarios. Period doubling was found by Grossman and Thomae, tls) Feigenbaum (16) and Coulett and Tresser. (17) F r o m considering various simple difference equations, many of which can be reduced to simple one dimensional logistic maps, solutions have been found to oscillate between stable values, the period of which successively doubles at distinct values of the external control parameter. This continues until the number of fixed points becomes infinite at a finite parameter value, where the variation in time of the solution becomes irregular. These features are illustrated in Fig. 2 for a simple logistic map X.+

x = rX.(1

-

X,)

where X. is plotted as a function of the control parameter r. The bifurcation sequence is a single 2" cascade (period doubling) through which the attractor first becomes chaotic and eventually fills the interval via the pairwise merging of bands. The period doubling and band merging accumulates at a value rc after an infinite number of bifurcations. In the chaotic regime above r c small windows of higher period cascade are found with periods q2", with q an integer and where n denotes the degree of period doubling of a fundamental periodic orbit of period q. This scenario is referred to as a U sequence. Within each such window one also finds the associated reverse bifurcation of q2" + 1 bands merging into q2" bands. In 1979 Manneville and Pomeau (ls) discovered the so-called intermittency route to chaos.

P-2

P-4 P-8

P-7 P-8

P-4

t

(xn}

0 3.0

r

t t rc P-6

t

t

P-5 P-3

4.0

FIG.2. Period doublingbifurcation for the logisticmap X,+ 1 = r X . (1 - X,). 700 iterations plotted after an initial 500 iterations for each increment in the bifurcation parameter r. The parameter was incremented 1000 times in the interval [3,4]. For the sake of clarity and resolution, only the bifurcation diagram r in [3,4] is shown. For r in [0,1], X, = 0 is the stable behaviour; and for r in [1,3], one has a stable fixed point described by X = (r - 1)/r (Ref.8).

Pulsating instabilities and chaos in lasers

I IIIi]ll llllllUlltllltl}lllttllltl

151

tNt il

FIG. 3. Intermittent route to chaos (see text and Ref. 18).

Intermittency means that a signal which behaves regularly in time becomes interrupted by statistically distributed periods of irregular motion. The average number of these intermittent bursts increases with the external control parameter until the condition becomes completely chaotic. This temporal behaviour is illustrated in Fig. 3; the lower case showing chaos. The third route was discovered by Ruelte and Takens t19~ and Newhouse, Ruelle and Takens. ~2°) Previously Landau, ~21)and Landau and Liftshitz(22~considered turbulence in time as a limit of an infinite sequence of instabilities (Hopf bifurcations) each of which creates a new basic frequency. However, Ruelle, Takens and Newhouse showed that after only two instabilities in the third step the trajectory becomes attracted to a bounded region of phase space in which initially close trajectories separate exponentially; as such the motion becomes chaotic. An example of such behaviour is shown in Fig. 4. Although less common alternating periodic-chaotic sequences have also been observed. However the route by which the periodic state becomes chaotic has in general not been established although it is believed that transition occurs through period doubling or intermittency. Furthermore each chaotic regime can contain many subintervals that are periodic. These results along with earlier results of Lorenz, have dearly established that even simple nonlinear systems can exhibit quasi-random behaviour, and also that rather complex systems may have chaotic attractors contained in a low-dimensional subspace of their phase space. These notions are central to lasers because dissipation typically contracts the phase space available to a laser. Here the pioneer work of Haken t23) established a fundamental link between lasers and hydrodynamical systems by showing that the Maxwell-Bloch model of a simple two level single mode homogeneously broadened laser is homologous to the Lorenz model with only three degrees of freedom. The significance of this and our revised understanding of nonlinear dynamics to laser physics has caused in the last three to four years an extensive re-investigation of many lasers and re-evaluation of the instability phenomena known to occur in these systems. The predicted routes to chaos have provided important signatures by which deterministic chaos has been identified. Far from rare, such effects have been found to be quite common for various lasers; the parameter window for stable operation in some of these often being quite small

152

R.G. HARRISONand D. J. BlSWAS 10-~

1

i

10-3 ]0 5

16 3 --N

w~

i0 S w~

b)

IL,q E

a 10.3 c)

i0S

103 I0 ~ 107 0

d) l 01

,

0.2 [Hz]

03

FIG. 4. Ruelle-TakensRoute to chaos. (a) Periodicoscillationwithone frequencyand its harmonics.

(b)Quasiperiodic motion with two incommensurate frequencies and their linear combinations. (c)Non periodic chaotic motion with some sharp lines. (d) Chaos. H. L. Swinneyand J. P. Gollub, Phys. Today 31, 41 (1978). and in some instances the instabilities are found to prevail just when the lasing condition is optimum. Unexplained instabilities in lasing emission previously descarded as a nuisance are now the focus of attention in providing a more fundamental understanding of these systems and their deterministic chaotic behaviour within the framework of these new mathematical discoveries. Importantly the laser which has relatively few degrees of freedom provides one of the simplest nonlinear dynamical systems for study. Also, the instability phenomena such as the oscillation periods are on a short timescale (typically nanosecond to microsecond). Acquisition of data under essentially constant environmental condition is therefore assured; particularly important since even small extraneous perturbations may dramatically alter the subsequent temporal evolution of the instability process. In contrast, in fluid systems, historically the father figure for such studies, and in common with many other systems, instability behaviour occurs on a very slow timescale, often days to weeks. Data retrieval is therefore extremely laborious and the requirement placed on experimental system and techniques are necessarily very stringent. These attractive features of the laser have therefore identified it as a prime candidate for investigation in this multidisciplinary field. Somewhat ironically, however, it has only been very recently that instability phenomena have been observed in the simple single mode homogeneously broadened laser analysed by Haken, owing to the restrictive operating conditions required to reach the threshold for such effects. However, in the interim there has been a proliferation of beautiful results on alternative but substantially more complex albeit conventional laser systems, e.g. inhomogeneously broadened lasers, multimode lasers and lasers with external control parameters. In this review we hope to convey some of the important advances made in the last few years in

Pulsating instabilities and chaos in lasers

153

this exciting field. The interested reader is also referred to articles by Arecchi (24'25) and Lugiato t26'27) and an article by Casperson (28) for a historical survey of early studies in this field. Recent developments are to be found in the special issue of the Journal of the Optical Society of America B on Instabilities in Active Optical Media (Abraham, Lugiato and Narducci, eds) (29) and also in the most recent volume in the Springer Series in Synergetics (Arecchi and Harrison, eds). (3°) Comprehensive treatments on the more general principles of deterministic chaos are also to be found in the Synergetics Series: An Introduction ~31~ and Advanced Synergetics ~32~ written and edited by Haken. An excellent treatment is also given in a recent text on deterministic chaos by Schuster. (33) The reader is also referred to a number of review articles in this area {34-39) and also the proceedings of the International Conference on order in Chaos (Los Alamos, May 1982)t4°~ and the very useful reprint collection compiled by Cuitanoric. t41) Although we concentrate here on laser systems the reader should also be aware of significant contributions made in the complimentary area of passive (non-lasing) optical systems. Essentially a contemporary area of development, instability phenomena, notably period doubling routes to chaos, have been predicted and observed in these systems. Here the nonlinearity may be caused by saturation of an absorbing medium or that arising in a Kerr medium. Following the early predictions of instabilities in absorbing media by McCall t42) much of the pioneering theoretical contributions have been made by Ikeda (43) and subsequently extended by various workers/44-5°) Demonstration of predicted period doubling routes to chaos have been made in hybrid bistable devices,~51~Kerr medium ~52)and of special significance in simple two level nonlinear media, t53-55) This area has been recently reviewed by Gibbs t56~ and also in the various contributions in the recent volume of the Springer Series in Synergetics. t3°~

1.2. Overview of the Laser Instabilities Singularly perhaps the most important early development in the field of laser instabilities was the work of Haken t23) in establishing the mathematical analogy between lasers and fluids. In view of its special relevance to current activities in this area it is appropriate to highlight aspects of this work. 1.2.1. Hydrodynamic systems--Lorenz equations. The Lorenz model is an instructive model for describing turbulence in fluid dynamics;in particular how the initially thermalised states of a system, characterised by complete disorder may suddenly show pronounced order if one goes away from thermal equilibrium. By way of example we consider convection instabilities treated by Bernard. Here an infinitely expanded horizontal fluid layer is heated from below so that a temperature gradient is maintained (Fig. 5). This gradient if expressed in suitable dimensionless units is called the Rayleigh number R. As long as the Rayleigh number is not too large the fluid remains quiescent and heat is transported by conduction (Fig. 5a). If the number exceeds a certain value however the fluid suddenly starts to convect. Surprisingly the convection pattern is very regular and may either show rolls or hexagons (Fig. 5b). The fluid may for example rise in the middle of the cell and go down at the boundaries or vice-versa. The mechanism of the sudden

J/

/

T +Ar

/t

t

t (a)

t

t

/ /

ii

ill /Q /ll

/I /

/

i / //" /11

I

!

1/

(b)

FIG. 5. The Bernard Instability. (a) A fluid layer is heated to create a temperature gradient between the upper and lower surface. (b) Formation of rolls by the liquid beyond a critical temperature gradient (Ref.33.)

154

R.G. HARRISONand D. J. BISWAS

disorder-order transition and predictions of the stability of the cells are reasonably well described by the Lorenz model. Here the motion of the fluid is described by the Navier-Stokes equation which are nonlinear partial differential equations describing the velocity and temperature field of the fluid. To simplify the solution these fields are expanded into spatial Fourier series, the coefficients of which are still time-dependent variables. By retaining only three terms altogether of the infinite series three coupled differential equations are derived for the three variables. Since their physical meaning does not matter in the present context we may label the variables X, Y and Z, and the Lorenz equations then have the following form (1)

X = trY- aX ~'= - X Z

+rX-

= XY-

(2)

Y

(3)

bZ,

where a is the Prandtl number, the ratio of kinematic viscosity to thermo-metric conductivity, r = R/Rc (where R is the Rayleigh number, Rc the critical Rayleigh number), b = 47~2/(/t 2 + k2), and k is a dimensionless wave number. Numerical analysis of this apparently simple set of nonlinear differential equations shows that its variables can exhibit chaotic motion above a threshold value re. The Lorenz equations are nevertheless only applicable in the immediate vicinity of the transition of heat conductions to convection rolls due to the truncation imposed in the model. To describe experimentally observed chaos many more spatial Fourier components have to be retained. In contrast, the application of the Lorenz description through Haken to lasers may provide a much closer agreement with experimental observations since unlike real hydrodynamic systems lasers may be inherently low dimensional. The irregular form of a solution of these equations is shown in Fig. 6. A computer analysis of the equations shows the point X(t), Y(t), Z(t) in phase space to circle in one region for a while but then to suddenly jump into another region whereupon after some movement it jumps, seemingly randomly back into the first region and so on (Fig. 7, see also Fig. 1). Such behaviour in which the trajectory of this point remains localised in phase space but nevertheless never intersects with itself is called, as discussed earlier, a strange attractor and describes the entirely irregular temporal form of chaos. 1.2.2. Laser s y s t e m s - - M a x w e l l - B l o c h equations. In considering the laser with continuously many modes the laser equations may be expressed as -AE

+ ~t32E/t~t 2 + ~2x- ~E/~t =

47z ~- P

(4)

~2P/t~t2 + 27t3P/Ot + 092p = -209o(]021[2/h)ED

(5)

dD/t3t = 7 II(Do - D) + (2/ho%)Et3P/Ot.

(6)

The first equation follows directly from Maxwell's equation and the remaining two are material equations described in laser theory using quantum mechanics, tsT) Here E refers to electric field strength, P the macroscopic polarisation and D the macroscopic inversion density of the gain medium, o9o is the atomic transition frequency, K the cavity loss, 7_ the atomic q

I A .AA tvxA A

J--

t

FIG. 6. Exampleof chaotic motion of a variable q vs time (Ref.32).

Pulsating instabilitiesand chaos in lasers

155

(a)

= X

(b)

;×

FIG.7. Trajectoriesofthe Lorenzattractor. (a)Projectiononthez-xplane. (b)Projectiononthey-x plane (Ref.151).

linewidth, 71/-1 = T1 the atomic inversion relaxation time. Do is unsaturated inversion due to pump and relaxation processes. The Maxwell equation describes the temporal change of the electric field due to different causes; the free oscillation of the field in the cavity, the damping (x) due to semitransparent mirrors and scattering losses and the generation by oscillating dipole moments (P). In turn the electric field influences the atoms as described by the Eqns (5) and (6). In Eqn. (5) the polarisation changes are due to the free oscillation of the atomic dipole moment, its damping (7±) and due to the field amplitude (ED). Finally inversion (Eqn. 6) also changes when light is emitted or absorbed. In the simplest case we may assume the field oscillates at frequency 090 resonant with the atomic transition frequency then the field may be described by E = ei°~'a-ik"xff~(x,t)

q-

c.C.,

(7)

where the amplitude/~(x, t) is generally regarded as a slowly varying with x and t. Analogous description of P is made. By normalising /2, /5 and a l s o / ) with respect to their steady state values, Ecw, Pc,, and Dthr, viz. E = /~//~cw, P =/~//~cw, D =/)//),hr and introducing an effective pump parameter A = (Do - - D t h r ) / D t h r ,

(8)

156

R.G. HARRISONand D. J. BISWAS

detailed analysis ~sT)shows that E, P and D are real (their phases are constant( so that Eqns (4), (5) and (6) take the well-known form + 71 P = 7 ± E D

&+?ll ~

(9)

I)=711(A+ 1)-7HAEP

(10)

+ ~c + c

(11)

E = •P.

If we assume single mode operation by putting ~?Efi?x = 0 then the above equations are formally identical with the Lorenz model. Equations (1-3) and (9 11) describe at least two instabilities which have been found independently in fluid dynamics and in lasers. For A < 0 (r < 1) there is no laser action (the fluid is at rest), for A ~> 0 (r ~> 1) laser action (convective motion starts) with stable timeindependent solutions occur. Besides this well-known instability a new one occurs provided laser: ~ > 71 + 711

fluid: a > b + 1

and

and

A > (71 + 71b + K) (71 ÷ ~)/71(~c - 71 - ?It)

r > a(a + b + 3)/(a - 1 - b).

(12)

(13)

This instability leads to the irregular motion, an example of which we have shown in Fig. 6. When numerical values are used in the above conditions (12) and (13) it turns out that the Prandtl number must be so high that it cannot be realised by realistic fluids. On the other hand, in lasers and masers it is likely that conditions (12) and (13) can be met although with some difficulty. The first of these conditions ~c > 71 + 711 was originally derived in 1964 by Korobkin et al. ~58) and is termed the bad cavity requirement, since it demands that cavity loss damp the field more strongly than the damping of either the polarisation or the population. These same authors also derived the instability threshold condition, the A expression of Eqn. (13). Consideration of the effect on the above expression (Eqn. 13) of varying the various decay rates shows that the right side of the above inequality will always be greater than nine, which implies a pumping rate at least nine times above the laser threshold. Furthermore the recent calculations of Lugiato et al. (59 61) have shown that the presence of a Gaussian transverse intensity profile inside the laser cavity, which is the case for a practical laser system, may raise this instability threshold even further. Upto quite recently this has been a major hurdle to the experimental realisation of the Lorenz system in lasers. One can get high gain by lengthening the laser, but then the free spectral range falls, and the laser will go multi-mode except close to threshold. To prevent this, a short laser is necessary, which means high density of atoms, which increases 71 and 7tl, which demands an increase in ~c,which raises the threshold (actual pump rate at threshold is proportional to ~c) which demands still higher density, etc.--a vicious circle. Ironically in lasers, the first electromagnetic wave devices for which equations of the Lorenz type were developed, the problem was that ~:in the then best-available cavities was so large compared to 71 and to 711 that A (which is ,-~K/71 in this limit) was again impossibly high. Subsequent development of superconducting microwave cavities may have changed the situation. Of particular interest however is the suggestion of operating in the intermediate regime--the far infrared. As discussed by Weiss and Klische t62~ in far infrared lasers 71 and 711 are in the MHz range, which readily allows optimisation of ~ so as to minimise the threshold value of Do. Recent experimental w o r k (96-1°2' 1 0 9 - - 1 1 1 , 1 1 6 ) along with some earlier observations ~103--1 0 6 . 1 0 8 ) has indeed provided evidence of pulsating instabilities and chaotic behaviour in such lasers: suggesting that the Lorenz system of equations may soon develop an experimental significance in laser physics to match their theoretical impact.

Pulsating instabilitiesand chaosin lasers

157

In discussing these points further it is important to note the general prerequisites for the onset of deterministic chaos, that apart from nonlinear interaction there is a sufficiently large phase space; the minimum requirement being that the system possesses at least three degrees of freedom. Various of these points have been discussed by Arrechi e t al. (63) and others, t64'65~ The Maxwell-Bloch equations described above for the single mode laser with field tuned to the centre of the gain line such that both field and polarisation are real quantities satisfy the minimum condition of three independent variable Eqns (9-11), each of which has its own relaxation. However if one variable relaxes much faster than the others the stationary solution for that variable may be taken which in fact is still slowly varying because of the coupling, so resulting in a reduced number of coupled differential equations (adiabatic elimination of the fast variablest31,3 2)). This is actually the case for a large number of lasers. In fact in many systems polarisation and population inversion have relaxation times much shorter than the cavity lifetime 0'l, ~ II < x) and both variables can be adiabatically eliminated. With just one variable describing the dynamics, the laser must show necessarily a stable behaviour (fixed point in phase space). This group of lasers comprises many common systems (He-Ne, Ar ÷, Dye...). In some cases only polarisation is fast (y± > 711,K)and hence two variables describe the dynamics. In this class we find ruby, Nd and CO2 lasers which exhibit oscillating behaviour under some conditions, although ringing is always damped. Evidently most lasers are not described by the full set of Equations (9-11), and so chaotic behaviour from these systems cannot be obtained; as noted far infrared lasers appear to be an exception since its three relaxation constants may be of the same order of magnitude. For systems with less than three variables the addition of independent external control parameters to the system have been extensively considered as a means to provide the extra degrees of freedom. This was indeed previously demonstrated for Benard convectiont66~ and first recognised for laser systems by Schotz e t al. tl so~ Here active modulation of a parameter such as population inversion, field, or cavity length as well as injection of a constant field detuned from the cavity resonance, and also the use of intracavity saturable absorbers have all been considered. For multimode rather than single mode lasers intrinsic modulation of inversion (or photon flux) by multimode parametric interaction ensures additional degrees of freedom. We also note that when the field is detuned from gain centre the field amplitude, polarisation and population inversion are complex, providing five rather than three nonlinear equations for single mode systems which is more than sufficient to yield deterministic chaos for suitable parameter values. Also of particular significance here is the remarkably low threshold found for the generation of instabilities and chaos in single mode inhomogeneously broadened laser systems. Compared with homogeneously broadened systems this is attributed to the increased number of independent gain packets available in inhomogeneous systems. Pulsating instabilities and routes to chaos have also been recently reported for Raman lasers where the instability threshold is againfound to be reduced. Before reviewing in more detail these various aspects, it is instructive to consider in some detail the fundamental physical mechanisms underlying the instability phenomena that occur in laser systems. 2. MECHANISM OF SINGLE MODE LASER INSTABILITY Pulsating instabilities in the emission from a laser implies the existence of oscillations ofmore than one frequency. It is therefore not obvious how a homogeneously broadened unidirectional ring laser sufficiently short to support only one cavity mode can give rise to such effects. The explanation lies in the fact that side modes of different frequency but with the same wavelength may build up in the presence of a sufficiently intense mode. This phenomena, now commonly referred to as mode splitting, is considered in some detail below for both homogeneously and inhomogeneously broadened laser systems.

158

R.G. HARRISONand D. J. BISWAS

2.1. Mode Splitting Mode splitting was first discussed by Casperson and Yariv. t6v) This occurs in a region of rapidly varying dispersion when the oscillating cavity mode splits into more than one frequency all of which fill the same number of half-wavelengths between the cavity mirrors. Coupling between these several frequencies having common mode index is a prerequisite for single mode pulsating instabilities. Dispersion changes its value rapidly near the wings of the gain curve and the splitting that occurs in this region is known as passive mode splitting. ~67) An oscillating cavity mode can also induce large variation in dispersion if it can locally saturate the gain curve and consequently the splitting here is known as induced mode splitting, t68'69) The condition under which instability may occur, the so-called bad cavity (see Section 1 ), has been considered by many authors, t28'58'69'7°-81) The condition requires that the cavity decay rate x that is the effective decay rate of the electric field within the laser resonator is greater than the decay rate of polarisation memory, (7±) and also population inversion via energy transfer (7 bl)" F r o m the viewpoint of mode splitting this relation can be physically interpreted in the following way. t81~A short cavity lifetime is associated with high cavity loss which must be made up by high gain in the laser medium. Large gain if accompanied by narrow linewidth will lead to large dispersion. For high gain lasers and a narrow homogeneous linewidths, the index of refraction in the medium may vary so rapidly with frequency that it will be possible for more than one frequency to have the same wavelength in the laser cavity. The resonance condition under which a single longitudinal mode will split into several resonant mode is then that each of the modes have the same wavelength and gain so that selection of one mode over another cannot be made. 2.1.1. Spontaneous mode splitting. In order to understand spontaneous mode splitting we begin with the equation of the oscillating active cavity mode, viz.

mc

I ( L - l)" l + l'n(v)l

U£ = v

L

(14) '

where L is the length of the cavity, l is the length of the gain medium, m is the mode index and n(v) is the index of refraction at the laser oscillation frequency v. In terms of the empty resonator mode frequencies v,, = mc/2L the above equation becomes vm -

Iv

v = T In(v) -

1 ].

(15)

When the value ofn(v) either for a Lorentzian or Gaussian gain profile is substituted it takes the form

x,, - x = flF(x),

(16)

where x and x,, are respectively the laser oscillation frequency and empty cavity resonant frequency now normalised as a detuning from the atomic resonance. F(x) has the same functional dependence with x as refractive index n(v) has with frequency its form depending on whether the gain is Gaussian or Lorentzian. The dimensionless parameter fl is called the mode splitting factor and is given by cgl fl = K (Av)~'

(17)

where g is the peak small-signal incremental gain, Av is the F W H M value of gain-width, c is the velocity of light in vacuum and K is a numerical factor whose value is ~ - 3/2. ~/ln2 for Gaussian gain profile and ~-1 for Lorentzian gain profile. Equation (16) has been graphically solved for different detunings in Fig. 8 for a Gaussian gain profile. It can be seen that under certain detuning (near the wings of the gain) the equation is simultaneously satisfied by more than one

Pulsating instabilitiesand chaos in lasers

159

\

~x

×m -×

FIG. 8. Illustration of passive (or spontaneous)mode-splitting. value ofx. Physically this means that the cavity mode is split into more than one frequency and all of them correspond to the same number of half-wavelengths within the resonator cavity. Such effects occur spontaneously at the wings of the gain curve and hence the name spontaneous mode splitting.

2.1.2. Induced mode splitting. Induced mode splitting is explained in terms of distortion in the dispersion caused by hole burning in the gain profile by a single mode operating above lasing threshold. This is illustrated in the schematic of Fig. 9. The dispersion may be so distorted that several new frequencies now satisfy the laser cavity boundary conditions (the gain of which can be more than that at the parent oscillating frequency; see Fig. 9). Because the dispersive effects are caused by the oscillating mode itself, this splitting is termed "induced mode splitting". It should be strongest at line centre and is likely to exist over much of the lasing tuning range. Two line broadening situations common to most lasers are homogeneous and inhomogeneous broadening. The saturating characteristics of an inhomogeneously broadened gain medium is ideal for the realisation of mode splitting. In such a medium more than one frequency differing in gain can independently oscillate and thus passive mode splitting can be observed. The oscillating frequency also burns a spectral hole in the gain medium thus favouring induced mode splitting. Further, the side bands split again as they again burn spectral holes. This process will continue and eventually lead to chaos via the generation of successive sidebands, as illustrated in Fig. 10. On the other hand, when a homogeneously broadened gain medium saturates, all the atoms or molecules actually contribute at the oscillating frequency thereby making the possibility of spectral hole burning almost impossible. In such a system, therefore, only the frequency with highest gain eventually grows. However in an extremely high gain system survival of more than one frequency may be possible; the origin of instabilities in the Haken-Lorenz system considered earlier. This is perhaps best understood from a saturation spectroscopy viewpoint. In considering tlae situation in which the pump field resonantly excites a two level homogeneously broadened system Rabi splitting of the transition occurs which to a probe signal of frequency different to that of the pump appears as a spectral hole burnt in the gain; qualitatively similar in feature to that normally arising in inhomogeneously broadened systems. This, as noted above, is reflected in the distorted dispersion curve which for narticular parameter

160

R.G. HARRISON and D. J. BISWAS

FIG. 9. Illustration of active (or induced) mode-splitting.

OSCILLATION

7H°'J - SJNGLE ~ , ~

GAIN (r)

SPLITTING CHAOS '

FIG. 10. Mode-splitting approach to chaos (Ref. 81).

Pulsating instabilities and chaos in lasers

161

conditions can support mode splitting. As such, the pump signal together with probe signal have the same wavelength but different frequency. Under conditions where the probe frequency, here initially laser cavity noise, is such as to see gain in the distorted gain profile, this may occur either side of line centre, it will grow along with the existing mode at line centre, so giving rise to oscillatory instabilities. These arguments may be readily generalised to the situation in which the system is off resonantly pumped. Details of this generalised treatment have been given in a series of papers by Sargent, Ovadio and Hendow. t82-s 51Aspects of this will be considered later. We also note spatial hole burning effects which occur in a standing wave F a b r y - P e r o t resonator cavity should further help this process. Though the split frequencies are resonant with the cavity length for the same mode number yet their individual standing wave pattern may be shifted with respect to each other since normally the cavity length and active lengths are different. 2.2. Thresholds for Laser Instabilities It is appropriate here to summarise some of the theoretical results with regard to the threshold gain for single mode instabilities for both homogeneously and inhomogeneously broadened lasers. We recall that for homogeneously broadened systems at least a gain of nine times above lasing threshold is necessary to drive a single mode homogeneously broadened laser into chaos. For inhomogeneously broadened systems, however, instabilities can be realised as soon as gain exceeds lasing threshold. Indeed, as far back as 1969, Casperson discovered that the inhomogeneously broadened high-gain xenon laser (wavelength 3.51/~m) could, even with steady excitation, produce its output as an infinite train of pulses, t86~ Moreover, these pulses could, depending on the conditions, repeat regularly, alternate in height or be aperiodic. Casperson pursued two independent avenues of approach to a theoretical understanding of these phenomena, viz. coherent effects and inhomogeneous broadening. Only in 1978 did it become apparent that both effects are instrumental in producing the observed phenomena. In more recent work, Minden and Casperson have analysed the threshold conditions for homogeneously and inhomogeneously broadened lasers. (76~The details of the resulting stability criteria depend on the relative values of the homogeneous and inhomogeneous broadening. The instability threshold for a truly homogeneously broadened system as a function of 3 (3 being the ratio of the gain bandwidth to the cavity linewidth) is shown in Fig. 11. As can be seen the instability threshold has a minimal functional dependence with 3. There is a clear minimum threshold for a bad but not too bad cavity (3 ~ 0.4) when the ratio of homogeneous linewidth to natural linewidth (7ph/V) is unity. Under the conditions that the decay rates of upper and lower lasing rates are equal (i.e. 7~/Vb = 1 ), this region corresponds to a gain of nine times above lasing threshold and is consistent with earlier predictions. For situations in which 7ph/)' < 1 (i.e. natural linewidth dominates over collisional broadening), not often realised in practical laser systems, the threshold is seen to increase. In the more general situation where the decay rates of the upper lasing level (Ta) is much larger than that of the lowest level (Vb) the instabilities are realised at a much lower threshold. The corresponding curves for the inhomogeneously broadened situation are shown in Fig. 12 where parameters appropriate to a xenon laser were used. The series of curves are for varying inhomogeneous to homogeneous linewidth ratios. Comparing the limiting situations of inhomogeneous (lower curves) to homogeneous (upper curves) broadening dramatic decrease in instability threshold under inhomogeneous conditions is clearly evident. It should be noted that although the stability calculations'cannot predict the pulsations process in detail yet it is very useful as it is possible to represent a wide range of operating conditions on a single plot. Detailed analysis of the weak side mode treatment qualitatively discussed in the beginning of this section has been given by Hendow and Sargent ~82-84~ and Ovadia and Sargent. ~85~ This encompasses both resonant and detuned conditions for both homogeneous and inhomogeneously broadened systems. Here we briefly detail some of the major points of this

162

R.G. HARRISONand D. J. BISWAS 20

18

16

~ 2

'~- 10

I

0 0

02

0.4

0.6

0.8

1.0

5 FIG. 11. Instability threshold for a collision-dominatedhomogeneouslybroadened medium in the limits ~'~/)'b= 1 and ?~/~'b<< 1. Dashed lines connect boundary points having the same piJisation frequency.Dotted lines representpoints of minimumpulsation frequencyas the mediumrangesfrom collision-dominatedbroadening (?ph/7= 1) to life-timedominated (?ph/Y---0) (Ref. 76). analysis in which they show that the dynamic stark effect, alternatively referred to as population pulsation, is inherently responsible for mode splitting phenomena in homogeneously broadened systems. Significantly inclusion of these effects in inhomogeneously broadened system is shown to reduce further the instability threshold over that predicted by Casperson. The wave equation (Section 1, Eqn. (4)) is simplified here by assuming the field is composed of infinite plane waves having no transverse variations. The field can therefore be expressed in its Fourier components.

E(z,t) = ½~ o~.(z,t)exp(-iv.t)U.(z) + c.c.,

(18)

n

where g . = E. exp(-iq~.), the amplitude coefficient E.(z, t), and the phase q~.(z, t) vary little in an optical frequency period and v. + th. is the oscillation frequency of the mode. Typically, g . is complex, whereas E. is real. The spatial character of the field is represented by U.(z), which depends on the experimental configuration; for example,

U.(z) = exp(ik.z) = sin k.z

for RW's

(19)

for SW's,

(20)

where k. is the wave number. RW corresponds to the unidirectional ring laser case, whereas SW corresponds to the F a b r y - P e r o t - t y p e cavity. The polarization P induced in the medium has the same form as the field, that is,

,

P(z, t) = ½~ ~.(z, t ) e x p ( - iv.t)U.(z) + c.c., tl

(21)

Pulsating instabilitiesand chaos in lasers

163

I OOC

IOC

I

0

02

04

06

08

IO

8

FIG. 12. Ins~bility thresholds~r a xenon or helium-xenonlaser with mixedbroadening (Re[ 76).

where ~,(z, t) is the complex slowly varying component of the polarization for the nth mode. The real part of ~ is in phase with the field and represents dispersion resulting from the medium; the imaginary part is in quadrature with the field and gives rise to gain or loss. Analysis considers dilute media and slowly varying fields, the so called mean field approximation, justified for most experimental situations. The wave equation for the nth mode is then obtained as oe. +

- i(v. - f l . )

~. = i~o~.,

(22)

where the average round trip cavity losses are represented by ~r, where ~r = ~ o v / O . ,

with v ~ v. and Q. is the cavity Q for the nth mode. The amplitude and frequency determining equations for a laser mode can then be obtained from the complex wave equation by assuming that g. is real, v. >> q~., and 3a.'= 3~. exp(-i~b.) and by isolating the real and the imaginary parts of Eqn. (22). These equations are E,, + ~

v

E. =

v. + q6. = f~.

v, Im{~;},

(23)

1 v. Re{~,},

(24)

2go

E. 2~o

where the passive-cavity-resonance frequency is f~. = c K . , c is the speed of light in vacuum, and JPQE l O : 3 - B

164

R.G. HARRISONand D. J. BISWAS

k, is the wave number for the nth mode. Equation (23) can be used to define a net gain parameter: v

g" -

2Q.

v

1

2~o E. Im{~;}"

(25)

The field amplitude and frequency equations are self-consistent, since the polarization induced in the medium in turn radiates the field that induced it. Equations (22)-(25) are the basic, working laser equations. To investigate the stability of a particular laser mode at line center, one looks for side modes that satisfy the cavity-resonance condition [Eqn. (24)] and that have positive gain [Eqn. (25)]. A single-mode laser is stable if no side modes exist, and it is unstable if side modes grow, causing the total field intensity supported by the medium to fluctuate in time. The frequencies of these modes v, are assumed to be evenly spaced, i.e. v, = vl + (n - I)A, where, as shown in Fig. 13, A is the intermode beat frequency. This assumption is self-consistent with the laser instability problem, since the configuration of interest is that of a strong mode and two weak but mode-locked sidebands. The amplitudes of the sidebands are assumed to be much smaller than the strong-mode amplitude and the saturation intensity of the medium. Here analysis involves determining the polarization ~'~ of the medium, as observed by one side mode in the presence of a strong mode and possibly of a second side mode. The beat frequency A is assumed to be real here; hence the gain and the dispersion of the medium will show up in 8,. A generalization of this treatment, however, may be obtained by allowing A to be complex and has also been analysed. The role of the populations pulsations (PP) can be intuitively understood by considering the side mode 81 in the presence of the strong mode 82. The interference between ¢1 and 82 creates a beat note of frequency A = v2 - vl. Since the medium is interacting with the field, this beat note is impressed on the population difference, thereby creating the PP's. The gain of the medium then acquires a sinusoidal time variation that is proportional to the PP's amplitude and frequency. Because of the interaction of 82 with the population inversion, the medium now acts as a modulator for ~2, producing sidebands on both sides of the spectrum around v2. The energy of these sidebands is derived from o~2and is proportional to both ~t and 82. If81 is weak, then only two sidebands are produced with frequencies v, + A. These sidebands interfere with the side modes that produced them, causing appreciable changes in their absorption and dispersion coefficients; for example, side modes propagating in absorbers may experience gain as the PP's divert energy from the saturator wave. Inside a laser cavity, these side modes must satisfy the cavity boundary condition. They may either have the same wavelength as the main mode (single-wavelength instability) or belong to different passive-cavity modes (multiwavelength instability). The instability conditions for these cases are discussed below,

E2

E,

E~

( , - Ix --. --- A-.. 1

It

v,

v2

~

v

FIG. 13. The amplitude spectrum of three-frequency operation (Ref. 84).

Pulsating instabilities and chaos in lasers

165

The multiwavelength instability is one in which several modes with different wavelengths can oscillate simultaneously. In this case, the side modes belong to different longitudinal modes, and hence the resonance conditions are already established. The instability requirement for the multiwavelength case is that these side modes have positive net gain. Considering the simplest case of central tuning, the phase angle 4~in Eqn. (24) is set to zero so that the second term on the right-hand side of Eqn. 25 may be expressed for the specialized case of n = I as v

cq + ik* = i 2 ~ 1 ,

(26)

which describes the first side mode case at frequency vl. We note that line centre is at frequency v2. This gives the condition for multiwavelength instability, ciz. gl > 0 a s I-1 c%

-- (IBF) Re(~

+

ik*)

> 1,

(27)

where cto is a homogeneous-broadening linear absorption coefficient. FI is the relative excitation for the strong mode IBF is the inhomogeneous broadening factor. Re(cq + ik*) is the net effective gain coefficient for the side mode, and ~1 and xl are the complex absorption and the coupling coefficients, respectively, usually defined in problems of phase conjugation. To solve relation (27) graphically, the left-hand side of this equation is plotted versus the sidemode detuning (i.e. vl - v2). The effect of the PP's is to introduce gain to side modes detuned from line centre by roughly the Rabi flopping frequency. If the PP's are strong enough, relation (27) would be satisfied for these side modes, and the laser would be unstable if a passive-cavity mode exists at that detuned value. The single-wavelength instability, unlike the multiwavelength type, occurs in bad cavities. To show how the PP's lead to this instability, the frequency-determining equation for the laser cavity [Eqn. (24) ] is examined. By specializing Eqn. (24) for n = 1, by taking into account that the passive-cavity resonance frequencies for the side modes are ~1 = ~')2 = ~')3 = V2, one obtains, - (v~ - f ~ 2 ) 2 Q = ~_ (IBF) Im(e~ + v

ik•).

(28)

~o

The right-hand side of Eqn. (28) represents the normalized dispersion introduced by the medium and amplified by the relative excitation I]. The left-hand side of Eqn. (28) holds the properties of the passive cavity and represents the equation of a straight line having a slope of -2Q/v. This equation represents the resonance condition for side modes in the presence of a saturator wave. Both sides are plotted versus the side-mode detuning, i.e. vs v l - vz. Intersections between the dispersion curve and the straight line represent resonances for side modes and a possible instability if those side modes have positive net gain, i.e. if they satisfy relation (27). Hence as discussed qualitatively in the beginning of this section, there are two conditions for the onset of the single-wavelength instability: (l)The medium should be anomalously dispersive to the point at which the index reversals cross the cavity line, creating resonances for side modes, and (2)those side modes have positive net gain. The instability conditions represented by Eqns 27 and 28 have been solved graphically in terms of the effective gain and dispersion coefficients for the side modes by Hendow and Sargent for a variety of conditions. The simplest case of a single side band, i.e. two waves, under homogeneously broadened condition, is shown in Fig. 14 in which the passive cavity line, e.g. LHS of Eqn. 28 is also plotted. As seen there are four intersections leading to four resonances of the bands. Only the two sidebands furthest from line centre have gain; this means that only one of them can oscillate leading to a single side band instability. The instensity/2 at which

166

R . G . HARRISON and D. J. BISWAS

2

1

I

!

!

I

y,

~'_ 0

I

I

I

I

2 wove

HB,RW

l I l l

l

o

|

-2 -2

-15

I

I

-I0

-5

0

( ~,,-=)l'r (i)

I

I

5

I0

I

15

-15

-I0

I

-5

0

I

I

5

10

15

(.,-~)/7 e)l

FIG. 14. Normalised sideband (a) absorption and (b) dispersion coefficients for the two-RW configuration and for a homogeneously broadened medium. The incoherent (dashed) and coherent (dotted-dashed) contributions are shown separately along with their sum (solid lines). HB stands for homogeneously broadened. Note that (a) shows relation (27) satisfied for A ~- 9r whereas (b) satisfies Eqn. (28). The straight line in (b) is the passive cavity line of Eqn. (28) (Ref. 84).

this instability occurs is shown by them to be about four times that of the three wave (double side band) case. Figure 15 shows an example of the three wave case for an inhomogeneously broadened media. As discussed earlier the instabilities in these systems are readily obtained from spectral hole burning resulting in index reversals in the dispersion curve shown as dotted lines in this figure. The dot-dash curves of Fig. 15 show the dramatic additional contribution of PP's which as Hendow and Sargent show cause a substantial decrease of the saturation intensity at which instability occurs in these systems to that when only the contributions from spectral hole burning is considered. Hendow and Sargent have also considered the more general case of when the laser cavity mode is detuned from the line centre. As the strong laser mode is tuned to line centre, a homogeneously broadened medium is completely saturated, and therefore the gain of the side modes is primarily composed of diverted energy from the central mode. However, detuning leads to partial saturation of the medium. The result is a lower instability threshold and a substantial reduction in the role of the PP's in causing instability off line centre. Other effects of detuning are the noticeable change of the central mode intensity and the change of the relative phase angle between the side modes and their relative amplitudes. The decrease in intensity with detuning is a major factor that influences the pulsation frequency of the laser. Notably they find for a homogeneously broadened system population pulsations are alone responsible for line centre instabilities while off-line centre instabilities rely on the unsaturated media for gain and anomalous dispersion and as such is identified with passive mode splitting. Contrary to these findings, the work of Mandel and Zeghlache,(s7'88) in linear stability analysis of a single mode homogeneously broadened ring laser, shows the contribution of population pulsations is dominant and independent of detuning. In other analytical work on single mode inhomogeneously broadened laser with both Doppler and Lorentzian linewidths Mandel shows that such lasers lose their stability through a Hopf bifurcation in the bad cavity limit.{89-92) If this bifurcation is then the first step of a route to chaos directly then it is the Ruelle_Takens(19,20) scenario, which will be followed. Linear stability analysis was also used by Lugiato eta/. (93) to obtain instability boundaries and the corresponding threshold pulsing

Pulsating instabilitiesand chaos in lasers

167

1.5 ] wove

AM

3 wove

IHB

AM

.5

• !oe

@ e tt

o

I

-1.

- 15

i -7.5

I

! o

(,;-w)/'r (a)

7.5

-1.

t5

-15

I

I

-'~5

o (,u-u)/~,

7..5

1.5

(b) FIG. 15. Normalisedprobeabsorptioncoetiicient(a)and dispersioncoeificient(b)vsprobe detuning for the three waveAM case.The dashedand dot-dashedcurvesshowthe incoherentand populationpulsation contributions,respectively.IHB stands for inhomogeneouslybroadened (Ref. 82). frequencies under different detuned conditions of the single mode laser. It was shown that stability criteria are qualitatively similar for inhomogeneous and homogeneous systems although with a substantially higher gain threshold for instabilities in homogeneous systems consistent with earlier predictions. The results of a comparison of instability boundaries for fixed detuning and variable inhomogeneous linewidth are also consistent with those predicted by Casperson, viz. the pulsing threshold becomes progressively closer to the laser threshold as inhomogeneous contribution towards linewidth increases. Complementary to the weak sideband approach of Hendow and Sargent, Lugiato and Narducci (94) have formulated directly from linear stability analysis a correspondence between single mode and multimode instabilities in which a single mode instability implies necessarily a multimode instability in the good cavity limit and conversely existence of a multimode instability in the good cavity limit implies under an additional condition the existence of a single mode instability for a sufficiently bad cavity. Although their analysis is for a homogeneously broadened ring cavity system these conclusions are also shown to be justified for other systems, e.g. inhomogeneously broadened lasers, lasers with a non-uniform transverse profile Fabry-Perot cavity systems, lasers with an injection field and lasers whose dynamics are governed by rate equations. Adopting the weak sideband approach it is clear how this link occurs for the single mode system. If the sideband which lies under the power broadened gain line experiences enough gain to overcome the cavity losses the stationary state becomes unstable as discussed earlier. If at this point the cavity length is increased to bring neighbouring cavity modes closer to resonant modes until their positions coincide with those of the unstable sidebands these cavity modes become unstable (amplitude multimode instability). The same result can be obtained by increasing the width of the atomic line until the unstable sideband overlaps the neighbouring cavity mode. In both cases increasing cavity length or atomic linewidth changes the cavity configuration from bad to good. 3. S I N G L E MODE H O M O G E N E O U S L Y B R O A D E N E D SYSTEM As noted earlier the onset of instabilities in single mode homogeneously broadened lasers (Lorentz-Haken system) not only requires a bad cavity condition but also a gain of nine times above lasing threshold thus making the experimental realisation of this impossible for most lasers. However a recent theoretical re-examination of this system by Narducci e t al. t9s) predicts

168

R . G . HARRISON and D. J. BISWAS

that for certain operating conditions periodic and chaotic emission may be obtained from this system for lower values of laser gain. They analyse the single mode laser equations in the vicinity of the instability threshold for different ratios of atomic decay rates (~ = 7 II/Y±,~ I1 population decay rate and 7± polarisation decay rate) and find the existence of a fairly wide range of values of the laser excitation parameter, below the instability threshold, in which periodic and chaotic solutions coexist with a stable stationary state. Coexisting periodic and stationary solutions develop when the ratio 7 is sufficiently smaller than unity. Chaotic solutions and stationary states coexist for values of 7 >/0.15. Optically pumped far infrared lasers (NH3 in particular) have been identified by Weiss and Klische~62) as perhaps one of the very few ideal candidates for the manifestation of Lorenz instability phenomenon. These lasers were first observed to exhibit persistent and damped self pulsing behaviour by Lawandy and Koepf in 1980.~103)This behaviour was observed in standing wave systems in both the homogeneously broadened and Doppler broadened limits of operation. Several other observations of pulsing and damped oscillations in optically pumped cw molecular lasers are summarised in reference} 1°7) As discussed below, in these systems the bad cavity conditions is automatically satisfied without the usual requirement of high loss resonators due to their extremely narrow line broadening. Furthermore such systems normally exhibit the high gain necessary for the generation of instability phenomena in single mode homogeneously broadened laser systems. However their equivalence to tow-level systems must be approached with some caution since optical pumping as distinct from electrical excitation may involve coherent interaction between the pump and the lasing transition. The effect of a coherent excitation results in several factors which must be considered carefully when drawing conclusions concerning routes to chaos and use of certain atom-field models. The effect of coherent pumping may result in (1) inhomogeneous broadening. (2) coherent pump depletion, and (3) complicated velocity relaxation resulting in absorptive contributions to the dispersion relation for mode frequencies. Notwithstanding, these systems provide easy and sensitive control of operational parameters and are amenable to relatively straightforward analysis. Three modes of operation can be readily classified; that of resonant pumping which under suitable operating conditions is identifiable with the Lorenz system; off-resonant pumping which gives two photon Raman laser action and thirdly, specific to Fabry-Perot systems in which pump absorption is weak, instabilities may also arise from excitation of molecules belonging to two distinct velocity groups when the pump laser is slightly detuned from the absorption line centre. These various aspects have been recently considered with regard to instability phenomena in both far infraredt96-99) and mid infrared systems, t~°°-~°2) The reader is also referred to earlier reports in which instabilities in these systems have been observed by Lawandy et alp 03-106) and others. ~°8 ~11)

3.1. Optically-Pumped Far Infrared Lasers 3.1.1. Resonant pumping. The pressure at which these lasers operate optimally is very low because rotational relaxation that competes with the FIR laser emission is fast owing to the low transition energies (see Ref. 112). In the FIR the homogeneous linewidth is determined by pressure broadening alone since spontaneous emission plays no significant role. In these lasers the homogeneous linewidths then lie typically in the range of 100 KHz to a few megahertz. The pump laser does not excite a pure single-velocity group but rather pumps molecules with a spread of velocities that are determined approximately by the power-broadened homogeneous linewidth of the pump transition. Thus each velocity group corresponds to a spread of several homogeneous linewidths of the IR frequencies. However, when this velocity spread is translated into the broadening of the FIR transition, the relative broadening is reduced by the ratio of the FIR to the IR frequencies and is small. As the homogeneous linewidths of the FIR and IR transitions are essentially equal, the relative inhomogeneous broadening of the FIR transition is

Pulsating instabilities and chaos in lasers

169

negligible. Thus lasing occurs under essentially homogeneously broadened conditions. In such lasers bad cavity conditions are therefore automatically satisfied without the requirement of high loss resonators due to their extremely narrow line broadening. This fact was first noted by Weiss and Klische in their recent publicationJ 62) Particularly interesting is the case of the optically pumped far infrared NH3 laser, which due to the low partition function of the NH3 molecule, possesses a small signal gain 1-2 orders of magnitude larger than other FIR lasers and is thus ideal for the observation of single mode instability phenomena. Appropriate to ensuring line-centre pumping, Weiss and Klische considered the aQ(8,7) pump transition of 14NH a resonantly excited by the IO P(13) line of the N20 laser which leads to emission at 81.5 #m. The major source of inhomogeneous broadening in this laser is the ACStark effect brought about by the coherent pumping;(113) dynamic splitting varying for the different M components of the transition dependent on their dipole strength. It has been shown that the broadening due to this effect is less for the backward gain line and is of the order of 2MHz. However, this effect can be overcome by raising the NH3 pressure to 10Pa, corresponding to 2 MHz homogeneous linewidth. For experimental realisation of this Weiss and Klische also suggested the use of a FIR ring laser cavity which could be made to emit in the forward as well as backward direction. Tuning the pump-laser frequency allows one, via the Doppler effect, to choose the velocity group of molecules providing the laser gain. Pumping slightly off the absorption line centre creates a situation where only the forward- or backward-emitting laser mode interacts with the inverted molecules and the laser oscillates in a single travelling wave. Pumping close to or at pump line centre allows both modes to interact with the inverted molecules,tl ~4) Although, under cw conditions only one mode will oscillate due to competition, in the presence of instabilities both modes may emit. Thus, single-mode operation in the presence of instabilities requires off-centre pumping and a gain linewidth (which in general is AC-Stark broadened in addition to homogeneously broadened), substantially narrower than the FIR Doppler width. This gives an upper limit of the operating pressure and the pump intensity. Very recently they have confirmed these predictions with the observation of what may be Lorenz instabilities from this laser. ~96'97) Their scheme, a single mode travelling wave (ring cavity) FIR NH3 laser is illustrated in Fig. 16. Overlap between the forward and backward profiles was prevented by operating at 9 Pa pressure, when tuning the N2Oqaser frequency somewhat off line centre from the NH3(8,7) absorption. The FIR laser could therefore be tuned across its whole (backward) oscillation bandwidth (of the TEMo0 mode) at this pressure. Progression to chaos of their system, as the resonator is tuned towards gain line centre is sequentially shown in Fig. 17. Trace (a) shows the

N20 Pump Laser

I

\

I \l

Forward Emission

I Backward ,~ Emission

FIG. 16. F a r infrared N H 3 ring laser (Ref. 96).

170

R.G. HARRISONand D. J. BISWAS

(a)

f 5 0 dB

(c)

(d)

(e)

(f)

I

f

I

I

0

1

2

3

I

4 MHz

FIG. 17. Sequence ofsubharmonic instabilities leading to chaos observed for NH3 ring laser at 13 Pa NH3 pressure and 4 W N20-laser pump power. Spectra a-fare recorded when tuning the FIR laser resonator progressively towards the gain line centre (Ref. 96).

self-pulsation spectrum consisting of the fundamental frequency and its harmonics. Trace (b) shows a first series of subharmonics appearing when tuning is closer to line centre. Trace (c) shows a second series of subharmonics and trace (d) a third series of subharmonics. Trace (e) shows the appearance of noise, and trace (f) broadband noise on which the pulse frequency is superimposed. This corresponds to tuning of the FIR resonator gain line centre. Some subharmonic transitions to chaos were found when tuning the cavity mode towards the line centre from either side. At pressures above 13 Pa no instability was observed though the laser

Pulsating instabilities and chaos in lasers

171

output power was at a maximum at 25 Pa. This is attributed to the larger homogeneous broadening at higher pressure which ultimately precludes bad cavity conditions. These results are intriguing since the observed period doubling sequences to chaos appears to be at variance with most known solutions of the Lorenz model <2)which predict abrupt changes from regular to chaotic emission (crisis tl 15~).On the other hand, Zeghlache and Mandel ~aS~have shown that a model including detuning of the laser yields a transition to chaos via a period doubling sequence. However the recent observation of periodic instabilities by Weiss and co-workers (166) for this system deliberately operated under the conditions prescribed by Narducci et al. (95~ (described in the beginning of this section) suggests that their observed phenomena are indeed associated with the Lorenz system. In other work aimed at reducing the threshold for chaos in Haken type laser systems Lawandy and Plant" ~7~have recently analysed a system comprising two coupled unidirectional single mode homogeneously broadened lasers. Considering coupled return maps as a model similar to that previously used by Ikeda for Kerr media they predict a strong shift in the bifurcation structure and an indication of a dramatic lowering of the threshold for the onset of instability. Mention should also be made of the observation of self-pulsing instability in the emission from an optically pumped single mode CH2F 2 laser by Weiss and Klische. t98~Figure 18 shows a good example of period doubling observed in this system. These authors have also observed transition from this period doubled state to a chaotic state which displayed a broadband spectrum.

a)

T

65dB

I<

5MHz

0

b)

T 1

~d8

k

lops

:4

HHz

>1

FIG. 18. Period doubting of self pulsing osdllation of the CH2F2 laser, (a) time picture and power spectrum of the initial waveform, (b) time picture and power-spectrum of the period doubled state (Ref. 98).

172

R . G . HARRISON and D. J. BISWAS

3.1.2. Detuned pumping. Conventional FIR laser cavities in general comprise Fabry-Perot cavity systems within which for typical operating pressures absorption of the pump radiation is weak. For such systems consideration must then also be given to effects arising from both the forward and the backward propagation of the pump. As discussed by Lefebvre et al. tl°9-111) bidirectional pumping brings about a two-peaked gain when the pump frequency is detuned with respect to the absorption line centre ensuring that the active molecules have a significant velocity component along the laser axis. When the FIR laser operates under conditions in which the homogeneous broadening is smaller than or comparable to the Doppler width of the transition on which the laser oscillates and the cavity is tuned to the centre of the lasing 0 transition Ogwg, then the FIR gain curve exhibits two maxima when the FIR oscillation frequency is varied. This limit, which occurs when the Rabi frequency is not in excess of the narrowed forward gain spike width has been observed by harmonic mixing experiments by Lawandy and Koepf.~1°5) These maxima are both due to the two velocity groups mentioned above and occur approximately at frequencies +co given by

--

--

(JllR

] '

where A~om is the detuning between the IR-pump radiation at frequency com and the IRabsorption transition through which population inversion is generated. As discussed earlier each of the gain peaks is to an excellent approximation homogeneously broadened. From the associated unsaturated dispersion of the medium (Fig. 19) it is easy to see that several different steady state solutions are possible for the same number of wavelengths in the cavity. That is, several frequencies have the same wavelength in the medium. In such a system Abraham et al. 199) have recently reported oscillatory instabilities when the FIR laser resonator is tuned midway between the gain peaks. Their system, a H12COOH laser emitting at 742 #m wavelength, was optically pumped by the 9R(40) line of cw-CO2 laser. The variations of the oscillation period as a function of CO2 laser detuning, FIR cavity detuning, COz pump power and pressure in the FIR laser is shown in Fig. 20. It can be seen that the period decreases with decreasing CO2 laser detuning, increasing FIR detuning, increasing CO2 pump 6

L

-6

I

(b)

I

-4

~

4

I

-2

2

4

6

FIG. 19. (a) Unsaturated gain and (b) dispersion for two resonant groups of atoms with their resonant frequencies separated by a detuning parameter value o f f = 2, Solid curves show contributions of the two groups separately, and the dashed curves show the total gain and dispersion (Ref. 99).

Pulsating instabilities and chaos in lasers

"7" r7 O rr" W O..

173

8

(a)

7 6 I

5

I I CO 2 LASER

1 I DETUNING

(MHz)

::x.

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(b)

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CAVITY

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(c) I

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500 1000 co2 PUMP POWER (~ Watt)

15 v a O r,r itl ix

z

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(d)

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1

2

3

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7

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LASER

PRESSURE

(mTorr)

FIG. 20. Dependence of the oscillation period on (a) the CO2-1aser detuning (b)the FIR-cavity detuning (c) the CO2 pump power, and (d) the pressurein the FIR laser. For (a) the horizontal scale is 2 MHz/division. CO2 laser power and FIR laser pressure were 1.1 W and 8mtorr respectively (Ref.99). power, or increasing gas pressure. These authors have also considered a theoretical model that offers an explanation of their observed single mode laser instabilities. The model which places primary emphasis on the two symmetric resonances as an explanation for certain observed FIR instabilities is found to be in excellent agreement with the experiments, although no account is taken of the complications of detuning, the standing-wave cavity, or coherent pump effects. When the Rabi splitting of the forward gain spike is large, owing to a large pump matrix element (NH3) or large pump fields, the additional complicated dispersion must be included for accurate modelling of the problem. The splitting of the forward spike has also been observed in heterodyne experiments, t1° s~ The analysis further predicts, abrupt transitions from cw to largeamplitude pulsing which may be tested experimentally, perhaps by fixing the CO2-1aser frequency and varying the pumping intensity. For large values of cavity decay rates (x) and &2 = 1.5 where ~ is far infrared cavity detuning, chaos should be observed two to three times

174

R . G . HARRISON and D. J, BISWAS

A

'L

CHAOTIC

6

~

K= S

P0,SAT,0, I.....

\

.....-"

4 ~ '~VL r"\

i"

.PERIO01C PULSATIONS

I

=

0

2

4

6

8

6' 10

FIG.21. Phase diagram in the A, 62 spaceshowingregionsof zero intensityor constant non-zero(cw) intensity as well as regions of stable and chaotic pulsing for K = 5 (Ref.99). above the power levels required to see the initial laser action (which is unstable). The details are shown in the phase diagram plotted in Fig. 21. No intermediate biurcations were observed in numerical results from a first scan of the parameter space, and this is as would be expected for chaos of the form of Type III intermittency, t118) In fact some chaotic behaviour has recently been experimentally observed although not exactly in the predicted region. However, in the type III intermittency predicted in Ref. 99 chaos appears throughthe growth of the second harmonic content of the limit cycle while the experimental signals present no such second harmonic content but indicate rather a transition to chaos through a tangent bifurcation similar to what was recently observed in a laser containing a stark modulated saturable absorber. 3.2. Near Resonantly Pumped Mid Infrared System It has been noted earlier that a Raman process is responsible for the gain in most of the optically pumped molecular lasers where chance coincidence of the pump signal frequency with the pump transition seldom ensures resonant conditions. ~119) Recently Harrison and Biswas (1°°-1°2) demonstrated instabilities leading to chaos, in a single mode homogeneously broadened laser belonging to this class, viz. an optically pumped mid infrared N H 3 laser. These effects have been obtained on two independent emitting transitions over a wide range of operating conditions, including those for optimum lasing, suggesting that this behaviour is indeed general for this broad class of laser. The operating pressure of the system as for most MIR lasers is much higher than for FIR systems, ensuring good homogeneously broadened conditions providing the differences in AC-Stark shift the the different M components of the transition are small. Also at these pressures due to high absorption of the pump, the possibility of bidirectional pumping considered earlier for F a b r y - P e r o t systems, which is also used here, is ruled out. As in the work of Weiss and his collaborators N H 3 gas was selected on the basis of its well-documented spectroscopy and efficient lasing action. Midinfrared lasing involves vibration-rotation transitions, here the aP(8,0) transition at 812 c m - 1 optically pumped on the aR(6,0) transition 1.3 G H z below line centre by the 9R(16) CO2 lasing emission at 1076cm-1. This laser transition has been clearly experimentally identified as Raman in origin ~120)for N H 3 pressures ~ 1-20 torr and pump intensity of ~ 0.6 MW/cm 2. This is further theoretically supported by the recent work of Dupertuis et al. ~121-123) who show that

Pulsating instabilities and chaos in lasers

175

for three level schemes Raman oscillation always suppresses line centre oscillation despite an equal, and for a highly saturated case even higher, small signal gain for the line centre oscillation. The pressure broadened bandwidth of the pump and lasing transition are 17.76 and 17.049MHz/torr respectively, u24~ which for the typical operating pressure of ~ 8 t o r r , considerably exceeds the Doppler bandwidth (74 MHz). For a pump intensity of ~ 500kW/m 2 lasing was obtained up to a pressure of 11 torr corresponding to a pressure broadened gain-bandwidth of ~ 187 MHz, substantially smaller than the free spectral range (FSR) of the laser cavity, viz. 600 MHz, thus ensuring a single mode condition. This was further established from determining the cavity tuning range as a function of pressure over which lasing was obtained. The data of Fig. 22 are consistent with the predicted dependence of gain bandwidth on pressure shown as a solid line, based on a value of 17.049 MHz/torr. The displacement in experimental data shows lasing to occur over a range somewhat greater than the F W H M of the gain bandwidth: thus indicating a gain which is little over double the lasing threshold. Chaotic and periodic pulsation behaviour in the NH3 emission which was sensitive to cavity length tuning occurred over NH3 pressures of 5-9 torr, smaller than the total range 3-11 torr for lasing emission, most pronounced effects occurring at a pressure ~ 8 torr. Within the narrow tuning range over which instability prevailed, two fundamental pulsation periods occur for different PZT settings; one at -,~ 3.8 nsec period and the other of relatively long period ,-~ 18 nsec. Straightforward mode pulling considerations show that the high frequency pulsation which leads to eventual chaos on fine cavity tuning is consistent with intermode beating of two cavity modes (cavity roundtrip time 1.6 nsec). More interestingly the slow periodic modulation (Fig. 23(a)) which exhibits distinct period doubling (Fig. 23(b)) with fine cavity length tuning before going into high period chaos (Fig. 23(c)) was obtained under single

180

/ / /

n160 I g3 ~ c~ z < rn

Z

~ Z

140

120

100

< ~

80

60

I

I

I

I

l

I

3

4

5

6

7

8

NH 3 PRESSURE

{TORR1

FIG. 22. Lasing and gain bandwidth of N H 3 emission as a function of pressure: dashed line, experimental data for lasing bandwidth determined from the P Z T tuning range over which l a s i n g occurred; solid line, prediction for gain bandwidth based on a F W H M value of 17 MHz/torr (Ref. 100).

176

R. G. HARRISON and D. J. BISWAS

lOOns II

50ns II

a

b

C FIG. 23. Example of single mode instability: (a) long periodic modulation, (b)period doubling, (c)high period chaos. The correspondingtime-expandedtraces are shown with an arrow (Ref.100.)

mode conditions. Cavity linewidth considerations show that the phenomena are obtained only under bad cavity limit. The instabilities in this Raman system occur at a considerably lower threshold to that predicted for two level systems. From the data of Fig. 22 and for typical operating pressures ,-, 10torr a gain of approximately two times above threshold for lasing is estimated; to be compared with a value of nine, the minimum value considered necessary for the onset of instabilities in an equivalent two level system. The mechanism of the instabilities observed here have been identified by Biswas and Harrison as arising from mode splitting, tl°~ From cavity tuning across the region of the FSR over which lasing persisted instabilities were found to occur only when the mode was detuned by >~30 MHz from the line centre; suggesting that induced mode splitting is not responsible for the observed phenomena. Rather passive or spontaneous mode splitting (the coexistence of multiple steady states), which occurs far away from the line centre for a sufficiently large value of the mode splitting factor (fl) are instrumental in generating the instabilities in their experiments (see Section 2). Theoretical considerations of mode splitting effects in Raman systems are indicated in the recent work of Dupertuis et al. ~125~ Quantitative understanding of Raman lasers is in a somewhat embryonic state. Aspects of their operation have been recently discussed by Lawandy tl°7~ and earlier various authors have

Pulsating instabilities and chaos in lasers

177

quantified the gain dynamics of these systems under steady state conditions. Extension of such approaches to real time analysis of these molecular pumped systems contained within optical resonators will enable characterisation of the instability phenomenon experimentally observed. In general investigations of laser instabilities in optically pumped mid and far infrared lasers is in its infancy. The recognition that bad cavity conditions are readily satisfied in most of these systems without the requirement of extremely lossy resonators and given the high gains of these lasers they have nevertheless rapidly established themselves as key candidates in this field. As relatively simple systems they provide highly versatile operation and are amenable to theoretical analysis. The two photon coherent interaction implicit to the operation of many of these systems identify them as distinct in many respects from more conventional lasers involving, e.g. electrical excitation. As such they provide an exciting new class of system for continued investigation. Pertaining to the simpler Haken system this is perhaps most effectively realised in these lasers using collisionally induced (and perhaps cascade) transitions which ensure resonant lasing emission and in which coherent interaction between the pump and lasing signal is automatically eliminated. Although a yet there has been no report of such operation, this will undoubtedly prove a profitable avenue for further investigation in far infrared and perhaps also mid infrared systems. 4. I N H O M O G E N E O U S L Y B R O A D E N E D SYSTEM The bulk of theoretical work on single mode laser instabilities in inhomogeneously broadened systems, were considered earlier with regard to mode splitting. Summarising, single mode laser instabilities are realised with relative ease in an inhomogeneously broadened system though the onset of instability still requires a bad cavity but with pumping rates considerably reduced to those needed for the corresponding homogeneous case. In fact way back in 1973, Idiatulin and Uspenski ~26~ demonstrated that even slight inhomogeneous broadening of the resonant frequencies of the medium is sufficient to change the nature of the laser stability in a qualitative way and to reduce the threshold for the observation of the pulsating instabilities. Their theoretical consideration of inhomogeneity arising from two closely spaced resonance species has since provided a qualitative explanation of pulsing observed in certain N d - Y A G lasers~127~ and this treatment has recently been extended to account for pulsating instabilities in the low pressure far-infrared laser of Abraham et al) 99) considered in the previous section. However, by far the most common form of inhomogeneous broadening in laser systems is a smoothly varying profile, such as the one caused by the thermal motion in a gaseous medium. Other kinds of inhomogeneous broadening do however occur for different reasons, for example, because of the presence of different isotopes in gaseous mixtures or of local variations in the crystal field of a solid or because of defects and impurities. These complications do not seem however to change the qualitative aspects of the instability phenomena in these various systems. Recently detailed numerical analysis has determined well defined routes to chaos in such a system/79.12s-3o) In these treatments, the Maxwell-Bloch equations describing a single mode travelling wave laser are used which are generalised to include inhomogeneous (Doppler) broadening, off-line centre operation and different relaxation rates of upper and lower levels of the laser transition. Here we concentrate on some of the results by Shih e t alJ 12 s.~a9) Figure 24 shows a progression to chaos via a period doubling route with pump rate as a control parameter (Doppler broadening is ~ 110 MHz and homogeneous width is ~ 10 MHz). Trace (a) shows computed intensity as a function of time for the lowest pump rate, viz. ~8.5 × 10 -2 sec- 1. A period doubling bifurcation on the base period of trace (a) is evident from trace (b), obtained at a slightly higher pump rate, viz. ~ 9 × 10- 2 sec- 1. Traces (c) and (d) for pump rates of 9.3 × 10- 2 and 9.4 × 10 -2 s e c - 1 respectively, reveal further period doubling. Slight further increase in pump rate result in more period doubling and eventually chaos as the period doubles ad infinitum according to the Feigenbaum sequence. I161 Trace (e), for instance, shows the power spectrum of the electric field when the pump rate is 9.6 x 10 -2 sec- i. From this trace a broad

178

R.G. HARRISONand D. J. BISWAS

(a)

(b) 4O

!

[l[I lttttl tllttlt iLtL LLLI ILtLtLtLI[t lttttlI JtiltlJlJ E u

30

o

2o

~ 20

E

"~

E

113

c

0

0

6

4

5

0

Time(microsec)

Time(rnlcrosec)

(c)

(d)

!

'

40

4O

(J ~- 30 o

E

20

c

10

c

0

0

2

0

4

4

6

Time(microsec)

Time(rnicrosec)

(e)

o

ck -5

10

20

30

Frequency(mHz)

FIG. 24. Computed period doubling route to chaos for a single mode inhomogeneously broadened laser with pumping rate as a control parameter,(a) initial periodicoscillation; (b), (c) and (d) example of period 2, 4 and 8 respectively; (e) power spectrum of the chaotic state (Ref. 128).

band spectrum, characteristic of chaotic dynamics, is evident. Similar results have been obtained by Bandy et a l ) 13o~ although in their work they find a far more regular period doubling sequence for off-line centre operation compared to line centre operation where in the latter there is also evidence of intermittency. However, in this model for the parameters chosen, chaos is only obtained for line centre operation. For small values of 7 = (711/7i) the output oscillation takes the form of a train of well resolved single pulses with a spacing that grows progressively on lowering the value of 7; again existence of detuning favours the occurrence of period doubling bifurcation and stable periodic patterns. The appearance of significant ringing in the pulses is a feature in qualitative agreement with some of Casperson's earlier experimental results ~v°'86~ and numerical simulations/68'69'76 81~

Pulsating instabilities and chaos in lasers

179

Returning to the results of Shih et al. (128' 129) they find a two frequency route to chaos when the field frequency is detuned from the atomic line centre. This is illustrated in the data of Fig. 25, which show the power spectra for increasing values of detuning. As seen, the initial appearance of two incommensurate frequencies progress ultimately to a broad band spectrum characteristic of chaotic time evolution. Under other control settings intermittency is observed whereby initial intermittent bursts of chaos becomes more frequent and lengthy on variation of the control parameter. Notably metastable chaos is observed whereby a long chaotic period is followed by an abrupt transition to order. Since, as we have shown, instabilities in single mode lasers are more accessible in inhomogeneously broadened systems, most experimental efforts up to recently have been directed to such lasers. However, the bad cavity requirement, even for this broadening, restricts the suitable laser media to only a few high-gain systems. So far H e - X e at 3.51/~m in particular and also H e - N e at 3.39/am have emerged as the prime laser candidates for the manifestation of instability phenomenon under single mode inhomogeneous conditions. 4.1. H e - X e

Lasers

Along with the early work of Casperson much of the contributions here have come from Abraham and collaborators. Here we highlight some of their results obtained with the H e - X e

(e) !

!

(b)

.,(

I

-2

2

!

!

fl f2 |2

o

,[..9o

I

fl*f2 fl

2fl 2fl "f2

2fl "f2

v

E

-S

Q.

vt

US

-10

-10 0

I0

20

0

.~0

I0

20

FrequencyM/Nz)

Freq ueney(n~lz)

(a)

(e) i

i

I

-2

i

i

!

I0

20

30

~'-2

g

0 -6

2 ~ -6 n

US --10

-8

0

t0

20

rrequency(mHz)

FIG. 25.

30

0

Frequency(mHz)

Computed two frequency route to chaos for a single mode inhomogeneously broadened laser

with cavity detuning as a control parameter; (a)initial single frequency and higher harmonies, (b) emergence of the second frequency, (c) broadening of the peaks, (d)fully developed chaos (Ref. 128). JPQE I0:3-C

180

R . G . HARRISON and D. J. BISWAS

system. Depending on the operating conditions this laser may possess a value of fl (mode splitting factor; see Section 2) much higher than the cut-off value of 3.51 for mode splitting to occur. The first major effort to observe passive mode splitting in this laser was however unsuccessful, ~131) perhaps due to the too high value of fl ( ~ 50) in this particular system; resulting in extremely strong mode pulling so preventing tuning of the single mode sufficiently far from the line centre necessary to observe this effect. First experimental observation of self pulsing instability arising out of mode splitting was appropriately made by Casperson. In Xe lasers Casperson observed self pulsations with frequencies much smaller than the axial mode separationff °'86) This indicates that mode splitting was instrumental in generating instabilities in this system. Some typical experimental results are represented in Fig. 26. The main pulses are usually followed by weaker echo pulses. Under some conditions, the successive dominant pulses alternate in height as shown in this figure providing possible evidence of period doubling. The variation of pulsation frequency with discharge current is shown in Fig. 27. It can be seen that pulsation frequencies vary from about 2 to 10 M H z while the free spectral range of the laser cavity was ~ 100 MHz. Initial analysis by Casperson based on a semi-classical model gave qualitative agreement with several of these observed effects,t ~o) Significantly this work showed for the first time that the Lamb equation for an inhomogeneously broadened laser possesses low instability threshold (see Section 2). More recently a Maxwell-Schrodinger semiclassical analysis has been applied to the ring laser oscillation in which the effect of direct spontaneous relaxation between the two laser states as well as the possibility of spectral cross-relaxation due to velocity changing collisions are considered.(79, 8o) With these additions the model gives exact agreement with experimental data. Interestingly, as noted earlier, a simplified model of Bandy et al. t13°) is also in qualitative agreement with some of the results of Casperson. More recently Abraham and his associates have extended this work to the H e - X e system and have obtained significant new experimental data concerning mode-splitting, spontaneous pulsations and chaos) 1°'132-140) I n their system the helium partial pressure allows the degree of inhomogeneous broadening to be varied as a control parameter with beautiful results showing

0

0.2

0.4

0.6 t(pS)

0.8

1.0

1.2

0.8

1.0

1.2

(a)

(b)

0.2

0.4

0.6

t (ps) (b)

FIG. 26. Experimentalplots of the pulsation instability for (a) discharge current of 40mA and (b) a discharge current of 50mA. A slight intensity alternation is observed in (b) between consecutive bursts. (Ref.70).

Pulsating instabilities and chaos in lasers

181

12

i

'2

o

l o

20

40

60

80

100

120

i(rna)

FIG.27. Experimentalpulse repetitionfrequencyas a functionof discharge current for a xenon laser (Ref.70).

that the pulsation frequency range increases as the ratio between homogeneous to inhomogeneous broadening reduces. In other words the laser instability threshold increases with the degree of homogeneous broadening. The extreme inhomogeneous broadening achievable in their system corresponds to an inhomogeneous to homogeneous broadening ratio of 22:1; the instabilities in this case appear within a few percent of the lasing threshold, while when the inhomogeneous-to-homogeneous linewidth ratio reduces to 2: 1, a gain of almost twice the lasing threshold is required for the onset of instabilities. Quantitative measurements on both passive mode splitting and active mode splitting in this laser were made by Bentley and Abraham. ~132~When the cavity mode of the laser (free spectral range of 910 MHz) is sufficiently detuned (passive mode splitting) a 10-15 MHz beat frequency was observed in the detected output of the laser suggesting two modes spaced by this frequency. The observed pulsation is therefore attributed to phase-locking of the two output frequencies of the split mode. The mode pulling data displayed an enhanced pulling of the strongest mode closer to the line centre frequency as if there was an effective repulsion between the two frequencies as is known to occur in multimode lasers. The mode splitting vanished by lowering the gain and it was also weakened when the homogeneous contribution towards the linewidth increased. Near the line centre (induced mode splitting), the output appeared to be a regular sawtooth oscillation of fundamental period in the 15-20 MHz range. This is consistent with the predicted pulsation frequencies. Like the case of detuned instability, this instability also disappears for higher helium pressure (larger homogeneous broadening). It should be noted here that these effects were observed for a value of fl which was lower than the predicted cut-off value of 3.51 by Casperson and Yariv ~67) confirming the important role of population pulsations in lowering the instability threshold (see Section 2). Abraham and collaborators have experimentally quantified the operating conditions over which self pulsing instability and routes to chaos, obtained in the X e - H e system. Figure 28 shows a sequence of graphs of the inter-laser beat frequency (obtained by standard heterodyne techniques) vs cavity detuning for different admixtures of helium. Each curve is drawn for the discharge current giving the maximum output. The region of detuning for which pulsing frequencies were observed is shown in each case. For helium pressures of 3.7 torr or more, no instabilities were observed though there was extreme mode pulling. As the helium pressure is reduced the appearance of a region of instability near line-centre can be seen. The region of

182

R.G. HARRISONand D. J. BISWAS HELIUM

3 7 Torr

HELIUM

3 0 Torr

HELIUM

2,7 Torr

HELIUM

2.0 Torr

7o \

o ¢,j

>W

o

W rr

N\\\\\\N\N\N\N

W O0

CC W .< ,_1 1

HELIUM

1.3 Torr

HELIUM

1.0 Tort

W I.Z

..........

.............

LASER CAVITY FREQUENCY

I~\\\\\\\\\\\\\\\\\\\\~ 80. M H z / d i v

FIG. 28. Laser operating frequency (inter-laser beat note) shown on vertical scale versus cavity detuning for peak output setting of dischargecurrent and 175 microns of Xe with amounts of helium as shown. Pulsing region t ~ ~ ; chaotic region ~ . Data is shown for laser operation in the vicinity of line centre tuning (Ref. 137).

observed instability widens as the helium pressure is reduced. The side-bands were nearly equal in strength near line-centre while under detuning the pulsing appeared to involve only some sideband. It is significant to note here that while theoretical and experimental evidence indicates that the line-centre instability represents a H o p f bifurcation the single sideband evidence suggests the bifurcation may be different for the detuned case. For the lowest helium pressure studied (1 torr), the pulsing frequency is observed to vary from 16 to 21 M H z with cavity tuning. Routes to chaos has also been extensively studied in this system. Depending on operating conditions they have observed all the three k n o w n routes to chaos in dissipative systems, viz. Feigenbaum (period doubling), ~15-17) Ruelle-Takens (two frequency) ~19'2°) and PomeauManneville (intermittency) (18) scenarios. Period-doubling route to chaos. The traces of Fig. 29 show: (a) frequencies corresponding to periodic pulsations, (b) and (c) two period doublings or frequency halvings and (d) a chaotic spectrum. The control parameter here is the cavity detuning. Above the chaotic threshold a region of period-three behaviour is also observed which is represented by trace (e) of the same figure. This figure is representative of those obtained at different discharge currents. They also have observed parts of similar sequences for fixed cavity length as the gain was varied.

Pulsating instabilities and chaos in lasers c

5

183

d

e

o '.Gf 3

- 20

~-a_

f

½f

- 30

~

....

- _ ?T

i

20

40

i

i

0

J

i

20

i

40

i

,

i 20

FREQUENCY,

J

~ 40

i

i

,

0

, 20

.

.

. 40

.

.

t 0

i

20

,

,

40

f (MHz)

FIG. 29. Power spectra of He-Xe laser output showingperiod doubling route to chaos with cavity detuning as a control parameter (175mtorr Xe and 0.7torr He) (Ref.133). R u e l l e - T a k e n s route to chaos. For some ranges of detuning the laser showed the coexistence of two initially incommensurate frequencies. Figure 30 shows a sequence of power spectra in such a region obtained for fixed cavity detuning and with discharge current as the control parameter. The initial pulsation frequency and its harmonics are shown in trace (a). As the gain is increased by increasing the discharge current, a second frequency appears which is at a ratio of 4.3 with the first (trace (b)). For higher discharge current the spectrum is enriched by the harmonics and combination frequencies of the two pulsation frequencies (traces (c) and (d)). At sufficiently high gain the broadband noise, characteristic of chaos, emerges (trace (e)). Intermittent route to chaos. Figure 31 illustrates another distinct route to chaos which has been observed for certain laser parameters (described in the figure). The initial instability here is characterised by a narrow spectral peak which is broader than the instrumental linewidth (trace (a)). Increasingly frequent, intermittent bursts of chaos for increasing gain results in a steady broadening of the peaks (evident from the traces of this figure).

Although F a b r y - P e r o t cavity systems as used above and in most other experiments to date provide experimentally simple single mode systems the standing wave formation in such systems considerably complicates theoretical analysis of their operation. Consequently, analysis of instabilities has concentrated on uni-directional ring lasers. Furthermore, systematic studies of standing wave lasers show that the instabilities are particularly sensitive to cavity tuning which can be associated with interaction of the counter-propagating parts of the standing wave, making it difficult to establish the dependence of laser output characteristics on parameter variations. Though more complex in design uni-directional ring laser systems do not suffer from these problems. Such a system based on the xenon emission at 3.51/~m and in which a Faraday isolator is used to ensure uni-directional lasing has been recently reported by Hoffer et al3139) in preliminary experiments. For increasing excitation beyond a well defined second

-t~

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FIG. 30. Powerspectra of He-Xe laser output intensityfor differentvalues of the discharge current. (175 mtorr Xe and 0.3 torr He). This is an example of Ruelle-Takens route to chaos (Ref.133).

40

184

R.G. HARRISONand D. J. BISWAS

-10-20 " -30 " -40

-

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FIG. 31. Intermittent route to chaos with increasing discharge current for 175mtorr Xe and no helium. The initial periodic behaviour (trace (a)) becomes increasingly chaotic as evidenced by the significant broadening of the peaks as the gain is increased (Ref. 133).

threshold, above the threshold for cw lasing, at which spontaneous self-pulsing occurs, transitions exhibit quasiperiodic behaviour and also occur through a sequence of period doublings. This contrasts to results obtained in the Fabry Perot systems (see above). Detuning the laser from resonance did not lead to significant complications in the pulsing structures. Instead with increasing detuning a smooth reduction in complexity of the dynamical behaviour was observed and ascribed to the reduction in gain and laser intensity on detuning. As we have noted earlier key features of deterministic chaos are irregular time evolution and broad band power spectra. These together with observation of one or other routes to chaos have been extensively used as the signatures of deterministic chaos in the experimental systems investigated so far. However, it is recognised that such measurements are not able to distinguish unambiguously between low dimensional deterministic chaos and broadband stochastic noise originating from the contributions of many uncorrelated sources. Such noise behaviour may arise from direct multi-source contributions or because the deterministic states of the system are relatively unstable and act as sensitive amplifiers of noisy perturbations.

Pulsating instabilities and chaos in lasers

185

In order to clearly distinguish stochastic noise from deterministic chaos, quantitative methods of analysing digitised time series have been reported ~°) from which reliable estimates of Lyapunov exponents, metric entropies, and attractor dimensionalities can be made. As discussed by Albano e t al. ~1°~ the most important advantage of these techniques is that recording of a single digitised variable of the system permits the reconstruction of the topological character of the attractor in the total variable space. This is accomplished by embedding the time series in an N-dimensional vector space in which each vector has components that are data values taken with equal time-delay separations (see also Section 1). For a sufficiently large embedding space, the attractor thus constructed will have the same Lyapunov exponents, metric entropies and dimensionalities. Albano et al. ~ ° ) have recently applied this technique to the laser transition at 3.51 #m from high gain xenon described earlier. The result of this study clearly demonstrates (i) that the irregular emission of this inhomogeneously broadened, single mode laser in time and the corresponding power spectra are consistent with their representing deterministic motion on a strange attractor of low fractal dimensionality, and (ii) that these broadband spectra do not arise from stochastic noise sources. Recognising the need for further test of other dimensionalities, entropies and Lyapunov exponents in establishing complete certainty of this argument their results nevertheless identify the importance of this approach in more fully quantifying deterministic chaotic behaviour. As discussed later the power of this approach has been beautifully demonstrated by Puccioni e t al. ~ 1) with regard to the formation and evolution of chaos in a CO2 laser with modulated losses. 4.2. H e - N e L a s e r a t 3.39 #m Using gain and linewidth considerations, Casperson t28) has recently computed a figure of merit factor to determine the relative accessibility of regions of unstable single-mode operation. On a scale where a value of one indicates the threshold for instabilities, the xenon laser at 3.51 #m was rated at 150, more than ten times the rating for any other cw laser while rating on He-Ne at 3.39 #m was only at 5. Because of relatively tight constraints on gain, linewidth and cavity loss it has seemed unlikely that instabilities would be observed at all in the H e - N e laser, more so because of the extensive use of this lasing transition in the past without report of unstable single mode operation. However, from the recent studies ~141-144) it now appears that the single mode instability is reasonably accessible in this laser whenever the "bad cavity" criterion is satisfied. First reports on instability in this system was made by Weiss e t al. ~4~ 142) The laser was tunable to within ~ 20 MHz around the gain line centre. They obtained different routes to chaos as a function of detuning when one resonator mirror was progressively tilted. These are summarised in Fig. 32. In a range of 6 MHz, centred at + 5 MHz from the gain line centre period doubling approach to chaos was observed (Fig. 33). As the tilting angle is increased the initial oscillation at 6.8 MHz (trace (a)) suddenly halves (trace (b)). As the angle is increased further two more period

'NTEM'TTENC I' IM -10 MHz

A T 0

PERIODDOUBLING

E, 11 MHz

He-2ONe-LASER FREQUENCY OFFSET FROM GAIN LINE CENTRE

FIG. 32. Different routes to chaos appearing at different laser frequency settings when all other laser parameters are held constant. (R stands for Ruelle-Takens route) (Ref. 141).

186

R.G. HARRISONand D. J. BISWAS

7o

(e) 70-

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FIG. 33. Starting from single mode oscillation, tilting of one resonator mirror leads to an oscillation [fl, (a)], period doubling [fl/2, (b)], period doubling [fl/4, (c)], period doubling [fl/8, (d)] and chaos [(e)]. Laser frequency offset from the gain line centre + 5 MHz (Ref. 141).

doubling appears (traces (c) and (d)) before the laser goes into an eventual chaotic state (trace (e)). In the range - 1 to - 3 M H z from the gain line centre chaos is a p p r o a c h e d t h r o u g h R u e l l e - T a k e n s sequence (Fig. 34). The sequence starts with the appearance of a 3 . 2 M H z oscillation (trace (a)) (notably different from the starting oscillation of the period doubling) which increases in amplitude with increasing tilting angle until a second incommensurate frequency of 1.4 M H z appears. C h a o s sets in with further increase in tilting angle. Within the range of - 3 to - 10 M H z from the gain line centre an intermittent route to chaos is observed.

Pulsating instabilities and chaos in lasers

S

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A second Ruelle-Takens route to chaos is observed in the frequency range of + 8 to + 11 M H z from the gain line centre. In the range - 1 to + 2 M H z the sequence starts with the 3.2 M H z oscillation of Fig. 34(a) but then switches to the 6.8 M H z of Fig. 33(a) and continues the route to chaos via period doubling as in the range of + 2 to + 8 M H z (see Fig. 33). This change between two sequences is probably caused by the small frequency change accompanying the resonator mirror tilting. In spite of the observation of these wealth of phenomena the use of mirror tilt as a control parameter for studying single mode laser instability may nevertheless also result in the undesirable coupling of transverse modes which may in turn lead to, or contribute to, the observed low frequency beats.

188

R.G. HARRISONand D. J. BlswAs

Notably, Gioggia and A b r a h a m ~143,144) have reported self-pulsing instabilities in this system with cavity length tuning rather than mirror tilt tuning as a control p a r a m e t e r - - under a truly single mode condition. The free spectral range of the cavity was nearly three times the gain broadening which was dominantly of Doppler origin. (Inhomogeneous to homogeneous ratio in their system was 9:1.) Figure 35 shows a sequence of power output curves versus cavity length for various discharge currents, Figure 35(a) is taken for the initial onset of the instability at 26 M H z at the bottom of the L a m b dip. This occurs when the laser is 2.9 times above threshold at line centre. The evolution of the L a m b dip and the pulsing frequencies is shown in Figs 35(b)-(f) for increasing discharge current. Figure 36 shows sample rf power spectra of the laser intensity from the regions of instabilities shown in Fig. 35. With small changes in detuning the pulsation character changes from high Q pulsations to two-frequency operation or gives evidence of a "subharmonic bifurcation" with other cases showing broadening of the peaks and the addition of a broadband spectral component characteristic of deterministic chaos. Their observation of three frequency quasiperiodicity [Fig. 36(e)] is of considerable interest against the background of recent discussion in the theoretical literature 12o,33,145.146)concerning the possibility of three frequency quasiperiodicity in a typical nonlinear dynamical system. Newhouse et al. ~2°~ has predicted that such quasiperiodic motion should occur though it may be destroyed by arbitrarily small amounts of perturbations. According to Grebogi et al) 145' 146) such a scenario is rather common in a typical nonlinear dynamical system and perturbation of some specialised form is needed to destroy the three frequency quasiperiodicity. It should be noted that the three frequency quasiperiodicity has recently been reported in a Benard experiment with mercury in a magnetic field ~147) and in the voltage spectrum of a ferroelectric-barium-sodium-nitrate crystal) I4s) Recent reports by Biswas and Harrison (289'29°1 and Hollinger and Jung (287) have also demonstrated a three frequency route to chaos in multi-transverse mode lasers (see Section 6).

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Pulsating instabilities and chaos in lasers

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FREQUENCY (MHz) FIG. 36. Sample rf power spectra from the regions of instability indicated in Fig. 35. Samples show (b) typical high-Q period doubling with a broadband chaotic background, (c)period doubling with chaos, (d-f) the appearance of two distinct frequenciesfrom an initial near degeneracyin the sequence from quasiperiodic (d) and multiple periodic (e) behaviour to stable two-frequencyoperation (d). (Ref. 143). 5. S I N G L E M O D E LASERS W I T H E X T E R N A L C O N T R O L P A R A M E T E R As we discussed in Section l, if in a laser system one or more variables relax much faster than the others a stationary solution for the fast relaxing variables may be taken; these are in fact still slowly varying because of the coupling. The number of coupled differential equations describing the system may therefore be reduced; so-called adiabatic elimination of the fast variables. Indeed this situation applies to a large number of existing lasers where polarisation and/or population inversion have relaxation times much shorter than the cavity lifetime (~±, ~ II >> x). Consequently with adiabatic elimination of these two variables the dynamic behaviour of the laser is described by just one equation which must show necessarily a stable behaviour. Lasers here include H e - N e , Ar +, Dye, etc. and adopting the classification scheme of Arecchi et al. t6a) are termed Class A. In other lasers (Class B) only polarisation is fast and hence two variables describe for the dynamics. Examples here are Nd and CO2 lasers which exhibit oscillating behaviour under some conditions although ringing is always damped. The third class (Class C) of lasers are those for which the three relaxation constants may be of the same order of magnitude so precluding adiabatic elimination of any of the variables; an important practical example considered earlier being the far infrared laser (see Section 3). Since nonlinear systems require a minimum of three degrees of freedom to exhibit deterministic chaos such behaviour may only normally occur in Class C systems. We nevertheless note that for Class A and B systems in which lasing may still be obtained for an increased cavity decay rate, perhaps for example in gas dynamic as well as electron beam pumped CO2 laser systems both of which have very high gain, the degrees of freedom may be restored. For more conventional lasers, the necessary degrees of freedom for deterministic chaos are restored to Class A and B systems by the inclusion of external control of the laser systems such as modulation of an external field or population inversion of the gain medium. These have been considered by various authors aspects of which are discussed below. 5.1. Modulation o f an E x t e r n a l Field or Population Inversion In the theoretical systems considered by Y a m a d a and G r a h a m ~ 4.9)full diabatic elimination is assumed resulting in the reduction of the Maxwell-Bloch equations to one field equation as fully describing the system. Inclusion of an external field which is both detuned from cavity resonance

190

R . G . HARRISON and D. J. BISWAS

and is modulated provides the necessary additional two degrees of freedom for the system to generate chaos under particular parametric conditions. Alternatively modulation of population inversion in the presence of an external field of constant amplitude t15°) equally well satisfies this condition. 5.1.1. Modulated external field. In analysing these problems we recall the Eqns 9-11 for a single mode'homogeneously broadened two level laser which may be more generally written to include an external field term/~E,xt as (see Ref. 151) d E / d t = - K(E

Eext)

-

-

igP

(29)

d P / d t = - 7 ± P + igEO

(30)

dD/dt = 711(Do - D) + 2ig(PE* - P ' E ) .

(31)

Here the variables are left in their complex form and have not been rescaled as for Eqns 9-11. Here g is a coupling constant between the electrons of the atom and the electric field. The simplest situation to consider is one in which P and D are adiabatically eliminated, i.e. dP

dt

dD -

dt

= 0 (see Section 1).

where we assume x << ~ll << 7±. Under these approximations Eqns (29) to (31) reduce to the single field equation, viz. dE/dt = -~c(E - Eext) + g2E

,, Do ~;±(1 + 4g2[E[2/T±~ll)"

(32)

For Eext = 0, the solution of Eqn. (32) relaxes to-a time independent constant, E = Eo. Therefore, in order to produce chaos the external field is important. Calling the frequency of the

external field ta,xt we introduce the quantity (o9~, - co)/x = 3~,

(33)

which measures the detuning in units of x. To perform numerical calculations it is advisable to introduce a dimensionless time z by t = r/x,

(34)

E =/~(T) (~±711)1/2 exp [i&oz ]/(2g)

(35)

E~t = A (z) (~711)1/2 exp [i6o9T ]/(2g).

(36)

and to rescale the variables according to

Using furthermore the abbreviation R - g2D°,

(37)

7 we arrive at the basic equation

dE

_ 1 ) / ~ + A(T). d ~ = - i 5 ~ ° 1 ~ + ( 1 + RIE,2-

(38)

In the limiting case of constant rather than modulated external field such that A(z) = a, from Eqn. (38) steady state solution Es is obtained from the reduced equation - if~/~ + (zs - 1)Es + a = 0.

(39)

Pulsating instabilities and chaos in lasers

191

where we have used the abbreviations R z~ = 1 + ~ '

[2 = 6o~.

(40)

Linear stability analysis of the equation shows that the steady state becomes always unstable provided R is sufficiently large resulting in a spontaneously modulated output. Physically the detuned external field interacts with the laser cavity field resulting in a modulation in the population inversion. This therefore increases the degrees of freedom of the system from its original one to two. Nevertheless this is still insufficient for the generation of chaotic output. This is obtained with the addition of low amplitude modulation to the external field, viz. A(z) = a + a' cos(D'z),

a > a'/> 0

(41)

With increasing amplitude of the external field the system behaves quasiperiodically with two characteristic frequencies, one at the frequency of the external modulation and the other at a value the same as obtained without any modulation (constant input field). Further increase in modulation strength eventually culminates in chaotic emission. The power spectra of the periodic and chaotic state are shown in Fig. 37. A broad peak is clearly seen in the chaotic state.

70.0

0.03

20£

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1(

10.0

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0.0

0.036

(.d 0.5

o.0

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FIG. 37. The power spectrum of the periodic (a~ = 0.03, left part of figure) and the chaotic (a I = 0.036, right part of figure) states. The sharp peaks at the frequencyto = 0.45 in both figures correspond to the frequency of the external modulated amplitude (Ref. 149).

192

R.G. HARRISONand D. J. B1SWAS

5.1.2.

Modulation oJ inversion. Because

the inversion Do enters Eqn, (38) via R we put

R = Ro + R'cos(f~'t).

(42)

Thus the model equation is provided by d/~ _ dz

i3o/~ +

(Ro + R' cos(~'t) 1 + IEI 2

)

- 1 /~ + A0.

(43)

This has been solved numerically for specific parameter values. In order to have a threedimensional phase space of the system, it is necessary that both ~ ' and Ao in Eqn. (43) are different from zero. For t)' = 0 or Ao -- 0, Eqn. (43) m a y be reduced to two equation of first order. For R' = 0, Eqn. (43) has a stationary state/~,. A limit cycle appears when this stationary state becomes unstable for sufficiently large Ro. Choosing R0 = 2, o9 = 0.5, Ao = 0.4, f2' = 0.4 and varying R', one finds a bifurcation scheme. Slightly above R' = 0.16, the transition from a three-periodic state into a chaotic one takes place via an intermittent mechanism (see Fig. 38). At R' = 0.1610 one observes only a few chaotic bursts interrupting the periodic motion. The number of chaotic bursts increases as the parameter R' becomes greater. The chaotic regime ranges at least up to R' ~, 0.22 where a complicated subharmonic bifurcation behaviour is observed. With increasing R' the intermittent region is followed by a fully chaotic one. That chaos confined here to the single mode case, may be c o m m o n to multimode lasers without external modulation is also noted by Scholz et al) 15°~ in their paper. In such systems the population inversion is itself internally modulated at the difference frequency between different modes; as such the amplitude of one mode provides a driving field for other modes. As discussed in Section 6 the mechanism has been analysed in some detail for two modes systems and subsequently demonstrated in a CO2 laser. 5.1.3. Experimental observation of instabilities in lasers with external modulation. The experimental realisation of routes to chaos in systems with external modulation has been demonstrated in various systems tl ~' 152-165~ Most notable is the comprehensive work of Arecchi and co-workers ~11,152,153~ on a CO2 laser with loss modulation provided by an intracavity electro-optical modulator to induce a sinusoidal time dependent perturbation of the cavity decay rate (~c).As discussed in Ref. 63 only adiabatic elimination of polarisation alone is possible

b

FIG. 38. Time evolution of the real part of the electric field for (a)R ~ = 0 . 1 6 1 0 , (e)R 1 = 0.1620. The intermittency is clearly visible (Ref. 151).

(b)R 1 =

0.1615,

Pulsating instabilities and chaos in lasers

193

in this system resulting in two degrees of freedom which in the absence of external control gives rise to relaxation of oscillations. Here the third degree of freedom is obtained by making the system non-autonomous with x(t) = ~o (1 + m cosog,,t), where m is the amplitude of the modulation, COrnits frequency and x0 the unperturbed cavity relaxation rate. If ~o,, is far from the relaxation oscillation frequency f~ the system follows the sinusoidal variation of K, but if ~o,, is near to f~ (or to one of its harmonics) nonlinear resonances are excited even for small m. By analogy with electric nonlinear oscillators which are known to exhibit chaotic behaviour it is expected that a laser relaxation oscillator, where loss is modulated at a frequency comparable with the laser relaxation oscillation frequency could also exhibit chaotic emission. Their experiments show that the first unstable region occurs around 64 KHz, and that new windows appear at higher harmonics, with smaller modulation depths. {13~ Various attractors, each with its proper periodicity, are found to coexist in the same parameter region and the system can enter one or another depending just on the one it has followed. Noise induces jumps away from less stable attractors or between equivalent attractors. With fixed modulation frequency (191 K Hz) and with increasing m (1-20 ~o) a period doubling cascade:f, f/2, f/4, f / 8 , chaos was observed--providing a well reproduced Feigenbaum's scenario; credited here to the high experimental reliability and control accuracy of the system. From digitising the time evolution of the signal and synchronising the sampling with the modulation frequency to obtain the projection of the Poincare section the embedding technique, t3 5) as discussed earlier, was used to reconstruct the attractor allowing evaluation of its dimension." 1) As predicted, a dimension near to 1 (0 for the Poincare section) for periodic behaviour. Fig. 39(a, b) was obtained. When the system entered the chaotic region the fractal dimension jumped to a higher value (between 2 and 3 for the time series and between 1 and 2 for the Poincare section) Fig. 39(e, f), agreeing with the general theory of strange attractors. ~154~In Fig. 39(c, d) we see that for the J/8 subharmonic the dimension is near 1.5 (0.5 for the Poincare section). This result even though not readily understandable because the time signal appears to be periodic nevertheless agrees with the theoretical prediction of the dimension at the accumulation point of the logistic map.(155) A lower bound of the Kolmogoroy entropy was also determined to be zero and 35 K H z for the chaotic attractor. Numerically integrating the model and processing the data obtained in the same manner results in agreement with the experimental ones were obtained. As an alternative to the loss modulation these authors have assessed pump modulation as discussed in Section 5.1.2 but without the need of an external field for this system for which the same temporal behaviour in the lasing emission is expected. ~153) However, they found that modulation depths much larger than in the case of loss modulation are required, typically ~ 3 ~o loss modulation is equivalent to ~ 54 ~o pump modulation. Recently Dangoisse et al) ~56~have considered cavity length modulation using an elasto-optic modulator which they show to provide a very efficient and progressive method for exploring instability phenomena in a CO2 laser system as shown in Fig. 40. A sequence of period doubling bifurcation culminating in chaos occurs as the driving voltage (v) is increased. For further increase in v periodic windows are reached through sharp sequences of inverse bifurcations and a different kind of chaos is observed for still higher voltages. Dependent on laser operating conditions, periodic behaviour at periods 3 T, 4 T and 5 T were recorded and a spectral analysis of the laser output power indicated that different chaotic behaviour was observed. Figure 41 shows a bifurcation diagram of the laser output obtained by sampling at the same frequency to that of the modulation. Displaying the signal amplitude as a function of the driving voltage, the period doubling route to chaos is beautifully demonstrated. Further multistability is observed on the upper part of the bifurcation diagram. In other reported work a homogeneously broadened solid state laser emitting at 1.3 #m with periodically modulated pump, modulated at a rate comparable with the relaxation oscillation

194

R . G . HARRISON and D. J. BISWAS

*,;., ,,,!

. : i ) ~ f t f l ~°';'' -'.il

7

4-

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FIG. 39. Plots of log N (e) vs log (e) for different values of the embedding dimension n calculated from the time series (a, c a n d e) and from stroboscopic sections (b, d a n d f) for f/4 (a a n d b), f/8 (c a n d d) and chaos (e and f) (Ref. 11).

frequency of the active medium NdPsO~4 has been driven to chaos. {15 7) On decreasing the modulation frequency successively the route to chaos was clearly identifiable as a period doubling sequence up to period eight beyond which chaotic emission was obtained• The wellknown period three window in the chaotic regime was also observed and for a different average pump power a period five window was also identified• With regard to practical applications in optical communications the authors note that since their system is exactly that encountered in semiconductor lasers which are rapidly modulated for high data transmission rates the occurrence of chaotic emission will clearly pose limits on this rate. In other reports sinusoidal modulation of the pump intensity of a laser diode pumped single mode LiNdP40~ 2 lasers resulted in regular spiking pulse oscillation, together with resonant relaxation oscillation and several subharmonic modes such as J/2 and f / 3 spike modes and f~2 relaxation oscillation (15s) on varying the modulation frequency. Recently self sustained relaxation oscillations in a GaA1As has also been reported by Tatah and Garmire (159) where the frequency of relaxation

Pulsating instabilities and chaos in lasers

195

T

2T 9v

4T 25v

8T 27v

~

CH 3Or

CH 32v

3T 35 v

CH 45 v

FIG. 40. Modulation of the laser output power (NT) periodic or (CH) chaotic for various driving voltages applied to the modulator (Ref. 156).

oscillation is found to depend on the drive current, the minimum current for oscillation being ~ 20 ~ above that for threshold. Analysis of the system with modulated drive currents at the relaxation oscillation frequency shows the onset of bifurcation up to period 4 with increasing modulation depths. Bifurcations are also indicated from experimental results for lasers modulated at the cavity round trip time. Period doubling routes to chaos have also been predicted by Lee et alJ 16o) in modulated semiconductor lasers. In other recent work Chen et al." 61,162) have experimentally investigated this diode laser with periodic modulation of the injection current. In the absence of rf modulation the laser output is characterised by intrinsic fluctuation whose frequency is centred around the relaxation oscillation frequency. The fluctuation increases in amplitude and evolves into an intermittent period two pattern as the modulation index increases. In the frequency domain the fluctuating period two gives rise to an induced noise peak at the subharmonic of the modulation frequency. With further increase in modulation index the amplitude of the fluctuation decreases and the JPQE IO:3-D

196

R.G. HARRISONand D. J. B1SWAS

FIG. 41. Oscilloscopedisplay of the bifurcationdiagram (Ref.156). laser response is characterised by fluctuations superimposed on a period one oscillation. Evidently these results indicate the significant role of quantum noise caused by random fluctuation in affecting the otherwise deterministic chaotic behaviour of the system. This is confirmed in the analysis of the system in which the inclusion of noise source term is shown to inhibit more than a single subharmonic bifurcation. Though not a conventional laser in the sense of the systems considered here, beautiful and comprehensive results with theoretical simulations have been recently reported by Brun et a/. (163-165) for a solid state nuclear-spin-flip, ruby nuclear-magnetic-resonance (NMR) laser(166) that operates in the rf range. This system comprises a single crystal ofA1203: Cr 3 ÷ in a magnetic field Bo inside the N M R coil of a tunable L - C circuit. The resonance circuit acts as a cavity for the laser and it provides the magnetic radiation field B(t) that acts on the A12 spin. The third important ingredient is the microwave pump that provides the necessary spin inversion for the N M R laser activity. By tuning the coil to a selected N M R transition single mode N M R lasing is possible. Specifically the strong (½,-½) N M R transition of frequency ~oa = gB0 is considered (g is the gyro magnetic ratio). The output of the N M R laser is the voltage across the L - C circuit. General considerations show the N M R system to be the rf analogue of a singlemode laser model where the macroscopic variables are the radiation field, the precessing nuclear magnetisation and the longitudinal magnetisation. Since for this laser the field is adiabatically eliminated external modulation is required to augment the dimension of the phase space to three again. The forced periodic modulation adds two new control parameters, viz. the modulation frequency (f~) and the modulation amplitude (F). Figure 42 shows the output of the free running N M R laser in the absence of modulation showing characteristic slow relaxation oscillation, frequency ~376sec -1. Selecting a modulation frequency similar to this to ensure strong nonlinear destabilisation of the system the recordings shown in Fig. 43(a-d) show a period doubling sequence up to period 8 followed by a chaotic response on increasing the modulation strength. Under most experimental conditions the control parameters are swept and as such the dynamics of the system may be affected by the sweep time and sweep speed. This is demonstrated in Fig. 44 which shows a bifurcation diagram where the output has been recorded at equal time intervals while the modulation amplitude has been swept either up from zero or down to zero. The up and down

Pulsating instabilities and chaos in lasers

197

FIG. 42. Real-time N M R laser output voltage after a perturbation of the cw laser activity (Ref. 163).

scans a and b, c and d show hysteresis effect that depend strongly on selected operating conditions. Further, an incomplete cascade of period doubling bifurcation exist that ends in chaotic domains with periodic windows and noisy bands and also transitions occur between coexisting basins of attraction (traces (a) and (b)). That the characteristics of a chaotic attractor can change abruptly (trace (e)) may be indicative of crises (1 ~5~where collision between a chaotic attractor and a coexisting unstable periodic orbit takes place. Pertinent to this the effect of time dependent parameter variation on the period doubling route to chaos have been analysed for the laser Lorenz equations by Mandel and Erneaux (~" ~ and also by Kapral and Mandel (168~ for a one dimensional map and a differential equation model of the physical system. They find that this effect is analogous in many respects to that of noise; both acting as a disordering field on the dynamics.

(o)

(b)

(c)

(d)

FIG. 43. Real-time N M R laser output with forced linewidth modulation,f = 74 Hz and for increasing F (Ref. 163).

R. G. HARRISONand D. J. BISWAS

198

)

.~t,.

(o)

(b) r

(c)

_

(d)

~:.;>

(e)

(f)

FIG. 44. Experimental bifurcation diagrams for the NMR laser with forced linewidth modulation: (a), (b) for up and down scan withf = 72 Hz; (c), (d) for up and down scan withf = 70 Hz; (e) and (f) for up scan with f = 108 and 66 Hz respectively (Ref. 163). 5.2. Single Mode Laser with Injected Signal: Theory and Experimental Observations F r o m a more general standpoint it is of interest to consider the full set of laser Eqns (29-3 l) without adiabatic elimination and in the presence of an injected signal. Although indeed in the absence of injection the system has sufficient degrees of freedom to manifest chaotic motion (see Section 1) the influence of injection is to generate other interesting instability behaviour. Lasers with injected signal have been considered quite early in theoretical studies (169 171)and re-analysed later in numerous contributions. (157-194) Physically, there are two types of laser response. There is the "spontaneous" laser action, cavity resonant, and we will assume resonant also with the atomic transition on which the population inversion exists. The second response is a regenerative amplification of the injected signal, which is detuned from resonance. For a linear drive oscillator, these responses correspond to the complimentary function (here a growing exponential because of the gain) and the particular solution respectively. In the laser the nonlinearity leads to competition between these two types of behaviour, and gives rise to unstable behaviour as the injected signal is increased from zero (stable self-excited laser, as discussed above) to a value high enough to slave the laser to the injected signal frequency, where the response is again stable. The self-excited instabilities obtained in this system are to some extent a limiting case of those considered earlier in which under total adiabatic eliminatiQn the

Pulsating instabilitiesand chaos in lasers

199

external pumping rate (detuned injected signal) was necessarily modulated providing the additional two independent variables necessary for deterministic chaos in the system; described by one field equation. Generally the instability threshold for single mode lasers is reduced with injection, and the restrictions on decay rates less severe. We note that the injection of a low power continuous signal is routinely used to induce single-longitudinal mode operation in, e.g. high power CO2 lasers, but this application corresponds only to the small-injected signal limit of the system under discussion. The Lorenz-Haken system may be generalised by introducing an injected amplitude Cex,with the same scaling as E (see Eqns (29-31), but assumed real and positive, while E and P are now complex, in general, because of the detuning of the injected field frequency from the atomic line centre (A) and the mistuning of the injected field frequency from that of the laser field (0). Following Lugiato et al. the resulting sets of equations are /~= [ - i(~o - co0) - tc]/~+ ~cA - i g f f

(44)

/~= [ - i(cb - ~Oo) - y±]/6+ igl~D

(45)

1) = 7jr(Do - D , ) + 2i(g*P-E-* - gP"*E"),

(46)

where A = (co -

~o)/7±

0 = (co -

O~o)/~.

The steady state solution of Eqns (44-46) can be found analytically. Scaling the incident field amplitude A to the square root of the saturation intensity and calling this new quantity y and similarly scaling the laser field amplitude, denoted by the quantity x, the relation between the input and output is then given by y=lx[

1

1 +Az+X 2

+

2CA )211/2, 0 + 1 + A 2 + Ixl 2

(47)

where C = ~Ldo/(2T),

(48)

where c~is the unsaturated absorption coefficient per unit length, L the length of the sample, and T the intensity transmission coefficient of the mirrors, -c

:

7±t,

: ~C/7± = C T / L ~ 7 ± ,

where ~ is the total length of the ring cavity. This is plotted in Fig. 45; the segment OAB turns out to be unstable against fluctuation, the upper branch instead is stable over its entire length down to the injection locking threshold (B). For y = 0, the output intensity is stationary and the system oscillates at the frequency co. With the external field on the output intensity oscillates with the frequency 09 - 09o. The signal amplitude increases as yl/2, while the period increases slightly with increasing in y until around y - 120 the system begins to display irregular self-pulsations (Fig. 46a). For y of the order of 250, highly chaotic emission develops (Fig. 46b). For further increase in y the system emerges from the chaotic domain through a sequence of inverse period doubling bifurcation (Fig. 46c, d) until regular oscillatory self-pulsing sets in again. Yet higher input intensity leads to a "breathing" behaviour-- a slow modulation of the oscillation (Fig. 47a), which develops into an output consisting of"spikes" followed by quiescent intervals (Fig. 47b). The latter increase in duration until they evolve quickly into the stable solution for y greater than Yth, which is close to 312 for these parameters. For other parameters, including the experimentally accessible C = - 2 0 , similar behaviour is seen, except that the breathing and spiking seem to be absent.

200

R . G . HARRISON a n d D. J. BISWAS

I×1 IO0

50 B e

m

I 111 Yth,

O 200

~ A 400

A

1 y

FIG. 45. E q u a t i o n (47) p l o t t e d for C = 500, A = 0 = 5, K = 1. T h e s e g m e n t O A B is u n s t a b l e (Ref. 181).

It should be emphasised that these phenomena are distinct from the Lorenz instability; though the system possesses the necessary 3 degrees of freedom for such instability since 711 = k = 7 it is nevertheless absolutely stable for y = 0, the Lorenz limit. Injection into an already unstable laser stabilises it at high enough y values: at low values o f y the behaviour is, unsurprisingly, irregular. In fact Lugiato et al. find empirically that 7c ~ F is necessary for chaos: V can be much smaller than F without hindrance. As discussed more recently by Bandy e t a / . (182-184~ and Arecchi e t a/. (63'185"186) a characteristic feature of the lasers with injected signal is the occurrence of distinct coexisting

a

b

C

|

I

i

1.

.

•

a_

~

1

•

FIG. 46. T i m e e v o l u t i o n o f t h e n o r m a l i s e d e m i t t e d field x for C = 500, A = 0 = 5, K = 1 a n d 7 = 1. T h e h o r i z o n t a l axis is m e a s u r e d in z u n i t s : (a) e r r a t i c b e h a v i o u r , y = 117; ( b ) b u r s t i n g , y = 250; ( c ) 4 P - t y p e s o l u t i o n , y = 2 7 9 ; ( d ) I P - t y p e s o l u t i o n , y = 300 (Ref. 181).

Pulsating instabilities and chaos in lasers

201

FIG.47. Time evolution of the normalised emitted field x with the same operating parameters as in Fig. 46. For the chosen values of the drivingfield,the systemdisplays: (a) a marked modulation of the self-pulsing envelope (heavy breathing), y = 310.3; (b) spiking action, y = 311 (Ref. 181). basins of attraction in its multidimensional phase. This follows from Eqn. 47 which, as a function of injection field strength y may be single or triple valued depending on the system's parameters. The curve of Fig. 45 is but one example. However the presence of bistability between the low and high transmission branches of the triple valued state equation is in general precluded by instabilities in one or other of these branches here the lower branch. Three distinct types of instability domain can be realized dependent on parameter values and the associated form of the bistable curves. The entire lower branch and a section of the upper branch, beginning at the upper turning point and terminating at the injection-locking threshold, are unstable; only the lower branch is unstable from y = 0 to the lower turning point; and thirdly a segment, but not the entire lower branch, is unstable beginning at y = 0. When the state equation is single valued the instability domain usually covers a single strip of the state equation from y = 0 to the injection locking threshold. F o r the system described above the similar decay rates (•, 7± and 711) ensure the necessary three dynamic variables for chaos and as noted by Arecchi e t al. ~63~ this should therefore relax the requirement of atomic and cavity frequency detuning used in the analysis. This is not so however for the CO2 system they consideP ~s6~ for which polarization is adiabatically eliminated and where detuning between external and internal frequencies is therefore required in providing both real and imaginary parts of the field amplitudes as dynamic variables. Analysis of the system, described by two coupled equations shows the lower branch to have a stable locked region and an unstable region where the laser oscillates regularly or irregularly. It is found that the locking regime is reached by either decreasing the oscillation frequency or by decreasing the amplitude of the external field (Fig. 48) for values less than that required to obtain a stable output period doubling. These characteristics are summarised in the phase diagram of detuning versus amplitude of the external field Fig. 48 for values less than that required to obtain a stable output locked in frequency to the external signal. Region 1 describes the situation, when the injecting amplitude is too low to lock the system steadily, the laser operates for most part of the time at COo (external frequency) but it regularly unlocks going to o~ (internal frequency). During the oscillation at co the energy of the injecting field enhances the population inversion so that it gives rise, with a delay related to the injecting intensity, to a giant pulse. At the same time this pulse acts as a perturbation of the stationary value and the intensity relaxes back to that value via damped oscillations. The frequency of the giant pulses is proportional to the frequency of the external field with the m a x i m u m value proportional to the detuning in frequency between the external field and the laser field. When the frequency of injection is sufficiently detuned from the laser frequency the frequency of the giant pulse can be of the order of that for the damped oscillation for suitable values of the external amplitude y. Then when the detuning is larger than

202

R.G. HARRISONand D. J. BISWAS 6

5 4

3

"(f)-6)

I

A

6

FIG.48. Phase diagram in the parameter space; detuning versusamplitude of the externalfield.The limits in the unstable region indicate the differentkinds of solutions (see text and Ref.186). or approximately the frequency of the relaxation oscillations this becomes undamped because the giant field cannot relax back to the steady state value before another giant pulse arises (Fig. 48 subregion 2). Further, when the two frequencies involved are approximately equal resonant destabilisation of the system is obtained. As represented in subregions 5 and 6 of Fig. 48 for slightly different cavity detuning a third low frequency emerges incommensurate with the others. However, for increasing injection field these two frequencies lock at a given ratio and for further increase in field the system enters an intermittency region and eventually approaches a chaotic regime. Further, when the absolute value for the detuning is increased subharmonic bifurcation involving the giant pulses and transition to chaos through period doubling are also possible as shown in Fig. 49 and represented by subregion 6 of the previous figure. Recently experimental investigation of this system, a single mode unidirectional ring lasers by Vanlerberghe e t al. (187) provides some experimental evidence in qualitative agreement with the above analysis. Experimental results have also been obtained for the ruby nuclear magnetic resonance laser discussed in Section 5.1.3 under conditions ofa detuned injected signal. Results and simulations appear to be at least qualitatively similar to those discussed above. Period doubling bifurcations on cavity detuning and chaotic self-pulsations have also been predicted by Otsuka and Kawaguchi. (1 s s.189) They also predict optical bistability and self pulsations with bifurcations here occurring at the edge of the injection locking region for low pump and also in the regime of small injection signal due to the anomalous dispersion effect at the lasing frequency. Figure 50 shows the injection locked output behaviour as a function of detuning and Fig. 51 shows the steady state stability diagram for the laser as a function of relative pump rate. In Fig. 50 the assymetric nature of the tuning curve is apparent and detuning features were devisable into the regions I (bistability with hyseresis), II (stable locking region), III (dynamically unstable region having pulsation solutions) and IV (self modulated region outside the lock-in range). In Fig. 51 the bistable region I exists for a pump rate 1.13. For a higher pump rate region II becomes narrower and region I disappears, the dynamic unstable region III appearing instead. For a pump rate 1.4, subharmonic bifurcations occur in region III although for lower pumping only self-sustained pulsations were obtained in these regions. Figure 52 shows the output response for various detuning for a stepwise increase of the pump rate in

Pulsating instabilities and chaos in lasers

203

20-2 I

q

I-

L A =0 0 2 2

= 0.021

o-

(b)

(O) I

A =0 . 0 2 3

o"

A : 0 02553

(C)

(d)

I=

- - 1 2 0.j

,-I

6al

I.,---

A :

0 024-

A : 0.0242

o-

0. ('f)

(e)

FIG. 49. (Subregion 6 in Fig. 48). A transition to chaos through period-doubling bifurcations. Intensity here is plotted as a function of time for different values of A (Ref. 186).

Loser L ~ I.-- E,~ I~ect,m

"

REGION I

_1

Eo-

REGION IV" I ......

0

I

- 0.05

I

I

-0025

Q025

Normclized Frequency

i

005

fl075

Oetuning , ,,

(b) FIG. 50. (a) Conceptual model of a detuned laser with external light signals, (b)injection locking curve (Ref. 188). JPQE IO:3-D*

204

R . G . HARRISON and D. J. BlSWAS 0075

Self Modulohng REGION I~ 005 ,q

#

== d3

0025

Stable REGION I] -0 0 2 5

0

z

-0.05

Bistoble REGION I -0075 0

Ol

02

03

04

05

Excess Pump RoTe,(P-PIh)/Pth FIG. 51.

Stabilitydiagramof injectionlockingin detuned lasers (Ref. 188).

regions II-IV. A steady state output is obtained in the stable lock-in region II and in region III period doubling bifurcation occurs until chaotic self pulsation results (b-e). In the self modulating region pulsations at a frequency corresponding to the difference frequency between the injected and lasting signal occur. Such modulation has been reported both theoretically~l 58,172,~9o) and experimentally.~ 75,~91) Experimentally confirmation of these instability phenomena, the first reported for an injection locked system showing period doubling and self modulating pulsations are also given by the authors using LiNdP40 ~2 (LNP) optically pumped by an Ar laser detuned from the gain peak. Recently Otsuka and Kamaguchi~192'193) have also considered a laser cavity configuration comprising two coupled Fabry-Perot resonators with one common mirror and which share the laser gain medium. For the short cavity sufficiently small for single mode conditions the system is shown to undergo successive inverse subharmonic modulation of sustained relaxation oscillations leading to intermittent turbulence with increasing detuning of the cavity frequencies from the gain centre frequency. This work has been extended to an experimengal investigation of a cw AIGaAs diode laser with a gold coated external mirror (see insert of Fig. 53). The light output power versus injection current for the conventional cavity and the compound cavity are shown in Fig. 53. The external mirror reduces the threshold current and gives rise to a kink in the curve. In this region instabilities were obtained with a fundamental frequency (fe) corresponding to the separation between the external and internal mirrors. For an increase in injection current in the kink region successive subharmonic oscillations occur through to chaotic emission. It is argued that the frequency dependent optical feed back enhances the amplified spontaneous emission (ASE) component which is separated by fe" These ASE peak frequencies are determined only by the external cavity length. The observed instabilities originate from nonlinear interaction b e t w e e n the compound cavity lasing mode and the ASE mode governed by the intensity dependent refractive index of the active layer.

Pulsating instabilities and chaos in lasers

205

(o)

(d)

(b)

(e)

(c)

(f)

FIG. 52. Output response in the stable lock-in region II [(a)], unstable locking region III [(b)-(e)], and self-modulating region IV [(f)]; (a)detuning Ato=-0.025, (b)0 (period 1), (c)0.0125 (period 2), (d)0.0156 (period 4), (e)0.0188 (chaotic), and (f)0.0625 (Ref. 188).

Instabilities in laser injected amplifier systems under delayed feedback conditions have also been analysed,t194) Feedback is provided by an external mirror which together with the entrance face of the amplifier, a semiconductor is considered here, forms a Fabry-Perot cavity. Period doubling bifurcation to chaos is obtained when the cavity photon lifetime is considerably longer than the gain medium relaxation time; here the carrier lifetime. These findings compliment the earlier significant contribution of Ikeda and co-workers for absorptive, t44'~2) rather than gain, media in the same dispersive, i.e. detuned limit and also in nonlinear Kerr media. In such systems instabilities arise in absorptive (gain) media as a result of saturation of absorption (decrease in population inversion) by the pump (injection) signal giving rise to an increase or decrease in the refractive index of the medium owing to the dispersion effect in the detuned cavity surrounding the medium. Here the polarity depends on whether the laser peak frequency is below or above line centre of the absorptive (gain) medium. For the gain system analysis of the chaotic regime which evolves through period doubling reveals successive changes in the shape of the strange attractor and in the appearance of an attractor which complies with 1If noise.

206

R.G. HARRISONand D. J. BISWAS

Mirror

3

Fabry-Perot Att. L~'I L4,2 ex,

~_= ~°z 2

Analyzer

S,_APD~ / Analyzer / ,(0) L~,I GHz / Oscilloscoae/

/ / /

Compound cavity E D /,/ ,

.G

1

/

/

~Solitary LD

Ith I

0

50 100 Injection current I (mA)

Fro. 53. Light output vs DC injectioncurrent curveswith and without the externalcavity(Ref.193).

Further in c o m m o n with the absorptive systems analysed by Ikeda a spiral attraction occurs although the predicted circular and doughnut-shaped attractors for the gain system are considered peculiar to active systems alone. 5.3. Lasers with Saturable Absorbers Another form of external control is the use ofintracavity saturable absorbers. Conventionally these have been used in single mode lasers to induce repetitive giant pulse mode operation and in multimode systems to cause mode locking. Passive Q-switching arises from an interplay between the slow response of the population difference of the amplifying and absorbing media and the fast response of the field. For pumped conditions sufficient to provide net gain in the presence of absorber loss the field builds up resulting in saturation of the absorber and subsequently greatly enhanced emission which then saturates the amplifier gain which results in a decrease in the field which finally decays to zero. The process repeats itself when the pump mechanism brings the atomic population back to the original condition. Unlike a normal laser this system may become unstable along the zero intensity branch of the steady state solution even before reaching the first laser threshold. ~195,196) As for other instability phenomena the full Maxwell-Bloch equations are required to describe this behaviour and have shown the existence of periodic and quasiperiodic pulsations - - as well as multiple steady state solutions. Along with experimental and theoretical studie# 195-2°9) of the dynamical behaviour of Q-switching, efforts have also concentrated on bistability and numerous other unstable modes of operation. ~2~°-z16) (For a recent review see Ref. 217.) Although of considerable interest in their own right the pulsating instability characteristics of this system do not appear to fall within the various scenarios typical of the other laser systems considered in this review. We therefore do not discuss this in further detail but refer the interested reader to the references cited here. 6. M U L T I M O D E LASER Multimode operation involving longitudinal and/or transverse mode is perhaps the most common mode of operation of conventional laser systems. Evidently beating between these various modes generate pulsations in the output of these systems. In the limiting condition of

Pulsating instabilities and chaos in lasers

207

equal frequency spacing and fixed phases for the excited mode the output signal comprises a train of ultrashort pulses. Particularly because of its practical application mode locking as it is commonly known has been extensively investigated and quantified in a very wide variety of laser systems almost since the inception of the laser, t218-2s2) (For reviews on the general nature of mode locking see refs 253-255). Analysis of such systems was originally made by Bennett(219) and Lamb t22o)for a three-mode laser. Other notable earlier contributions were made by Statz and Tang ~221,222) and Haken. t256'257) On increasing the excitation of some lasers such pulsations may breakdown into more complex temporal behaviour and in some instances give chaotic emission. However due to the many coupled variables which describe such systems, analysis has proved prohibitively difficult in most cases. For homogeneously broadened lasers of moderate gain lasing occurs on the cavity mode at or near line centre for which gain is highest and oscillates at the expense of other cavity modes lying within the gain profile. However, as the gain is increased a threshold is reached at which additional modes become excited resulting in a complex time dependent laser output. In at least qualitatively understanding such behaviour we refer to the discussions given in the section on mode splitting. As we saw, for homogeneously broadened systems, under resonant lasing conditions, population pulsation, or otherwise Rabi splitting is induced in the lasing transition to produce a double peaked gain distribution. Evidently for a multimode system in which the cavity mode spacing is of the order of the Rabi splitting, then multimode instabilities are possible for sufficient gain as determined by Eqn. (27) (see Section 2). First treatment of this situation based on laser stability analysis was given in the classic papers by Risken and Numendal, 1258'259) Haken ~26°'261) and Graham and Haken. ~262) In contrast to the single mode laser for which the intensity distribution is uniform within the cavity, for the multimode system the electric field evolves as a pulse localized in space arising from the interference of the ensemble of modes. As such the Maxwell-Bloch equation describing this system must include the spatial dependence of the electric field given in the full set of Eqns 9-11 (Section 1); otherwise neglected in the reduced set of equations discussed earlier for the single mode case. Stability analysis of these equations in the on-resonance case shows that the emission from the homogeneously broadened laser system becomes unstable if the unsaturated inversion density expressed as before in terms of an effective pump parameter A = (Do -

Dthr)/Dthr,

satisfies A > Ac - 4 + 3711/7± + 2x/2(1 + TII/7±)(2 + 711/7±), providing the cavity modes defined for a ring cavity as o~ = 2 r c ( C / L ) n ,

n = 0, ___1, 4-2 ....

lie in the region ~min < Ice.I < ~ . . . .

where

O~max

/ 711(37±l-

7pl ---R) 1 - - T ± ( l _ 2 ) Z711 + R

'

R = x//27 2 - 2(47i + 37tl)/7 + ~)~. These are sufficient conditions for the generation of instabilities with no requirement placed on the cavity Q. Thus unlike the case of a single mode system instabilities here are obtained in both good and bad cavity limits.

208

R. G, HARRISON and D. J. BISWAS

2

fo.

l 0

20

~0

60

O0

100 120

IZO 150

180 200 270 N

FIG. 54. The transient build-up of the intensity from a small Gaussian disturbance as a function of the time (Ref. 258).

The instability leads to a build up of a pulse, the transient evolution of which is shown in Fig. 54. After a few round trips the variables reach the approximate cw solution

but for the parameter of Fig. 54 these values are not stable. After the transient build up the field variables have the form of a travelling wave pulse f ( t - z/v, t), which slowly changes its amplitude, its velocity and its pulse shape. For initial conditions lower than the threshold value similar transient behaviour is obtained which however leads in this case to cw emission. Interestingly the steady state pulse has a velocity greater than the velocity of light in the host material, v > c. However, as discussed by Risken and Nummedal the photons are produced at the front edge of the pulse and travel with velocity (v - c ) through the pulse to be finally absorbed at the trailing edge of the pulse; thus never exceeding the velocity of light. As expected, in the fimiting situation of a single resonant mode, analysis reduces to that described earlier, (Sections 1 and 2) giving rise to the same conditions for the onset of instabilities. In a series of papers by Haken and O h n o ~263-265) multimode instabilities have been analysed analytically and the results of the temporal form of the laser output found to be in good agreement with the numerical solutions of Risken and Nummedal. Recently the treatment of Risken and Nummedal has been generalized by Narducci e t a/. ~266) t o consider the more general case when the cavity mode is detuned from line centre. Under this condition they find hysteric behaviour with self pulsing instabilities similar in character to those discussed above. However, whereas phase is constant for the on-resonance situation, here the output pulsations are a direct result of a phase instability. Furthermore the reduced gain requirement for the onset of phase instabilities suggest that these should be more readily observable than those arising under resonant condition (amplitude instability). Also significant is the work of Lugiato e t a / . t267) in which they generalise the earlier treatment of Risken and Nummedal ~25s'259) and Graham and Haken t262) to allow arbitrary values for the cavity transmission and unsaturated gain, hitherto considered as small to permit the mean field approximation. Results show that the multimode laser possesses a rather complicated phenomenology that includes periodic and chaotic attractors, soft and hard mode instabilities and even the appearance of square wave pulsations for high density of cavity modes. Other work by Narducci et al. ~z68) have also considered multimode lasers with injected signal. The mechanism giving rise to instability in this system is essentially the same as that found for single mode systems. However, here the incident field frequency competes not only with the modepulled frequency of the resonant mode, as for the single mode case, but also with the mode pulled frequencies of the off-resonant modes that are above threshold. They find the multimode instability develops for values of the side mode frequency which are of the order of a few 7±. The immediate consequence of this is that the instability here requires a good cavity (as discussed in Section 2), apart from the requirement of a long cavity. Furthermore, and unlike other systems, e.g. lasers with saturable absorbers and ordinary free running lasers, the resonant mode is found to become unstable simultaneously with off resonant side modes.

Pulsating instabilitiesand chaos in lasers

209

The complexity of multimode systems has restricted analysis almost exclusively to systems with homogeneously broadened gain media,t94'258-268) Recently, however, Mandel ~269) has extended analytical treatments to a ring system with inhomogeneous broadening. As in the case of single mode instabilities (Sections 3 and 4) the threshold for unstable behaviour is dramatically reduced compared to that for the homogeneous system of Risken and Nummedal.~258,259) Indeed it is shown that there always exists a domain of instability even when the intensity goes to zero. However, for gain broadening conditions of both types it is found that the good cavity modes furthest from the line centre becomes unstable whereas in the bad cavity limit the unstable modes are located near line centre and are therefore more accessible. Further recent analysis of this system by Narducci et al. ~27°) predicts various dynamical scenarios; long term time independent solutions are predicted when the selected parameters provide only one stable steady state solution, if more than one steady state is stable the long-term configuration of the system exhibits hysteresis and finally if all the steady states are unstable for the same control parameters long term output pulsations are expected. These behaviours are largely governed by the frequency separation between the centre of the gain line and the selected cavity mode. A limiting form of multimode lasers is the much simpler two mode system which are more easily analysed;11°°'271-279~ particularly the bidirectional ring laser. ~271-278) Notably, in the recent work of Lippi e t a / . ~277~ for a CO2 laser and Khadokin and Khanint278~ for a solid state system, periodic and aperiodic switching and chaos have been observed. For simplicity most of the treatments on multimode instability neglect transverse mode effects since the plane wave approximation is used. It has however been shown quite early that modes having different transverse indices may also lock together resulting in both spatial and temporal beat patterns. ~28°-285~ Recently attention has been given to the effect of transverse mode interaction on laser instabilities which have been found to proceed through established routes to chaotic mption.~137,286-29°) These various aspects of multimode operation are discussed below. 6.1. Multiaxial M o d e Systems Most of the experimental work in this area may be found in the early literature particularly pertaining to mode locking. Motivated here by the desire to generate clean signals with high contrast ratio, irregularities in such behaviour in many cases undoubtedly testifying to more complex instability phenomena, were carefully avoided. Consequently these reports are unfortunately of reduced value in the context of present interest. However one recent example which addresses the question of instabilities in multiaxial systems is that given by Hillman et al. ~291~ Here they consider homogeneously broadened ring dye laser system pumped by a cw-Argon ion laser of cavity lengths 25 cm and incorporating broadband highly reflective mirrors. The dependence of the dye laser output power on Argon pump power is shown in Fig. 55. In region AB lasing occurs at gain centre whereas at the point C lasing switches to simultaneous two frequency operation symmetrical about the gain centre-with some evidence of small bistable action in this region. For further increase in pump power, the two lasing wavelengths further splits with a parabolic dependence on pump power. The dependence is consistent with that produced by Rabi oscillations which strongly couples the two fields. At still higher pump power discontinuity again occurs proceeded by the growth of a mode at the gain centre. This mode rapidly grows and bifurcates with the simultaneous extinction of the outer two modes. Largest hysteresis effects occur in this region. Again a parabolic dependence of the splitting with power is observed (region FG) although the dependence indicates atomic" oscillations at a sub-multiple of the Rabi frequency. Beyond G, further power jump, hysteresis and spectral bifurcation occur leading to what appears to be a chaotic state. These general features have been analysed by the authors by considering three AM phase locked fields of arbitrary.strength interacting with a two level atomic medium; their results indeed show the appearance of gain maxima at both Rabi frequency and the sub-harmonic of it. More recently these investigations have been extended in which the instabilities are studied as a

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function of pump power, intracavity power, cavity Q and intracavity dispersion for different cavity configurations. C292) Milloni and Shih C293) have also analysed this system and find that following the observed bifurcation of the single mode steady state and the onset of the two modes symmetrically displaced about the line centre there is a period doubling route to chaos which so far has not been experimentally identified. The results of Hillman eta/. (291) are nevertheless not readily understood in relation to the earlier models (Risken and Nummedal (258'259) and G r a h a m and Haken (262)) since the gain threshold for the onset of instabilities is only slightly larger than that for lasing and secondly the discontinuous suppression of the laser mode with the simultaneous evolution of the side bands for increasing values of pump parameter is at variance with these models which predict the persistent presence of the central laser components. Nevertheless the recent analysis of Lugiato et al. (267) goes someway towards resolving this behaviour although their results are unable to identify all the main signatures of the instabilities. They find for example that although unstable side bands emerge symmetrically at a distance from the line centre which is a monotonically increasing function of Rabi frequency, consistent with experimental findings, the threshold gain for this instability is several times larger than that required for laser action, whereas experimentally instabilities are observed just above the lasing threshold. At variance with observations they further find that the central mode is never quenched although noise injected into the model considerably helps in its suppression. Finally although discontinuous behaviour associated with the generation of the two symmetric side bands is predicted, rather special initial conditions are required and the pulsation frequency is.fixed and independent of Rabi frequency. 6.2. Two Mode Systems As perhaps first noted by Scholz et al. ~5°) (see section on external control) a multimode homogeneously broadened system intrinsically provides internal population inversion modulation as an independent variable for coupling modes which under suitable parameter conditions will generate instabilities. The limiting case of the relatively simple two mode laser is especially illuminating and has been the subject of considerable analysis. (27 ~ 279)Application of this to experimental systems is to be found in the work of Mandel, Roy and S i n g h ~272'274'275) for a bidirectional ring dye laser which can support two stable states, switching between one or the other being induced by noise. Particularly attractive here is also the work of Lippi et al. ~27v) for a bidirectional CO2 ring system. The system they describe is a CO2 ring laser in which both directions of propagation are allowed. Since the linewidth is homogeneously broadened, the two counter-propagating beams can not work at the same time since they must compete for the same

Pulsating instabilities and chaos in lasers

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population inversion. Moreover they are slightly detuned from each o t h e r - - and with respect to line centre--because, for intrinsic asymmetries, cavity losses are different for the two propagation directions (k~ and k2); this results in a different mode pulling and thus different lasing frequencies. The detuning has been shown to be essential for breaking the symmetry between the two directions. As such it may be inferred that the population inversion is modulated at the difference frequency so providing the necessary additional degree of freedom for this laser for parametric interaction between the modes and the resultant generation of instabilities. Modelling of the system necessarily includes the two cavity mode fields which are complex since they are detuned from line centre together with the uniform component of population inversion and the complex amplitude of the population grating, induced by the two fields. (277) Numerical solutions of this seven equation system closely matches all experimental results. Three different regions are identified, in the first self pulsing is obtained very similar to that for a laser with injected signal. One mode exhibits spikes superimposed on cw emission, while the other exhibits spikes alone in phase with the main mode, which occur at a repetition rate (o~) of the order ofYlr. In fact, as in the case of injection, the cw working mode injects some energy into the other one letting population inversion increase up to a level at which a giant pulse takes place. During the pulse both modes go above threshold and spike in phase. Superimposed on the decay, relaxation oscillations typical of the CO2 lasers with a frequency (09o) very near to f~ are observed; they are out of phase because of competition between the two modes. For higher excitation currents a deterministic switching is observed due to competition between the two fields with low frequency ( ,-~30 Hz). The transition between these two regimes is not abrupt and it takes place through a region which shows chaotic behaviour. Here both phenomena related to population inversion, spiking (lower currents) and oscillation (higher currents), take place; effective output frequency results also as a combination of the two others (~o, + ~o0). At the same time if the cavity mirror position is so adjusted that f~ ~ (o~s + ~o) then competition of two different variables (population inversion and field) are obtained on the same time scale. The result is a fully developed chaos as shown in Fig. 56. Similar two mode instability phenomena has also been reported (279) in an optically pumped ring cavity laser system. As noted in Section 3, pumping substantially offthe pump-absorption line centre creates a situation where only the forward or the backward emitting laser mode interacts with the inverted molecules and the laser oscillates in a single travelling wave. Pumping close to or at pump line centre allows both modes to interact with the inverted molecules. In this case, under cw conditions only one mode will oscillate due to competition. However, in the presence of instabilities both modes may emit. The system is a 15NH3 laser pumped by 10R(42) CO2--1aser line and emitting at 376#m wavelength. The laser first reaches the threshold for continuous single-mode emission and then, as the pump intensity is increased, the threshold for two-mode oscillation is reached

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accompanied by a high frequency self-pulsing. With still higher pump power a second lower frequency appears. The ratio of the two frequencies can be chosen by tuning of the pump frequency. Further increase of pump power results in locking of the two frequencies at the nearest harmonic of the lower frequency. For additional increase in pump power a perioddoubling cascade eventually leading to chaos was observed. Figure 57 shows a time picture and the corresponding power spectrum of the laser emission in the chaotic case. The AC-stark broadening or splitting of the gain lines by the pump laser is smaller here than the pressure broadening. This homogeneously broadened two mode laser possesses the same dynamics as reported for a Bernard convection cell. t294) The dynamics of this system is, however, different from the two-mode CO2 ring laser previously discussed, since unlike it, the three relaxation times of this laser are of the same order of magnitude thus precluding adiabatic elimination of any of them. Pulsating instabilities arising out of the coupling between two modes has also been reported in a homogeneously broadened Raman laser. ~1°°~ The system, a 12.8/~m pulsed NH 3 laser optically pumped by 9P(16) CO2 laser radiation, is identical to the one described in Section 3.2 in regard to single mode instabilities in homogeneously broadened system. For typical operating pressures, the free spectral range of the cavity was about four times the FWHM value of the gain broadening. Yet for certain cavity lengths conditions mode pulling effects forced lasing on two axial modes resulting in avariety of interesting observations as shown in Fig. 58. Figure 58(a) shows a kind of intermittency where there is a direct transition from an orderly state, here initial low frequency modulation, into metastable c h a o s , (128'129) viz. a chaotic burst followed by an abrupt transition to a steady output. Intermittent chaotic bursts are evident in the trace (b) of this figure. The beating effects seen in Fig. 58(c) can be identified with heavy breathing as discussed recently by Lugiato et al. (181) Traces (d) and (e) respectively typify examples of weak and fully d e v e l o p e d c h a o s . (122)

6.3. Transverse M o d e Induced Instabilities As discussed in the early literatures by Lamb ~22°) and Haken (256'2sT) and others,~219'253-25s) mode-locking which occurs under conditions of equal frequency spacing and fixed phase for all the excited modes may be destroyed, or at least modified, by various nonlinear mechanisms in the dispersion and the mode coupling, which cause variations in phase and/or intermode frequency spacing, giving rise to pulsating instabilities of multiaxial mode systems. Practical laser systems, may however provide emission on more than one transverse mode for which the frequency spacing is inherently unequal. This in conjunction with the different gain experienced

Pulsating instabilities and chaos in lasers

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by the various modes due to their different spatial field structures are factors which may greatly aid the onset of pulsating instabilities in systems even operating on only one axial mode. In the work of Shih and Milonni ~286) they have considered the effect of spatial inhomogenity in the gain of the system supporting two transverse modes and where population inversion is adiabatically eliminated in their analysis. Considering parameters characteristics of the Ruby laser they show the system to exhibit chaotic emission for sufficient spatial inhomogeneity; the data is shown in Fig. 59 (traces (a-d)). In the absence ofinhomogeneity a steady state solution is obtained. When the degree of inhomogeneity is ,-~1.5 ~ , the intensity settles into a regular oscillation at a frequency close to the mode difference frequency (trace (a)). For increased inhomogenuity, regular oscillations now at a period of twice that of trace (a) are obtained (trace (b)) which evolved to chaotic emission (trace (c)) when the degree of inhomogeneity is ~ 10 Y/o. The spectrum of the time series of trace (c) is shown in trace (d). For larger values of inhomogeneity, the emission returns to regular time evolution. Transitions to chaos therefore occur from both sides of the regular regime. From a practical standpoint the inhomogeneity considered here in terms of the degree of pumping nonuniformity is a quite realistic situation encountered in various laser types. In other interesting work Hollinger and Jung ~287~ theoretically show a solid state laser running on a single longitudinal mode to exhibit chaotic time behaviour if several transverse

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modes are excited. The temporal development of the laser is approximated by Kirchoffintegral together with an equation for the inversion, and analysis considers a homogeneously broadened ring cavity system. The Kirchoffintegral maps the transverse electric field distribution on a two dimensional section through the resonator, onto the field distribution here, after one round trip. The iterated Kirchoffintegral therefore defines a discrete time evolution of the electric field on this surface. For simplicity the active medium is concentrated in a thin slab lying directly beside this section. The numerical calculations are done using parameters typical of Nd-glass laser systems, where the control parameters are the pump rate and the "g" parameter of the resonator.

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For increasing pump rate, steady, quasiperiodic and irregular time behaviour for the output power is obtained. The sample plots shown in Fig. 60 (a,b) are of the time dependence of the spatial energy distribution in the beam in the steady state (transients are already damped out) for two different pump rates, other parameters being constant. Trace (a) shows a periodic superposition of TEM00 and TEM0~ modes and correspondingly the intensity fluctuates (beats) with one discrete frequency shown in trace (a) of Fig. 61. However at the higher pump rate (trace (b), Fig. 60) three modes are excited where however the phase shift between modes are irrational JPQE IO:3-E

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secondary beat-note region increased abruptly from 10 KHz to 1 MHz. This broadband state persisted over a narrow range of increasing current and then more stable pulsing was observed. This was followed by another broadband spectrum typical of deterministic chaos, which appeared close to the mode-locking threshold. However as the current was decreased, chaotic regions were skipped either partly or in full, the system persisting in the mode-locked condition before abruptly making the transition to secondary beats. As noted before, the studies of Hallas et al. are limited to a lasing transition which is predominantly inhomogeneously broadened. However, the relative simplicity of the analysis of multimode instability for system with homogeneously broadened gain medium is well recognised. The first observation of similar effects for a predominantly homogeneously broadened lasing transition was recently made by Biswas and Harrison. tzs9) A commercial cw-CO2 laser was used for this investigation in which a maximum homogeneous to inhomogeneous broadening ratio of 2:1 was achieved. The transverse nature of the emission was confirmed by taking a spatial intensity scan of the laser beam by a pyroelectric array detector. A Gaussian distribution was obtained for TEMoo operation whereas with wide aperture this was degraded to some extent. On cavity tuning the distribution expanded spatially and became complex with no readily definable mode pattern

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indicating an admixture of transverse modes. Similar observations have also been made by Hauck et al. (295) for a pulsed solid state laser. A typical pyroelectric array recording is shown in Fig. 65. A large variation in oscillation period (67-360 nsec; cavity round trip time tR is ~ 12 nsec) was found to occur with coarse cavity length tuning across the FSR. This is attributed to change in gain seen by individual transverse modes on cavity tuning thus bringing different modes into lasing; corroborated also by the transverse intensity pattern of the laser beam. This in turn changes the beating period. Figure 66 shows an approach to chaos with fine cavity tuning. Undamped periodic oscillation (trace (a)) as a result of beating between two modes first develops into quasi-periodic oscillation (trace(b)) involving the excitation of three transverse modes. On further cavity tuning irregularities in the pulsating instabilities are clearly evident (trace (c)). This ultimately develops to the chaotic behaviour shown in trace (d). This behaviour is completely consistent with the recent predictions of Hollinger and Jung. (287) This data was taken for an operating pressure of 20 mb. The amount of tuning required for going from trace (a) to trace (d) was about ~ 6 MHz compared to the cavity FSR of ,-~85 MHz. The routes to chaos proceeds via a Ruelle-Takens sequence involving initially two coupled transverse modes developing to chaos through the further coupling of a third mode. Quasiperiodic motion involving oscillations of three modes precedes chaos and as noted in Section 4 such motion has also been observed in a single mode H e - N e laser. (~43) These effects were obtained under normal operating conditions of a standard commercial laser. Undesirable for many applications of such systems these effects are eliminated by operation on the TEMoo mode alone; conventionally obtained using an intracavity aperture. However in maximising the efficiency of operation, design may be most effective by ensuring better matching between the TEMoo mode and the gain cross-section. More recently Biswas and Harrison (29°) have extended this result to a truly homogeneously broadened albeit a pulsed system where the homogeneous to inhomogeneous broadening ratio

220

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FIG.65. A typical pyroelectricarray detector scan showing the complicatedtransverse nature of the CO2 laseremission. Pitch of elementsis 0.5 mm. For clarity,adjacent elementsare joined by thin white line (Ref.289). is increased to 75:1. The system is a T E A - C O 2 laser coupled with a low pressure section in the hybrid configuration t296~ for longitudinal mode control. TEMoo to higher order transverse mode selection was achieved by intracavity aperture control. In order to obtain lasing on single longitudinal and transverse mode from the T E A - C O 2 laser it was necessary that gain exist in the low pressure section and the aperture be shut appropriately. The first operation selects modes with the same axial mode index while the latter suppresses the higher order transverse modes. This is evident from the steady output as shown in trace (a) of Fig. 67. At this point if the aperture is fully open the emission becomes oscillatory with a period of 100nsec (trace (b)) of Fig. 67), compared to the cavity round trip time (tR) of ~ 13 nsec. This attributed to beating between the two transverse modes separated by ~ 12 MHz. Cavity tuning was found to have no effect on the oscillation period suggesting that the discharge configuration and the cavity geometry favoured oscillation of only these particular transverse modes under all cavity length conditions. A third mode was never excited here even when the pumping was increased. In order to couple an additional mode, having a different axial mode index, into the system the discharge in the low pressure section was switched off; the system then operates as a conventional multiaxial mode laser. Trace (c) of Fig. 67 shows the quasiperiodic oscillation in this particular case. Two distinct oscillation periods are evident here: one at close to 100 nsec and the other at close to tR. This latter period is the result of beating between two adjacent axial modes. This trace was obtained for a gain which is ~ 2 times the first laser threshold. When this value is increased to ~ 3 by increasing the energy dissipation into the discharge load, the emission becomes chaotic. Trace (d) of the same figure shows evidence of such chaotic behaviour. The chaotic motion develops into regular periodic oscillation when the aperture is shut down to limit oscillation to the TEMoo modes. This is illustrated in the trace (e) of Fig. 67. For the highest pumping possible in this system (this is limited by the arcing and eventual degradation

Pulsating instabilities and chaos in lasers

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lossy cavity and nine times above lasing threshold, have up to recently been considered to be impracticable. In the interim there has been a proliferation of beautiful results on alternative but m o r e complex laser systems; inhomogeneously broadened lasers, injected systems, systems with external m o d u l a t i o n s and multimode lasers. Indeed re-examination of m a n y conventional lasers has revealed a wealth of instability phenomena. Furthermore, recently far infrared optically p u m p e d molecular lasers have been identified as unique systems in providing readily achievable operating conditions by which instability p h e n o m e n a m a y be described by the H a k e n - L o r e n z model. Investigations here will u n d o u b t e d l y be a major area of future activity. U n d e r s t a n d a b l y in this y o u n g field experimental findings, which have been abundant, in providing evidence of routes to chaos and chaotic m o t i o n are nevertheless in general far from being sufficiently comprehensive and rigorous to permit quantitative analysis necessary to test the theoretical models used to describe their behaviour. O n the other hand, theoretical modelling of instabilities of these systems has also met with mixed success. It is clear from theoretical investigation of transition to chaos in various physical contexts that ad hoc truncation of the full equation describing physical problem should be avoided. While these

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N O T E A D D E D IN P R O O F

The reader is also referred to a recent complimentary review article published during the preparation of this manuscript: J. R. ACKERHALT,P. W. MILONNI and M. L. SHIH, Chaos in quantum optics, Phys.Reports 128, 205 (1985).