Self-pulsing and chaos in inhomogeneously broadened single mode lasers

Volume 47, number 1

OPTICS COMMUNICATIONS

1 August 1983

SELF-PULSING AND CHAOS IN INHOMOGENEOUSLY BROADENED SINGLE MODE LASERS R. GRAHAM and Y. CHO 1,2 Fachbereich Physik, Universitiit Essen, Fed. Rep. Germany Received 11 May 1983

Self-pulsing and chaos in inhomogeneously broadened single mode lasers is investigated in the framework of two models with 4 and 6 degrees of freedom, respectively. These models axe derived as the first two steps of an infinite hierarchy of approximations of increasing sophistication. Numerical examples of self-pulsing and chaotic dynamics axe presented.

Homogeneously broadened single mode lasers are well known to exhibit self-pulsing instabilities and chaotic dynamics under the combined conditions of large ratio of gain over losses and low cavity quality [1,2]. In fact, a homogeneously broadened single mode laser in resonance has essentially 3 degrees of freedom only and is realistically described by the Lorenz model [3,4], which has served as a prototype model for investigating chaos in continuous dynamical systems [5]. However, the above mentioned conditions for the occurrence of chaos have not yet been realized, experimentally. Recently, bad-cavity instabilities have also been discussed for inhomogeneously broadened lasers [6, 7], and seem to be more easily accessible, experimentally [8,9], but models of comparable simplicity as the Lorenz model have not yet been proposed for inhomogeneously broadened lasers. It is the purpose of the present note to show how such models can be constructed. We find that an infinite hierarchy of models exist which increase in accuracy and complexity. We present a 4-dimensional model and a 6dimensional model, which are the two simplest members of this hierarchy. The results of a linear stability analysis of the time-independent states and some numerical solutions are given to show the various

types of dynamical behavior which may occur in these models. The dynamical behavior is found to be much richer than in the homogeneously broadened case and is obtained under physically more realistic conditions. The equations of motion of an inhomogeneously broadened laser in single mode action are given by [10] = -~b + ~

&u = -(iuu + 3'±)°tu + g u o u b ' b u = 711(d0u - ou) - 2gu(ot*ub + otub+ ) .

1 Permanent address: The Institute of Scientific and Industrial Research, Osaka University, Yamadakami, Suita, Osaka 565, Japan. Supported by the Alexander yon Humboldt-Stiftung.

52

gu~,

(1)

For the details of the derivation and notation in these equations see ref. [10]. We merely note that b is the slowly varying complex mode amplitude at the line center of the inhomogeneously broadened transition, e~u, o u are, respectively, the complex polarisation and population inversion of the group of atoms with frequency uu off the line center,g u is the dipole coupling constant of this group of atoms, 711d0u their pumping rate, 7j_, 3'11are transverse and longitudinal relaxation rates of the atomic transition, 2~ is the photon decay rate of the empty cavity. In the following we assume a running mode, in which case we may takeg u =g, see ref. [10]. Under the assumption of K ,~ 711, 7± and moderate pumping rate eqs. (1) may be simplified by applying the "adiabatic approximation". Since we are here interested in the "bad cavity" case K ~> 71,711, and in

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OPTICS COMMUNICATIONS

the case of strong pumping, we shall avoid this type of approximation in the following. The form of eqs. (1) then suggests a different strategy for simplification. Since the single mode couples only to the combined dipole moments of all groups of atoms, it is useful to introduce this quantity as a new variable. The equation of motion for this new variable will again contain new variables, which are represented by sums over the different groups of atoms. Repeating this process one generates an infinite hierarchy of equations of motion, which is equivalent to eqs. (1). Approximations of increasing accuracy can be made by dosing this hierarchy at increasingly higher levels. In this way one arrives at approximations of (1) by a sequence of finite dimensional systems. In the following we present the two simplest approximations obtained in this way and discuss their dynamical behavior. We introduce the scaled variables

P = (2g2197j.)~

E = (2g/"/±)b,

D=(g2/r~±) ~] ou ,

n'u ,

r=7±t,

(2)

dE/dr = - ( r / 7 ± ) ( E - P ) , -P +

DE

+

(602/7 2 )s,

dD/dr = (Tn/7±)(r - D) -- ~(PE* +e ' E ) ,

(3)

where r = (g2ficTj_)~

dou

A = ( g 2 1 r 6 0 2) ~

(4)

e o=(2921r~,) 2] %(I #

(7)

-

2 2 v;,1601),

dA/dr = - ( . r d v D A - (i/2)(SE* - S ' E ) , dPo/dr = -Po + DoE + [( 602 - 602)172]S+R,

dDo/dr = -(711/7±)(D 0 + Xr) - ~ t( E P 0• + E*Po) .

@2/602_

~, ~ 1)do~ = AV2/602 -- 1 , Y-udo#

k =-u

(10)

and the new variables

R =(292/g72) ~

(1 - v2/6012)o•

,

[ ( ~ 2 - v 2 )/6o 2I ] (-ivu) % .

(12) t

/.L

~"

Using eqs. (1) we may rewrite 601 as

where

dS/dr = - S - iEA - (e- Co).

[E°I 2 =(711/7±)(r- 1 - c o 2)

(6)

(11)

The hierarchy of equations of motion can now be continued by writing an equation of motion for R. It should be noted that the parameters co1 , 602, are yet arbitrary. We now consider the two simplest approximations, which close the hierarchy, but still contain effects due to inhomogeneous broadening. The simplest approximation isl obtained by putting P0 = 0. We may chose the free parameter 601 in such a way that this approximation becomes exact at least in the steady state a u = tx0 , E = E 0 0 2

(5)

(9)

Here, we have assumed symmetrical pumping ~ vudou = 0 and introduced the parameter

is a new dynamical variable. We note that by putting S = 0, eqs. (3) are reduced to the equations of motion of a homogeneously broadened single mode laser, which may be reduced to the Lorenz model. Inhomogeneous broadening is taken into account by writing down the equation of motion of S obtained from (1)

Here we introduced the new variables

(8)

which satisfy the equations of motion

is the pumping parameter, and

23

v#o.,

Do=(g2/rTj.) ~/z

and obtain the exact equations

deld =

1 August 1983

D. w(1) =

"

.i-;wJ 1),

(13)

do~ 3,Z+v , 2 2 +(~3/7,)1E012'

=0

(14)

ferry> 1 +602, r~l

+60 2 . 53

Volume 47, number 1

OPTICS COMMUNICATIONS

In this approximation, the model consists of eqs. (3), eq. (6) with P0 = 0, and eq. (7). It still has 8 real degrees o f freedom. However, it is fairly easy to show that the solutions in this 8-dimensional space are attracted to a 4-dimensional subspace in which E, P, S have equal and constant phases and may be taken as real without loss o f generality, and A = 0. Thus we arrive at a minimal real 4-dimensional model of the form

1 August 1983

E = -@/3",)(E- e), P =-P

+ D E + (002/3'2)S ,

D = (3'1113'1)(r- D ) - P E , S =-S-P+P

o ,

Po = - Po + E D o - XrS ,

= -(tc/3"j_)(E - P ) ,

D O = -(3'11/3'1)(D0 + ~ ) - E P 0 . 1~ = - P + D E

+(002/3"2)S,

/) = (3'11/3'1)(r- D ) - P E ,

d =- s- e,

(15)

which contains inhomogeneous broadening via S. The model contains 4 independent parameters. For strong inhomogeneous broadening Av >> 3'± (1 + ]E012T±/3'tl)l/2 we have 001 ~ (AY3'I)l/2" A more sophisticated approximation is obtained by closing the infinite hierarchy via the assumption R = 0. We keep the value o f 002 as given by eq. (12) and chose 002 in such a way thati R = 0 is satisfied at least in the steady state and obtain 002 =

.

w.

. 2. /i v g"

,

(16)

where ( 2w(1) wt,2) = via

(17)

From the requirement that P0 = 0 in the steady state, which is imposed by our choice o f 002, we obtain the condition

(00 -

:-.

(18)

For strong inhomogeneous broadening Av >> 3'±(I + 3'±[E012/711)1/2 we have 002 ~ Av. The resulting model consists of eqs. (3), (6), (9) and has 11 real degrees of freedom. However, the solutions in the 11-dimensional phase-space are attracted to a 6-dimensional subspace, where the phases o f E , P , S , PO are all constant and equal and may be taken as 0 without loss of generality. In addition A = 0 in this subspace. Thus, we arrive at the following 6-dimensional model 54

(19)

It contains 5 independent parameters. The minimal 4dimensional model is reobtained by taking P0 = 0, and omitting in (1 9) the equation for P0" It is now clear, how an infinite hierarchy of models of increasing dimensionality may be constructed along these lines. We now present an investigation of the dynamical behavior of the two models we have derived. We begin by noting that both, the 6-dimensional model and the 4-dirnensional model reduce to the Lorenz model for the case 002/7l = 0, which is the special case of homogeneous broadening. For X = 0, the 6-dimensional model is reduced to the 4-dimensional model. In this special case the characteristic frequencies of the system, Av, co 1 , w 2 a r e all equal. The analysis of the full set of equations (1) and the two models show that there are two time independent solutions, the trivial one, E = P = S =P0 = 0, D = - D o / X = r , and a nontrivial one, P 0 = 0 , D 0 = -Xr, D = 1 +002,13"2,E=P=-S = D,,(r- 1 -007/ 3'1)/3'±] 1/2, whichexists only for r ~>"1 + 0012/3'2. linear stability analysis of the f u l l set of equations (1) shows that the trivial time-independent solution is linearly stable in the whole domain 0 ~< r ~< 1 + 0021/ 3,2 and becomes unstable for r > 1 + 002i3'2.,_ The trivial solution of the 4-dimensional model is also stable for 0 ~< r ~< 1 + 002/3" 2 provided t¢ < 3'± or 002/3'2 < (~ + 3'±)/(K - 3'1) if K > 3'±, and we restrict ourselves to these cases in order to avoid a spurious instability which is not present in the full set of equations (1). A linear stability analysis of the non-trivial timeindependent solution o f the 4-dimensional model is straightforward but involves tedious algebra, which can be largely avoided by considering the special case 3'.t = 3'11"The result o f the stability analysis for this special case is that the non-trivial time-independent

occurs in an inhomogeneously broadened laser, but at a greatly reduced threshold. This is the central analytical result of the present paper. Finally, we present some numerical results which illustrate the dynamical behavior of the 4-dimensional and the 6-dimensional model and allow a comparison with the Lorenz model. In fig. 1 we present projections of typical non-transient trajectories on the P - D plane in arbitrary units which are scaled with the parameter r in order to preserve the size of the attractors. The fixed parameter values Tllh'± = 1, ~o2/ 72 = 2 and K/T± = 2.8.were selected in order to satisfy the bad cavity condition and in order to avoid a spurious instability for ~o2/72 > (~ + 7±)/(K - 71) which exists in the 4-dimensional model but is not present in the exact full equations (1), as was mentioned above.

solution is stable, provided ~o2/72 < (r + 7z)/(r - 7±) and either

< 2,y,,

(20)

or

>27±, (K + 47.L)r

r < v±(K - 270

1<2002

73 (K - 27,)

= rth .

(21)

Instability occurs in the case of a bad cavity >2T±,

1 August 1983

OPTICS COMMUNICATIONS

Volume 47, number 1

r>rth.

(22)

In the limit w 2 = 0, this is the self-pulsing instability of a homogeneously broadened laser for K > 7± + 711. Our result shows that a corresponding instability still

(C)

(B)

(A)

LORENZMODEL

6-DIMENSIONALMODEL

4-BIMEN$1ONALMODEL

,t© :@

@

J

loc

t -

,,@ i

,@

q t

8c / 6C ~

~ @

I

4C-

40

- :"im k.

1'+

12c

UNSTABLEREGION

f

i

I

20-

, I

15

t

i

10

lc

REGION

5 I

0 0

0.5

1.0 ---A

1.5

2.0

0

Fig. 1. Trajectories projected on the P - D plane of (a) the 4-dimensional model, (b) the 6-dimensional model, and (c) the Lorenz model. Used parameter values: 711/7± = 1, wt2/'~ = 2, and g/'r± = 2.8. For detail refer to the text.

55

Volume 47, number 1

OPTICS COMMUNICATIONS

Fig. 1a shows typical trajectories of the 4-dimensional model along the r-axis. The threshold of instability of the time-independent state is rth = 4.2 according to eq. (21), and much lower than the corresponding threshold rth = 23.8 o f the Lorenz model, whose trajectories (for 602 = 0) are shown in fig. 1 c. Immediately above threshold a limit cycle appears in the 4-dimensional model, which, for somewhat larger values o f r, bifurcates to a strange attractor similar to the Lorenz attractor. For still larger values o f t there are windows o f periodic behavior and period doubling of symmetrical and asymmetrical limit cycles, which are qualitatively similar to corresponding results in the Lorenz model. For r ~ 100 the two models are very similar. In fig. lb we present typical trajectories of the 6dimensional model for different values of the parameters r (vertical axis) and k defined in eq. (10). For = 0 the 6-dimensional model is reduced to the 4dimensional model. The threshold of instability rth of the time-independent state is also shown in fig. 1 b as a function o f k. It increases monotonically with and, for large ~, comes close to the threshold of the Lorenz model. For large values o f r (r ~> 60) the trajectories again show a pronounced qualitative similarity with corresponding states in the Lorenz model. For smaller values of r, but still above threshold, there exist limit cycles without counterparts in the Lorenz model if k is sufficiently small (~ ~ 0.5), which disappear for larger values of k, where the dynamical behavior, also near threshold, becomes more similar to the Lorenz model. It is clear from fig. 1 that changing the parameters of the 6-dimensional model along various different curves in the parameter space (?~, r) one may see a variety of routes from the time-independent state to a chaotic state. We briefly summarize our results in physical terms as follows. The characteristic frequency 6Ol, according to eq. (12), arises from an inhomogeneous saturation of the inhomogeneously broadened line. Its effect is to generate an out-of-phase perturbation of the

56

1 August 1983

radiating dipole moment as described by eq. (15) which greatly reduces the self-pulsing threshold of the homogeneously broadened laser. However, the more detailed 6-dimensional model shows that this effect is somewhat exaggerated in the 4-dimensional model, since the out-of-phase perturbation of the polarization is smeared out if k is large, as is described by the last three eqs. of (19). One may suspect that, again, the latter effect is somewhat exaggerated in the 6dimensional model due to the truncation which was necessary for the derivation of this model. For large values of the pumping parameter the out-of-phase perturbation of the radiating dipole moment is relatively weak, and the system always behaves similar to the Lorenz model. We wish to acknowledge useful discussions with Michael D6rfle and Axel Schenzle concerning both physical and numerical aspects of this work.

References [1] H. Haken, Z. Physik 190 (1966) 327. [2] A.Z. Grasyuk and A.N. Oraevskii, Radiotekh. Elektron. 9 (1964) 524. [3] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [4] H. Haken, Phys. Lett. 53A (1975) 77. [5] R. Graham and H.J. Scholz, Phys. Rev. A22 (1980) 1198; M. D6rfle and R. Graham, Phys. Rev. A, to appear. [6] L.E. Casperson, Phys. Rev. A21 (1980) 911 ; A23 (1981) 248. [7] S.T. Hendow and M. Sargent III, Optics Comm. 40 (1982) 385. [8] C.O. Weiss and H. King, Optics Comm. 44 (1982) 59: C.O. Weiss, A. Godone and A. Olaffson, Phys. Rev. A (1983), to appear. [9] J. Bentley and N.B. Abraham, Optics Comm. 41 (1982) 52; M. Maeda and N.B. Abraham, Phys. Rev. A26 (1982) 3395. [10] H. Haken, Laser theory, Encyclopedia of Physics 25/2c (1970).