Coupling of two chaotic lasers

8 August 1994 PHYSICS LETTERS A

ELSEVIER

PhysicsLetters A 191 (1994) 134-138

Coupling of two chaotic lasers Yudong Liu ~,a, J.R. Rios Leite Departamento de Fisica, Universidade Federal de Pernambuco, 50739 Recife, PE, Brazil

Received 4 January 1994;acceptedfor publication 27 May 1994 Communicatedby A.R. Bishop

Abstract

Numerical solutions for two sets of equations describing lasers with intracavity saturable absorbers were studied for their instabilities associated to passive Q-switching. For realistic values of parameters of CO2 lasers, we have demonstrated that two independent chaotic lasers can to into synchronized periodic oscillation after 3% of coupling. If both lasers start from periodic oscillation, after coupling they can go into either synchronized periodic or synchronized chaotic oscillations depending on the coupling strength. The coupling can be made either through absorber, laser gain or both.

Coupling chaotic systems has been studied while searching for their synchronization [ 1-3 ] which may have important applications in signal processing. The mathematics of such a high dimensional nonlinear set of equations corresponding to dynamical systems is not yet understood by fundamental theorems [4 ] and thus numerical solutions of specific cases are actively pursued both for pure and practical interest. Here we demonstrate numerically the coupling of two chaotic lasers. The chaotic systems were two COa lasers with intracavity saturable absorber SF6. Each laser was described by a set of four differential equations with practical values of each parameter. Such equations have been demonstrated to have periodic and chaotic oscillations for a wide range of parameters [ 58 ]. The coupling was introduced either through the absorber or through the gain tube. The initial states of two systems can be either periodic or chaotic. We have studied the case that the initial states were pe1Present address: DFESCM, Instituto de Fisica Gleb Wataghin, UNICAMP,Caixa Postal 6165, Campinas 13081-970,SP, Brazil. 2 E-mail:[email protected].

riodic+periodic, periodic+chaotic and chaotic+chaotic. The coupled system shows synchronized periodic oscillations or synchronized chaotic oscillations depending on the strength of coupling. The equations describing one laser with saturable absorber can be written in normalized form as fol-

lows [ 8 ], (1)

I=I(U-W-1), U=eI[V--U(I+I)] V=e~(d~ + b U -

,

V),

W = e 2 [ d 2 - W(1 - a I ) ] ,

(2) (3) (4)

where I stands for the laser intensity, U corresponds to the population inversion in the gain medium, V is the sum of the population in the coupled transition of the three level model for the gain and W the population difference of the saturable absorber. The coefficients e~ and e2 describe the degree of population change, dl and d2 are the equilibrium populations of gain and absorber without oscillation while a and b give the population saturation by absorption

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Y. Liu, J.R. Rios Leite / Physics Letters .4 191 (1994) 134-138

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Fig. 1. Calculated laser power versus time, in: units of cavity relaxation time, without coupling for (a) laser A and (b) laser B; (c) correlation between laser A and laser B.

and stimulated emission in the absorber and gain media, respectively. The laser is thus described by a mean intensity inside the oscillator, the optical polarization both in the amplifier and in the absorber being

eliminated adiabatically. Numerically solving Eqs. ( 1 )- (4) gives chaotic solutions for a certain set of parameters [ 5-8 ], as shown in Fig. 1a. The coupling of two CO2 lasers can be realized in

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Y. Liu, J.R. Rios Leite / Physics Letters A 191 (1994) 134-138

136

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Fig. 3. Calculated laser power after gain coupling with C~A= CAa ~ ---- C a~A -- - C A a = 0 and C~A = C ~ = 0 . 0 3 . (c) laser A versus laser B. All other parameters are the same as in Fig. 1.

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Fig. 4. Bifurcation maps of absorbing coupling, C~A =CAa=CBA t 2 =C~ 2 =O and C~A =C~=CBA. (a) Chaotic+chaotic, (b) Periodic 3 +periodic 3.

Y. Liu, J.R. Rios Leite / Physics Letters A 191 (1994) 134-138 three manners. One is by sending a small amount of one laser beam to another laser's absorber and vice versa, and this will be called absorber coupling. The second is by sending a small amount of laser beam to another's laser's gain and vice versa, which will be called gain coupling. The third way is to send each laser beam into the other's cavity and hence the injected beam will be the cavity mode. This later case will be difficult in experiment but is possible. All three ways of coupling with different initial states of two lasers have been studied. To save for clarity, we only show examples of absorber and gain coupling with initial states of chaotic + chaotic. The coupled equations can be written in the following form, I A = I A ( U A - - WA-- 1 ) + C~A/B,

(5)

UA =e,A [ VA-- UA( 1 +IA + C2AIB) ] ,

(6)

VA----elA(dlA +bA UA -- VA) ,

(7)

WA=e2A{d2A--WA[1--aA([A +CaBAIB)]} ,

(8)

IB =In( UB - WB -- 1 ) + C~AslA ,

(9)

UB = era[

Ve -

U , ( 1 +In + C~nlA) ] ,

(10)

VB= eln(dla -- bB UB -- VB) ,

( 11 )

WB=e2B{dzn--WB[I--an(IB+C3ABIA)]},

(12)

where the parameters are chosen as follows: exA=0.137, e2A= 1.6, bx=0.85, aA=4.17, dlA=l.81042, d2A=l.81042, era=0.137, e2B=4.6, bB=0.85, an=4.17, daB= 1.81686, d2a= 2.01774. Absorber coupling. The absorber coupling was realized by setting C~A = c a b = C2A = C2B = 0 and C~A = C3An= 0.03. Before coupling both laser oscillations are chaotic as shown in Figs. la and lb. The phase map of two lasers is shown in Fig. 1c which indicates that the two lasers are not correlated before coupling. Fig. 2 shows the results after 3% of coupling through the absorber while all other parameters are kept the same as in Fig. 1. Fig. 2c was plotted only for t > 400 in order to eliminate initial points. A clear orbit was obtained and indicates that the two lasers are synchronized. Gain coupling. The gain coupling was realized by setting C~A = C / B ---~CBA 3 3 = CAB = 0 and CgA=C~B=O.03. Before coupling both lasers are chaotic as shown also in Fig. 1. Fig. 3 shows the os-

137

cillations of both lasers after 3% of gain coupling. Fig. 3c was plotted only for t> 200 in order to eliminate initial points. Cavity mode. The cavity mode coupling was realized by setting CBA 2 = CAB 2 = CaaA= C3AB= 0 and I = CAB. 1 We have also tried the cavity mode couCBA pling with different initial states of both lasers. The results are very similar to the absorber coupling but with much less coupling strength. In Fig. 4 we show the bifurcation maps of absorber coupling of two lasers. The maps were obtained by selecting the maxima of laser intensity. Fig. 4a is chaotic+chaotic, that is, the parameters for both lasers are chosen so that they are in different chaotic regions. At the beginning, both lasers are still in chaotic oscillation. As the coupling increases, they go into synchronized periodic, synchronized chaotic, a big range of periodic, and finally to continuous lasing (no oscillations). Fig. 4b is periodic 3 +periodic 3, that is, the parameters for both lasers are chosen so that they are in periodic oscillation. The bifurcation map in this case is more complicated. When the coupling is small, the map is similar to Fig. 4a. When the coupling is larger, the whole system bursts into chaos again and experiences all the way inverse period double bifurcation. This new order and chaos are all synchronized. All numerical calculations described here correspond to parameters compatible with experimental values for CO2 lasers with saturable absorber [6,8 ]. The verification in our laboratories where we already verified new features on the dynamics of a single mode ring CO2 laser with saturable absorber [ 9 ] will be pursued.

References

[ 1] L.M. Pecoraand T.L. Carrol, Phys. Rev. Lett. 64 ([990) 821. [2 ] J.M. Kowalskiand G.L. Albert,Phys.Rev.A 42 (1990) 6260. [3 ] F. Mossayebi,H.K. Qammar and T.T. Hartley, Phys. Len. A 161 (1991) 255. [4] J. Guckenheimer and P. Homes, Nonlinear oscillations, dynamical systemsand bifurcations of vector fields, 3rd Ed. (Springer, New York, 1990). [5 ] M. Tachikawa, F.-L. Hong, K. Tanii and T. Shimizu, Phys. Rev. Lett. 60 (1988) 2266. [6] M. Tachikawa, K. Tanii and T. Shimizu, Appl. Phys. B 39 (1986) 83; J. Opt. Soe. Am. B 4 (1987) 387; B 5 (1988) 1077.

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Y. Liu, J.R. Rios Leite / Physics LettersA 191 (1994) 134-138

[7] F. de Tomasi, D. Hennequin, B. Zambon and E. Arimondo, J. Opt. Soc. Am. B 6 (1989) 45. [ 8 ] M. Lefranc, D. Hennequin and D. Dangoisse, J. Opt. Soc.

Am. B 8 (!991) 239. [9] Y. Liu and J.R.Rios Leite, Opt. Commun. 88 (1992) 391; Y. Liu, L. B. de Oliveira and J.R. Rios Leite, to be published