Chaos in coherent two-photon processes in a ring cavity

Volume 47, number 1

OPTICS COMMUNICATIONS

1 August 1983

CHAOS IN COHERENT TWO-PHOTON PROCESSES IN A RING CAVITY Surendra SINGH Department o f Physics, University o f Arkansas, FayettevUle, AR 72701, USA and

G.S. AGARWAL School o f Physics, University o f Hyderabad, Hyderabad 500134, India

Received 2 November 1982 Revised manuscript received 16 March 1983

The output of a ring cavity containing a resonant medium undergoing two photon transitions is shown to become chaotic, after following a series of bifurcations involving 2n cycles, as the strength of the driving field is increased. The chaotic regime is followed by a sequence of period doubling bifurcations in reverse order.

The resonant interaction of the atoms, contained in a cavity, with the self-consistent fields in the cavity leads to a variety of instabilities [ 1 - 6 ] . For example, Ikeda et al. [ 1] predicted the chaotic behavior of the output of a ring cavity for certain values of the input field strength and the parameter of the system. The type of the chaotic character predicted by Ikeda et al. has been observed by Gibbs et al. [7] in experiments involving nonlinear feedback mechanisms such as those involving nonlinear electrooptic modulators. The transition to chaos has been shown [1,4] to follow the Feigenbaum scenario [8 ~9]. This interrelation of the "optical turbulence" and the chaotic character exhibited by various mathematical models [8-10] is quite exciting and one is led to think that other types of nonlinear interactions of atoms with the fields in the cavity might exhibit similar turbulent or chaotic behavior. In order to explore this possibility, we have analyzed the problem of coherent two photon transitions [6,11-13] in a ring cavity. We show that the path leading to chaos in such transitions again follows Feigenbaum's scenario in spite of the very complicated two dimensional map that characterizes the coherent two photon transitions in a ring cavity. Our treatment is based on the coupled MaxwellBloch [13] equations for coherent two photon pro0 0304018/83/0000_0000/$ 03.00 © 1983 North-Holland

cesses with appropriate boundary conditions. We use a method similar to that of Ikeda et al. in order to obtain the two dimensional dynamical map that characterizes the output of the ring cavity produced in coherent two photon transitions. The Maxwell-Bloch equations for two photon absorption processes [11-13] are given by ~S/ar = -3"±S + i(¢o12 - 26o)S + ig ~ , 2 W,

(1 a)

a w/a1- = -3,11(i4/+ 1) + ~i(g* 6 2 s - c.c.),

(lb)

a C/Sz = OrwNvll/c 2) igS* C * •

(1 c)

In writing (1) we have ignored the Stark shifts. We have also introduced the reduced time r = t - z/o. We assume that 3,± ~, 3,11' SO that the variable S relaxes quickly to the equilibrium value S = ig 6"2 W/3,±(1 - i 8 ) ,

5 = (~o12 - 26o)/3't .

(2)

We then obtain a 6 _ 7r60Nfi Igl2 ,¢ az c 3"/1+1+~ ''126w'-

o~c

,

( Ig12 ~ 1~14I¢

~-~= -3,,(w+ 1) - 3,, \~--~, I ~ "

(3a) (3b)

In terms of an auxiliary quantity ~k defined by 73

1 August 1983

OPTICS COMMUNICATIONS

Volume 47, number 1 Z

$(r, z) = 71 f

(4)

dz' W ( t - z ' / v , z ' ) ,

600

o

we can integrate (3) to lead to 4oc

I6 ( t - z/v, z) l2 = I6(t, 0) 12 X

I

1-

231g12z

16(t,O)12~(r,z)

T±(1 + 8 2)

1-1

(5a)

,

2OO

~b(t - z / v , z ) = ~ ( t , O)

- ~-ln (1

~

Y

23 lgl2z I6(t,0)12 ) , 3,1(1 +8 2)

(5b) (5 c)

(3 = IrwN]l/c .

Here ~b is the phase of the slowly varying complex field amplitude. The Stark shifts were important in an earlier work [12] dealing with bistability in coherent two photon processes. However in the numerical work in this paper, we will deal with the monostable regime. The Stark shifts are not crucial for the existence of a monostable regime. Therefore, to simplify the analysis in this paper we drop these. Then, the equation for ~ is given by

0

f/J ~ ~ - - ~ 1

=

I

2

,

I

3

,

Fig. 1. The steady-state relation between IXI and the input field Y. The curves marked A s and B s are the real and the imaginary parts o f the field amplitudes X s. Y1 denotes the onset o f the bifurcation sequence and C 1 denotes the onset o f the chaotic instability. The region beyond C 1 is characterized by chaotic trajectories and period halving sequences. For this figure R =

0.99,~L = 10,0 = 6, and6 = 10. = 23(Ig1271117±) 112,

t R = (2-

2L)/c~--Z?/c, (9b)

where ~b(t) obeys the following equation: d ~ ( t ) / d t = -711(~b + l)

0~ =--TII(~ + 1)-(1g[21 I 6 ( t ' O ) 1 4 ~ / ( r ' z ) 0--7 711 \~--~I ! (1 +6 2)

7, I X ( t - tR)14 ~(t)

(10)

(1 + 8 2 _ a L I X ( t - tR)12ff(t)) X (1

-

-

7±(12/3[g[2z+82) '~(t, 0),2~(r, z)) -1

.

(5)

In this paper we further take the limit "),lltR ~. oo so that ~k(t) can be adiabatically eliminated, i.e., ~(t) = 0,

The boundary condition connecting the incident field ¢i to the field in the cavity is ( (t, O) = X/~-ei¢ ~ i

+

(1 - T ) ~ (t - (.£2 - L)/c,L)e -i:ff),.

where T is the transmission coefficient of the output mirror, 0 is the cavity detuning, Z? is the cavity length and the length of the medium is L. On introducing X(t) and Y(t) defined by

30

I

I

I

X ( t ) = (Igl 2/3'113',)1/4 6, Y ( t ) = (Igl2/7, 7,)1/4 ei~° t~ i / v / T- ,

(8)

and on using (5), we can write (7) as X ( t ) = T Y ( t ) + R X ( t - tR)

X e x p l --i0 +(~57-~) In ( 1 -- -1 +- 8 2 [ X ( t - t R ) 1 2 ~ t (9a) 74

0

100

200

t

Fig. 2. The output intensity [X(t)l2 versus time t measured in units of t R for Y = 400. For other parameter values see fig. 1.

Volume 47, number 1 1+52 -aLIX(t-

=

OPTICS COMMUNICATIONS tR)12+ IX(t - tR)l 4 [

2aL ~X(t - tR)l 2 4aL(l + 6 2 ) l X ( t - t R ) l 2

-

3

I

2

/i7

1

1112],

( 1 + (1 + 82 - - ~ ~ t R ) 1 4 ) 2 !

.J (11)

Here we have picked up that root o f ~ that goes to - 1 in the limit aL -+ 0, T ~ 0, 1)(]2 -~ 0. On substituting (11) in (9) we obtain quite a complicated equation that gives the time dependence of the field in the cavity for a given input field Y. We will refer to the resulting equation as the two dimensional map characterizing the behavior o f the field produced in coherent two photon processes. We first note that such a map in the steady state and in the limits [ 14] aL ~ O, T ~ O, ctL/T = constant leads to the standard mean field relation [11,12] between X and Y. The stability of steady-state solutions can be studied by writing (9) as

X n = T Y n + X n _ l f ( l X n _ 112).

1 August 1983 I

(a)

;I ~'.

.v,'~= . . . . . . . . .

"- ~ ,4.

1

B

J~ -1

-2 0

I 1

I 2

i 3

I 4

5

A .4

I

I

I

(b)

(12)

The steady.state values o f X satisfy the relation,

X = TY + Xf(IXl2).

The steady state solution is stable provided that the eigenvalues A of the matrix

( f + [Xl2f' M = \ X * 2 f *'

X2f'

~

(14)

f* + IXi2f,, / ,

are such that IAI < 1. The eigenvalue equation can be written as A 2 - A(1 + b - alYI2/a[xI 2) + b = 0 , b = ( a / a lXl 2) (iXl2ff,).

i'i

(13)

(15)

In view o f the complexity o f our two dimensional map * a, we have investigated the stability numerically - a complete discussion o f which will be given elsewhere. Here we investigate the behavior for values o f the system parameters for which the steady.state value o f X is a single valued function ,2 of Y. ,i It may be noted that in the model of ref. [1], b = (a/alXI 2) x (IXI2ff *) = const. ,2 This should be contrasted with the optical turbulence produced in coherent single photon transitions, where the steady-state is a multiple valued function of the external field Y.

.2

I 1.2

1.1

"J " ' I

1.3

A

Fig. 3. (a) A phase space plot of 3000 successiveiterations of X for Y = 400. See fig. 1 for other parameters. The plot was insensitive to the initial conditions except for first two or three points. (b) An expanded version of the rectangle in fig. 3(a) after 30000 iterations of the map. The steady.state behavior o f X as a function o f Y is shown in fig. 1. The curves marked A s and B s represent the real and imaginary parts o f the field amplitude X in the steady-state. It can be seen that for the parameter values used here * a, the variables Xs, A s are single valued functions of the input field Y, but, B s is multiple valued. A stability analysis, as outlined above, shows that the steady-state response o f the system is not stable for all points on the curves in fig. 1. The steady.state becomes unstable at point marked ,s The parameter aL essentially characterizes the strength of two photon transitions. The parameter C used in the mean field theories of bistabflity (refs. [ 14,12]) corresponds to

aL/2T. 75

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We will, however, not go into the details of the dynamics in this region here. In conclusion we have for the first time shown the existence of chaotic behavior in atom field interaction underlying multiphoton transitions. The details of this work and the situation where time variation of W during one round trip is important will be published elsewhere.

I

The work of one of the authors (SS) was supported in part by a grant from the University of Arkansas. 0

.1

.2

.3

.4

.5

Fig. 4. Power spectrum of IX(t)l 2 for Y = 400. The spectrum was obtained by averaging over 100 sample sequences. A cosine window was applied to each sequence before taking the fast Fourier transform. The resolution is 27r/1024.

Y1 with the appearance of a 21-cycle. On increasing Y further, a sequence of period doubling bifurcations appears, which ultimately terminates into a chaotic regime at point marked C 1 . We have traced the sequence {Yn}, where various 2 n -cycles appear and found that the bifurcation velocity seems to converge to the same value as predicted by Feigenbaum. It must be mentioned that the Y versus IXsl curve does not have a maximum. A typical time sequence for the output light intensity IX(t)l 2 is shown in fig. 2, in the chaotic regime. The appearance of period doubling bifurcations and chaos suggests the existence of an attractor. A phase space plot of X(t) is shown in fig. 3 which was obtained after 5000 iterations. We have also examined the spectrum o f the output field. On the regular side of the chaos, the output instensity IX(t) 12 has a spectrum which consists of discrete peaks. The onset of chaos is accompanied by the appearance o f a broad spectrum which is shown in fig. 4. Well beyond the chaotic regime following the point C I , the chaotic behavior terminates and a sequence of 2 n-cycles appears. In this region, as Y is increased, the period o f the stable point cycle is halved as opposed to the case described previously where the period of the stable cycle doubled as Y was increased.

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References [ 1] K. Ikeda, H. Daido and O. Akimoto, Phys. Rev. Lett. 45 (1980) 709. [2] L.A. Lugiato, L.M. Narducci, D.K. Bandy and C.A. Pennise, Optics Comm. 43 (1982) 281 ; R. Bonifacio and L.A. Lugiato, in: Optical bistability, eds. C.M. Bowden, M. Ciftan and H.R. Robl (Plenum, New York, 1981) p. 31. [3] E. Abraham, WJ. Firth and J. Carr, Phys. Lett. 91A (1982) 47. [4] H.J. Carmichael, R.R. Snapp and W.C. Schieve, to be published. [5] K. Ikeda and O. Akimoto, Phys. Rev. Lett. 48 (1982) 617. [6] J.A. Hermann, Phys. Lett. 90A (1982) 178. [7] H.M. Gibbs, F.A. Hopf, D.L. Kaplan and R.L. Shoemaker, Phys. Rev. Lett. 46 (1981) 474. [8] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25. [9] R.M. May, Nature 261 (1976) 459. [10] M. Henon, Comm. Math. Phys. 50 (1976) 69. [ 11 ] F.T. Arecchi and A. Politi, Nuovo Cimento Lett. 23 (1978) 65; G.P. Agrawal and C. Flytzanis, Phys. Rev. Lett. 44 (1980) 1058; J.A. Hermann and B.V. Thomson, in: Optical bistability, eds. C.M. Bowden, M. Ciftan and H.R. Robl (Plelaum, New York, 1981) p. 199. [12] G.S. AgarwaL Optics Comm. 35 (1980) 149. [13] M. Takatsuji, Phys. Rev. A l l (1975) 619; L.M. Narducci, L.G. Johnson, E.J. Seibert and W.W. Eidson, in: Coherence in spectroscopy and modern physics, eds. F.T. Arecchi, R. Bonifacio and M.O. Scully (Plenum, New York) p. 131. [ 14] Cf.R. Bonffacio and L.A. Lugiato, Nuovo Cimento Lett. 21 (1978) 505.