Volume 59, number 4 OPTICS COMMUNICATIONS 15 September 1986 DYNAMIC BISTABILITY IN PARAMETRIC RESONANCE IN CRYSTALS W. GADOMSKI and B. RATAJASKA-GADOMSKA Laboratory of Physicochemistry of Dielectrics, Department of Chemistry, Universityof Warsaw, 02-089 Warsaw,Poland Received 18 March 1986 We report the dynamic bistability of the amplitude of the overtone lattice vibration with respect to the time-dependent bihar- monic optical field interacting with a crystal. The bistable parametric resonance was shown to occur in a crystal interacting with a biharmonic op- tical field, E(t) = El cos oglt + E2 cos(o92t - ~, when the difference between the optical frequencies oJ~ - o92 equals the double vibrational frequency OgR of the lattice mode [ 1,2]. In the stationary case, the square of the amplitude of the lattice vibrations is resonantly enhanced and exhibits bistable behaviour with respect to both the difference frequency and the field amplitudes El and E2 [1,2]. The parametric character of this enhancement is due to the external field induced modulation of the vibrational fre- quency oj R. In this paper we present the dynamic bistability [3] occurring in a crystal if one of the ex- ternal fields is being swept linearly in time. The dy- namics of the crystal is described by a set of three equations for the second order products of the am- plitude QR and momentum PR of a chosen mode, av- eraged over all other modes [ 1]. The chosen mode is resonantly coupled with the external field. For the optical q = 0 crystal mode of frequency 09R we have [1] :r= 2z, 3) = --2z[1 - ~cos(~z- o")] - 2exz - 4 3,y + 43,/(1 +r/), = y - x[1 - ~cos(w - 0")] - ex 2 - 23,z + 2 .4/(1 + ~/), (1) 0 030-4018/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) where x= <Q2)sl(~oo)s, y= <e2>B/of<Qo2>s, z = <QP)B/oJ<Q~o)s, r = o)t, are the dimensionless variables. < Qo 2 )s denotes the mean value of the square of the harmonic amplitude of the chosen mode, ( • ) B denotes the average over the bath of all other modes. The following coeffi- cients: ~/= gtlE~12 + K2IE2[ 2 the vibrational fre- quency shift; o9 = COR(I + t/) I/2, 0 = (OJ~ -- 092)/O9; 3, = (3'0 + 3,~lEtl 2 + y21E212)/(1 + r/) and d = (A0 + At[Etl 2 + J2IE2t2)/(I + ~l), the inverse lifetime and the frequency shift, respectively, both related to the third order anharmonicity; e = (Co + e,IE~I 2 + e21E212)/(l + if), the coefficient of the fourth order nonlinearity are field intensity dependent. = K3IE~IIEzl/(I + rl) is the modulation coeffi- cient of the vibrational frequency ohm.All the param- eters Kj, K2, K3, 3,1, 3,2,A~, A 2 and ¢1, ¢2 are dependent on the field polarization towards the crystalographic axes and are given in [1 ]. The set of equations (1) corresponds to the third order differential equation for x(r). We seek its solution in the form [1,2,4]: x(T) = b(¢) + a(¢)cos(oz + ~(~)) (2) if o: = 4 + e0 where 0 is a smal ! detuning. The sta- tionary amplitudes as and bs exhibit a bistable behav- iour with respect to the external field intensities [ 1,2]. By way of example we shall discuss the amplitude a. 313