Volume 51, number 6 OPTICS COMMUNICATIONS 15 October 1984 NONLINEAR THEORY OF SELF-OSCILLATIONS IN A PHASE-CONJUGATE RESONATOR E.M. WRIGHT and P. MEYSTRE Max-Planck lnstitut ]fir Quantenoptik, D-8046 Garching, Fed. Rep. Germany and W.J. FIRTH Department of Physics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK Received 23 July 1984 We present an exact, nonlinear analysis of self-oscillations in a phase-conjugateresonator. Optical bi- and multistability, as well as period doubling to chaos are predicted. 1. Introduction There is currently much interest in the four-wave mixing (FWM) process [1], as well as in phase-conju- gate resonators (PCR) [2], in which one end mirror is a FWM cell. This interest derives from the ability of the FWM cell (or phase-conjugate mirror) to re- duce or cancel phase distortions within the resonator [2]. The mode characteristics of PCR's in the unde- pleted pump beam approximation have been discussed by several authors [3,4]. In particular, half-axial modes [2] are predicted to arise and indeed, these have been observed experimentally [5]. The theories given by these authors can, however, give only an in- dication of the possible modes of PCR's and cannot predict the dynamic behaviour above the threshold for self-oscillations [2]. Large signal analyses of PCR's have been published [6,7], but these studies consider only the static solutions, that is, the spontaneous sig- nals that build up at the frequency of the pump beams. In this paper, we present an exact, nonlinear analy- sis of the self-oscillations in a PCR, in which the phase- conjugate mirror (PCM) is a Kerr medium having in- stantaneous response. Firstly, we consider the static solution and show that optical multistability occurs. We then study the dynamic behaviour of the system 428 in the limit of a very short medium length (compared to the external resonator length). Here, we find that for certain parameter ranges, the static solution be- comes unstable and oscillations follow. Initially, these oscillations correspond to the excitation of the half- axial modes predicted by the linear theories [2]. How- ever, further change of parameters can lead to a period- doubling sequence which evolves to chaotic response [8]. The basic formalism and static solution are dis- cussed in sections 2 and 3, and the dynamic behaviour in section 4. Summary and conclusions are given in section 5. 2. The phase-conjugate mirror The phase-conjugate mirror we consider is a lossless, isotropic Kerr medium, characterised by an intensity- dependent index of refraction n(1) = n o + n21 , that is irradiated by two counterpropagating pump fields of equal frequency w and intensities I F and IB, as shown in fig. 1. For the moment, we ignore the normal mir- ror. We are then left with the problem of finding the backscattered field B 1 (z) exp(iAkx) at an angle 0 with respect to the z-axis, given an incident field E(z) X exp(-iAkx) (B 1 and E being slowly varying field 0 030-4018/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)