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Physics Letters A

Routes to chaos in a delay-differential equation system modelling a passive optical resonator

F. Kaiser, D. Merkle

Published Jul 1, 1989• 9 citations
Publisher
DOI
10.1016/0375-9601(89)90347-2

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There has been considerable interest over the last few years in transitions to chaos in passive optically bistable systems. Ikeda1 showed in 1979 that an optically bistable ring resonator containing the simplest of nonlinear systems, a two-level media, can show a period doubling route to chaos. This was demonstrated by us2 using NH3 gas near resonantly pumped by pulsed CO2 laser radiation. More generally we identified and demonstrated the unique advantages of molecular gases in this field, providing in particular a vast range of laser-molecular coincidences together with control of response time, and therefore switching time and modulation frequency, by varying gas pressure. Subsequent extension of this work to all optical Fabry-Perot resonators revealed huge oscillatory effects3 leading to chaos in the transmitted laser signal. Under other parameter conditions for both NH3 and SF6 a rich variety of optically bistable effects were observed obtained in systems as small as 1 mm for SF64,5.
Dispersive optical bistability and instabilities leading to period doubling, higher harmonics, and chaos in an all-optical passive quantum system are discussed theoretically and experimentally. These phenomena have been observed for a wide range of operating conditions in ring and Fabry-Perot resonators containing ammonia molecular gas as the nonlinear medium. Laser input intensity, gas pressure, and cavity tuning are the main control parameters in observation of these effects. The experimental results are well modeled by our generalization of standard two-level system theory to include standing-wave, reservoir, and transverse effects.
The aim of present paper is to propose several dynamical models of nonlinear Fabry-Perot resonators. The models are all described by sets of difference-differential (delay-differential) equations involving multiple time delays, and they are extended versions of the difference-differential equations proposed for the ring-type resonators by one of the present authors. We show in detail how these model equations are derived from the Maxwell-Bloch equations, assuming: 1) fast transversal relaxation, 2) weak nonlinearity, and 3) dispersive limit. We propose a hybrid optical bistable device as an analog system of a model equation.