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Optica Acta International Journal of Optics

Periodic Oscillations and Chaos in a Fabry-Pérot Cavity Containing a Nonlinearity of Finite Response Time

E. Abraham, W. J. Firth

Published Nov 1, 1983• 25 citations
Publisher

Abstract

We present a detailed analysis showing that positive-branch instabilities occur in nonlinear resonators in the Fabry-Pérot (folded) configuration. Responses in the unstable region can be unstable or chaotic, depending on parameter values. Values of the response/round-trip time ratio in excess of unity can still yield instabilities, albeit on progressively higher branches as the ratio increases. The analysis proceeds from the basic optical bistability equations through the Maxwell-Debye equations to an explicit expression for the nonlinear phase shift. Linearization leads to thresholds in good agreement with direct computation. A simple expression for the accessible domain of phase-space is derived. The fundamental oscillation is interpreted as a cavity-resonant sideband instability.

DOI
10.1080/713821091

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We present a detailed analysis showing that positive-branch instabilities occur in nonlinear resonators in the Fabry-Pérot (folded) configuration. Responses in the unstable region can be unstable or chaotic, depending on parameter values. Values of the response/round-trip time ratio in excess of unity can still yield instabilities, albeit on progressively higher branches as the ratio increases. The analysis proceeds from the basic optical bistability equations through the Maxwell-Debye equations to an explicit expression for the nonlinear phase shift. Linearization leads to thresholds in good agreement with direct computation. A simple expression for the accessible domain of phase-space is derived. The fundamental oscillation is interpreted as a cavity-resonant sideband instability.
Instabilities are predicted for absorptive optical bistability in a nonideal Fabry-P\'erot cavity on the basis of Maxwell-Bloch theory. We show that the stability problem of the Maxwell-Bloch equations can be formulated in terms of a Riccati differential equation with boundary conditions. We have integrated this equation numerically. Our results crucially depend on the value of the transmission coefficient of the mirrors. For finite values of this quantity we find that the nearest side mode is responsible for two disconnected instability domains in the plane spanned by the output intensity and the medium response time. One of these domains can generate instabilities in the upper and the lower branch of the steady-state curve if the cooperation parameter is sufficiently large. Higher side modes can give rise to positive-slope instabilities as well. From our findings it can be understood why a recent experimental search for instabilities in absorptive optical bistability has led to a negative result. We finally demonstrate that the instability spectra of a Fabry-P\'erot cavity and a so-called equivalent ring cavity differ considerably.
We start from the Maxwell-Bloch equations that govern the dynamics of Fabry-Perot lasers and, considering a doubled cavity, generalize our traveling wave formalism to the case in which the two-level medium does not fill the cavity but occupies only a portion of it. We linearize these equations and focus on the adiabatic elimination of both atomic fluctuations. Next, we restrict our attention to free running lasers under resonant conditions and analyze amplitude and phase instabilities. The examination of the unstable domain leads to the conclusion that the multimode Fabry-Perot instability arises near threshold only when the ratio of the longitudinal to the transverse atomic relaxation rate is substantially smaller than unity. This result agrees with our previous study in the limit of adiabatic elimination of the atomic polarization fluctuations only. We describe the self-pulsing behavior that arises from the multimode instability, and exhibits hysteretic behavior when the pump parameter is swept forward and backward. Finally, we investigate the single-mode instability and show that in the Fabry-Perot case there is no longer the correspondence between single-mode and multimode instabilities that is well known in the case of ring lasers. We confirm that in the case of resonant Fabry-Perot lasers the single-mode amplitude instability arises very far from threshold.
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.
Absolute instabilities of counterpropagating pump beams in a dispersive Kerr medium, placed inside a Fabry–Perot cavity, are analytically studied by use of the analysis and the results of part I [J. Opt. Soc. B14, 607 (1998)]. Our approach allows characterization of such a complicated nonlinear system in terms of a doubly resonant optical parametric oscillator. We consider the growth of modulation-instability sidebands associated with each pump beam when weak probe signals are injected through one of the mirrors of the Fabry–Perot cavity. The results are used to obtain the threshold condition for the onset of the absolute instability and the growth rate for the unstable sidebands in the above-threshold regime. As expected, the well-known Ikeda instability is recovered at low modulation frequencies. The effects of the group-velocity dispersion are found to become quite important at high modulation frequencies. Although the absolute instability dominates in the anomalous-dispersion regime, it exists even in the normal-dispersion regime of the nonlinear medium. Below the instability threshold, our analysis provides analytic expressions for the probe transmittivity and the reflectivity of the phase-conjugated signal that is generated through a four-wave-mixing process.
We consider nonlinear dynamics of a system of interacting photons and excitons driven by a resonantly exciting light field. Using the linear stability analysis for nonlinear coupled equations of photon and exciton operators we investigated that, besides the usual unstable domain which is a negative slope branch, there are other unstable ones in the leftmost and rightmost sides of the lower and upper branches of the frequency-density bistable curves. Numerical study shows that the system has very rich information on the instability nature and a self-oscillation process leading to chaos can occur in these unstable domains.