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Optics Letters

Absolute minimum instability thresholds in a nonlinear ring cavity containing a Kerr medium

W. J. Firth, Scott Sinclair

Published Mar 1, 1988• 1 citation
Publisher

Abstract

Stability analysis of an arbitrary-finesse ring cavity containing a Kerr medium of arbitrary response time reveals an absolute minimum threshold for the driving intensity necessary for realization of self-pulsing instabilities. This threshold is minimized not only with respect to cavity mistuning but also with respect to the medium's response time. The instability threshold is typically six times that for optical bistability.

DOI
10.1364/ol.13.000218

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Stability analysis of an arbitrary-finesse ring cavity containing a Kerr medium of arbitrary response time reveals an absolute minimum threshold for the driving intensity necessary for realization of self-pulsing instabilities. This threshold is minimized not only with respect to cavity mistuning but also with respect to the medium's response time. The instability threshold is typically six times that for optical bistability.
By considering a passive ring cavity with linear absorption in the dispersive limit, a unified analysis is presented for tuning-optimized minimum instability thresholds; the treatment is general, allowing consideration of arbitrary material response times, cavity tuning and finesse. In the adiabatic limit (τ → 0) the results reduce to the usual Ikeda map. The case of long medium-response time (τ → ∞) is treated in some detail in the high-finesse limit; these conditions being of particular relevance to semiconductor laser amplifiers. It is found analytically that in the high-finesse limit, both optical bistability and self-pulsing instabilities should exhibit the same low-threshold behaviour. An analytical expression is derived for the ‘instability threshold curve’, A 2 B against B for minimum A 2, in the mapping limit.
We consider the usual ring cavity with a nonlinear medium of which the response time is much longer than the period of the light frequency. It is assumed that two polarization modes affect this medium incoherently, i.e. on an additive power basis. We will show by heuristic physical reasoning, based partly on graphical construction, that a sufficient condition for jump type instabilities is that the sum of the inverse relative differential gains for the two modes not exceed unity. We will then briefly sketch how the same result can be obtained from a formal mathematical stability analysis.
We study the propagation of electromagnetic radiation through a ring-cavity device containing a dispersive nonlinear medium. Under conditions appropriate for introducing the slowly varying envelope approximation and for adiabatically eliminating the atomic variables, we calculate analytical expressions between the steady-state-output intensity and the input-field amplitude for two models of the nonlinear susceptibility, the two-level atomic medium and the Kerr medium. The stability of these curves is determined, with special attention given to changes in the thresholds by varying the return-trip phase shift, the linear absorption, and the reflection coefficients of the mirrors. We investigate the positive-branch instabilities resulting in a pulsing output intensity and calculate the bifurcation rate for the period-doubling hierarchy and the fractal dimension of our two-dimensional maps for the Kerr medium. The convergence of the bifurcation rate to a constant is slow, and we find, as in the publications of Snapp et al. [ Opt. Commun.40, 68 ( 1981)] and Carmichael et al. [ Phys. Rev. A26, 3408 ( 1982)], that this constant is consistent with that found for the class of one-dimensional maps with a quadratic maximum. This universal constant is observed over a large range of fractal dimensions for the chaotic attractor. Discussions of pulse-output bistability, intermittency, and anomalous switching are given.