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Optics Letters

Pitchfork bifurcation using a two-beam nonlinear Fabry–Perot interferometer: an analytical study

Marc Haelterman, Paul Mandel

Published Dec 1, 1990• 29 citations
Publisher

Abstract

A recent modal theory of nonlinear planar resonators is applied to the analysis of the steady transmission characteristics of the two-beam nonlinear Fabry-Perot interferometer. An analytical study, based on the planewave approximation, is presented that shows the potential applications of the device. In particular, a flip-flop mode of operation, derived from a pitchfork bifurcation, is predicted. Its salient feature is to require only positive triggering pulses.

DOI
10.1364/ol.15.001412

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Journal of the Optical Society of America B•Jul 1, 1990•53 citations
A recent modal theory of nonlinear planar resonators is applied to the analysis of the steady transmission characteristics of the two-beam nonlinear Fabry-Perot interferometer. An analytical study, based on the planewave approximation, is presented that shows the potential applications of the device. In particular, a flip-flop mode of operation, derived from a pitchfork bifurcation, is predicted. Its salient feature is to require only positive triggering pulses.
A unified nonlinear modal theory that is valid not only for waveguide couplers but also for nonlinear Fabry–Perot resonators has recently been proposed. The validity of this new approach is numerically demonstrated by comparison with the well-known bidimensional theory. The unified modal theory appears to be a powerful tool for studying transverse effects in a broad class of nonlinear resonators. It has been used to derive the main characteristics of the stationary behavior of the optical resonators in the case of a local nonlinearity. In particular, the disappearance of optical bistability when the resonator is tilted is predicted. Optical bistability at nearly normal incidence is interpreted by the coupling between two resonantly excited counterpropagating modes that are at the origin of the required transverse feedback.
A Fabry-Perot interferometer filled with a medium whose refractive index depends upon intensity has a multiple-valued transmission-vs-intensity characteristic. When the incident light is circularly polarized, and the nonlinearity is only cubic in the fields, Maxwell's equations may be solved exactly for the plane-parallel resonator field in terms of elliptic functions. An accurate approximate analysis is given for more general cases, including resonators with spherical mirrors and finite beams, where self-focusing is important. The theory is developed to yield design information for operation as an optical switch, "transistor," and power limiter.
On the basis of a pole analysis, we derive a simple and powerful model of the nonlinear Fabry–Perot resonator. Our theory extends the modal approach, which was already developed for the nonlinear prism coupler, to the Fabry–Perot resonator for any angle of incidence. The main advantage of the modal approach is that it greatly simplifies the numerical study of transverse effects and provides a deep understanding of physical phenomena related to optical bistability. By means of a simple analysis of the equations, our model permits us to discuss the role of diffraction and the influence of the incident angle on the nonlinear Fabry–Perot behavior.