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IEEE Journal of Quantum Electronics

Modeling of nonlinear Fabry-Perot resonators by difference-differential equations

Kensuke S. Ikeda, Masataka Mizuno

Published Sep 1, 1985• 25 citations
Publisher

Abstract

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.

DOI
10.1109/jqe.1985.1072824

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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.
In Chapters 8–13 we always assumed that the nonlinear medium is contained in a unidirectional ring cavity. The case of a Fabry–Perot cavity (Fig. 8.1) is substantially more complex because the interference of the two counterpropagating fields gives rise to a standing-wave pattern that modulates the population difference and the atomic polarization to form gratings with a spatial frequency equal to the inverse of the half-wavelength and its harmonics. Therefore the atomic variables exhibit, in addition to the slow spatial variation which characterizes the envelopes, also a fast spatial variation on the wavelength scale, and the coexistence of these two spatial scales poses a highly nontrivial problem that must be treated with care. In Chapters 9 and 11 we solved exactly the Maxwell–Bloch equations with the ring-cavity boundary conditions for both the active and the passive configuration. Exact solutions of this kind have been obtained also for the equations and boundary conditions of a Fabry–Perot cavity [117–119]. However, for the sake of simplicity we will pass directly to models that are valid in the low-transmission approximation. Again for the sake of simplicity, in this chapter we assume that n B = 1, so that the light velocity in the empty part of the cavity is the same as that in the atomic sample. By following the same line of reasoning as that which we pursued in Chapter 12, in Section 14.1 we introduce a spatial transformation of the electric-field envelopes, which leads to boundary conditions without parameters, while the parameters present in the boundary conditions of the original variables appear in the envelope equations. As in the ring-cavity case, this approach allows one to introduce the cavity modes in a natural way, and to derive a set of modal equations coupled to the equations for the atomic variables. Sets of equations of this kind are common in the laser literature, but are not derived from the true boundary conditions for the cavity.
Abstract We formulate the dynamics of nonlinear optical Fabry–Perot (FB) cavities in terms of a model in which only one of the two counter-propagating electric field envelopes appear. Thus, the model is simpler than the standard ones but still exact. The field envelope which propagates in the opposite direction is expressed in a very simple way in terms of the envelope that appears in the model. Thus we generalize in the simplest way to the FB case the set of Maxwell–Bloch equations for the ring cavity. The boundary condition for the field envelope that appears in the model is a simple periodicity condition. This feature allows for expanding the variables of the model in terms of traveling waves instead of standing waves as it is customary, which implies noteworthy simplifications in the calculations. On the basis of the modal equations which arise from the model, we discuss the adiabatic elimination of the atomic variables and of the atomic polarization only.
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.
In this paper we present a theory of optical subharmonic generation in a nonlinear Fabry-Perot resonator. Our approach, based on classical electrodynamics, leads to a Mathieu-Hill-type equation. This enables us to describe the generation of optical subharmonic oscillation in terms of a general formalism used to account for parametric phenomena, and to outline the possibility of using a nonlinear Fabry-Perot resonator as a memory device able to store the phase of a signal. The oscillation threshold condition is discussed and the rate of amplitude rise at start-up is evaluated.
Most of the intrinsic optical devices which have been studied in recent years consist of a Fabry-Perot resonator containing a nonlinear medium. The response time of such devices is determined by the transients associated with the high-Q resonator and the response time of the nonlinear medium. The fastest devices must be free of any Fabry-Perot cavity. The excitation of either surface plasmons at a nonlinear interface or guided waves in thin nonlinear films is a way to obtain an ultrafast bistable optical device. These two kinds of device have been studied.