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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Optical Extinction Theorem In The Nonlinear Theory Of Optical Multistability

R. Bullough, Saad S. M. Hassan

Published Jun 16, 1983• 7 citations
Publisher

Abstract

The optical extinction theorem due to P. P. Ewald, C. W. Oseen and C. G. Darwin is used completely to describe the action of a dielectric slab as a Fabry-Perot interferometer. The analysis is very comparable to that given by Born and Wolf. It is then shown that by generalising the optical extinction theorem to the nonlinear regime essentially the same Fabry-Perot cavity action is regained but in this case the refractive index and hence the cavity tuning depends on the input intensity. This way a multistable output/input relation is derived. It is shown further how the extinction theorem will also handle the problem of 'standing waves' in the cavity. The possibility of optical turbulence in a Fabry-Perot cavity is briefly mentioned.

DOI
10.1117/12.934377

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The optical extinction theorem due to P. P. Ewald, C. W. Oseen and C. G. Darwin is used completely to describe the action of a dielectric slab as a Fabry-Perot interferometer. The analysis is very comparable to that given by Born and Wolf. It is then shown that by generalising the optical extinction theorem to the nonlinear regime essentially the same Fabry-Perot cavity action is regained but in this case the refractive index and hence the cavity tuning depends on the input intensity. This way a multistable output/input relation is derived. It is shown further how the extinction theorem will also handle the problem of 'standing waves' in the cavity. The possibility of optical turbulence in a Fabry-Perot cavity is briefly mentioned.
We have reported two different aspects of this work already [1,2]. Nevertheless as the point of view is unconventional we review it here strictly within the context of the theory of optical bistability and optical multistability. We are concerned to connect the envelope Maxwel1-Bloch equations with optical bistability (multistability) in a Fabry-Perot (FP) cavity in a rigorous and potentially quantitative way. One problem in this connection is an adequate statement about standing waves. We present methods which derive the standing wave equations of motion completely as a part of a comprehensive non-linear refractive index theory of multistabi1ity inside the FP cavity. The theory is a c-number one — but a comparable quantum theory seems possible. A key feature of the argument is generalisation of the famous 'optical extinction theorem' [1,2,3] to this non-linear regime. In practice it means we do not invoke any boundary conditions at the surfaces of the FP cavity — only conditions at infinity — and this offers advantages for the quantitative description as we show.
We study the nonlinear optical bistability and multistability of a nonlinear ferroelectric (FE) Fabry-Perot (FP) interferometer assumed coated with partial dielectric mirrors on its interfaces. The FE material is considered to have an intensity-dependent refractive index where the third order nonlinear susceptibility χ ilmn (3) acts like Kerr coefficient. The nonlinear response of the FE medium is modeled by the Landau-Khalatnikov (LK) dynamical equation. Within a single frequency approximation, the nonlinear wave equation is written in terms of polarization P rather than the electric field E as the dependent variable. The resulting nonlinear polarization equation is numerically integrated across the FP film using the fourth order Runge-Kutta method. The behavior of the resulting polarization and transmission as a function of the field input intensity is investigated.
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.
A theoretical investigation is made of the polarization characteristics of the optical polarization and amplitude multistability of a Fabry?Perot interferometer filled with a nonlinear resonant medium. In the case of linear polarization of the incident radiation it is shown that an amplitude hysteresis differs considerably from a hysteresis predicted by a scalar theory; moreover, the degree of polarization exhibits a hysteresis. A calculation is made of the critical input intensity of elliptically polarized radiation for which the ellipticity at the output rises strongly even when it is negligible at the input.
We present a theory of optical bistability for a Fabry-Perot cavity containing a medium with nonlinear refraction but linear absorption with only plane-wave and slowly varying envelope approximations. An analytic expression for the critical intensity Icfor the onset of bistability is derived and used to compare cavity designs. It is shown that 1) the important material parameter for minimum Icis the ration_{2}/\alpha 2) the limit to Icis set by limitations on finesse due to inhomogeneities rather than the absorption itself, and 3) the cavity design which gives lowest Icfor a given finesse is that for which the mirror transmissivity equals the absorption per pass; at high finesse this design leads to a total peak cavity transmission of 1/4 when mirror reflectivities are equal.