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Topical Meeting on Optical Bistability (OB)

Optical extinction theorem theory of optical multi-stability bifurcations and turbulence in the Fabry-Perot interferometer

R. Bullough, Saad S. M. Hassan, G. P. Hildred, Ravinder Rupchand Puri

Published Jan 1, 1983• 0 citations

Abstract

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.

DOI: 10.1364/obi.1983.thb43

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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.

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.

The theory of the optical multistability by two-photon resonant interaction is investigated. The nonlinear medium inside a Fabry-Perot cavity is in two-photon resonance with the laser beam. The two-photon equation of state including dynamical Stark shift and relaxation parameters is obtained and studied under mean-field approximation. The equations for both Fabry-Perot cavity and ring cavity take basically the same form. The physics mechanisms resulting in optical multistability are also studied. It has been shown that the combined effects of Kerr switching and saturation absorption switching can lead to two-photon optical double-bistability, tristability as well as bistability. Furthermore, the criterion conditions under which double-bistability or tristability can occur are given and discussed. Finally, the phase-conjugated reflectivity by two-photon four-wave mixing is also shown to exhibit multistability as well as hysteresis.

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.

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In this work a theoretical model treating the optical bistability in a two state dissipative atomic medium inside a Fabry-Perot cavity is presented.The model is based on a semiclassical point of view to explain the field-matter interaction.The atoms of the medium are described by the "quantum polarization equation of motion" while the field is described by the "classical Maxwell's field equations".Applying the boundary conditions to the fields inside the Fabry-Perot cavity, the occurrence of optical bistability is deduced between the incident field intensity and both the intracavity and transmitted field intensities.Also the effect of building standing and nonstanding waves inside the cavity on bistability is discussed.

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