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Nonlinear Dynamics in Optical Systems

Maxwell-Bloch Theory in A Nonideal Fabry-Perot Cavity

A. J. van Wonderen, L.G. Suttorp

Published Jan 1, 1990• 0 citations
Publisher

Abstract

In recent years it has become clear that the Maxwell-Bloch theory is capable of providing reliable predictions on instabilities in nonlinear media. For instance, it is successful in describing the single-mode instability of optical bistability [1]. Also for active media important results have been achieved on the basis of the Maxwell-Bloch theory [2].

DOI
10.1364/nldos.1990.ob236

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In recent years it has become clear that the Maxwell-Bloch theory is capable of providing reliable predictions on instabilities in nonlinear media. For instance, it is successful in describing the single-mode instability of optical bistability [1]. Also for active media important results have been achieved on the basis of the Maxwell-Bloch theory [2].
In this chapter we outline the general scenario for optical temporal instabilities, starting from the standard Maxwell–Bloch equations. With the exception of Section 22.7, in Part II we will always consider the ring-cavity configuration because only in the case of the ultrathin-medium case considered in Section 22.7 is the treatment of instabilities in Fabry–Perot cavities not substantially more complicated than that in the ring-cavity configuration. Some authors have introduced special approaches in order to study temporal optical instabilities, exploiting the concepts of gain and of dispersion, which are natural in optical systems and therefore help our intuition. In particular, a hint of the mechanism that produces instabilities in optical systems is offered by the so-called “weak-sideband” approach or “gain-feedback approach” [245–249], whose validity can be justified rigorously with the help of the linear-stability analysis [250]. The central point of this method is the response of the system to a weak probe passing through a collection of two-level atoms driven by a strong field [251–253]. If the strong signal with carrier frequency ω 0 saturates the medium, we can find regions of the frequency axis, which are far removed from the carrier frequency of the signal, where the weak probe can experience gain instead of absorption even if the medium has no population inversion. In this book we follow the approach of [250], which is rigorously rooted in the linear- stability analysis and, in addition, establishes a general, useful and, in a sense, surprising connection between single-mode and multimode instabilities. We classify as single-mode a temporal instability that involves the resonant mode only, and classify as multimode a temporal instability that involves only off-resonance cavity modes. The discussion in this chapter is mainly focussed on this connection. In Section 20.1 the multimodal equations derived in Section 12.2 are reformulated by introducing a modal expansion also for the atomic variables. In this way one obtains a complex hierarchy of equations that couple the modal amplitudes for the field and for the atomic variables.
We propose a new scheme for the classification of the optical instabilities that emerge within the framework of the Maxwell-Bloch model. Our scheme is based on an appropriate ex-tension of the notions of amplitude and phase instability advanced in earlier publications and generalized in this work to include the free running laser, optical bistability and the laser with an injected signal. We address our considerations to a unidirectional ring cavity model, in the plane wave approximation and with a homogeneously broadened, two-level active medium.
The theory of optical bistability can be developed at various levels of sophistication.(1) In this overview I have chosen an approach which I hope will make the simplicity and generality of the phenomenon clear, and be suited to the emphasis of this meeting. I will describe a deterministic, semiclassical theory of optical bistability, based on coupled Maxwell-Bloch equations. Statistical and quantum statistical theories will not be included. I also limit my attention to questions concerning steady states and their stability, leaving the theory of higher bifurcations to other speakers.