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Quantum Electronics

Eigenfrequencies of a nonreciprocal nonuniformly filled ring optical resonator

В. Ф. Судаков

Published Aug 31, 2008• 0 citations
Publisher

Abstract

A nonreciprocal ring optical resonator with the simplest type of nonuniformity in the form of two interacting scatterers is considered. A boundary problem of a periodic type allowing one to write the characteristic equation for determining eigenfrequencies is formulated. Analytic expressions for eigenfrequencies and the distribution coefficient of travelling waves in the mode are derived under the assumption of relative smallness of the reflection coefficient and the nonreciprocity parameter. The peculiarities of splitting these frequencies and the structure of the eigenfrequencies (resonator modes) corresponding to them are discussed.

DOI
10.1070/qe2008v038n08abeh013296

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A nonreciprocal ring optical resonator with the simplest type of nonuniformity in the form of two interacting scatterers is considered. A boundary problem of a periodic type allowing one to write the characteristic equation for determining eigenfrequencies is formulated. Analytic expressions for eigenfrequencies and the distribution coefficient of travelling waves in the mode are derived under the assumption of relative smallness of the reflection coefficient and the nonreciprocity parameter. The peculiarities of splitting these frequencies and the structure of the eigenfrequencies (resonator modes) corresponding to them are discussed.
The mutual interaction of counterpropagating waves in a ring resonator containing a medium with second-order dispersive nonlinearity is studied. The theoretical treatment of nonreciprocal propagation is based upon the novel coupling of counterpropagating and mode-locked first- and second-harmonic waves. The experimentally observed transfer characteristics of a high-frequency model system reveal a directionally asymmetric bistability. This nonreciprocity can also be attributed in the time domain to the occurrence of dissipative structures, i.e., to solitons traveling within the resonator.
We consider a ring resonator with two resonant modes undergoing dynamic modulation. Using the Floquet analysis, we show that with a proper choice of modulation profile, such a resonator can achieve a direction-dependent Rabi splitting of its Floquet eigenstates, which results in direction-dependent dissipation and nonreciprocal transmission for waves. We numerically demonstrate this nonreciprocal effect and show that a realistic refractive index modulation of the ring can result in significant transmission in one direction and near complete dissipation in.
With the exception of Chapter 1, up to this point we have considered cavityless optical systems. With this chapter, instead, we start focussing on the case that the atomic medium is located in an optical cavity. We describe in detail the case of a unidirectional ring resonator, the boundary condition which characterizes it and the relations between its output fields and the intracavity field (Sections 8.1–8.3). We discuss the transmission of an empty cavity (Section 8.4) and of a cavity that contains a linear medium (Section 8.5). Optical cavities An optical cavity or optical resonator is formed by at least two mirrors, appropriately arranged in such a way that the radiation beam follows a closed path. The beam splitters allow the radiation to escape from the cavity (output) and allow one to inject radiation into the cavity (input). In order to confine the radiation transversally with respect to the direction of propagation, the mirrors are usually spherical (see Fig. 1.2). The case of cavities with spherical mirrors will be described in Part III of this volume. On the other hand, an analytical treatment of the problems is possible only within the plane-wave approximation which we will use in the remainder of Part I and in Part II. In accord with this, we assume here that the mirrors are planar. The two most common types of optical resonator are the Fabry–Perot cavity and the ring cavity . The planar Fabry–Perot cavity consists in two beam splitters in a parallel configuration as shown in Fig. 8.1. Within the cavity there are two counterpropagating fields, E F propagating in the forward direction and E B propagating in the backward direction with respect to the input field E I . Because we have derived the Maxwell–Bloch equations for the unidirectional configuration, in this and the following chapters we focus on the case of a ring cavity, which allows unidirectionality. The Fabry–Perot case will be considered in Chapter 14.
In the previous chapters we considered the Maxwell–Bloch equations with the boundary condition appropriate for a unidirectional ring cavity and calculated the stationary solutions for laser and optical bistability, respectively. Next, we showed that the exact stationary solutions simplify appreciably in the low-transmission limit. In this chapter we want, first of all, to apply the low-transmission approximation on the dynamical equations directly, beyond the stationary case. We start from the Maxwell–Bloch equations (4.35)–(4.37) and the boundary condition (8.36). The latter is complex and includes the retardation Δ t and some parameters such as the transmissivity and reflectivity coefficients T and R , the input field y and the cavity detuning δ 0 . We introduce a transformation of the coordinates z and t and a transformation of the variables F and P [121–123]. As a result, the boundary condition reduces to a simple periodic boundary condition, and the electric-field equation incorporates the parameters which previously appeared in the boundary condition. The virtue of the periodic boundary condition is that it allows one to expand the electric field in terms of modal amplitudes and to recast the field equation in the form of a set of time-evolution equations for the modal amplitudes. All of this is shown in Section 12.1. In Section 12.2 we apply the low-transmission approximation and the Maxwell–Bloch equations assume a form that is a generalization of a set of equations commonly used in the literature to describe the ring laser [124]. Together with the Bloch equations for the atomic variables, the modal equations provide a model that lends itself ideally to a numerical resolution of the equations. In Section 12.3 we show that, if we assume that only the resonant cavity mode is active, this model reduces to a simple single-mode model consisting in three coupled ordinary differential equations, which will be analyzed in Part II of this book. Next, in the same section we derive the single-mode model in a more direct way, without passing through the multimodal equations.