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Nonlinear Optics

Polarization patterns in a passive ring-cavity

J.B. Geddes, Jerome V. Moloney, Ewan M. Wright, W. J. Firth

Published Dec 17, 2002• 0 citations
Publisher

Abstract

A ring-cavity system, filled with an isotropic nonlinear medium and driven by a linearly polarized coherent input field, has been the subject of intense study over the last decade. This system can show a number of non-equilibrium phase transitions, including optical bistability (OB), and transverse pattern formation. The instability responsible for pattern formation then simply serves to drive the system from the unstable lower branch to the stable upper branch. This conclusion, however, is based on the assumption that the internal field in the nonlinear ring-cavity preserves its polarization state as that of the input field. Here we extend the mean field model to include polarization effects.>

DOI
10.1109/nlo.1994.470859

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A ring-cavity system, filled with an isotropic nonlinear medium and driven by a linearly polarized coherent input field, has been the subject of intense study over the last decade. This system can show a number of non-equilibrium phase transitions, including optical bistability (OB), and transverse pattern formation. The instability responsible for pattern formation then simply serves to drive the system from the unstable lower branch to the stable upper branch. This conclusion, however, is based on the assumption that the internal field in the nonlinear ring-cavity preserves its polarization state as that of the input field. Here we extend the mean field model to include polarization effects.>
Pioneered by Moloney and collaborators [1], transverse effects of nonlinear passive systems in a ring cavity have been the object of extensive study. Analytical investigations with diffractive coupling in one transverse dimension have demonstrated [1] that static and dynamic transverse structures occur in this system. Extension of the mean-field dispersive model developed from optical bistability to include diffraction [2] independently demonstrated qualitatively similar features, again in one transverse dimension. Investigations in two transverse dimensions have been largely confined to numerical studies by the Moloney group [1], showing for gaussian beam illumination ring structures which develop through azimuthal instability to a pattern of spots which typically execute a slow chaotic motion with a progressive loss of symmetries.
We study the kinetics of transverse pattern formation in a passive ring cavity containing two-level atoms. Parameters are chosen such that the Turing instabilities are on the upper and lower branches of a bistable domain and close to the limit points, so that the interaction between the long-wavelength modes includes the homogeneous mode. This leads to the formation of a localized structure which connects the two stable branches of the homogeneous bistable solution. The kinetics of this pattern formation is described by simple scaling laws.
In Chapter 9 we discussed the case of a nonlinear cavity containing an active medium. In this framework, the outstanding phenomenon which arises is laser emission, and we calculated the stationary solutions for the ring laser. In this chapter, instead, we turn our attention to the passive nonlinear cavity and in this framework the outstanding phenomenon is optical bistability , i.e. the existence of two distinct stable stationary states that coexist for the same fixed values of the parameters in play. This phenomenon can arise also in the case of an active system contained in a cavity, as we have already noted in Section 9.1 and as we will discuss further in connection with the laser with injected signal, but it became popular from the studies of nonlinear passive optical cavities in the 1970s and 1980s. In contrast to the laser case, in the passive configuration we use an input coherent field, which is injected into the cavity. We discussed this case in Chapter 8 and derived a general equation for the transmission of the cavity (Eq. (8.23)), which we used for an empty cavity and for a linear cavity in Sections 8.4 and 8.5, respectively. In this chapter we will discuss the full nonlinear configuration. It is customary to identify two extreme cases of optical bistability called absorptive optical bistability and dispersive or refractive optical bistability . In both cases an essential ingredient to obtain the bistable behavior is the feedback action exerted by the mirrors of the cavity. In the purely absorptive configuration, in which there is exact resonance among the frequencies of the input field, of the atoms and of the cavity, the other essential ingredient is the saturable nonlinearity of the absorption. In the dispersive case, instead, the main feature is the nonlinearity of the refractive index. These two cases will be illustrated in Sections 11.1 and 11.2, respectively, both “exactly” and in the low-transmission limit.