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Annual Meeting Optical Society of America

Dynamic effects of nonuniqueness of cavity loss

Albert T. Rosenberger

Published Jan 1, 1990• 0 citations
Publisher

Abstract

Optical resonators, usually in the form of spherical-mirror interferometers, are employed in a wide variety of experiments, including studies of steady-state and dynamic behavior in lasers and optical bistability. To make meaningful comparisons between experiment and theory, a resonator must be characterized in terms of its losses; that is, the behavior of the empty, or linear, resonator must be well understood if one wishes to understand the behavior of the filled, nonlinear resonator. In general, however, it is not possible to characterize an optical cavity in terms of a single loss parameter. Doing so results, for example, in a different state equation for optical bistability.

DOI
10.1364/oam.1990.tup3

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Optical resonators, usually in the form of spherical-mirror interferometers, are employed in a wide variety of experiments, including studies of steady-state and dynamic behavior in lasers and optical bistability. To make meaningful comparisons between experiment and theory, a resonator must be characterized in terms of its losses; that is, the behavior of the empty, or linear, resonator must be well understood if one wishes to understand the behavior of the filled, nonlinear resonator. In general, however, it is not possible to characterize an optical cavity in terms of a single loss parameter. Doing so results, for example, in a different state equation for optical bistability.
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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.
For many contemporary physics experiments, the use of microwave and optical resonators has become a powerful tool for enhancement in detection sensitivities, nonlinear interactions, and quantum dynamics. Most often, the term cavity is used to describe such electromagnetic resonators. This term has been taken over from microwave technology, where resonators really look like closed cavities, whereas optical resonators traditionally have an open kind of setup. That difference in geometry is related to the fact that optical resonators are usually very large compared with the optical wavelength, whereas microwave cavities are often not much longer than a wavelength. As discussed in the course of this book, microwave and optical cavities allow one to extend the interaction length between matter and field, to build up the optical power, to impose a well-defined mode structure on the electromagnetic field, to implement extreme nonlinear optics, and to study manifestly quantum mechanical behavior associated with the modified vacuum structure and/or the large field associated with a single photon confined to a small volume [100].
In the first section we discuss the stationary solutions for a laser with an injected signal, the active counterpart of the optical bistability analyzed in Chapter 11. Next, in Section 13.2 we consider another very interesting system, namely a laser with a saturable absorber, a free-running laser in which the optical cavity contains a passive medium in addition to the active medium. The single-mode model for this system can be constructed by combining the single-mode models for the free-running laser and for optical bistability. The discussion is then focussed on its stationary states. In Section 13.3 we return to the single-mode model for optical bistability and, by introducing an appropriate “cubic limit” on the parameters in play, we derive a cubic model for dispersive optical bistability with two-level atoms. The stationary equation for such a model formally coincides with that derived and discussed for a Kerr medium in Section 11.2. In the last section we start from the equations for a quadratic nonlinear medium in the degenerate configuration derived in Chapter 6 and combine them with the appropriate boundary conditions for the two field envelopes in play, assuming that the quadratic medium is contained in a ring cavity as happens in an optical parametric oscillator or in second-harmonic generation in a cavity. Next, by introducing the low-transmission limit and the single-mode limit we derive a two-mode model that describes the dynamics of a degenerate optical parametric oscillator or of second-harmonic generation in a cavity. Finally, for the case of a parametric oscillator we illustrate extensively the stationary solutions of the equations. A laser with an injected signal From a technical point of view, this system is interesting because of the possibility of stabilizing the frequency and the phase of separate lasers by taking advantage of the known affinity of nonlinear oscillators to lock onto a periodic forcing element. In 1972 Spencer and Lamb [125] predicted the possibility of bistable behavior in a laser with an injected signal before the surge of interest in optical bistability in the passive configuration.