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Optics Express

Bragg localized structures in a passive cavity with transverse modulation of the refractive index and the pump

A. G. Vladimirov, Dmitry V. Skryabin, Gregory Kozyreff, Paul Mandel, Mustapha Tlidi

Published Jan 1, 2006• 48 citations
Publisher

Abstract

We consider a passive optical cavity containing a photonic crystal and a purely absorptive two-level medium. The cavity is driven by a superposition of two coherent beams forming a periodically modulated pump. Using a coupled mode reduction and direct numerical modeling of the full system we demonstrate the existence of bistability between uniformly periodic states, modulational instabilities and localized structures of light. All are found to exist within the conduction band of the photonic material. Moreover, contrary to similar previously found intra-band structures, we show that these localized structures can be truly stationary states.

DOI
10.1364/opex.14.000001

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We consider a passive optical cavity containing a photonic crystal and a purely absorptive two-level medium. The cavity is driven by a superposition of two coherent beams forming a periodically modulated pump. Using a coupled mode reduction and direct numerical modeling of the full system we demonstrate the existence of bistability between uniformly periodic states, modulational instabilities and localized structures of light. All are found to exist within the conduction band of the photonic material. Moreover, contrary to similar previously found intra-band structures, we show that these localized structures can be truly stationary states.
Summary form only given. Stationary localized structures (LS) and localized patterns have been predicted in passive resonators in regimes where the homogeneous steady states exhibit a subcritical Turing (or modulational) instability. These solutions consist of bright or dark spots in the transverse profile of the intracavity field amplitude. They can be either spatially independent and randomly distributed or clusters forming well-defined spatial patterns in the transverse plane orthogonal to the propagation axis. We consider an optical cavity with plane mirrors filled with resonant two-level atoms and driven by a coherent plane-wave injected signal. We focus on the purely absorptive limit where the two-level atoms are in resonance with the injected field. In this system, two type of LS may be generated, depending on the magnitude of the critical wave number associated with the modulational instability: circular localized bright spots or a stationary localized single stripe. Both types of LS can form for the same value of parameters.
We discuss a novel type of optical instability, which leads to the spontaneous formation of a stationary spatial structure, instead of to the onset of oscillations or pulsations. A passive system is contained in an optical cavity, to which we add two lateral mirrors. Under appropriate conditions, diffraction of radiation can induce the onset of a transverse-stripe structure in an initially uniform plane-wave field configuration. In this paper, we consider the case of a homogeneously broadened two-level system. The model on which our analysis is based is derived from the Maxwell-Bloch equations in the paraxial approximation, using the mean-field limit and assuming a large Fresnel number. We show that, with respect to the bistability parameter C, the threshold of the spatial instability coincides with the bistability threshold for absorptive, resonant optical bistability. The spatial instability requires, however, a detuned configuration and can arise both in the absence and presence of bistability. In the latter case, it can occur both in the lower and in the upper branch of the steady-state curve. The instability is caused by competition between different transverse modes, which realize a situation of spatial coexistence, different from the state of temporal coexistence which characterizes the spontaneous oscillations that arise from the previously known multimode instabilities. Finally, we discuss some points relevant for an experimental observation of this phenomenon.
We consider a model recently proposed, describing a multiple quantum well semiconductor microcavity driven by a coherent holding beam. We discuss here some general features of stationary modulational instabilities affecting the system. By generalizing a method introduced in a previous paper on two-level atoms in optical cavities, we give a detailed description of the stationary instabilities in the parameter space. The modulational instability can be related to the presence of stable cavity solitons, which have been predicted in this system. Therefore we have a helpful tool to research the best parametric conditions for stable cavity solitons, a very demanding task due to the large number of parameters. Finally, we show that our analysis can be extended to more general models, describing a nonlinear diffusive material inside an optical cavity with a general form of the interaction terms.
We investigate superradiant lasing in 1D photonic crystal taking into account both homogeneous and inhomogeneous broadening in a two-level active medium. The latter may be based on, e.g., the quantum-well heterostructures and the fibers activated by color centers where Bragg structure can be fabricated artificially by means of a periodic modulation of the cladding layers. The truncated Maxwell-Bloch equations for the counter-propagating waves, the polarization and inversion of an inhomogeneously broadened two-level medium are used to analyze numerically the interplay between the back-scattering, amplification, dispersion, and de-phasing of waves. It is found that, if the photonic band gap is less than the coherent amplification bandwidth, there is a wide range of the pumping and relaxation parameters of a two-level medium where superradiant lasing exists and results in generation of a quasi-periodic and/or chaotic series of powerful and short pulses similar to those of Dicke superfluorescence. It is the cavity mode selection due to distributed feedback caused by the resonant Bragg structure which is responsible for the existence of superradiant lasing. Typical parameters of superradiant photonic-crystal lasers, including lengths, photonic band gaps and relaxation rates, are indicated.
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.