Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity

Numerous studies have been devoted to photonic crystals, i.e., periodic dielectric devices, made of linear materials. These structures could allow us to obtain full gaps, to realize laser microcavities with a very high efficiency, multiplexers, or directive antennas. However, the analogy between semiconductors and linear photonic crystals cannot be pushed too far inasmuch as photons, being not submitted to Coulomb interaction, are not as easily controlled as electrons are. In particular, linear photonic crystals are not easily tunable. For these reasons, it seems natural to turn to photonic crystals made of nonlinear media. Some explicit computations have been made in case of one-dimensional ~1D! nonlinear photonic crystals ~note that these are just Bragg mirrors, and that bistability is known to occur in nonlinear Fabry-Perot resonators!. In particular a very interesting physical phenomenon, that of ‘‘gap soliton,’’ has been demonstrated. The subject of twoand threedimensional solitons has also been dealt with by John and Akozbek with some approximations ~media with smallcontrast, slowly varying envelope approximation, etc.!. Finally, bistability near a bandedge has been numerically investigated. However, all these studies, apart from in the case of 1D structures, deal with the four-wave approximation and use the Bloch-waves decomposition. For our part, we are interested in the computation of the energy transmitted through a finite-size nonlinear photonic crystal, so as to be as close as possible to an experimental situation. The crystal is doped with a microcavity ~we have removed a rod at the center of the crystal!. In the linear theory, such a microcavity generates a deep acceptor mode. We demonstrate that this property may be used, in the case of a nonlinear material, to induce optical switching and bistability in the transmission ratio. We use the doped crystal depicted in Fig. 1, made of 26 infinitely long parallel rods made of a material with x (3) nonlinearity. An s-polarized plane-wave impinges on the structure from the upper face and the transmission coefficient T is defined as the ratio between the flux of the Poynting vector of the total field collected on a segment situated below the crystal to the flux of the incident Poynting vector calculated on the same segment ~see Fig. 1!. All the computations are done using a rigorous linear multiscattering theory of diffraction and an iterative scheme. More precisely, the permittivity of the rods is given by «r(x ,y)5«r (x ,y) 1x uE(x ,y)u. For a fixed relative permittivity «r(x ,y), it is possible to compute the scattering matrix S(l) of the