Transfer-matrix modeling of four-wave mixing at the band edge of a one-dimensional photonic crystal

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Abstract
We present a modeling of a degenerate four-wave-mixing nonlinear process in one-dimensional photonic crystals. The model is based on the nonlinear extension of the transfer-matrix description of propagation in the structure. The influence of light localization, near the band edge of the structure, on the enhancement of the phase-conjugate reflectivity is studied. The phase-conjugate reflectivity is shown to increase as the eighth power of the number of layers with an additional large dependence on the index contrast of the structure. In both cases the enhancement is accompanied by a strong reduction of the resonance width, which may lead to some limitation of the enhancement when ultrashort pulses are used. A strong influence of the losses on the nonlinear efficiency of the structure is also predicted with a great importance of scattering losses at the multiple interfaces of the structure.