Any succesfull experimental exploration of the huge parameter space provided by PCs has to be accompanied by a quantitative theoretical analysis in order to identify the most interesting cases and to help interpret the data. To date, only a few works along these lines has been carried out for either Kerr-nonlinearities [1] or sum-frequency and second harmonic generation [2]. Moreover, the approximations involved seriously limit the applicability of these theories to real PCs. For instance, the study of Kerr-nonlinearities in two-dimensional PCs [1] has been limited to the nearly free photon case, i.e., weak modulations in the linear index of refraction. Similarly, the recent investigation of second harmonic generation in two-dimensional PCs [2] failed to reproduce the well-known results for the limiting case of a homogeneous material.
In this presentation, we will apply the multi-scale analysis to the case of sum-frequency generation in two-dimensional PCs and discuss some of the most interesting results. In particular, we show how the PC allows to access experimentally a parameter range that has been considered theoretically by Zakharov [3]. For instance, as there are no stable soliton solutions in two dimensions, a two-dimensional PC would allow the complete radiation conversion from one wavelength to another through nonlinear interactions. This may have important applications for wavelength shifters in telecommunication technology.
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