Theory of Parametric Photonic Crystals
A photonic crystal is defined by its band-gap, but it is possible to have more than one bandgap. In this chapter, we focus on cases of two bandgaps where one has twice the frequency of the other. This allows coupling to occur between the bandgaps, provided the dielectric itself has a quadratic or parametric nonlinearity. Since Bloembergen’s time, it has been known that a modulated quadratic nonlinearity can enhance the prospects for phase-matching in frequency-doubling. Here, we consider phase-matching obtained through a modulated linear refractive index, giving the advantage that shortened interaction lengths are possible. In the following sections the equations for electromagnetic propagation in a grating structure with a parametric nonlinearity are derived using Maxwell’s equation as well as a coupled mode Hamiltonian analysis.
KeywordsPhotonic Crystal Solitary Wave Pulse Energy Displacement Field Carrier Frequency
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