Enhancement of nonlinear effects at the degenerate band edge of two-dimensional photonic crystals

The ability of two-dimensional (2D) photonic crystals (PC) for high enhancements of nonlinear processes is analyzed in the case of a degenerate band edge when two symmetrical diffracted beams are generated by Bragg diffraction in the 2D-PC. Calculations are performed using the very simple Bragg-coupled wave theory which only involves three coupled waves (the incident wave and the two diffracted waves) for the linear interaction. The validity of the approximation is proved for wavelengths lying at the neighborhood of the band edges of 2D-PC. Very large local-field intensities are predicted around the band-edge wavelengths, in particular for the upper band edge. Nonlinear propagation is studied through the analysis of degenerate four-wave mixing. For counterpropagating pump beams orthogonally sent onto the 2D-PC huge improvement of the phase conjugate reflectivity are predicted at least for small incidence angles of the signal beam. These results represent an improvement by a factor of 20 when compared to the case of a one-dimensional PC of the same thickness made of the same materials. As three intense phase conjugate beams are generated in the four-wave mixing interaction, the 2D-PC could be very interesting for the purpose of dense parallel optical signal processing. Moreover, the simple theoretical analysis developed in the paper can be used for any kind of 2D-PC.