Sum-frequency generation in a two-dimensional photonic lattice

We have formulated the Green's-function method for describing nonlinear optical processes in an arbitrary two-dimensional photonic lattice with particular regard to sum-frequency generation. In addition to the derivation of the generalized phase-matching condition, we have shown that the field intensity and the average Poynting's vector of the sum-frequency component are proportional to its (group velocity)^-2 and (group velocity)^-1, respectively. Therefore, an enhancement is expected for both of them at photonic band edges, where the group velocity tends to zero. This method was applied to a square lattice composed of circular air-rods formed in a LiNbO₃ crystal, and the average intensity of the electric field of the second harmonic and the effective nonlinear susceptibility were numerically calculated. {} 1996 The American Physical Society.