Generalized Expressions of Effective Nonlinear Optical Coefficient for Non-collinear Phase Matching in Uniaxial and Cubic Media

The conditions of non-collinear phase matched frequency conversion are analyzed and the corresponding expressions of the effective nonlinear optical coefficient (deff) for 13 classes of uniaxial crystals and 3 classes of cubic crystals are derived. The discussed cases correspond to the situation when dijk not equal dikj, when dijk = dikj, and when Kleinman symmetry condition holds, with the general consideration that the extraordinary ray (e-ray) is not perpendicular to the phase propagation vector k in the uniaxial medium.