Invariants of the Third-Rank Cartesian Tensor: Optical Nonlinear Susceptibilities

Optical nonlinearities in crystals are for the first time analyzed from the rotational-invariance point of view. Several previous theoretical results are discussed on the basis of the decomposition of a tensor into irreducible parts. A proportionality between the spontaneous polarization and the vector parts of the nonlinearities is established, leading to new relations for the second-harmonic generation and linear electrooptic-effect coefficients in the $\delta$ formulation.