Reduction of angular momentum coupling trees and the polarization dependence of nonlinear optical processes

Diagrams representing angular momentum coupling trees with vector (j=1) external labels in a Cartesian basis and with even degree (up to 8) are expressed as sums of products of Kronecker delta functions. Integrity basis theory is used to verify the possibility of such a representation, and to determine the number of independent invariants. The results are of use in the analysis of irreducible Cartesian tensors and thus, e.g., in nonlinear optics. As examples, the polarization dependences of the hyper-Raman and three-photon Raman effects are discussed.