Complete classification of the direction loci for three-wave collinear quadratic parametric interactions in biaxial acentric crystals

One new topological classification of the phase-matching direction loci for sum/difference frequency generation in biaxial acentric crystals is presented. Based on nine fundamental axial wave constant mismatches, 12 locus indices are used to distinguish permissible direction loci. Totally 66 kinds of combined loci belonging to 30 main classes are found and categorized by a set of ternary-digit class symbols or equivalently by sequence numbers. They are well illustrated in a classification diagram, indicative of 170 possible transitions among the loci. By an analysis of the dispersion of refractive indices, the historic “optically well-behaved crystals” are redefined and improved with simplified classification diagrams and corresponding schematic stereographic projections.