Counterpropagating-pump phase matching (${\mathbf{k}}_1+{\mathbf{k}}_2=0$)

For opposite equal pump wave vectors, a conjugate mixing source has wave vector $-{\mathbf{k}}_3$. In an isotropic nondispersive medium this exactly matches a free backward wave at the generated frequency only in the degenerate case. Away from degeneracy the longitudinal mismatch has magnitude $2n|{\omega }_P-{\omega }_3|/c$; a finite interaction length supplies a tolerance, not exact vector matching.