Degenerate four-wave mixing

All four interacting waves have the same frequency. Counterpropagating equal-frequency pumps can give a source proportional to the conjugate signal field, providing phase conjugation. The pump product must have the appropriate uniform spatial phase, and finite apertures and phase mismatch limit reconstruction.

Because one wave travels backward, its incident boundary condition belongs at the opposite end of the medium. With $u(0)=a$ and $v(L)=0$, differentiating once gives $u^{\prime \prime }=-{\kappa }^{2}u$ and the solutionThe invariant $|u{|}^2+|v{|}^2$ is constant along $z$, while the input/output powers obey transmission minus reflection equal to the incident power in this normalization. A proposed solution must satisfy both first-order equations, not just their second derivatives.

The first zero of $\cos \kappa L$ makes the homogeneous opposite-end boundary problem singular. Its undepleted-pump treatment admits a nontrivial field without an imposed signal, indicating a parametric oscillation threshold. Depletion, saturation and the actual noise-seeded dynamics determine finite fields near threshold.

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.