Backward four-wave-mixing boundary problem ($u^{\prime }=i\kappa v^{\ast },v^{\prime }=-i\kappa u^{\ast }$)

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.