Manley–Rowe relations

Remarks

The coupled amplitude equations of the three waves have coupling constants that, by permutation symmetry in a lossless medium, are all the same effective coefficient. Substituting them into $dI_j/dz$ gives the same expression for $I_1'/\omega_1$ and $I_2'/\omega_2$ and its negative for $I_3'/\omega_3$. The relations hold for any phase mismatch. Adding them gives conservation of total intensity, $I_1 + I_2 + I_3$ constant, since $\omega_3 = \omega_1 + \omega_2$. In terms of photons, $I_j / (\hbar \omega_j)$ is a photon flux, so each photon at $\omega_3$ that disappears produces one photon at $\omega_1$ and one at $\omega_2$. Manley and Rowe first derived such relations in 1956 for lossless nonlinear reactances in electrical circuits.

Going deeper

For a student

In an optical parametric amplifier, the relations show that the signal and the idler gain equal numbers of photons. In sum-frequency generation they give the maximum conversion efficiency from the lower frequencies to $\omega_3$. For second-harmonic generation, with $\omega_1 = \omega_2$, they reduce to conservation of energy between the fundamental and the harmonic. They are a first integral of the coupled amplitude equations and help to solve them in terms of Jacobi elliptic functions.