= Manley-Rowe relations
{c}
{title2=$I_1'/\omega_1=I_2'/\omega_2=-I_3'/\omega_3$}
{wiki=Manley–Rowe_relations}
Conservation relations for resonant, lossless wave mixing. For <sum-frequency generation> with $\omega_3=\omega_1+\omega_2$, the intensities obey $I_1'/\omega_1=I_2'/\omega_2=-I_3'/\omega_3$. Thus $I_1/\omega_1+I_3/\omega_3$ and $I_2/\omega_2+I_3/\omega_3$ are constant. Their difference gives a third, dependent invariant. Adding the intensity derivatives yields <conservation of energy>.
The exchange interpretation divides each intensity by its <photon> energy $\hbar\omega_j$. One conversion removes a photon from each lower-frequency mode and adds one to the upper mode. For classical coupled-wave equations, differentiating the squared <complex amplitudes> gives the same common exchange rate multiplied by $(\omega_1,\omega_2,-\omega_3)$. Loss or externally replenished pumps require corresponding source terms.
Back to article page
