Manley-Rowe relations ($I_1^{\prime }/{\omega }_1=I_2^{\prime }/{\omega }_2=-I_3^{\prime }/{\omega }_3$)

Conservation relations for resonant, lossless wave mixing. For sum-frequency generation with ${\omega }_3={\omega }_1+{\omega }_2$, the intensities obey $I_1^{\prime }/{\omega }_1=I_2^{\prime }/{\omega }_2=-I_3^{\prime }/{\omega }_3$. 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 $\hslash {\omega }_j$. 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.