Three-wave mixing

An interaction among three waves mediated by a second-order nonlinear optical susceptibility. In sum-frequency generation, two input angular frequencies satisfy ${\omega }_3={\omega }_1+{\omega }_2$. The same coupling permits reverse conversion. Phase matching determines whether successive generated contributions reinforce; the Manley-Rowe relations express the associated lossless exchange constraints.

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.

A nonlinear optics process producing a field at the sum of two input angular frequencies. The driven electric polarization contains a term proportional to the product of the input complex amplitudes. With undepleted plane-wave inputs, integration along a crystal gives an output amplitude proportional to $L{e}^{i\Delta kL/2}\mathrm{sinc}(\Delta kL/2)$; the sinc function therefore determines the ideal length and phase mismatch dependence. A nonzero initial output field also interferes with this generated field.