Show explicitly that the diffraction grating made by two beams interfering in a medium will give diffracted output of a third beam consistent with momentum conservation of the photons. In other words, first calculate the interference pattern  for two beams with the same wavelength and wave vectors k(vectors)₁ and k(vectors)₂ interfering in a planar medium. Assume that the medium then has index of refraction proportional to the intensity of the interference pattern in the plane. Last, calculate the Fraunhofer (far-field) diffraction pattern for a third beam with the same wavelength, and wave vector k(vectors)₃ impinging on this medium. Show that the output pattern corresponds to beams with 

One can also show this for non-equal wavelengths, but then one must perform a volume interference pattern instead of just an interference pattern in a two-dimensional plane.

+(k1k2) + k3. k4 = (k