The diffraction of light by superposed parallel supersonic waves general theory
The theory of the diffraction of light by two superposed parallel supersonic waves, consisting of the fundamental tone and then-th harmonic is developed, starting from the wave equation for the electric field of the light. The Fourier series method, first used in Raman and Nath’s generalized theory, is here employed to derive a system of difference-differential equations for the amplitudes of the diffracted light waves. Ther-th order spectrum makes an angleθr = -Arc sinrγ/γ* with the direction of the incident light and presents a change of frequency —rv*. In the case that the right-hand side of the difference-differential equations may be neglected, the exact solution is obtained by means of a complex function method. From the structure of the general system of difference-differential equations it is shown that the intensities of the ordersr and —r are always different for an even as well as for an odd ratio of the sound frequencies, excepted for some special values of the phase angle of the sound waves. A solution of the general system forn=2 is written in the form of a power series in ζ, the terms of which are calculated till the third ones included; the asymmetry of the intensities of opposite orders is not due to the terms in ρ. In the casen=3 a series solution also leads to an asymmetric pattern with respect to the zero order line; the terms in ρ are here responsible for the asymmetry, so that the symmetry property reappears for ρ=0, in accordance with the simplified theory based on Raman and Nath’s preliminary theory.