The diffraction of light by progressive supersonic waves. Oblique incidence of light: II. Exact solution of the Raman-Nath equations

Abstract

The partial differential equation associated with the system of difference-differential equations of Raman-Nath for the amplitude of the diffracted light waves, in the case of oblique incidence of light, is solved exactly by the method of separation of variables. The complete solution is presented as an infinite series containing Mathieu functions of integral order, if the angle of incidence corresponds with a Bragg angle, of fractional order, otherwise. Considering those solutions as Laurent series in one of the variables, the Laurent coefficients immediately give the exact expressions for the amplitudes of the diffracted light waves, from which the intensities may be calculated. From the obtained formulae, asymmetry and symmetry properties of the spectrum may be derived. It is also found that the nth order intensity has an extremum for the corresponding nth order Bragg angle. Numerical calculations show that those extrema can be maxima as well as minima. For ϱ ⪢ 1 the approximate solutions of David, Bhatia and Noble and Phariseau are found.