Diffraction of light by ultrasonic waves. II. Approximate expressions for the intensities and comparison with experiment

In part I of this investigation the problem of calculating the intensities of the various orders in the diffracted spectra was reduced to one of solving three infinite sets of linear algebraic equations I (4$\cdot $15) to (4$\cdot $17). This paper concerns itself with the approximate solutions of these equations and the calculation of intensities for some specific cases. It is shown that it is possible to solve these equations (and thence calculate the intensities) to any degree of accuracy over the entire range of experimental conditions, i.e. for any $\delta $ = $\Delta \Lambda ^{2}$/$\lambda ^{2}$ and angle of incidence $\theta $. Here $\Delta $ is the ratio of the maximum density change to the average density of the medium, $\Lambda $ is the wave-length of the sound wave in the medium and $\lambda $ the vacuum wave-length of the incident light. For $\delta \ll $ 1 or $\delta $/$\xi \ll $1, where $\xi $ = $\Lambda $ sin $\theta $/$\lambda $, the solution of these equations is particularly simple. Under these conditions only one or two orders are likely to appear on either side of the transmitted beam and expressions for their intensities are obtained. For larger $\delta $ ($\delta \sim $ 1 or $\gg $ 1) and $\delta $/$\xi $ not $\ll $ 1, the solution of the equations I (4$\cdot $15) to (4$\cdot $17) is complicated, but it is shown that for any given $\delta $ and $\theta $, only a finite number (at most $\sim $ 20 from each set and ordinarily considerably less) of these equations suitably chosen need be considered. One is also able to deduce the number of orders likely to appear under given experimental conditions. In particular, the well-known asymmetry in the diffracted spectra for oblique incidence receives a natural explanation. These and various other results of the theory are in satisfactory agreement with experimental data. Finally, if one makes approximations similar to those made by Raman & Nath in their theory for large $\delta $, the present theory also gives for the intensity of a given order J$_{l}^{2}$, where J$_{l}$ is the Bessel function of order l. This approximation, it is shown, overestimates the intensities in higher orders.