• Oswald Leroy

Articles written in Proceedings – Section A

• Diffraction of light by supersonic waves: The solution of the raman-nath equations—I

In the problem of the diffraction of light by a supersonic wave, at normal incidence of the light, the solution of the system of difference-differential equations of Raman and Nath, for the amplitudes of the diffracted light beams, is reduced to the integration of a partial differential equation. The coefficients of the Laurent expansion of the solution of the latter equation yield the expressions for the amplitudes of the diffracted light waves. The partial differential equation has been integrated for two approximations.

(1) Forρ=0, the well-known results of Raman and Nath’s preliminary theory are re-established.

(2) Forρ≪1 a power series inρ, the terms of which are calculated as far as the third one, leads to the solution of Mertens and Berry obtained by a perturbation method.

• Diffraction of light by two superposed parallel ultrasonic waves, having arbitrary frequencies. General symmetry properties; method of solution by series expansion

In this study about the diffraction of light by superposed parallel ultrasonics, with frequency ration1:n2, we deduce a general symmetry property for the intensities of the diffraction pattern: if the intensities of the ordersn and −n are equal the phase-difference must be of the form:$$\delta = \frac{{n_1 - n_2 }}{{n_1 }} \begin{array}{*{20}c} \pi \\ 2 \\ \end{array} + p \frac{\pi }{{n_1 }}$$.

Further we discuss the general form of a series expansion ofψn as a solution of the system of difference-differential equations for the amplitudes of the diffracted waves, and find that the amplitude of the ordern=mn2+p(m⩾1, 0&lt;pn2) expanded in series inζ=(2πμ1z)/λ, can only begin with the termζm+1. Finally, we have calculated the first three terms in the series expansion for the intensities of the diffracted light beams, in the special casen1=2,n2=3. Here we establish that only the second term is responsible for the asymmetry and that the general symmetry property is fulfilled.

• Diffraction of light bv two parallel superposed supersonic waves, one being the second harmonic, the other the third harmonic of the same fundamental. Description of two methods leading to approximate solutions in finite form

The diffraction of light by superposed parallel ultrasonic waves, with frequency ratio 2:3, is solved by the NOA-method (N−th order approximation method). Explicit calculations of the intensities are given forN = 2. The same problem is also treated with the SA-method (method of successive approximations). The latter results are improved by iteration. In both methods the general symmetry property of the diffraction pattern is verified, namely, that symmetry with respect to the zero order occurs when the phase-difference of the supersonic waves δ = (2k + 1) π/4.

• Diffraction of light by superposed parallel supersonic waves, being harmonics of the same fundamental. Solution of the system of difference-differential equations for the amplitudes

Starting from the general system of difference-differential equations for the amplitudes of the diffracted beams of light, given by Mertens, and using the method of Kuliasko, Mertens and Leroy for the diffraction of light by one supersonic wave, it is possible to reduce the solution of the system of difference-differential equations, to the solution of a partial differential equation. In this way it is possible to calculate the intensities of the ordern and −n, as a series expansion in ρ. Here we only considered terms up to ρ2. It was also possible to verify the general symmetry properties for the intensities studied by Leroy and Mertens.

• Diffraction of light by superposed parallel supersonic waves, one being then-th harmonic of the other. Solution of the system of difference-differential equations for the amplitudes by a series expansion method

In the problem of the diffraction of light by two parallel supersonic waves, consisting of a fundamental tone and itsn-th harmonic, the solution of the system of difference-differential equations for the amplitudes has been reduced to the integration of a partial differential equation. The expressions for the amplitudes of the diffracted light waves are obtained as the coefficients of the Laurent expansion of the solution of this partial differential equation. The latter has been integrated for two approximations:

Forρ = 0, the results of Murty’s elementary theory are reestablished.

Forρ ≤ 1, a power series inρ, the terms of which are calculated as far as the third one, leads to a new expression for the intensities of the diffracted light waves, verifying the general symmetry properties obtained by Mertens.

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