• Robert Mertens

Articles written in Proceedings – Section A

• On the diffraction of light by progressive and standing supersonic waves

• 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.

• The diffraction of light by two superposed parallel supersonic waves being harmonics of the same fundamental

Starting from the wave equation for the electric field of the light, the theory of the diffraction of light by two superposed supersonic waves is developed for sound waves, the frequency ratio of which isn1:n2, wheren1 andn2 are simple incommensurable whole numbers, different from one. A system of difference-differential equations is derived, using the Fourier series method of Raman and Nath. The diffracted light waves make angles defined by$$\sin \theta _{rs} = - \left( {\frac{r}{{\lambda _1 ^* }} + \frac{s}{{\lambda _2 ^* }}} \right)\lambda$$ with the direction of the incident light and have frequency changes −(1*+2*), (r, s) being all the couples of integers satisfying the relationn=rn1+sn2, wheren is the number of the diffraction order. Neglecting the right-hand side of the difference-differential equation, approximation which corresponds with very intense ultrasonic waves having great wavelengths, the exact solution of the problem is obtained by using a complex integral method. In this special case the diffraction pattern is symmetric with respect to the zero order ifn2n1 is even; forn2n1 odd, the pattern is asymmetric, except whenn1 is odd and the phase angle of the sound wavesδ=(k+1/2)π. It has further been shown, that in the general case the intensity of the ordersn and −n are always different, excepting for some special values of the phase angles of the supersonic waves.

• Diffraction of light by two parallel superposed supersonic waves, one being then-th harmonic of the other: A critical study of the methods leading to approximate solutions in finite form

The system of difference-differential equations for the amplitudes of the light diffracted by two parallel superposed supersonic waves, consisting of a fundamental tone and itsn-th harmonic, is treated by two approximative methods giving solutions in finite form:

The method of successive approximations (SA method) and the method of theN-th order approximation (NOA method). The SA method reduces the system of difference-differential equations to a series of differential equations for each of the amplitudes; the integration of each differential equation becomes possible through the knowledge of the foregoing ones. In the NOA method the system of difference-differential equations is replaced by a system of 2N+1 simultaneous ordinary linear differential equations, the characteristic equation of which has only purely imaginary roots. Explicit calculations are made when the fundamental tone is accompanied by the second or the third harmonic; for the NOA method the approximationsN=1 andN=2 are considered. For values of the parameterρ=λ2/μ0μ1λ*2 large with respect to unity both methods lead to nearly the same results. If, however,ρ is of the order unity, the intensities of the diffracted orders obtained by the SA method are only valid for values ofζ=2πμ1z/λ small with respect to unity. For the same order of magnitude ofρ, the NOA method leads to more acceptable results; for values ofan=μn /μ1 (n=2, 3) of the order 10−1 there exists only a small difference between the intensities obtained in the first and in the second approximation; those differences become more appreciable for growing values ofan.

• 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 supersonic waves: The solution of the raman-nath equations - II. The exact solution

The partial differential equation associated with the system of difference-differential equations of Raman-Nath for the amplitudes of the diffracted light-waves is solved exactly by the method of the separation of the variables. The solution is presented as a double infinite series containing the Fourier coefficients of the even periodic Mathieu functions with periodπ and the corresponding eigenvalues. Considering this solution as a 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 formulae for the intensities are calculated. The connection between the Raman-Nath method and Brillouin’s Mathieu function method has thus been achieved.

• 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.

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