On amplitudes of Fraunhofer diffractions of waves by 3D objects. Applications to half-spaces and oblique pyramids

Abstract

Thank to the Dirac’s fundamental quantum conditions between the position and momentum operators, we derive a formula showing that in a Fraunhofer diffraction of a plane wave with wave vector ${\overrightarrow{\mathrm{k}}}_0$ by a 3D object, the amplitude of the diffracted wave with wave vector $\kern0.1em \overrightarrow{\mathrm{k}}$ is the product of the values of the Fourier transforms with respect to $\overrightarrow{\Delta \mathrm{k}}\equiv \overrightarrow{\mathrm{k}}-{\overrightarrow{\mathrm{k}}}_0$ of two functions, one of them represents the effect of the interaction causing by the object onto the wave and the other the form of the object. If the object is delimited by planes, we show that its representative function may be obtained from the Dirac and Heaviside functions of linear transforms $\mathrm{U}\overrightarrow{\mathrm{r}}$ of $\kern0.1em \overrightarrow{\operatorname{r}}$. The Fourier transform of the representative function may then be calculated thank to a simple formula involving the reciprocal vectors extracted from the matrices U and   U^− 1. As results, we find again, without utilizing the Huygens-Fresnel principle nor the Maxwell’s equations, the principal laws concerning Fraunhofer diffractions and interferences of plane waves in optics including the Fresnel Formulae. In obtaining the latters we find at the same time the factor representing the interaction effect between the waves and the medium, characterized by its index of refraction. As further examples of applications, the calculations for obtaining the amplitudes of diffractions by tetrahedra and oblique pyramids are given in details.