Propagative and Evanescent Waves Diffracted by Periodic Surfaces: Perturbation Method

Abstract

Abstract-The propagation equation, written in a curvilinear coordinate system, is solved by using a perturbation method inspired from quantum physics and extended to imaginary eigenvalues and evanescent waves. The parameter of perturbation is the groove depth which is small compared to the period. The method is expanded up to second order for the non-degenerate problem. In this way the solutions have analytical form compared to a numerical method. They present the advantage to put in evidence the evolution of the energy distribution for different diffraction orders as a function of the magnitude of the perturbation. The efficiencies which are deduced from these analytical solutions are compared of those obtained by the curvilinear coordinate method. The good agreement between the two methods occurs for a groove depth with respect to the wavelength less than or equal to 0.16. Thus, this new approach opens a new range of applications for inverse problems.

Fraunhofer Diffraction by a Strip: Perturbation Method

“…-incidence angle: θ 0 = 1 • (close to the normal incidence that is a degenerated point given that the perturbation method is developed for undegenerated cases), -magnitude of grooves: h c /λ = 0.08 (chosen in the validity domain of the perturbation method compared to the rigorous numerical method [18]), -period: d/λ = 3.9 (electromagnetic optics domain).…”

“…m , are estimated with a phase term exp(±ikβ m v) for propagative waves and with an amplitude term exp(∓kβ m v) for evanescent waves which decrease quickly when v increases respectively by positive or negative values. We have shown in [18] that they form an orthogonal basis of 2(2N + 1) dimension. But for v = 0 on the perfectly conducting surface where the boundary conditions are written, they form an orthonormal basis identical to the B ± basis: {|exp(ikα m u) , |exp(ikα m u) }.…”

“…Then the calculation of the amplitude A + d (α) given in relation (34) requires the calculation of the amplitudes A + m = A + (α m ) of the diffracted fields by the periodic surface associated to the strip. It is the object of next sections where we give the essential steps of the calculation largely developed in [17,18] in the case of a strip weakly deformed with respect to its width.…”

“…Section 5 presents the essential steps of the perturbation method [17,18]. They lead to eigenvalues and eigenvectors according to expansions up to the second order in series of powers of the amplitude of the perturbation.…”