Guided waves by electromagnetic gratings and non‐uniqueness examples for the diffraction problem

Abstract

Communicated by J. C. NedelecWe consider the two-dimensional scalar problem of the diffraction of a plane wave by an infinite grating of conducting bodies immersed in a periodical dielectric medium. A Fredholm-type formulation is derived and studied. The existence of a solution is proved and some uniqueness results are established. A detailed description of the guided modes of the grating is carried out. Finally, various non-uniqueness examples for the diffraction problem are exhibited.

A new integral representation for quasi-periodic scattering problems in two dimensions

“…The above conditions result in the existence of a solution u for all frequencies ω and incident angles θ [9,Thm. 3.2].…”

“…Finally, u satisfies the following radiation condition [9]. Let y 0 > 0 be sufficiently large that the rectangle (−d/2, d/2) × (−y 0 , y 0 ) contains .…”

“…The solution is unique (i.e. the problem wellposed) at all but a discrete set of frequencies ω, for each fixed horizontal wavenumber κ i = ω cos θ [9,Thm. 3.4].…”

“…(There may also exist robust guided modes with frequencies below the continuous spectrum corresponding to the mode's horizontal wavenumber, which are therefore inaccessible in the scattering setting.) If n < 1 and is shaped such that the refractive index is non-decreasing with |y| then no guided modes can exist [9,Thm. 3.5].…”

On the excitation of a closely spaced array by a line source

“…These RB waves, as we will henceforth refer to them, decay exponentially away from the array but travel along the array undamped and our main objective is to develop a method by which the amplitude of the RB waves that are excited can be determined without having to solve the full scattering problem. Note that if the boundary condition on the cylinders was changed to φ = 0 then no such waves exist [4].…”