On the Wiener–Hopf Method for Surface Plasmons: Diffraction from Semiinfinite Metamaterial Sheet

Abstract

By formally invoking the Wiener-Hopf method, we explicitly solve a one-dimensional, singular integral equation for the excitation of a slowly decaying electromagnetic wave, called surface plasmon-polariton (SPP), of small wavelength on a semiinfinite, flat conducting sheet irradiated by a plane wave in two spatial dimensions. This setting is germane to wave diffraction by edges of large sheets of single-layer graphene. Our analytical approach includes (i) formulation of a functional equation in the Fourier domain; (ii) evaluation of a split function, which is expressed by a contour integral and is a key ingredient of the Wiener-Hopf factorization; and (iii) extraction of the SPP as a simple-pole residue of a Fourier integral. Our analytical solution is in good agreement with a finite-element numerical computation.1 Here, we use the term "metamaterial" to mean any atomically thick, conducting material whose optical conductivity allows the generation of SPPs with sufficiently small wavelength; see (2) later. 2 The terms "conducting" and "resistive" sheet or half plane are used interchangeably. The "top Riemann sheet," which is invoked below for a particular branch of a multiple-valued function in the complex plane, should not be confused with the physical sheet causing wave diffraction. 3 The placement of the 1D Helmholtz operator, (d 2 /dx 2 ) + k 2 , outside the integral of (1) affords a kernel, K, that is (logarithmically) integrable through x = x ; cf. Pocklington's integral equation for wire antennas [9,10]. The problem can be stated in alternate forms, e.g., via direct application of the Fourier transform to a boundary value problem [11], which circumvents (1).

Generation of surface plasmon-polaritons by edge effects

“…Our numerics for finite sheets show how the diffracted electric field transcends from a singular behavior near each edge to the SPP away from the edge. We validate our numerical approach by use of a semiinfinite sheet: in this case, our numerical results are in excellent agreement with the analytical prediction [29]. This study places on a firm foundation our finite element approach for SPPs generated and sustained by defects and finite-size effects.…”

“…We should add that the SPP wavenumber, k m , obeys the dispersion relation k ⊥ := k 2 − k 2 m = −2k 2 /(ωµσ Σ ), which furnishes the wavenumber, k ⊥ , of propagation transverse to the sheet; thus, k m /k = 1 − 4k 2 /(ωµ 0 σ Σ ) 2 , and |k m /k| 1 in the nonretarded regime (Definition 2.1) [29]. By imposing Im k ⊥ > 0 according to the radiation condition at infinity vertically to the sheet, we obtain Im σ Σ > 0 which expresses the metamaterial character of the sheet.…”

“…Hence, as x 0, the solution, u, of integral equation (6) cannot be separated into the distinct (physical) contributions that are evident in the field far from the edge. Instead, we analytically compute the asymptotic behavior [29] u…”

Edge plasmon-polaritons on isotropic semi-infinite conducting sheets

“…5,9 The roots ξ for each case are present in the top Riemann sheet under suitable conditions on the phase of σ (see Section VI B). 8,9,54,55 By an elementary calculation, eigenvectors of Λ(ξ; q) are given by which depend on the material parameters through q if the latter satisfies a dispersion relation.…”

“…Because P TM (ξ) and P TE (ξ) are even functions of ξ, we can assert that ν = 0 (18) which implies that splitting (17) makes sense and can be carried out directly via the Cauchy integral formula. 26,55 In contrast, for certain strictly anisotropic conducting sheets, the index for the underlying Wiener-Hopf integral equations in the quasi-electrostatic approach may be nonzero, which implies distinct possibilities regarding the existence, or lack thereof, of the EP. 23,25 This material anisotropy lies beyond the scope of the present paper.…”