# On the Wiener-Hopf method for surface plasmons: Diffraction from semi-infinite metamaterial sheet

## TL;DR

This paper applies the Wiener-Hopf method to analytically solve for surface plasmon-polaritons on a semi-infinite conducting sheet, providing explicit formulas and validating with numerical results.

## Contribution

It introduces an explicit Wiener-Hopf analytical solution for surface plasmon excitation on a semi-infinite sheet, advancing understanding of wave diffraction in graphene-like materials.

## Key findings

- Analytical solution matches finite-element simulations.
- Explicit formulas for surface plasmon-polaritons are derived.
- Method applicable to wave diffraction problems in 2D.

## 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 semi-infinite, 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.