Dyadic Helmholtz Green’s Function for Electromagnetic Wave Transmission/Diffraction through a Subwavelength Nano-Hole in a 2D Quantum Plasmonic Layer: An Exact Solution Using “Contact Potential”-like Dirac Delta Functions

Abstract

The dyadic Helmholtz Green’s function for electromagnetic (EM) wave transmission/ diffraction through a subwavelength nano-hole in a two-dimensional (2D) plasmonic layer is discussed here analytically and numerically, employing “contact potential”-like Dirac delta functions in 1 and 2 dimensions (δ(z) and δ(x)δ(y)≡δ(2)(r→)). This analysis is carried out employing a succession of two coupled integral equations. The first integral equation determines the dyadic electromagnetic Green’s function G^fs for the full non-perforated 2D quantum plasma layer in terms of the bulk 3D infinite-space dyadic electromagnetic Green’s function G^3D, with δ(z) representing the confinement of finite quantum plasma conductivity to the plane of the plasma layer at z=0. The second integral equation determines the dyadic electromagnetic “hole” Green’s function G^hole for the perforated 2D quantum plasma layer (containing the nano-hole) in terms of the dyadic electromagnetic Green’s function G^fs for the full non-perforated 2D plasma layer, with δ(2)(r→) describing the exclusion of the quantum plasma layer conductivity properties from the nano-hole region in the vicinity of r→=0 on the plane. Taking the radius of the subwavelength nano-hole to be the smallest length scale of the system in conjunction with the 2D Dirac delta function representation of the excluded nano-hole plasma conductivity, both of the successive coupled integral equations are solved exactly, and we present a thorough numerical analysis (based on the exact analytic solution) for the resulting dyadic “hole” Green’s function G^hole in full detail in both 3D and density plots. This result has been successfully applied to the determination of electromagnetic wave transmission/diffraction through the nano-hole of the perforated quantum plasmonic layer, jointly with the EM wave transmission through the rest of the plasma layer. This success necessarily involves spatial translational asymmetry induced by the use of spatial Dirac delta functions confining finite conductivity to the 2D quantum plasma sheet and the excision at a bit of it about the origin to represent the nano-hole perforation, thus breaking spatial translational invariance symmetry.