Electromagnetic Wave Transmission Through a Subwavelength Nano-hole in a Two-dimensional Plasmonic Layer

Electromagnetic Wave Transmission Through a Subwavelength Nano-hole in a Two-dimensional Plasmonic Layer

Norman J. M. Horing Corresponding author: nhoring@stevens.edu    Désiré Miessein Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA    Godfrey Gumbs Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10065, USA
July 19, 2019
Abstract

An integral equation is formulated to describe electromagnetic wave transmission through a subwavelength nano-hole in a thin plasmonic sheet in terms of the dyadic Green’s function for the associated Helmholtz problem. Taking the subwavelength radius of the nano-hole to be the smallest length of the system, we have obtained an exact solution of the integral equation for the dyadic Green’s function analytically and in closed form. This dyadic Green’s function is then employed in the numerical analysis of electromagnetic wave transmission through the nano-hole for normal incidence of the incoming wave train. The electromagnetic transmission involves two distinct contributions, one emanating from the nano-hole and the other is directly transmitted through the thin plasmonic layer itself (which would not occur in the case of a perfect metal screen). The transmitted radiation exhibits interference fringes in the vicinity of the nano-hole, and they tend to flatten as a function of increasing lateral separation from the hole, reaching the uniform value of transmission through the sheet alone at large separations.

1 Introduction

The description of electromagnetic wave transmission through an aperture in a screen involves the solution of Maxwell’s equations with complicated boundary conditions that are often not really known(or effectively implemented) for the materials involved. The diffraction of light by such subwavelength apertures (radius ) has attracted a great deal of interest by many researchers: In an elegant analysis 70 years ago, Bethe [1] showed that the classic Kirchoff method is probably approximately valid only for large apertures, but not for small ones, and Bethe developed a theoretical approach appropriate to the subwavelength regime. Important further developments involving variational principles and dyadic Green’s functions were presented by Levine and Schwinger [2], [3]. However, all of the works we have cited above assume the screen to be a perfect metallic conductor, whereas intense current interest is focussed on plasmonic semiconductors. To address such systems and incorporate the issue of boundary conditions we employ an integral equation formulation [4],[5],[6] describing electromagnetic wave transmission through a nano-hole in a thin plasmonic layer in terms of the dyadic Green’s function for the associated vector Helmholtz problem [7],[8],[9],[10]. Interesting electromagnetic phenomena, including enhanced transmission through a nano-hole in an opaque screen, have been discussed in the literature mostly for a metallic thick screen [11],[12],[13],[14]. The aim of the present work is to study electromagnetic wave field transmission and reflection at a thin semiconductor screen including the role of the embedded 2D plasmonic layer, and transmission directly through it as well as through the aperture, thereby improving our understanding of subwavelength electromagnetic phenomenology.

This paper is structured as follows: in Section , we discuss the dyadic Green’s function solution for a thin (2D) semiconductor plasma layer located at the plane in a 3D bulk host medium with background dielectric constant . Section deals with the description of a nano-hole in the dyadic Green’s function-integral equation for the 2D plasmonic layer following the techniques of references [4] and [5]. In Section , we address the electromagnetic wave transmission of a perforated thin screen. These results are then used in Section to obtain the transmitted/reflected electromagnetic field in terms of an infinite incident plane wave train. Calculated results for normal incidence are exhibited in Section for the near-field (), for intermediated-field () and for far-field () zones of diffraction.

2 Dyadic Electromagnetic Green’s Function for a Thin Semiconductor Layer

The description of electrodynamics in terms of dyadic Green’s functions has a long history in physics and in electrical engineering [7], [8], [9], [10]. The present interest in the electrodynamics of low-dimensional semiconductor nanostructures, which involve currents that are geometrically confined in narrow regions, offers a fertile ground for the application of dyadic Green’s functions. To deal with such problems we introduce the description of electromagnetic response of various semiconductor nanostructures in terms of a dyadic Green’s function propagator [2], [8], [9]. To start, we focus on a two dimensional plasmonic layer located on the plane , embedded in a three dimensional bulk host medium with background dielectric constant (Fig.1), and will examine the electromagnetic wave transmission through it using the dyadic Green’s function method.

Figure 1: Two dimensional plasmonic layer of thickness (small compared to the wavelength ) embedded at in a three dimensional bulk host medium with dielectric constant .

The Helmholtz equation for the electric field, based on the Maxwell equations, is given in frequency representation as

(1)

where is a unit dyadic tensor and is the speed of light in vacuum. This field is driven by the total electric current density at space-point at time , which includes the induced current density, (taken to be linear) composed of a term arising from the conductivity of the local bulk host medium,

(2)

( is the 3D Dirac delta function) and a term arising from the conductivity of the thin plasmonic layer under consideration. Including the role of the external driving current density , the electromagnetic Helmholtz equation can then be written as

(3)

In general, in electromagnetic theory for linear materials, the dyadic Green’s function is defined as the electric field at the field point generated by a radiating unit dyadic impulsive source located at the point , so that, the dyadic Green’s function relates to the current density as [3], [4], [7]

(4)

To fulfill the role of mandated by Eq.(4) with Eq.(2), we define the dyadic Green’s function of the homogeneous layer, as

(5)

Following the position-space inversion of the differential operator in the brackets on the Left Hand Side, carried out in the analysis of Refs. [4], [5] and [6], Eq.(2) may be written in the form of a full integral equation as

Because of spatial translational invariance in the plane of the plasmonic layer, the dyadic Green’s function may be written in terms of a Fourier transform in the parallel plane
, so that Eq.(2) takes the form

where . In this matter, the response of the uniform two dimensional plasmonic layer is described by the local conductivity tensor as

(8)

where

(9)

and represents the dielectric function of the 2D plasmonic layer of thickness .

With this, the solution of Eq.(2) for was determined in Refs. [4],[5] and [6] as

(10)

with

(11)

and

in mixed () Fourier representation with . Furthermore,

(13)

and

(14)

where . The sign before the radical has been chosen in such a way that the field in the host medium satisfies the radiation condition for when the radical is purely real, whereas it represents an evanescent field for when the radical is purely imaginary.

The closed form for of Eq.(10) requires evaluation of
at and

where the Dirac delta function evaluated at the origin (), is approximated by the inverse of the plasmonic layer thickness, , and . This approximation of is undertaken in the context of the integral equation for , Eq.(2) in which is involved in the kernel. In this regard, the zero width of the -integration that is mandated by Eq.(8), is recognized as artificial, and is more realistically smeared over a small (but finite) range .

The determination of requires the 3D matrix inversion of
and, specifically, is found to be:

Denoting the dyad to be inverted as

Noting that is block diagonal, its 22 block is readily inverted and its -element inverts algebraically. The matrix elements of are written out explicitly in Appendix A.

Forming the product

(18)

using the matrix elements provided in Appendix A, we employ a coordinate system in which , with the result(see Appendix C):

where

(20)
(21)
(23)
(24)
(25)
(26)
(27)
(28)

and ( note that defined here is not to be confused with in Appendix C)

(29)
(30)
(31)

with . It is interesting to note that Eq.(2) can be rewritten as a sum of diagonal and anti-diagonal dyads as follows

(32)

The dispersion relations for electromagnetic wave modes for the full non-perforated plasmonic sheet may be obtained by examining the vanishing determinant of the corresponding dyadic Green’s function . These normal mode frequencies defined by the vanishing denominators are given by

(34)
(36)

3 Nano-hole in the Dyadic Green’s Function Integral Equation of a 2D Plasmonic Layer

Figure 2: Two dimensional plasmonic layer (thickness , embedded at in a three dimensional bulk meduim) with a nano-hole of area at the origin of the plane.

We consider a two dimensional plasmonic layer perforated by a nano-scale aperture of area in the plane, representing the nano-hole by subtracting from the part of the full sheet conductivity associated with the hole (Fig.2),

(37)

Accordingly, the dyadic Green’s function for the perforated screen satisfies the integral equation in position-frequency representation given by:

(38)

where is the dyadic Green’s function for the 2D plasmonic layer in the absence of the nano-hole and is the kernel of the new integral equation. The excised part the conductivity defining the hole is described by the localized conductivity tensor

where Eq.(9)

(40)

and is the Heaviside unit step function confining the integration range on the 2D sheet to the nano-hole dimensions. Noting that

(41)

is a representation of the Dirac delta function, we write

(42)

where is the area of the nano-hole. Employing this in Eq.(38) to execute all the integrals, we find

where

(44)

Setting and in Eq.(3), we obtain as

(45)

so that

In particular, proves to be of special interest, and is given by

(47)

It is important to note that involves a divergent integral when all its positional arguments vanish. In terms of lateral wavevector representation

(48)

the divergence may be removed by introducing a cutoff of the -integration range, namely that for nano-holes of subwavelength dimensions. Considering the matrix elements of (Eq.(20)-Eq.(29)) with , and , is diagonal:

(49)

with matrix elements given by

(50)
(51)
(52)

and

(53)
(54)
(55)

where

(56)

The evaluation of the matrix elements of of Eq.(48) are evaluated with the - cutoff as follows [15]: