## Abstract

An array of flared rectangular holes pierced through a conducting screen is treated herein by a rigorous full-wave modal analysis using the moment method entailing Green’s functions for rectangular cavities and planar multilayer structures in the spectral domain as well as classical Floquet theorem and the mode-matching technique. In this way, flared holes with arbitrary taper profile that may each even be composed of different dielectric sections and which perforated metal films that may be sandwiched between multiple layers of dielectric slabs on both sides is herein treated. The eclectic permutations of geometrical, structural, and material attributes thus afforded by this generic topology facilitate correspondingly diverse investigations that may prove pivotal to the success of future explorations in search for new breakthrough discoveries and innovations in the subject of extraordinary transmission through subwavelength hole arrays, to which the herein-analyzed configuration is central. Oblique angles of incidence for both principal polarizations and metal losses incurred by imperfect conducting screens are also investigated in this work, all constituting crucial aspects that may often be neglected.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Reports of measurable surges in transmission through metal gratings [1–3], meshes [3,4], and circular hole arrays [5,6] date back to the 1960s. These resonances in wave penetration were however not deemed to be anything unusual back then. Early studies of this phenomenon were instead focused primarily on their abilities to serve as filters, having thus facilitated developments of frequency-selective surfaces and artificial dielectrics [7]. It was not until the experimental work of Ebbesen et al [8] in 1998 when such excessive levels of wave transmission through conducting screens perforated by arrays of subwavelength holes were recognized as extraordinary; in the sense that the diffusion can be orders of magnitude larger than expected from classical aperture theory [9]. Ever since then, intensive research had been carried out on such arrays of holes throughout the electromagnetic spectrum in terms of optical properties and transmission characteristics.

Over the years, numerous works on various topologies of grilles and gratings for the study of that unusual phenomenon had been reported. An interpretation of extraordinary transmission (ET) through metallic films pierced by arrays of subwavelength holes was offered by [10] via equivalent circuit models and impedance matching. Investigations of infinitely long slits then ensued in [11], which took conductor attenuation into consideration. Dielectric sheets covering the input and output sides of the pierced film were treated in [12] by the lumped-element method along with the study of surface-wave modes in grounded slabs, in the context of which ET was explored. The significant role played by the TE_{20} modes of the virtual waveguides has been highlighted. Effects of the permittivities and thicknesses of the dielectric slabs on the attributes of the transmission spectra were also discussed, such as, locations of peaks, bandwidths about them, how many of them, and so on. With a physical illustration of the concept of standing waves within a unit cell, resonance phenomena were explained as being due to the coupling of the impingent plane wave to grounded slab surface-wave modes via Floquet space harmonics of the periodic structure. Prospects of using stacked hole arrays to synthesize left-handed metamaterials or just ordinary right-handed ones were raised in [13], ways that are synonymous with the usual approach of modulating the effective medium parameters by tailoring the lattice properties of each layer of perforated screen. An extensive work was done in [14], one whose analysis bears full-wave rigor and which treats oblique incidences, provides field solutions, considers conductor losses, and investigates stacked screens. Demonstrated theoretically and experimentally, the influence exerted by the number of holes on the transmission through perforated screens was highlighted in [15]. A subwavelength hole array stashed between two dielectric slabs was examined in [16]. The attributes of its modal surface-wave dispersion diagram have been linked with the peaks in its transmission spectra. It was described there how resonances in transmission are attributed to the coupling of the incident wave with guided slab modes. This work was then extended to transient waves in [17].

There had also been works carried out on arrays of slits with tapered profiles [18–23]. However, not only are those papers concerned with gratings having just one dimensional periodicities, none of them has offered any theoretical, analytical, or numerical treatment of the structure. The same goes for those of [24–26], which albeit considered holes with two dimensional periodicities, did not provide any form of solution approach for the perforated screens that they reported on. This is not to further mention that circular holes had been considered in most of these prior studies.

Hence, this paper offers a theoretical solution approach for treating perforated screens with tapered or flared rectangular holes that begin at the input windows each with a certain size but get larger as they tunnel through the metal sheet to the exit aperture. The case of flared square holes is depicted in Fig. 1. Among the purposes and benefits of treating such 2D arrays of rectangular flared holes include the fact that they constitute the general case of the 1D array of tapered slits already established to be relevant in the field of optics, thus serving as its reference case, in addition to the polarization purity they offer as compared to circular holes. Moreover, no prior study of this structure had ever been conducted in the same modal fashion and neither had there been any treatment that is presented up to the same level of detail and rigor as herein.

A rigorous full-wave modal analytical method for solving the array of rectangular holes flared to any profile and pierced into a conducting sheet of finite thickness shall be presented, which entails the moment method using Green’s functions for rectangular cavities and planar stratified media in the spectral domain, as well as classical Floquet theorem and the mode-matching technique. The details of this technique, which bears similarities to the rigorous coupled-wave analysis (RCWA) method, are given in the upcoming Sections 2 through 10. Results of transmission spectra computed by this method shall be validated in Sections 11 and 12 with those simulated by a commercial software solver for various hole taper-profiles as well as different angles of plane-wave incidence (including obliquely impinging ones too) for both principal polarizations. Metal losses incurred by imperfect conducting screens shall also be investigated in Section 13. Convergence studies and comparisons between the proposed method and the simulation software are presented in Section 14. The paper is then drawn to a close with a final section that summarizes the key aspects of this work.

## 2. Main geometry of the problem structure

The two-dimensional array of flared-open rectangular holes with periods *d _{x}* and

*d*along

_{y}*x*and

*y*that is pierced into an infinite PEC (perfect electric conducting) planar sheet of finite thickness

*d*is schematized in Fig. 2. The width and height along

*x*and

*y*of each flared-open hole are respectively

*a*

_{0}and

*h*

_{0}on the input side at

*z*=

*b*= ${z}_{0}^{}$ and

*a*(≥

_{Z}*a*

_{0}) and

*h*(≥

_{Z}*h*

_{0}) on the output side at

*z*=

*b*+

*d*= ${z}_{Z}^{}$. Consider an incident plane wave that is impingent on this perforated screen in which a reference coordinate system is also shown, whose origin is at a perpendicular distance of

*b*from the screen. The direction of arrival relative to the structure is characterized by the angles denoted as

*θ*and

_{inc}*ϕ*measured respectively from the

_{inc}*z*and

*x*axes, the latter azimuth angle defining the plane of incidence as portrayed by the tilted sheet in the drawing.

## 3. Flared-open hole

Each flared-open hole is composed of an interconnected series of rectangular waveguide cavity sections, as schematized in Fig. 3. The length of the *ι*^{th} section is ${\mathcal{l}}_{\iota}={z}_{\iota}^{}-{z}_{\iota -1}^{}$ and the boundary at ${z}_{\iota}^{}$ is deemed as the outer (right) end bounding face of the *ι*^{th} section, separating it from the one [(*ι* + 1)^{th}] further outwards (to its right). The indexing of the sections goes as: *ι* = 1, 2, ..., *Z*−1, *Z*. Any *ι*^{th} section is deemed to be generally filled homogeneously with a material of parameters: (*ε _{ι}*,

*μ*), and its cross-sectional dimensions are

_{ι}*a*and

_{ι}*h*, being the width and height along

_{ι}*x*and

*y*, respectively. If the flared hole is homogeneously filled with a material with parameters, say (

*μ*,

_{univ}*ε*), as would often and likely be the case, then

_{univ}*μ*=

_{ι}*μ*and

_{univ}*ε*=

_{ι}*ε*for all

_{univ}*ι*= 1, 2, ...,

*Z*−1,

*Z*. Notice the terminal imaginary waveguides, indexed as 0 and

*Z*

_{Θ}to respectively represent the leftmost (innermost) and rightmost (outermost) sections, the purpose of which being the permissible treatment of iris-type input and output holes whose sizes are generally smaller than their associated terminal cavity-hole sections.

## 4. Rectangular waveguide modal field functions

The transverse waveguide modal field functions entailed throughout this paper must be in their orthonormalized forms, as required by the dyadic cavity Green's functions. For a coordinate-origin centered rectangular waveguide indexed by *ι* with width and height along *x* and *y* being *a _{ι}* and

*h*, respectively, and homogeneously filled with a medium of parameters $({\mu}_{\iota}^{},{\epsilon}_{\iota}^{})$, the following are stated.

_{ι}*p*

^{th}mode in waveguide cavity section

*A*and the

*u*

^{th}mode in waveguide cavity section

*B*, which is larger than section

*A*, as follows:

*p*,

*u*)

^{th}element is this ${C}_{{A}_{p}{B}_{u}}$, for

*p*= 1, 2, ...,

*N*and

_{A}*u*= 1, 2, ...,

*N*.

_{B}## 5. PEC-equivalent magnetic aperture currents

With reference to Fig. 2 and considering any *ι*^{th} cavity-hole section bounded by ${z}_{\iota -1}^{}$ and ${z}_{\iota}^{}$, the component of the electric field parallel with the transverse *xy* plane is expressed as:

*e*

_{t}^{(}

^{ι}^{)}represents the transverse modal electric field vector function of the rectangular waveguide with the same

*a*×

_{ι}*h*cross section along

_{ι}*x*and

*y*as the cavity apertures, and where

*pqr*denotes the modal triple-index,

*p*and

*q*being integers as of

*pπ*/

*a*and

_{ι}*qπ*/

*h*, while

_{ι}*r*signifies TE or TM. The total number of waveguide modes considered for representing the fields in the

*ι*

^{th}cavity-hole section is denoted as

*N*. Evidently, ${A}_{pqr}^{(\iota )}$ symbolizes the modal amplitude coefficients, being thus far unknown and to be solved for. Consequently, the PEC-equivalent magnetic current densities over the two aperture-faces bounding the

_{ι}*ι*

^{th}section are given by:

*k*,

_{x}*k*) domain are written as

_{y}*w*may be

*x*or

*y*.

## 6. Definition of pertinent notational operators

To aid the upcoming formulation of matrix algebra, a couple of pertinent notational vector and matrix operators shall now be defined. Let $\underset{\_}{D}$ = diag($\underset{\_}{V}$) be the operational notation for the placement of the vector $\underset{\_}{V}$ with length *N* along the diagonal of a square diagonal *N* × *N* matrix, $\underset{\_}{D}$. If each *k*^{th} element of $\underset{\_}{V}$, ${\underset{\_}{V}}_{k}$, is a matrix of size *P _{k}* by

*Q*, then $\underset{\_}{D}$ = diag($\underset{\_}{V}$) places ${\underset{\_}{V}}_{k}$ along the diagonal of $\underset{\_}{D}$ in a staggered fashion, i.e. $\text{diag}\left(\underset{\_}{V}\right)=\text{diag}\left(\left[\begin{array}{cccc}{\left[{\underset{\_}{V}}_{\text{1}}\right]}_{{P}_{1}\times {Q}_{1}}& {\left[{\underset{\_}{V}}_{\text{2}}\right]}_{{P}_{2}\times {Q}_{2}}& \cdots & {\left[{\underset{\_}{V}}_{N}\right]}_{{P}_{N}\times {Q}_{N}}\end{array}\right]\right)=\left[\begin{array}{cccc}{\left[{\underset{\_}{V}}_{\text{1}}\right]}_{{P}_{1}\times {Q}_{1}}& {\left[0\right]}_{{P}_{1}\times {Q}_{2}}& \cdots & {\left[0\right]}_{{P}_{1}\times {Q}_{N}}\\ {\left[0\right]}_{{P}_{2}\times {Q}_{1}}& {\left[{\underset{\_}{V}}_{\text{2}}\right]}_{{P}_{2}\times {Q}_{2}}& \cdots & {\left[0\right]}_{{P}_{2}\times {Q}_{N}}\\ \vdots & \vdots & \ddots & \vdots \\ {\left[0\right]}_{{P}_{N}\times {Q}_{1}}& {\left[0\right]}_{{P}_{N}\times {Q}_{2}}& \cdots & {\left[{\underset{\_}{V}}_{N}\right]}_{{P}_{N}\times {Q}_{N}}\end{array}\right]$

_{k}In addition, let the row vector $\underset{\_}{V}=\left[\begin{array}{cccc}{\text{V}}_{n}& {\text{V}}_{n+1}& \cdots & {\text{V}}_{N}\end{array}\right]$ be expressed concisely by: $\left[{\text{V}}_{n\to N}\right]$ in which the range indicated by the arrow spans, in increasing order, over integer values only. If each V* _{n}* element is itself a vector or matrix, then $\underset{\_}{V}$ is a juxtaposed string of those constituting vectors or matrices. Although shown as a subscript here, the

*n*→

*N*may also appear as a superscript or any other fashion.

Lastly, the *δ _{mn}* showing up throughout this paper represents the Kronecker delta, being unity when both

*m*and

*n*are equal, but vanishing otherwise.

## 7. Mode-matching of interconnected series of waveguide cavities composing flared hole

This section shall be concerned with the matrix operations associated with the mode-matching between the various interconnected series of waveguide cavities that together compose the flared hole.

Due to the presence of hyperbolic trigonometric functions in the waveguide cavity Green's functions that describe the back and forth propagation of waves via the sum or difference of two exponential phase functions with opposite signs of their generally complex exponents, the following quantities are first defined:

The continuity of the tangential electric field component across the boundary at ${z}_{\iota}^{}$ separating any two cavities [the *ι*^{th} and (*ι* + 1)^{th}] is already satisfied by the sign reversal of the PEC equivalent magnetic current density, ${\overrightarrow{M}}_{eq}^{({z}_{\iota}^{})}$ at that interface as it radiates into those two cavities on either side of it; see (12). Hence, what's left to be enforced is the continuity of the tangential magnetic field component across the numerous boundaries that separate various pairs of cavity sections. Since the fields within any *ι*^{th} cavity are radiated by ${\overrightarrow{M}}_{eq}^{({z}_{\iota}^{})}$ and ${\overrightarrow{M}}_{eq}^{({z}_{\iota -1}^{})}$, being the currents that flow over the two apertures (at ${z}_{\iota}^{}$ and ${z}_{\iota -1}^{}$) which bound that *ι*^{th} cavity, the boundary condition requiring the tangential *H*-field continuity across that same aperture at ${z}_{\iota}^{}$ will be in terms of the modal amplitude coefficients that expand ${\overrightarrow{M}}_{eq}^{({z}_{\iota -1}^{})}$, ${\overrightarrow{M}}_{eq}^{({z}_{\iota}^{})}$ and ${\overrightarrow{M}}_{eq}^{({z}_{\iota +1}^{})}$, the currents over three interfaces centered about that at ${z}_{\iota}^{}$. Consequently, three sub-matrices may be identified for the boundary condition imposed on each interface, the details of which are presented as follow.

Subsequently, the following matrices are established.

*ι*= 1, 2, ...,

*Y*, and

*ι*= 1, 2, ...,

*X*, whereas for

*ι*=

*Y*, we have instead:

As a result of mode-matching across the various interface boundaries, which progressively involves triplet sets of amplitude coefficients that expand magnetic currents further down the sequence of cavities (as explained above), the following diagonally-banded matrix is then produced, with each row of three sub-matrices being those just defined above:

*N*

_{0}+

*N*

_{Z}_{Θ}equations, which are acquired from the boundary condition across the two terminal apertures at ${z}_{0}^{}$ and ${z}_{Z}^{}$, as shall be done next.

## 8. Interfaces between terminal cavity-hole sections and exterior half-spaces

#### 8.1 PEC-equivalent magnetic current densities over terminal apertures interfacing with exterior half-spaces

The PEC-equivalent magnetic current densities over the terminal cavity apertures are given by:

*Z*.

#### 8.2 Fields radiated by PEC equivalent cavity-aperture magnetic currents into various regions

For the left semi-infinite (incident) region within which the plane wave impingent on the hole array emerges, where a single homogeneous medium is concerned, the pertinent multilayer configuration for the core-routine of the numerical spectral-domain multilayer Green's function (details of which are found in [27]) is simply a two-layer scenario with interface at *z* = *b* = ${z}_{0}^{}$ comprising a PEC semi-infinite upper half-space layer that extends up to *z* = + ∞ and a lower likewise semi-infinite half-space layer made of the material constituting the left semi-infinite (incident) region that stretches down to *z* = –∞. But in anticipation of possible subsequent analysis of multiple dielectric layers incorporated over the input side of the hole array for potential enhancement of transmission, a five-layer configuration shall instead be considered, as shown in Fig. 4(a). The central third layer is labeled as the *i*^{th} layer, with the other layer indices with respect to this *i* = 3, e.g. the first lowermost layer without a lower bound is the (*i* – 2)^{th} layer while the topmost fifth one (PEC) that is unbounded above is tagged as the (*i* + 2)^{th} layer. The *z*-level of the upper boundary of the *i*^{th} layer is indicated as *z _{i}*.

The scattered fields within this left-side incidence region are caused by the spectral current sources at the upper boundary *z* = *z _{i}*

_{+1}=

*z*

_{4}= +

*b*= ${z}_{0}^{}$ of Fig. 4(a), being ${\tilde{\tilde{M}}}_{w}^{{z}_{s}={z}_{0}^{}=b}$ as relayed by the preceding subsection, in which again

*w*may be

*x*or

*y*.

The following (*pqr*)^{th} modal basis functions for expanding the magnetic currents over the input cavity aperture at *z* = *b* = ${z}_{0}^{}$ with $-\widehat{z}$as the outward unit normal and their Fourier-transformed spectral forms are first defined:

*x*and

*y*being

*d*and

_{x}*d*, these above Fourier-transformed modal basis current functions are evaluated at discrete values of

_{y}*k*and

_{x}*k*governed by

_{y}*m*and

*n*being integers, and

*θ*

_{0}and

*ϕ*

_{0}are measured from the

*z*and

*x*axes respectively and they define the direction of the (0, 0)th dominant Floquet modal plane wave, being the incident plane wave in the present context, whose wavenumbers along

*x*and

*y*are ${k}_{{x}_{m=0}}$and ${k}_{{y}_{n=0}}$, constituting the forcing wavenumbers along

*x*and

*y*respectively.

The spatial-domain fields in the $\mathcal{l}$^{th} layer ($\mathcal{l}$ = 1, 2, 3 or 4) of Fig. 4(a) are represented by

*m*and

*n*indices is within the summation over the Floquet modal harmonic index

*mn*in (28). The field symbol

*F*is

*E*or

*H*,

*w*is

*x*,

*y*or

*z*. The symbol $\stackrel{\approx}{G}$ represents the spectral multilayer Green's function that operates on the spectral modal basis current functions of (24) which now have superscripted indications of their

*z*-locations (Fig. 4(a)), thereby giving the associated spectral field component

*F*(as subscripted in $\tilde{\tilde{G}}$) radiated by them. Notice that a '

_{w}*left*' superscript has been added to signify that the five-layer configuration of Fig. 4(a) for the left semi-infinite incidence region is used in the core routine, with the topmost layer made of PEC. Also note the distinction between indices $\mathcal{l}$ and

*i*, respectively for the index of the layer (in Fig. 4(a)) in which the field

*F*is observed and that of the one within which the (

*mn*)

^{th}spectral (

*pqr*)

^{th}basis magnetic current source ${\tilde{\tilde{m}}}_{{w}_{pqr,mn}}^{}$is located.

As was for the left-side (incidence) region, the pertinent multilayer configuration of the spectral multilayer core routine for a homogeneous right-side (transmission) region is also simply a two-layer scenario, this time with interface at *z* = *b* + *d* = ${z}_{Z}^{}$ comprising a lower semi-infinite half-space PEC layer that reaches down to *z* = –∞ (i.e. spans from *z* = –∞ to *z* = *b* + *d*) and an upper likewise semi-infinite half-space layer made of the material constituting the right semi-infinite (transmission) side region of the hole array that extends up to *z* = + ∞ (i.e. exists from *z* = *b* + *d* to *z* = + ∞). But retaining the same liberty of stratifying the right-side region with four different media as well, the five-layer configuration of Fig. 4(b) is applied to the core-routine of the spectral multilayer Green's function.

The PEC-equivalent magnetic cavity-aperture currents responsible for driving the core routine are now located on the lower boundary at *z* = *z _{i}*

_{–2}=

*z*

_{1}=

*b*+

*d*= ${z}_{Z}^{}$ just within the (

*i*– 1)

^{th}= 2nd layer. The relevant modal basis functions for expanding these currents with $+\widehat{z}$ as the outward unit normal and their Fourier-transformed spectral forms are given by:

^{th}layer ($\mathcal{l}$ = 2, 3, 4 or 5) of Fig. 4(b) are given by

#### 8.3 Boundary conditions: continuity of tangential magnetic field component across cavity apertures

Through the sign cases of (12), (13), (22) and (23), depending on whether the PEC-equivalent cavity-aperture magnetic current is right or left facing, i.e. with the unit normal pointing towards the region into which fields are radiated being $+\widehat{z}$ or $-\widehat{z}$, the continuities of the tangential electric field components across the various cavity apertures are already implicitly satisfied. What remains is then to enforce the continuity of the tangential magnetic field component.

The pertinent boundary condition at *z* = *b* is stated as:

*x*,

*y*) observation coordinates [albeit not explicit in the first two terms within parentheses on the left-hand side of this equation] being over the left-side input cavity aperture at

*z*=

*b*with –

*a*/2 <

*x*<

*a*/2, –

*h*/2 <

*y*<

*h*/2. The symbol ${\Psi}_{\sigma}^{{z}_{\tau =i}^{ex1\xb0}}$ (Ψ is

*J*or

*M*and

*σ*may be

*x*,

*y*or

*z*) represents the primary (1°) excitation source that is assumed to be located at $z={z}_{\tau =i}^{ex1\xb0}$ within the

*τ*

^{th}=

*i*

^{th}= 3rd layer (middle part) of Fig. 4(a) for the left-side incidence region. The subscript

*z*=

*b*attached to the square brackets signify that the

*H*-fields are observed at that interface.

The pertinent boundary condition at *z* = *b* + *d* is stated as:

*x*,

*y*) observation coordinates being over the right-end output aperture of the cavity at

*z*=

*b*+

*d*, and noticing this time the absence of the primary excitation source term on the left-hand side unlike (32) for the input side. As before, the functional dependencies (arguments) on the (

*x*,

*y*) observation coordinates of the

*H*fields on the left-hand side are not explicitly shown.

_{w}## 9. Weighting: generate system of equations

Respectively taking $\underset{y=-{h}_{0}/2}{\overset{y={h}_{0}/2}{\int}}{\displaystyle \underset{x=-{a}_{0}/2}{\overset{x={a}_{0}/2}{\int}}{\overrightarrow{e}}_{{t}_{rst}}^{(0)}(x,y)\times \_\_\_\cdot \widehat{z}dxdy}$ and $\underset{y=-{h}_{0}/2}{\overset{y={h}_{{Z}_{\Theta}}/2}{\int}}{\displaystyle \underset{x=-{a}_{0}/2}{\overset{x={a}_{0}/2}{\int}}{\overrightarrow{e}}_{{t}_{rst}}^{({Z}_{\Theta})}(x,y)\times \_\_\_\cdot \widehat{z}dxdy}$ throughout (32) and (33), for *rst* = 1, 2, …, *N*_{0}, and *rst* = 1, 2, …, *N*_{ZΘ}, the following two equations are obtained.

*N*

_{0}+

*N*

_{Z}_{Θ}equations, constituting the shortfall of row equations in (21), thus enabling the solution for the total number of unknown modal amplitude coefficients.

## 10. Construction of matrix equation

With the completion of Sections 7 and 8 that dealt respectively with the mode-matching amongst cavity sections making up the flared hole and the boundary conditions across the input and output apertures that interface the generally-stratified exterior regions on both sides of the screen, accompanied by the generation of as many equations as there are unknowns via the weighting procedure in Section 9, the next step would be the bridging of the treated interior cavities with the exterior regions on both sides. Only by doing so may the ultimate matrix equation be constructed and subsequently solved for the unknown amplitude modal coefficients, which are placed in the following column-vector:

We begin with the matrix representation of (34). Defining *κ* = 1/(*d _{x}d_{y}*), using underscores to signify that their associated symbols are matrices containing those quantities, engaging parenthesized row and column index symbols to explicitly indicate the structural forms of the matrices, and letting

*w*be

*x*or

*y*, the following matrices are first constructed.

With these above established, the two sub-matrices that represent (34) for the bridging of the flared hole (modeled by cavities) to the exterior generally-multilayered regime of the input side are given as follow:

Proceeding to the representation of (35) that bridges the interior cavities to the exterior region of the output side (that is likewise generally stratified), the associated two sub-matrices are stated by the following:

The corresponding ultimate vector containing the unknown modal amplitude coefficients is stated as:

## 11. Validation with independent commercial solver

Based on the theoretical formulation and modal analysis of Sections 4 through 9, computer program codes which treat and solve the periodic array of generally-flared rectangular holes penetrating a metallic sheet were written. The transmission coefficient of a plane wave that is impingent on this perforated screen may then be computed from the numerical data that are generated. Such results shall in this section be validated with those obtained from CST Microwave Studio, a commercial full-wave solver.

#### 11.1 Unflared ordinary square holes without flanged irises

With reference to the geometries of Figs. 1, 2 and 3, the parameters of the hole-array in the ambience of free space that have been computed for validation are as follow: square unit cell with period *d _{x}* =

*d*=

_{y}*d*= 5 mm, likewise square and unflared (ordinary straight-type) hole without flanged diaphragms at the input and output openings (

_{x&y}*a*

_{0}=

*a*

_{1},

*h*

_{0}=

*h*

_{1},

*a*=

_{Z}*a*

_{Z}_{Θ},

*h*=

_{Z}*h*

_{Z}_{Θ}), having dimensions

*a*=

_{ι}*h*= 2.5 mm for all

_{ι}*ι*= 0, 1, 2, ...,

*Y*,

*Z*,

*Z*

_{Θ}, and the thickness of the perforated PEC screen is

*d*= 1.25 mm. Referring to Fig. 2 for the angular coordinates, for a TM

*polarized plane wave impingent on the hole array with angles of incidence*

^{z}*θ*= 0 (normal incidence) and

_{inc}*ϕ*= 90° (

_{inc}*yz*plane of incidence), consequently entailing the

*E*and

_{y}*H*transverse field components, the graph of the power transmission coefficient versus frequency for the dominant zeroth Floquet modal order acquired from the modal analysis of Sections 4 through 9 is presented in Fig. 5, together with the corresponding plot generated by CST simulations. Evidently, excellent agreement between the modal approach and the commercial software is achieved.

_{x}#### 11.2 Linearly flared square holes without flanged irises

For the same screen thickness (*d* = 1.25 mm) and the same sizes of the unit cell (*d _{xy}* = 5 mm) and the entrance aperture opening at

*z*=

*b*=

*z*

_{0}on the input side (

*a*

_{0}=

*h*

_{0}=

*a*

_{1}=

*h*

_{1}= 2.5mm) as were of the previous subsection, a linearly flared-open hole is now considered, with an exit hole size of

*a*=

_{Z}*h*=

_{Z}*a*

_{Z}_{Θ}=

*h*

_{Z}_{Θ}= 4.5mm on the output side at

*z*=

*b*+

*d*=

*z*

_{Z}, modeled as a connected stepped series of five cavity-hole sections, each with the same thickness of

*d*/5. For the same normally-incident (

*θ*= 0) TM

_{inc}*polarized plane wave with*

^{z}*ϕ*= 90°, the zeroth-order power transmission coefficient for this flared-hole array as computed by the present modal analysis over a range of frequencies is compared with that simulated by CST in Fig. 6(a). Again, it is seen that fine agreement is attained.

_{inc}#### 11.3. Linearly flared square holes with flanged irises

Entrance and exit flanged-diaphragms at *z*_{0} and *z _{Z}* which are smaller than their associated input and output cavity-hole sections (1

^{st}and

*Z*

^{th}ones; see Fig. 3) are now considered. Two cases of iris sizes shall be demonstrated: one with

*a*

_{0}=

*a*

_{1}/2 &

*h*

_{0}=

*h*

_{1}/2, and another with

*a*

_{Z}_{Θ}= 3

*a*/4 &

_{Z}*h*

_{Z}_{Θ}= 3

*h*/4.

_{Z}For the former case, the same portrayal of how the transmission coefficient of a normally-incident TM* ^{z}*-polarized plane-wave varies with frequency is given in Fig. 6(b), in which one of the two traces is produced by the computer program code based on the present formulation while the other is simulated by CST. The corresponding comparison for the second case is presented in Fig. 6(c). Once more, the modal technique is well corroborated by the simulation software.

## 12. Oblique incidences

This section investigates hole arrays illuminated by plane waves with directions of incidence that are not perpendicular to the surfaces of the screen, i.e. *θ _{inc}* ≠ 0. With that same lattice size of

*d*=

_{x}*d*= 5 mm and total screen thickness of 1.25 mm as before, the square hole considered is one that is segmented into just two sections (single-step flare), of which the cavity on the input side has an aperture size of 2.5

_{y}^{2}mm

^{2}along with a thickness of 0.5 mm while the remaining 0.75 mm is bored through by the output hole with an exit portal area of 4.5

^{2}mm

^{2}on the output side. Both TE

*and TM*

^{z}*polarized plane waves are now assumed to arrive along oblique angular*

^{z}*θ*directions of propagation contained within a fixed azimuth plane of incidence

_{inc}*ϕ*= 0 (see Fig. 2). The plots of Fig. 7, as computed by the present modal technique, portray the variations with frequency of the zeroth-order transmission, within each of which the corresponding results simulated by CST are superposed. Evidently, this latter independent software again validates the herein proposed analysis. For Fig. 7(a), the impingent plane wave is

_{inc}*TE*polarized with

^{z}*θ*= 15°, whereas it is

_{inc}*TM*polarized with

^{z}*θ*= 15° in Fig. 7(b), and finally for Fig. 7(c), it is

_{inc}*TE*polarized incidence with

^{z}*θ*= 30°.

_{inc}## 13. Conductor losses and sub-efficiencies

With flared-open holes comes the risk of greater conductor losses due to larger exposed metallic surfaces, if imperfect metals with finite conductivities are considered. It is the intent of this section to look into this matter.

The transmission coefficient dealt with so far, being the ratio of the transmitted (*P _{trans}*) to the incident (

*P*) power per unit cell of the hole array, may also be perceived as the so-called transmission efficiency,

_{inc}*ε*=

_{trans}*P*/

_{trans}*P*. Consequently, by defining the conductivity efficiency as:

_{inc}*ε*= (

_{cond}*P*−

_{trans}*P*)/

_{cond}*P*where

_{trans}*P*is the power loss per unit cell due to finite metal conductivity, the total efficiency,

_{cond}*ε*is stated as the product of the latter two subefficiencies, i.e.

_{tot}*ε*=

_{tot}*ε*, indeed being (

_{trans}ε_{cond}*P*−

_{trans}*P*)/

_{cond}*P*as required. The conductivity power lost to the imperfect metal walls is given by:

_{inc}*μ*and

_{cond}*σ*are its permeability and conductivity, respectively, and where

*S*denotes all conducting surfaces inside that cell, comprising the interior walls of the flared hole as well as the exterior surfaces of the metallic flanges on both the input (incidence) and output (transmission) sides of the screen, with

_{metal}*H*being the tangential magnetic field component over those aforementioned metal surfaces made available by the full-wave modal analysis presented herein.

_{t}For normal incidence, the same unit cell size (*d _{x}* =

*d*= 5 mm) and total screen thickness (

_{y}*d*= 1.25 mm) as before, and with

*σ*= 5.8 × 10

^{7}S/m and

*μ*=

_{cond}*μ*

_{0}(copper assumed as the lossy metal), the variation of these aforesaid efficiencies with frequency is given in Fig. 8 for flared square holes each with an input size of 1.5

^{2}mm

^{2}that tapers linearly to an output one of 4.5

^{2}mm

^{2}. As can be seen, the conductivity efficiency is observed to commensurate with the transmission efficiency; as in, when

*ε*rises with frequency,

_{trans}*ε*climbs as well (albeit not in the same fashion), both of them peaking at the same frequency, and when the former encounters a sharp dip at a certain frequency, so does the latter. This is an important finding since extraordinary transmissions pertaining to summits in

_{cond}*ε*are thus in concert with the peaks in

_{trans}*ε*as well, being a favorable outcome. Another significant aspect learned is that, despite the flaring open of the holes resulting in increased metallic surfaces, the conductivity losses are not found to be severe. In fact, the metal attenuation are actually very low at most frequencies in the considered band. It is instead the transmission efficiency itself that is the limiting factor, or the 'weaker link' of the two.

_{cond}## 14. Further aspects of the modal analysis

More discussions about the numerical aspects of the presented modal analysis shall now be made, among which is the convergence of the number of modes used in the computation. As well, a comparison of the simulation time of the modal approach with that clocked by CST Microwave Studio shall also be carried out.

#### 14.1 Convergence of the number of modes

There are two types of modes: those of the cavity-hole sections, and Floquet harmonics associated with the periodicity of the structure. How the number of each kind, henceforth denoted as *N _{cav}* and

*N*respectively, affects the computed transmission spectra will be studied. The linearly-flared case of Fig. 6(a) under likewise normal incidence shall be assumed for this purpose.

_{Floq}Each subplot in Fig. 9 presents, for a certain *N _{Floq}*, the transmission spectra for six numbers of cavity modes, namely

*N*= 6, 16, 30, 48, 70 and 96 as annotated. The likewise six panels are for

_{cav}*N*= 9, 25, 49, 81, 121, and 169, given in Figs. 9(a) to 9(f) respectively. As can be seen, for any particular number of Floquet modes (any one subplot), while the uppermost two closely-separated traces pertaining to the two lowest

_{Floq}*N*are visibly renegade, all the others of

_{cav}*N*≥ 30 are bundled rather tightly together, meaning that good convergence is already achieved by 30 cavity modes, regardless of

_{cav}*N*.

_{Floq}Swapping the two types of modes, each panel in Fig. 10 presents, for a certain *N _{cav}*, the transmission spectra for six numbers of Floquet harmonics, namely

*N*= 9, 25, 49, 81, 121 and 169 as annotated. A similar observation as Fig. 9 can be made, as in, the topmost two or three curves belonging to the two or three lowest numbers of

_{Floq}*N*are distinctly dislodged from the rest that are tightly bundled, meaning that so long

_{Floq}*N*≥ 81, good convergence is assured to have been attained, regardless of what

_{Floq}*N*is.

_{cav}#### 14.2 Comparison with commercial solver

The computation time of the present modal technique and that of the transient solver in the CST solver, both run on an Intel Core i7-5820K CPU at 3.30-GHz clock speed and supported by 32 GB of RAM, were recorded for the same set of parameters as Fig. 6(a). For the CST software, a total simulation time of 1213 seconds to acquire numerical data at 1001 frequencies was noted, whereas the herein proposed method took just 721 seconds for the same number of frequency points. Therefore, the present approach is notably more computationally efficient.

As compared to the CST package or any other purely numerical brute-force mesh-and-solve simulation software in general, further advantages of the herein analytical method include its need for fewer unknowns and thus reduced taxation on computational resources, the liberty it enjoys with post-processing the data to any fancied spatial resolution upon a round of computation, as well as its exemption from heightened code complexity and programming effort associated with any increase in the required precision level of the solutions. Furthermore, the major setback of staircasing approximation suffered by the FDTD-based transient solver in the commercial simulator is not something the modal method has to bear.

## 15. Experimentation

As initiated in the Introduction, among the goals of this work is to provide an important extension of 1D arrays of slits with tapered profiles to 2D arrays of corresponding rectangular flared holes, of which the former is just a special version. In other words, the presently studied rectangular hole array is the general case of the 1D array of slits (gratings or corrugations) that has already been found in existing literature to be important and relevant to the world of optics. In fact, any manufactured realization of such topologies has to be an array of elongated rectangular holes as the slits cannot be infinitely long. Hence, the rectangular flared-hole array serves as an important reference case to slit-arrays with tapered profiles. This section thus seeks to demonstrate the congruence between the measured results of a manufactured 1D grating (with finite or truncated length of its slits) with those obtained from simulations by the code (based on the herein modal approach) for a 2D array of rectangular holes, whose gap-size and period along one dimension equal those of the 1D grating while the period of the other dimension takes on the same value as the truncated length of the fabricated slits, thereby being elongated along the orientation of the gratings.

Figure 11 displays the graph of measured transmission coefficient versus frequency of a manufactured 1D array of slits with period 20 mm, groove width 14 mm, and a truncated length of 500 mm. Plotted in this same figure is the corresponding theoretical curve computed by the present modal method, for a 2D array of elongated rectangular holes with unit cell size of *d _{x}* = 20 mm,

*d*= 500 mm, and hole dimensions of

_{y}*a*= 14 mm, and

*h*= 0.95

*d*. For both the experimental and computed cases, the screen thickness is 12 mm and the polarization of the impinging wave is

_{x}*TM*with

^{z}*θ*= 0 (normal incidence) and

_{inc}*ϕ*= 90°. As seen, decent agreement is achieved between theory and experiment despite the differences in their physical structures. Hence, the purpose of this measurement work along with its theoretical validation is two-pronged; to firstly exemplify how good a reference the 2D array is to the 1D slit array, and secondly to also inject this additional facet of experimental realization to the paper.

_{inc}## 16. Conclusions

A rigorous full-wave modal analysis of penetrable arrays of flared rectangular holes that get bigger as they pierce through a planar conducting screen of finite thickness has been presented. Based on the method of moments, Green’s functions for rectangular cavities and planar multilayer structures are used, those of the latter being in the spectral domain. Floquet theorem is also thus employed. The connected series of cavity-hole sections that model the flared holes in a staircase fashion is treated by the mode-matching technique.

With the numerical tool developed based on this modal formulation, computed results in the forms of transmission spectra (graphs of transmission versus frequency), modal surface wave dispersion diagrams, and field distributions can all be generated, although it is only the first type that has been presented in this paper. Results of transmission spectra computed by this proposed modal approach corroborate well with and are thus validated by those simulated by an independent commercial software solver: *CST Microwave Studio*. Yet, comparisons of computational durations between both solution tools show that the latter simulation software is notably slower and less efficient than the present method. Experimentations were also conducted on a manufactured array prototype, the measurement results acquired from which agree well with theoretical predictions of the corresponding results computed by the code that was developed based on the present modal formulation.

The analyzed and treated structure is of course central to the topic of extraordinary transmission (ET), which finds enormous range of applications. By virtue of the mode-matching between cavity sections that compose the flared hole, the tapering can take on any arbitrary profile and may even be inhomogeneously filled. Moreover, due to the usage of the spectral-domain Green’s functions for stratified media, multiple layers of dielectric slabs that cover both or just any side of the perforated screen can also be treated. These features afford flexibility and generality of the topology, an asset that is vital to further explorations in search for discoveries of new breakthroughs in the subject of ET. Furthermore, lossy metallic screens are also treated, an aspect that is especially important for flared holes with increased surface areas of metal but yet lacking in the literature. In addition, not only are normal incidences of plane-wave illumination considered in this paper but oblique ones as well, not to further mention that both principal polarizations are also considered, being another important element. Such a wholesome treatment of a hole-array configuration that offers this much generality for specializations to myriad sets of permutations of attributes is nowhere to be found. It is thus believed that this work can help to open doors to new horizons in the exciting and ever-expanding subject of extraordinary optical and microwave transmissions through arrays of subwavelength holes.

## Funding

Ministry of Science and Technology (MOST), Taiwan (107-3017-F-009-001,107-2221-E-009-051 -MY2.

## Acknowledgments

This work was partially supported by the “Center for mmWave Smart Radar Systems and Technologies” under the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan, and partially funded by the Ministry of Science and Technology (MOST) of Taiwan under Grant number MOST 107-3017-F-009-001.

## References

**1. **K. F. Renk and L. Genzel, “Interference filters and Fabry-Perot interferometers for the far infrared,” Appl. Opt. **1**(5), 643–648 (1962). [CrossRef]

**2. **R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. **7**(1), 37–55 (1967). [CrossRef]

**3. **G. M. Ressler and K. D. Möller, “Far infrared bandpass filters and measurements on a reciprocal grid,” Appl. Opt. **6**(5), 893–896 (1967). [CrossRef] [PubMed]

**4. **A. Mitsuishi, Y. Otsuka, S. Fujita, and H. Yoshinaga, “Metal mesh filters in the far infrared region,” Jpn. J. Appl. Phys. **2**(9), 574–577 (1963). [CrossRef]

**5. **C. C. Chen, “Diffraction of electromagnetic waves by a conducting screen perforated periodically with circular holes,” IEEE Trans. Microw. Theory Tech. **19**(5), 475–481 (1971). [CrossRef]

**6. **C. C. Chen, “Transmission through a conducting screen perforated periodically with apertures,” IEEE Trans. Microw. Theory Tech. **18**(9), 627–632 (1970). [CrossRef]

**7. **S. W. Lee, G. Zarrillo, and C. L. Law, “Simple formulas for transmission through periodic metal grids or plates,” IEEE Trans. Antenn. Propag. **30**(5), 904–909 (1982). [CrossRef]

**8. **T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature **391**(6668), 667–669 (1998). [CrossRef]

**9. **H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. **66**(7-8), 163–182 (1944). [CrossRef]

**10. **F. Medina, F. Mesa, and R. Marques, “Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective,” IEEE Trans. Microw. Theory Tech. **56**(12), 3108–3120 (2008). [CrossRef]

**11. **F. Medina, F. Mesa, and D. C. Skigin, “Extraordinary transmission through arrays of slits: a circuit theory model,” IEEE Trans. Microw. Theory Tech. **58**(1), 105–115 (2010). [CrossRef]

**12. **M. Beruete, M. Navarro-Cia, and M. S. Ayza, “Understanding anomalous extraordinary transmission from equivalent circuit and grounded slab concepts,” IEEE Trans. Microw. Theory Tech. **59**(9), 2180–2188 (2011). [CrossRef]

**13. **M. Beruete, I. Campillo, M. Navarro-Cía, F. Falcone, and M. Sorolla, “Molding left- or right-handed metamaterials by stacked cutoff metallic hole arrays,” IEEE Trans. Antenn. Propag. **55**(6), 1514–1521 (2007). [CrossRef]

**14. **V. Delgado, R. Marques, and L. Jelinek, “Coupled-wave surface-impedance analysis of extraordinary transmission through single and stacked metallic screens,” IEEE Trans. Antenn. Propag. **61**(3), 1342–1351 (2013). [CrossRef]

**15. **M. Beruete, M. Sorolla, I. Campillo, and J. S. Dolado, “Increase of the transmission in cut-off metallic hole arrays,” IEEE Microw. Wirel. Compon. Lett. **15**(2), 116–118 (2005). [CrossRef]

**16. **V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B Condens. Matter Mater. Phys. **71**(23), 235117 (2005). [CrossRef]

**17. **V. Lomakin and E. Michielssen, “Transmission of transient plane waves through perfect electrically conducting plates perforated by periodic arrays of subwavelength holes,” IEEE Trans. Antenn. Propag. **54**(3), 970–984 (2006). [CrossRef]

**18. **T. Søndergaard, S. I. Bozhevolnyi, S. M. Novikov, J. Beermann, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission enhanced by nanofocusing,” Nano Lett. **10**(8), 3123–3128 (2010). [CrossRef] [PubMed]

**19. **T. Sondergaard, S. I. Bozhevolnyi, J. Beermann, S. M. Novikov, E. Devaux, and T. W. Ebbesen, “Extraordinary optical transmission with tapered slits: effect of higher diffraction and slit resonance orders,” J. Opt. Soc. Am. B **29**(1), 130–137 (2012). [CrossRef]

**20. **J. Beermann, T. Sondergaard, S. M. Novikov, S. I. Bozhevolnyi, E. Devaux, and T. W. Ebbesen, “Field enhancement and extraordinary optical transmission by tapered periodic slits in gold films,” New J. Phys. **13**, 063029 (2011). [CrossRef]

**21. **H. Shen and B. Maes, “Enhanced optical transmission through tapered metallic gratings,” Appl. Phys. Lett. **100**(24), 241104 (2012). [CrossRef]

**22. **Y. Liang, W. Peng, R. Hu, and H. Zou, “Extraordinary optical transmission based on subwavelength metallic grating with ellipse walls,” Opt. Express **21**(5), 6139–6152 (2013). [CrossRef] [PubMed]

**23. **F. Medina, R. Rodriguez-Berral, and F. Mesa, “Circuit model for metallic gratings with tapered and stepped slits,” Proc. 42nd European Microwave Conference, Oct. 2012. [CrossRef]

**24. **J.-C. Yang, H. Gao, J. Y. Suh, W. Zhou, M. H. Lee, and T. W. Odom, “Enhanced optical transmission mediated by localized Plasmons in anisotropic, 3D nanohole arrays,” Nano Lett. **10**(8), 3173–3178 (2011). [CrossRef] [PubMed]

**25. **R. J. Moerland, J. E. Koskela, A. Kravchenko, M. Simberg, S. van der Vegte, M. Kaivola, A. Priimagio, and R. H. A. Ras, “Large-area arrays of three-dimensional plasmonic subwavelength-sized structures from azopolymer surface-relief gratings,” Mater. Horiz., Royal Soc. Chem **1**(74), 74–80 (2014).

**26. **H. L. Chen, S. Y. Chuang, W. H. Lee, S. S. Kuo, W. F. Su, S. L. Ku, and Y. F. Chou, “Extraordinary transmittance in three dimensional crater, pyramid, and hole-array structures prepared through reversal imprinting of metal films,” Opt. Express **17**(3), 1636–1645 (2009). [CrossRef] [PubMed]

**27. **M. Ng Mou Kehn, “Modal analysis of substrate integrated waveguides with rectangular via-holes using cavity and multilayer Green’s functions,” IEEE Trans. Microw. Theory Tech. **62**(10), 2214–2231 (2014). [CrossRef]