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 TE₂₀ 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.

 

Fig. 1 Periodic two-dimensional array of flared-open type of generally rectangular holes (although shown as square ones) pierced through a metallic sheet of thickness d, with each tunnel modeled as a connected stepped series of cavity sections. The width a_ι and height h_ι along x and y of an arbitrarily-selected section deemed as the ι^th section is as indicated. Two perspectives showing front and back of the screen are displayed.

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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_y along 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 a₀ and h₀ on the input side at z = b = $z_0^{}$ and a_Z (≥ a₀) and h_Z (≥ h₀) 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_inc and ϕ_inc measured respectively from the z and x axes, the latter azimuth angle defining the plane of incidence as portrayed by the tilted sheet in the drawing.

 

Fig. 2 Schematic and geometry of rectangular hole array illuminated by incident plane wave, with plane of incidence defined by (θ_inc, ϕ_inc).

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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_ι), and its cross-sectional dimensions are a_ι and h_ι, being the width and height along x and y, respectively. If the flared hole is homogeneously filled with a material with parameters, say (μ_univ, ε_univ), as would often and likely be the case, then μ_ι = μ_univ and ε_ι = ε_univ for all 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.

 

Fig. 3 Interconnected series of rectangular waveguide sections modeling a flared-open hole. Grayed portions represent the conducting parts of the metallic screen. Any ι^th section filled with a material of parameters: (μ_ι, ε_ι)

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

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:

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_k, then $\underset{\_}{D}$ = diag($\underset{\_}{V}$) places ${\underset{\_}{V}}_k$ along the diagonal of $\underset{\_}{D}$ in a staggered fashion, i.e. $\text{diag}\left(\underset{\_}{V}\right)=\text{diag}\left(\left[\begin{array}{cccc}{\left[{\underset{\_}{V}}_{\text{1}}\right]}_{P_1\times Q_1}& {\left[{\underset{\_}{V}}_{\text{2}}\right]}_{P_2\times Q_2}& \cdots & {\left[{\underset{\_}{V}}_N\right]}_{P_N\times Q_N}\end{array}\right]\right)=\left[\begin{array}{cccc}{\left[{\underset{\_}{V}}_{\text{1}}\right]}_{P_1\times Q_1}& {\left[0\right]}_{P_1\times Q_2}& \cdots & {\left[0\right]}_{P_1\times Q_N}\\ {\left[0\right]}_{P_2\times Q_1}& {\left[{\underset{\_}{V}}_{\text{2}}\right]}_{P_2\times Q_2}& \cdots & {\left[0\right]}_{P_2\times Q_N}\\ ⋮& ⋮& \ddots & ⋮\\ {\left[0\right]}_{P_N\times Q_1}& {\left[0\right]}_{P_N\times Q_2}& \cdots & {\left[{\underset{\_}{V}}_N\right]}_{P_N\times Q_N}\end{array}\right]$

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.

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:

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:

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.

 

Fig. 4 Five-layer configuration for (a) left semi-infinite (incident) region, and (b) right semi-infinite (transmission) region of rectangular hole-array used in core-routine of spectral multilayer Green's function.

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The scattered fields within this left-side incidence region are caused by the spectral current sources at the upper boundary z = z_i+1 = z₄ = + 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:

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

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 = z₁ = 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:

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:

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

9. Weighting: generate system of equations

Respectively taking ${\displaystyle \underset{y=-h_0/2}{\overset{y=h_0/2}{\int }}{\displaystyle \underset{x=-a_0/2}{\overset{x=a_0/2}{\int }}{\overrightarrow{e}}_{t_{rst}}^{(0)}(x,y)\times \_\_\_\cdot \widehat{z}dxdy}}$ and ${\displaystyle \underset{y=-h_0/2}{\overset{y=h_{Z_{\Theta }}/2}{\int }}{\displaystyle \underset{x=-a_0/2}{\overset{x=a_0/2}{\int }}{\overrightarrow{e}}_{t_{rst}}^{(Z_{\Theta })}(x,y)\times \_\_\_\cdot \widehat{z}dxdy}}$ throughout (32) and (33), for rst = 1, 2, …, N₀, and rst = 1, 2, …, N_ZΘ, the following two equations are obtained.

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_xd_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.