AN EXTENDED EQUIVALENT CIRCUIT MODEL FOR ANALYSIS OF ARRAYS OF SUB -WAVELENGTH HOLES PERFORATED IN A METALLIC FILM

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AN EXTENDED EQUIVALENT CIRCUIT MODEL FOR ANALYSIS OF

ARRAYS OF SUB-WAVELENGTH HOLES PERFORATED IN A

METALLIC FILM

Mohammad Pasdari-Kia

Department of Electrical Engineering

Sharif University of Technology

Tehran, Iran

mohammad.pasdarikia@sharif.edu

Mohammad Memarian

Department of Electrical Engineering

Sharif University of Technology

Tehran, Iran

mmemarian@sharif.edu

Amin Khavasi

Department of Electrical Engineering

Sharif University of Technology

Tehran, Iran

khavasi@sharif.edu

August 10, 2023

ABSTRACT

Periodic arrays of sub-wavelength holes, due to a variety of applications (sensors, polarizers, ﬁlters)

and unique abilities in manipulating different characteristics of impinging light, have been the subject

of many studies in recent years. This paper presents a new high-precision circuit model to investigate

these structures’ electromagnetic response, which provides higher accuracy than available literature.

To develop the model, a multi-mode approach is taken for inside the apertures, and the proposed

model is presented in three general forms which are explained in detail. Along with the proposed

circuit model, accurate spatial proﬁles for square and circular apertures are presented in which the

transverse components of the electric ﬁeld are approximated. In this paper, it is shown that structures

such as an array of PEC pillars can be analyzed similar to other arrays of apertures, and the circuit

model accurately predicts the behavior of these structures. In some practical applications, such as

sensors and TCEs (transparent conducting electrodes), the array is embedded in a layered medium,

and the circuit model with minor changes and few modiﬁcations can analyze these structures with

high accuracy.

Keywords Equivalent circuit model, Metasurfaces, Sub-wavelength holes

1 Introduction

The propagation of electromagnetic waves along periodic arrays of sub-wavelength apertures has been the subject of

many studies in the last two decades. These structures have various applications, including sensors [1

–

5], ﬁlters [6],

absorbers [7], waveguides [8, 9], lenses [10, 11], etc. Along with these novel applications, observation of the physical

phenomena such as extraordinary transmission (EoT) [12, 13] and broadband Brewster transmission [14

–

16] has

increased the importance of these structures.

In recent years, understanding and justifying the physical behaviors of the array of holes has been researched using

various analytical methods [17, 18]. For instance, the extraordinary transmission at optical frequencies may be

understood by coupling incident light with the surface plasmon polariton modes supported at the boundary between

metals and dielectrics [19]. On the other hand, this enhanced transmission at microwave frequencies through perforated

PEC structures, has recently been linked to the role of proper complex modes [20]. Providing an analytical method

arXiv:2210.00568v2 [physics.app-ph] 8 Aug 2023

Preprint

in the form of a circuit model can become a tool for designing and predicting these behaviors. This work focuses

on microwave and terahertz frequencies, where the permittivity of the metals becomes high enough to justify perfect

electric conductor (PEC) approximation. The transmission line model is considered in this paper due to its appropriate

accuracy and high ﬂexibility, which can be easily generalized to multilayer structures.

Modeling a waveguide structure in the form of a transmission line has been researched for a long time and has numerous

applications in obtaining scattering parameters [21, 22]. The transmission line model is used in a wide frequency range

from microwave to infrared [23

–

27]. With the help of the transmission line, the array of apertures in a PEC ﬁlm has

already been investigated [28

–

31]. In the ﬁrst developed circuit model, the array of holes was analyzed by considering

the thickness of the PEC ﬁlm. In this case, the circuit model was created by considering only the dominant mode inside

the apertures [32, 33].

In previous works, a circuit model has been developed for rectangular apertures by considering only

T E01

as the

dominant mode [33, 34]. In addition to two-dimensional arrays, one-dimensional arrays of slits have also been

investigated by considering the

T EM

mode as the dominant mode using the circuit model [32, 35]. If the thickness of

the aperture is large enough that the excited high-order modes inside the hole are damped from the ﬁrst opening to

the second opening of the hole, the model has high accuracy. In these models, if the structure’s thickness decreases,

the error increases due to high-order modes; additionally, the error increases more rapidly for circular-shape apertures

compared to rectangular apertures. Due to the widespread applications of metasurfaces and frequency selective surfaces,

we need to create models for thin arrays.

Frequency selective surfaces are thin periodic surfaces that are important both in terms of industrial applications as well

as in terms of fundamental science. The array of apertures perforated in a PEC thin ﬁlm is a kind of FSS, and many

powerful circuit models have been proposed for these structures that are suitable for better design and understanding

of the behavior of these structures [36, 37]. In these structures, the circuit model is created with the help of the

spatial proﬁle of the transverse electric ﬁeld at the opening of the aperture. Previous works have provided proﬁles

for rectangular [37], cross [28], and annular holes [37, 38]. Initially, the circuit model was developed for a structure

consisting of a single layer of the intended FSS [25]; then, the circuit model was improved for the case where the FSS is

embedded in a layered medium [39, 40]. Structures with multiple dielectric layers and multiple FSS layers were then

examined using circuit models [28, 37], but all of these circuit models were developed regardless of the FSS’s thickness.

It should be noted that effect of thickness can cause a considerable error; however, the thickness is about one-ﬁftieth of

the structure’s period.

In this paper, an extended circuit model is developed, which is much faster than full-wave simulations and is much

more accurate than its predecessors. The circuit model is formulated in such a way that it takes into account high-order

modes inside the holes; thus, it has high accuracy in low thickness structures. A closed-form expression is presented for

the spatial proﬁle of the transverse electrical ﬁeld on the circular aperture which means that, unlike previous models, we

do not need to extract the ﬁeld proﬁle from the full-wave simulators. It is shown that the previous electric ﬁeld proﬁle

for the square hole has a large error when the width of the square hole is large (the width of the square is greater than

half the period) [28], and this problem is well solved by presenting the new proﬁle. Unlike square and circular holes, an

array of PEC pillars always supports propagating mode that makes interesting properties [16], and these structures are

also analyzed analytically with the help of the proposed circuit model in this work.

The paper is organized as follows: Section 2 is devoted to describing how the proposed circuit model is developed,

and three different cases: single-mode, multi-mode, and modiﬁed single-mode circuit models are described. In section

3, the low-thickness structures are investigated. In section 4, the proposed circuit model is extended to multilayered

structures; ﬁnally, a conclusion is described in section 5.

2 The proposed circuit model

Periodic structures can be analyzed similar to other waveguide structures by considering a unit cell as a waveguide;

thus, the formulation presented can be extended to any waveguide structure, but we focus only on the square array of

sub-wavelength holes. In addition to the previous circuit models, environmental models for the desired structure have

also been investigated [16, 41]. Similarly, the isotropic/anisotropic environmental model is created only by considering

the dominant mode inside the aperture, which means that these methods also have a large error for a thin array of holes.

The following formulation is presented by considering all the modes inside the hole.

Figure 1 shows the schematic of the structures which are illuminated by a normal incident plane wave. The upper

environment of the structure is region I (

), inside the holes is region II (

), and the lower environment is region III

(

). For simplicity’s sake, we ﬁrst assume the structures depicted in Fig. 1 is semi-inﬁnite, i.e.,

L→ ∞

. Impinging of

a plane-wave on a semi-inﬁnite structure excites aperture modes in region II, and the Floquet modes in region I. The

Preprint

Figure 1: Figure (a) shows the array of perforated square apertures in a PEC ﬁlm, and (b) shows the array of circular

holes. In two structures, the period is P in both directions, the width of the square is W, and the circle’s diameter is D.

Refractive index of the upper and lower (yellow layer) environments are

and

, respectively, and the holes are ﬁlled

with refractive index n2.

Floquet expansion of the transverse (x, y components) electric and magnetic ﬁeld at the discontinuity (

z= 0

) can be

written as follow [28,40]:











E1(x, y) = (1 + R)e1

0(x, y) + P

′

hV1

he1

h(x, y)

H1(x, y) = (1 −R)Y1

0(ˆz×e1

0(x, y)) −P

′

hV1

hY1

h(ˆz×e1

h(x, y))

(1)

where the expression

0(x, y)

is the tangential component of the incident wave whose reﬂection is R. The superscript

refers to the region, and the prime in the series of Eq. (1) indicates that the incident wave is excluded from the

summation.

h(x, y)

is the normalized transverse electric ﬁeld of

th Floquet harmonic (

is associated with a pair of

integer numbers mn). Similarly, expansion of aperture modes can be written as











E2(x, y) = V2

0e2

0(x, y) + P

′

hV2

he2

h(x, y)

H2(x, y) = V2

0Y2

0(ˆz×e2

0(x, y)) + P

′

hV2

hY2

h(ˆz×e2

h(x, y))

(2)

where e2

0(x, y)is the tangential component of the dominant mode inside the aperture. It should be noted that we want

to use the dominant mode to create the transmission line; thus, we have written this mode separately from the other

modes. Similarly,

h(x, y)

is the transverse electric ﬁeld of harmonic

inside the aperture. Modes inside the aperture

can be used in equations depending on the type of aperture but the Floquet modes in region I can be written as [28, 40]:

h(x, y) = e−jK1

th.ˆρ

P P ˆeh1ˆρ=xˆx+yˆy(3)

th =k1

xm ˆx+k1

yn ˆy= (k1

m+k1

x0)ˆx+ (k1

n+k1

y0)ˆy(4)

x0=k1sin(θ) cos(ϕ)k1

y0=k1sin(θ) sin(ϕ)(5)

m=2πm

Pk1

n=2πn

P(6)

ˆeh1=









kth

1T M

(ˆ

kth

1×ˆz)T E

kth

1=k1

|k1

th |(7)

The modal admittances Y1

hare given by

h=1

η1











T M

k1T E

zh =q(k1)2− | k1

th |2(8)

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ANEXTENDEDEQUIVALENTCIRCUITMODELFORANALYSISOFARRAYSOFSUB-WAVELENGTHHOLESPERFORATEDINAMETALLICFILMMohammadPasdari-KiaDepartmentofElectricalEngineeringSharifUniversityofTechnologyTehran,Iranmohammad.pasdarikia@sharif.eduMohammadMemarianDepartmentofElectricalEngineeringSharifUniversityofTechnologyTehra...

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AN EXTENDED EQUIVALENT CIRCUIT MODEL FOR ANALYSIS OF ARRAYS OF SUB -WAVELENGTH HOLES PERFORATED IN A METALLIC FILM

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